|
|
|
|
 |
|
|
|
|
|
|
|
Probability Abstracts 95
This document contains abstracts 4722-5092 from
Oct-1-2006 to Dic-31-2006.
They have been mailed on Jan 1st, 2007.
Author(s): Pablo A. Ferrari and Claudio Landim and Valentin V. Sisko
Abstract: A family of m independent identically distributed random variables indexed by
a chemical potential \phi\in[0,\gamma] represents piles of particles. As \phi
increases to \gamma, the mean number of particles per site converges to a
maximal density \rho_c<\infty. The distribution of particles conditioned on the
total number of particles equal to n does not depend on \phi (canonical
ensemble). For fixed m, as n goes to infinity the canonical ensemble measure
behave as follows: removing the site with the maximal number of particles, the
distribution of particles in the remaining sites converges to the grand
canonical measure with density \rho_c; the remaining particles concentrate
(condensate) on a single site.
http://arXiv.org/abs/math/0612856
http://front.math.ucdavis.edu/math.PR/0612856
(alternate)
Author(s): Alan J. Bray and Richard Smith
Abstract: We calculate the exact asymptotic survival probability, Q, of a
one-dimensional Brownian particle, initially located located at the point x in
(-L,L), in the presence of two moving absorbing boundaries located at
\pm(L+ct). The result is Q(y,\lambda) = \sum_{n=-\infty}^\infty \cosh(ny)
\exp(-n^2\lambda/4), where y=cx/D, \lambda = cL/D and D is the diffusion
constant of the particle. The results may be extended to the case where the
absorbing boundaries have different speeds. As an application, we compute the
asymptotic survival probability for the trapping reaction A + B -> B, for
evanescent traps with a long decay time.
http://arXiv.org/abs/cond-mat/0612563
http://front.math.ucdavis.edu/cond-mat/0612563
(alternate)
Author(s): Emmanuel J. Candes and Paige A. Randall
Abstract: This paper discusses a stylized communications problem where one wishes to
transmit a real-valued signal x in R^n (a block of n pieces of information) to
a remote receiver. We ask whether it is possible to transmit this information
reliably when a fraction of the transmitted codeword is corrupted by arbitrary
gross errors, and when in addition, all the entries of the codeword are
contaminated by smaller errors (e.g. quantization errors).
We show that if one encodes the information as Ax where A is a suitable m by
n coding matrix (m >= n), there are two decoding schemes that allow the
recovery of the block of n pieces of information x with nearly the same
accuracy as if no gross errors occur upon transmission (or equivalently as if
one has an oracle supplying perfect information about the sites and amplitudes
of the gross errors). Moreover, both decoding strategies are very concrete and
only involve solving simple convex optimization programs, either a linear
program or a second-order cone program. We complement our study with numerical
simulations showing that the encoder/decoder pair performs remarkably well.
http://arXiv.org/abs/cs/0612124
http://front.math.ucdavis.edu/cs.IT/0612124
(alternate)
Author(s): Francis Comets and Serguei Popov and Gunter Sch\"utz and Marina Vachkovskaia
Abstract: We study stochastic billiards on general tables: a particle moves according
to its constant velocity inside some domain ${\mathcal D} \subset {\mathbb
R}^d$ until it hits the boundary and bounces randomly inside according to some
reflection law. We assume that the boundary of the domain is locally Lipschitz
and almost everywhere continuously differentiable. The angle of the outgoing
velocity with the inner normal vector has a specified, absolutely continuous
density. We construct the discrete time and the continuous time processes
recording the sequence of hitting points on the boundary and the pair
location/velocity. We mainly focus on the case of bounded domains. Then, we
prove exponential ergodicity of these two Markov processes, we study their
invariant distribution and their normal (Gaussian) fluctuations. Of particular
interest is the case of the cosine reflection law: the stationary distributions
for the two processes are uniform in this case, the discrete time chain is
reversible though the continuous time process is quasi-reversible. Also in this
case, we give a natural construction of a chord ``picked at random'' in
${\mathcal D}$, and we study the angle of intersection of the process with a
$(d-1)$-dimensional manifold contained in ${\mathcal D}$.
http://arXiv.org/abs/math/0612799
http://front.math.ucdavis.edu/math.PR/0612799
(alternate)
Author(s): Olle H\"{a}ggstr\"{o}m and Johan Jonasson
Abstract: This paper is an up-to-date introduction to the problem of uniqueness versus
non-uniqueness of infinite clusters for percolation on ${\mathbb{Z}}^d$ and,
more generally, on transitive graphs. For iid percolation on ${\mathbb{Z}}^d$,
uniqueness of the infinite cluster is a classical result, while on certain
other transitive graphs uniqueness may fail. Key properties of the graphs in
this context turn out to be amenability and nonamenability. The same problem is
considered for certain dependent percolation models -- most prominently the
Fortuin--Kasteleyn random-cluster model -- and in situations where the standard
connectivity notion is replaced by entanglement or rigidity. So-called
simultaneous uniqueness in couplings of percolation processes is also
considered. Some of the main results are proved in detail, while for others the
proofs are merely sketched, and for yet others they are omitted. Several open
problems are discussed.
http://arXiv.org/abs/math/0612812
http://front.math.ucdavis.edu/math.PR/0612812
(alternate)
Author(s): Svante Janson and Malwina J. Luczak
Abstract: We study the $k$-core of a random (multi)graph on $n$ vertices with a given
degree sequence. In our previous paper `A simple solution to the k-core
problem' we used properties of empirical distributions of independent random
variables to give a simple proof of the fact that the size of the giant
$k$-core obeys a law of large numbers as $n$ tends to infinity. Here we develop
the method further and show that the fluctuations around the deterministic
limit converge to a Gaussian law above and near the threshold, and to a
non-normal law at the threshold. Further, we determine precisely the location
of the phase transition window for the emergence of a giant $k$-core. Hence we
deduce corresponding results for the $k$-core in $G(n,p)$ and $G(n,m)$.
http://arXiv.org/abs/math/0612827
http://front.math.ucdavis.edu/math.PR/0612827
(alternate)
Author(s): N. K. Jana and B. V. Rao
Abstract: A unified treatment for the existence of free energy in several random energy
models is presented. If the sequence of distributions associated with the
particle systems obeys a large deviation principle, then the free energy exists
almost surely. This includes all the known cases as well as some heavy-tailed
distributions.
http://arXiv.org/abs/math/0612836
http://front.math.ucdavis.edu/math.PR/0612836
(alternate)
Author(s): Evarist Gin\'{e} and Vladimir Koltchinskii and Wenbo Li and Joel Zinn
Abstract: About forty years ago it was realized by several researchers that the
essential features of certain objects of Probability theory, notably Gaussian
processes and limit theorems, may be better understood if they are considered
in settings that do not impose structures extraneous to the problems at hand.
For instance, in the case of sample continuity and boundedness of Gaussian
processes, the essential feature is the metric or pseudometric structure
induced on the index set by the covariance structure of the process, regardless
of what the index set may be. This point of view ultimately led to the
Fernique-Talagrand majorizing measure characterization of sample boundedness
and continuity of Gaussian processes, thus solving an important problem posed
by Kolmogorov. Similarly, separable Banach spaces provided a minimal setting
for the law of large numbers, the central limit theorem and the law of the
iterated logarithm, and this led to the elucidation of the minimal (necessary
and/or sufficient) geometric properties of the space under which different
forms of these theorems hold. However, in light of renewed interest in
Empirical processes, a subject that has considerably influenced modern
Statistics, one had to deal with a non-separable Banach space, namely
$\mathcal{L}_{\infty}$. With separability discarded, the techniques developed
for Gaussian processes and for limit theorems and inequalities in separable
Banach spaces, together with combinatorial techniques, led to powerful
inequalities and limit theorems for sums of independent bounded processes over
general index sets, or, in other words, for general empirical processes.
http://arXiv.org/abs/math/0612726
http://front.math.ucdavis.edu/math.PR/0612726
(alternate)
Author(s): Ou Zhao and Michael Woodroofe
Abstract: There has been recent interest in the conditional central limit question for
(strictly) stationary, ergodic processes $... X_{-1}, X_0,X_1,...$ whose
partial sums $S_n = X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is
a square integrable martingale with stationary increments and $R_n$ is a
remainder term for which $E(R_n^2) = o(n)$. Here we explore the Law of the
Iterated Logarithm (LIL) for the same class of processes. Letting
$\Vert\cdot\Vert$ denote the norm in $L^2(P)$, a sufficient condition for the
partial sums of a stationary process to have the form $S_n = M_n+R_n$ is that
$n^{-{3\over 2}}\Vert E(S_n|X_0,X_{-1},...)\Vert$ be summable. A sufficient
condition for the LIL is only slightly stronger, requiring $n^{-{3\over
2}}\log^{3\over 2} (n)\Vert E(S_n|X_0,X_{-1},...)\Vert$ to be summable. As a
by-product of our main result, we obtain an improved statement of the
Conditional Central Limit Theorem. Invariance principles are obtained as well.
http://arXiv.org/abs/math/0612747
http://front.math.ucdavis.edu/math.PR/0612747
(alternate)
Author(s): Nathanael Enriquez (PMA)
Abstract: A detailed combinatorial analysis of lattice convex polygonal lines of N^2
joining 0 to (n,n) is presented. We derive consequences on the line having the
largest number of vertices as well as the cardinal and limit shape of lines
having few vertices. The proof refines a statistical physical method used by
Sinai to obtain the typical behavior of these lines, allied to some Fourier
analysis. Limit shapes of convex lines joining 0 to (n,n) and having a given
total length are also characterized.
http://arXiv.org/abs/math/0612770
http://front.math.ucdavis.edu/math.PR/0612770
(alternate)
Author(s): Evarist Gin\'{e} and Vladimir Koltchinskii
Abstract: Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension
$\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$
with uniform distribution. We study the random operators $$
\Delta_{h_n,n}f(p):=\frac{1}{nh_n^{d+2}}\sum_{i=1}^n
K(\frac{p-X_i}{h_n})(f(X_i)-f(p)), p\in M $$ where
${K(u):={\frac{1}{(4\pi)^{d/2}}}e^{-\|u\|^2/4}}$ is the Gaussian kernel and
${h_n\to 0}$ as ${n\to\infty.}$ Such operators can be viewed as graph
laplacians (for a weighted graph with vertices at data points) and they have
been used in the machine learning literature to approximate the
Laplace-Beltrami operator of ${M,}$ ${\Delta_Mf}$ (divided by the Riemannian
volume of the manifold). We prove several results on a.s. and distributional
convergence of the deviations
${\Delta_{h_n,n}f(p)-{\frac{1}{|\mu|}}\Delta_Mf(p)}$ for smooth functions ${f}$
both pointwise and uniformly in ${f}$ and ${p}$ (here ${|\mu|=\mu(M)}$ and
${\mu}$ is the Riemannian volume measure). In particular, we show that for any
class ${{\cal F}}$ of three times differentiable functions on ${M}$ with
uniformly bounded derivatives $$ \sup_{p\in M}\sup_{f\in
F}\Big|\Delta_{h_n,p}f(p)-\frac{1}{|\mu|}\Delta_Mf(p)\Big|=
O\Big(\sqrt{\frac{\log(1/h_n)}{nh_n^{d+2}}}\Big) a.s. $$ as soon as $$
nh_n^{d+2}/\log h_n^{-1}\to \infty and nh^{d+4}_n/\log h_n^{-1}\to 0, $$ and
also prove asymptotic normality of
${\Delta_{h_n,p}f(p)-{\frac{1}{|\mu|}}\Delta_Mf(p)}$ (functional CLT) for a
fixed ${p\in M}$ and uniformly in ${f}.$
http://arXiv.org/abs/math/0612777
http://front.math.ucdavis.edu/math.PR/0612777
(alternate)
Author(s): stefano m. iacus and nakahiro yoshida
Abstract: The telegraph process $\{X(t), t>0\}$, is supposed to be observed at $n+1$
equidistant time points $t_i=i\Delta_n,i=0,1,..., n$. The unknown value of
$\lambda$, the underlying rate of the Poisson process, is a parameter to be
estimated. The asymptotic framework considered is the following:
$\Delta_n \to 0$, $n\Delta_n = T \to \infty$ as $n \to \infty$. We show that
previously proposed moment type estimators are consistent and asymptotically
normal but not efficient. We study further an approximated moment type
estimator which is still not efficient but comes in explicit form. For this
estimator the additional assumption $n\Delta_n^3 \to 0$ is required in order to
obtain asymptotic normality. Finally, we propose a new estimator which is
consistent, asymptotically normal and asymptotically efficient under no
additional hypotheses.
http://arXiv.org/abs/math/0612784
http://front.math.ucdavis.edu/math.PR/0612784
(alternate)
Author(s): Greg W Anderson and Ofer Zeitouni
Abstract: We consider the spectral properties of a class of {\em regularized
estimators} of (large) empirical covariance matrices corresponding to
stationary (but not necessarily Gaussian) sequences, obtained by {\em banding}.
We prove a law of large numbers (similar to that proved in the Gaussian case by
Bickel and Levina), which implies that the spectrum of a banded empirical
covariance matrix is an efficient estimator. Our main result is a central limit
theorem in the same regime, which to our knowledge is new, even in the Gaussian
setup.
http://arXiv.org/abs/math/0612791
http://front.math.ucdavis.edu/math.PR/0612791
(alternate)
Author(s): Janos Englander
Abstract: In a previous paper of Winter and the author the Law of Large Numbers for the
local mass of certain superdiffusions was proved under a spectral theoretical
assumption, which is equivalent to the ergodicity (positive recurrence) of the
motion component of an $H$-transformed (or weighted) superprocess. In fact the
assumption is also equivalent to the property that the scaling for the
expectation of the local mass is pure exponential.
In this paper we go beyond ergodicity, that is we consider cases when the
scaling is not purely exponential. Inter alia, we prove the analog of the
Watanabe-Biggins Law of Large Numbers for super-Brownian motion (SBM).
We will also prove another Law of Large Numbers for a bounded set moving with
subcritical speed, provided the variance term decays sufficiently fast.
http://arXiv.org/abs/math/0612797
http://front.math.ucdavis.edu/math.PR/0612797
(alternate)
Author(s): Zhenxin Liu
Abstract: In this paper, stochastic inertial manifold for damped wave equations
subjected to additive white noise is constructed by the Lyapunov-Perron method.
It is proved that when the intensity of noise tends to zero the stochastic
inertial manifold converges to its deterministic counterpart almost surely.
http://arXiv.org/abs/math/0612774
http://front.math.ucdavis.edu/math.DS/0612774
(alternate)
Author(s): E. Ben-Naim and N.W. Hengartner
Abstract: League competition is investigated using random processes and scaling
techniques. In our model, a weak team can upset a strong team with a fixed
probability. Teams play an equal number of head-to-head matches and the team
with the largest number of wins is declared to be the champion. The total
number of games needed for the best team to win the championship with high
certainty, T, grows as the cube of the number of teams, N, i.e., T ~ N^3. This
number can be substantially reduced using preliminary rounds where teams play a
small number of games and subsequently, only the top teams advance to the next
round. When there are k rounds, the total number of games needed for the best
team to emerge as champion, T_k, scales as follows, T_k ~N^(\gamma_k) with
gamma_k=1/[1-(2/3)^(k+1)]. For example, gamma_k=9/5,27/19,81/65 for k=1,2,3.
These results suggest an algorithm for how to infer the best team using a
schedule that is linear in N. We conclude that league format is an ineffective
method of determining the best team, and that sequential elimination from the
bottom up is fair and efficient.
http://arXiv.org/abs/physics/0612217
http://front.math.ucdavis.edu/physics/0612217
(alternate)
Author(s): Paavo Salminen and Pierre Vallois and Marc Yor
Abstract: We present a number of important identities related to the excursion theory
of linear diffusions. In particular, excursions straddling an independent
exponential time are studied in detail. Letting the parameter of the
exponential time tend to zero it is seen that these results connect to the
corresponding results for excursions of stationary diffusions (in stationary
state). We characterize also the laws of the diffusion prior and posterior to
the last zero before the exponential time. It is proved using Krein's
representations that, e.g., the law of the length of the excursion straddling
an exponential time is infinitely divisible. As an illustration of the results
we discuss Ornstein-Uhlenbeck processes.
http://arXiv.org/abs/math/0612687
http://front.math.ucdavis.edu/math.PR/0612687
(alternate)
Author(s): Przemys{\l}aw Repetowicz and Peter Richmond
Abstract: We model the logarithm of the price (log-price) of a financial asset as a
random variable obtained by projecting an operator stable random vector with a
scaling index matrix $\underline{\underline{E}}$ onto a non-random vector. The
scaling index $\underline{\underline{E}}$ models prices of the individual
financial assets (stocks, mutual funds, etc.). We find the functional form of
the characteristic function of real powers of the price returns and we compute
the expectation value of these real powers and we speculate on the utility of
these results for statistical inference. Finally we consider a portfolio
composed of an asset and an option on that asset. We derive the characteristic
function of the deviation of the portfolio, \mbox{${\mathfrak D}_t^{({\mathfrak
t})}$}, defined as a temporal change of the portfolio diminished by the the
compound interest earned. We derive pseudo-differential equations for the
option as a function of the log-stock-price and time and we find exact
closed-form solutions to that equation. These results were not known before.
Finally we discuss how our solutions correspond to other approximate results
known from literature,in particular to the well known Black & Scholes equation.
http://arXiv.org/abs/math/0612691
http://front.math.ucdavis.edu/math.PR/0612691
(alternate)
Author(s): Wei Biao Wu
Abstract: We obtain an almost sure bound for oscillation rates of empirical
distribution functions for stationary causal processes. For short-range
dependent processes, the oscillation rate is shown to be optimal in the sense
that it is as sharp as the one obtained under independence. The dependence
conditions are expressed in terms of physical dependence measures which are
directly related to the data-generating mechanism of the underlying processes
and thus are easy to work with.
http://arXiv.org/abs/math/0612692
http://front.math.ucdavis.edu/math.PR/0612692
(alternate)
Author(s): Paul Deheuvels
Abstract: For $\gamma>-{1/2}$, we provide the Karhunen-Lo\`{e}ve expansion of the
weighted mean-centered Wiener process, defined by \[W
_{\gamma}(t)=\frac{1}{\sqrt{1+2\gamma}}\Big\{W\big(t^{1+2\gamma}\big)-
\int_0^1W\big(u^{1+2\gamma}\big)du\Big\},\] for $t\in(0,1]$. We show that the
orthogonal functions in these expansions have simple expressions in term of
Bessel functions. Moreover, we obtain that the $L^2[0,1]$ norm of $W_{\gamma}$
is identical in distribution with the $L^2[0,1]$ norm of the weighted Brownian
bridge $t^{\gamma}B(t)$.
http://arXiv.org/abs/math/0612693
http://front.math.ucdavis.edu/math.PR/0612693
(alternate)
Author(s): Vladimir Dobri\'{c} and Francisco M. Ojeda
Abstract: In this paper the whole family of fractional Brownian motions is constructed
as a single Gaussian field indexed by time and the Hurst index simultaneously.
The field has a simple covariance structure and it is related to two
generalizations of fractional Brownian motion known as multifractional Brownian
motions. A mistake common to the existing literature regarding multifractional
Brownian motions is pointed out and corrected. The Gaussian field, due to
inherited ``duality'', reveals a new way of constructing martingales associated
with the odd and even part of a fractional Brownian motion and therefore of the
fractional Brownian motion. The existence of those martingales and their
stochastic representations is the first step to the study of natural wavelet
expansions associated to those processes in the spirit of our earlier work on a
construction of natural wavelets associated to Gaussian-Markov processes.
http://arXiv.org/abs/math/0612694
http://front.math.ucdavis.edu/math.PR/0612694
(alternate)
Author(s): Raouf Ghomrasni
Abstract: We show that for a wide class of functions $F$ that: $$ {\lim_{\epsilon
\downarrow 0} {\frac{1}{\epsilon}} \int_0^t \Big\{F(s, X_s) - F(s, X_s -
\epsilon)\Big\} d\big_s} = - \int_0^t\int_{\R} F(s, x) d L_s^x $$
where $X_t$ is a continuous semi-martingale, $(L_t^x, x \in \R, t \geq 0)$ its
local time process and $(\big_t, t \geq 0)$ its quadratic variation
process.
http://arXiv.org/abs/math/0612699
http://front.math.ucdavis.edu/math.PR/0612699
(alternate)
Author(s): Dongsheng Wu and Yimin Xiao
Abstract: Let $\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$ be a random string taking values
in ${\mathbb{R}}^d$, specified by the following stochastic partial differential
equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial
t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\] where $\dot{W}(x,t)$ is
an ${\mathbb{R}}^d$-valued space-time white noise. Mueller and Tribe (2002)
have proved necessary and sufficient conditions for the ${\mathbb{R}}^d$-valued
process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$ to hit points and to have double
points. In this paper, we continue their research by determining the Hausdorff
and packing dimensions of the level sets and the sets of double times of the
random string process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$. We also consider
the Hausdorff and packing dimensions of the range and graph of the string.
http://arXiv.org/abs/math/0612700
http://front.math.ucdavis.edu/math.PR/0612700
(alternate)
Author(s): Shahar Mendelson and Joel Zinn
Abstract: We show that a modified Empirical process converges to the limiting Gaussian
process whenever the limit is continuous. The modification depends on the
properties of the limit via Talagrand's characterization of the continuity of
Gaussian processes.
http://arXiv.org/abs/math/0612703
http://front.math.ucdavis.edu/math.PR/0612703
(alternate)
Author(s): Richard Nickl
Abstract: We give several conditions for pregaussianity of norm balls of Besov spaces
defined over $\mathbb{R}^d$ by exploiting results in Haroske and Triebel
(2005). Furthermore, complementing sufficient conditions in Nickl and
P\"{o}tscher (2005), we give necessary conditions on the parameters of the
Besov space to obtain the Donsker property of such balls. For certain parameter
combinations Besov balls are shown to be pregaussian but not Donsker.
http://arXiv.org/abs/math/0612706
http://front.math.ucdavis.edu/math.PR/0612706
(alternate)
Author(s): Magda Peligrad and Sergey Utev
Abstract: In this paper we give simple sufficient conditions for linear type processes
with short memory that imply the invariance principle. Various examples
including projective criterion are considered as applications. In particular,
we treat the weak invariance principle for partial sums of linear processes
with short memory. We prove that whenever the partial sums of innovations
satisfy the $L_p$--invariance principle, then so does the partial sums of its
corresponding linear process.
http://arXiv.org/abs/math/0612707
http://front.math.ucdavis.edu/math.PR/0612707
(alternate)
Author(s): Jean Bertoin (PMA and DMA)
Abstract: A homogeneous mass-fragmentation, as it has been defined in \cite{RFC},
describes the evolution of the collection of masses of fragments of an object
which breaks down into pieces as time passes. Here, we show that this model can
be enriched by considering also the types of the fragments, where a type may
represent, for instance, a geometrical shape, and can take finitely many
values. In this setting, the dynamics of a randomly tagged fragment play a
crucial role in the analysis of the fragmentation. They are determined by a
Markov additive process whose distribution depends explicitly on the
characteristics of the fragmentation. As applications, we make explicit the
connexion with multitype branching random walks, and obtain multitype analogs
of the pathwise central limit theorem and large deviation estimates for the
empirical distribution of fragments.
http://arXiv.org/abs/math/0612710
http://front.math.ucdavis.edu/math.PR/0612710
(alternate)
Author(s): Adrian P.C. Lim
Abstract: A typical path integral on a manifold, $M$ is an informal expression of the
form \frac{1}{Z}\int_{\sigma \in H(M)} f(\sigma)
e^{-E(\sigma)}\mathcal{D}\sigma, \nonumber where $H(M)$ is a Hilbert manifold
of paths with energy $E(\sigma) < \infty$, $f$ is a real valued function on
$H(M)$, $\mathcal{D}\sigma$ is a \textquotedblleft Lebesgue measure
\textquotedblright and $Z$ is a normalization constant. For a compact
Riemannian manifold $M$, we wish to interpret $\mathcal{D}\sigma$ as a
Riemannian \textquotedblleft volume form \textquotedblright over $H(M)$,
equipped with its natural $G^{1}$ metric. Given an equally spaced partition,
${\mathcal{P}}$ of $[0,1],$ let $H_{{\mathcal{P}}%}(M)$ be the finite
dimensional Riemannian submanifold of $H(M) $ consisting of piecewise geodesic
paths adapted to $\mathcal{P.}$ Under certain curvature restrictions on $M,$ it
is shown that \[
\frac{1}{Z_{{\mathcal{P}}}}e^{-{1/2}E(\sigma)}dVol_{H_{{\mathcal{P}}}%
}(\sigma)\to\rho(\sigma)d\nu(\sigma)\text{as}\mathrm{mesh}%
({\mathcal{P}})\to0, \] where $Z_{{\mathcal{P}}}$ is a \textquotedblleft
normalization\textquotedblright constant, $E:H(M) \to\lbrack0,\infty)$ is the
energy functional, $Vol_{H_{{\mathcal{P}}%}}$ is the Riemannian volume measure
on $H_{\mathcal{P}}(M) ,$ $\nu$ is Wiener measure on continuous paths in $M,$
and $\rho$ is a certain density determined by the curvature tensor of $M.$
http://arXiv.org/abs/math/0612711
http://front.math.ucdavis.edu/math.PR/0612711
(alternate)
Author(s): Jos\'{e} E. Figueroa-L\'{o}pez and Christian Houdr\'{e}
Abstract: Estimation methods for the L\'{e}vy density of a L\'{e}vy process are
developed under mild qualitative assumptions. A classical model selection
approach made up of two steps is studied. The first step consists in the
selection of a good estimator, from an approximating (finite-dimensional)
linear model ${\mathcal{S}}$ for the true L\'{e}vy density. The second is a
data-driven selection of a linear model ${\mathcal{S}}$, among a given
collection $\{{\mathcal{S}}_m\}_{m\in {\mathcal{M}}}$, that approximately
realizes the best trade-off between the error of estimation within
${\mathcal{S}}$ and the error incurred when approximating the true L\'{e}vy
density by the linear model ${\mathcal{S}}$. Using recent concentration
inequalities for functionals of Poisson integrals, a bound for the risk of
estimation is obtained. As a byproduct, oracle inequalities and long-run
asymptotics for spline estimators are derived. Even though the resulting
underlying statistics are based on continuous time observations of the process,
approximations based on high-frequency discrete-data can be easily devised.
http://arXiv.org/abs/math/0612697
http://front.math.ucdavis.edu/math.ST/0612697
(alternate)
Author(s): Philippe Berthet and David M. Mason
Abstract: We demonstrate the strength of a coupling derived from a Gaussian
approximation of Zaitsev (1987a) by revisiting two strong approximation results
for the empirical process of Dudley and Philipp (1983), and using the coupling
to derive extended and refined versions of them.
http://arXiv.org/abs/math/0612701
http://front.math.ucdavis.edu/math.ST/0612701
(alternate)
Author(s): Miguel A. Arcones
Abstract: We find the Bahadur slope of the Lilliefors and Cram\'{e}r--von Mises tests
of normality.
http://arXiv.org/abs/math/0612708
http://front.math.ucdavis.edu/math.ST/0612708
(alternate)
Author(s): Kenneth S. Alexander
Abstract: We consider a polymer, with monomer locations modeled by the trajectory of an
underlying Markov chain, in the presence of a potential thatinteracts with the
polymer when it visits a particular site 0. Disorder is introduced by having
the interaction vary from one monomer to another, as a constant $u$ plus i.i.d.
mean-0 randomness. There is a critical value of $u$ above which the polymer is
pinned, placing a positive fraction (called the contact fraction) of its
monomers at 0 with high probability. When the excursions of the underlying
chain have a finite mean but no finite exponential moment, it is known that the
depinning transition (more precisely, the contact fraction) in the
corresponding annealed system is discontinuous. One generally expects the
presence of disorder to smooth transitions, and it was proved by Giacomin and
Toninelli that when the excursion length distribution has power-law tails, the
quenched system has a continuous transition even if the annealed system does
not. We show here that when the underlying chain is transient but the finite
part of the excursion length distribution has exponential tails, then the
depinning transition is discontinuous even in the quenched system, and the
quenched and annealed critical points are strictly different. By contrast, in
the recurrent case, the depinning behavior depends on the subexponential
prefactors on the exponential decay of the excursion length distribution, and
when these prefactors decay with an appropriate power law, the quenched
transition is continuous even though the annealed one is not.
http://arXiv.org/abs/math/0612625
http://front.math.ucdavis.edu/math.PR/0612625
(alternate)
Author(s): Tatyana S. Turova and Thomas Vallier
Abstract: We study a random graph model which is a superposition of the bond
percolation model on $Z^d$ with probability $p$ of an edge, and a classical
random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous}
random graph, has a natural relation to the so-called "rank 1 case" of {\it
inhomogeneous} random graphs. This allows us to use the newly developed theory
of inhomogeneous random graphs to describe the phase diagram on the set of
parameters $c\geq 0$ and $0 \leq p
http://arXiv.org/abs/math/0612644
http://front.math.ucdavis.edu/math.PR/0612644
(alternate)
Author(s): Aur\'{e}lien Alfonsi (CERMICS) and Benjamin Jourdain (CERMICS)
Abstract: It is well known that in models with time-homogeneous local volatility
functions and constant interest and dividend rates, the European Put prices are
transformed into European Call prices by the simultaneous exchanges of the
interest and dividend rates and of the strike and spot price of the underlying.
This paper investigates such a Call Put duality for perpetual American options.
It turns out that the perpetual American Put price is equal to the perpetual
American Call price in a model where, in addition to the previous exchanges
between the spot price and the strike and between the interest and dividend
rates, the local volatility function is modified. We prove that equality of the
dual volatility functions only holds in the standard Black-Scholes model with
constant volatility. Thanks to these duality results, we design a theoretical
calibration procedure of the local volatility function from the perpetual Call
and Put prices for a fixed spot price $x_0$. The knowledge of the Put (resp.
Call) prices for all strikes enables to recover the local volatility function
on the interval $(0,x_0)$ (resp. $(x_0,+\infty)$).
http://arXiv.org/abs/math/0612648
http://front.math.ucdavis.edu/math.PR/0612648
(alternate)
Author(s): Aur\'{e}lien Alfonsi (CERMICS) and Benjamin Jourdain (CERMICS)
Abstract: In this paper, we investigate the generalization of the Call-Put duality
equality obtained in [1] for perpetual American options when the Call-Put
payoff $(y-x)^+$ is replaced by $\phi(x,y)$. It turns out that the duality
still holds under monotonicity and concavity assumptions on $\phi$. The
specific analytical form of the Call-Put payoff only makes calculations easier
but is not crucial unlike in the derivation of the Call-Put duality equality
for European options. Last, we give some examples for which the optimal
strategy is known explicitly.
http://arXiv.org/abs/math/0612649
http://front.math.ucdavis.edu/math.PR/0612649
(alternate)
Author(s): Marie F. Kratz
Abstract: This paper presents a synthesis on the mathematical work done on level
crossings of stationary Gaussian processes, with some extensions. The main
results [(factorial) moments, representation into the Wiener Chaos, asymptotic
results, rate of convergence, local time and number of crossings] are
described, as well as the different approaches [normal comparison method, Rice
method, Stein-Chen method, a general $m$-dependent method] used to obtain them;
these methods are also very useful in the general context of Gaussian fields.
Finally some extensions [time occupation functionals, number of maxima in an
interval, process indexed by a bidimensional set] are proposed, illustrating
the generality of the methods. A large inventory of papers and books on the
subject ends the survey.
http://arXiv.org/abs/math/0612577
http://front.math.ucdavis.edu/math.PR/0612577
(alternate)
Author(s): Charles Bordenave and Serguei Foss and Vsevolod Shneer
Abstract: We analyse an aloha type access protocol where users have local interactions.
We establish that the fluid model of the system workload satisfies a
differential equation. We exhibit a sufficient condition on the stability of
this differential equation and deduce a sufficient condition for the stability
of the protocol. We discuss the necessary condition.
http://arXiv.org/abs/math/0612583
http://front.math.ucdavis.edu/math.PR/0612583
(alternate)
Author(s): David Croydon
Abstract: In this article, we prove global and local (point-wise) volume and heat
kernel bounds for the continuum random tree. We demonstrate that there are
almost-surely logarithmic global fluctuations and log-logarithmic local
fluctuations in the volume of balls of radius $r$ about the leading order
polynomial term as $r\to0$. We also show that the on-diagonal part of the heat
kernel exhibits corresponding global and local fluctuations as $t\to0$
almost-surely. Finally, we prove that this quenched (almost-sure) behaviour
contrasts with the local annealed (averaged over all realisations of the tree)
volume and heat kernel behaviour, which is smooth.
http://arXiv.org/abs/math/0612585
http://front.math.ucdavis.edu/math.PR/0612585
(alternate)
Author(s): Igor S. Borisov and Alexander A. Bystrov
Abstract: In the first part of the paper we study stochastic integrals of a nonrandom
function with respect to a nonorthogonal Hilbert noise defined on a semiring of
subsets of an arbitrary nonempty set. In the second part we apply this
construction to study limit behavior of canonical (i.e., degenerate) Von Mises
statistics based on weakly dependent stationary observations.
http://arXiv.org/abs/math/0612594
http://front.math.ucdavis.edu/math.PR/0612594
(alternate)
Author(s): Tom Kennedy
Abstract: The scaling limits of a variety of critical two-dimensional lattice models
are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the
parameter kappa. These lattice models have a natural parametrization of their
random curves given by the length of the curve. This parametrization (with
suitable scaling) should provide a natural parametrization for the curves in
the scaling limit. We conjecture that this parametrization is also given by a
type of fractal variation along the curve, and present Monte Carlo simulations
to support this conjecture. Then we show by simulations that if this fractal
variation is used to parametrize the SLE, then the parametrized curves have the
same distribution as the curves in the scaling limit of the lattice models with
their natural parametrization.
http://arXiv.org/abs/math/0612609
http://front.math.ucdavis.edu/math.PR/0612609
(alternate)
Author(s): Yoshiyasu Ishigami
Abstract: We show that the Ramsey number is linear for every uniform hypergraph with
bounded-degree. This is a hypergraph extension of the famous theorem for
ordinary graphs which Chv\'atal et al. showed in 1983. Our proof is simple,
contains the multicolor case, and provides a strong embedding lemma.
http://arXiv.org/abs/math/0612601
http://front.math.ucdavis.edu/math.CO/0612601
(alternate)
Author(s): Jonathan E. Taylor and Robert J. Adler
Abstract: We consider vector valued, unit variance Gaussian processes $y$ defined over
piecewise $C^2$ stratified manifolds $M$ and consider the geometry of their
(random) excursion sets $M\cap y^{-1}D$ for $D$ a stratified subset of
Euclidean space. In particular, we develop an explicit formula for the
expectation of all the Lipshitz-Killing curvatures of these sets. This formula
has an interpretation as a version of the classic kinematic fundamental formula
of Integral Geometry, in which integration over the isometry group with respect
to Haar measure is replaced by integration over a function space with respect
to an appropriate Gaussian measure.
Particularly novel is the method of proof, based on approximating the
Gaussian processes by processes on spheres, the orthonormal expansions of which
have (random) coefficients on the $n$-sphere. The $n\to\infty$ limit is handled
via recent extensions of the classic Poincar\'e limit theorem.
http://arXiv.org/abs/math/0612580
http://front.math.ucdavis.edu/math.DG/0612580
(alternate)
Author(s): Hari Bercovici and Jiun-Chau Wang
Abstract: We give a streamlined proof of the limit theorems for the free additive
convolution of infinitesimal triangular arrays of probability measures on the
real line. The result was first proved by Chistyakov and G\"otze using analytic
subordination.
http://arXiv.org/abs/math/0612599
http://front.math.ucdavis.edu/math.OA/0612599
(alternate)
Author(s): Dimitrios Cheliotis
Abstract: For a diffusion X_t in a one-dimensional Wiener medium W, it is known that
there is a certain process b_x(W) that depends only on the environment W, so
that X_t-b_{logt}(W) converges in distribution as t goes to infinity. We prove
that, modulo a relatively small time change, the process {b_x(W):x>0}is
followed closely by the process {F_X(e^x): x>0}, with F_X(t) denoting the point
with the most local time for the diffusion at time t.
http://arXiv.org/abs/math/0612533
http://front.math.ucdavis.edu/math.PR/0612533
(alternate)
Author(s): V. P. Maslov and T. V. Maslova
Abstract: The notions of real and user cardinality of a sign are introduced. Rank
distributions can be extended to arbitrary sign objects, i.e., semiotic
systems. The dynamics of the distribution of consumer durables, such as
automobiles, is studied.
http://arXiv.org/abs/math/0612540
http://front.math.ucdavis.edu/math.PR/0612540
(alternate)
Author(s): David Gamarnik and Sean Meyn
Abstract: One of the key performance measures in queueing systems is the exponential
decay rate of the steady-state tail probabilities of the queue lengths. It is
known that if a corresponding fluid model is stable and the stochastic
primitives have finite moments, then the queue lengths also have finite
moments, so that the tail probability \pr(\cdot >s) decays faster than s^{-n}
for any n. It is natural to conjecture that the decay rate is in fact
exponential.
In this paper an example is constructed to demonstrate that this conjecture
is false. For a specific stationary policy applied to a network with
exponentially distributed interarrival and service times it is shown that the
corresponding fluid limit model is stable, but the tail probability for the
buffer length decays slower than s^{-\log s}.
http://arXiv.org/abs/math/0612544
http://front.math.ucdavis.edu/math.PR/0612544
(alternate)
Author(s): Thomas M. Liggett and Rinaldo B. Schinazi and and Jason Schweinsberg
Abstract: Consider the following stochastic model for immune response. Each pathogen
gives birth to a new pathogen at rate $\lambda$. When a new pathogen is born,
it has the same type as its parent with probability $1 - r$. With probability
$r$, a mutation occurs, and the new pathogen has a different type from all
previously observed pathogens. When a new type appears in the population, it
survives for an exponential amount of time with mean 1, independently of all
the other types. All pathogens of that type are killed simultaneously. Schinazi
and Schweinsberg (2006) have shown that this model on $\Z^d$ behaves rather
differently from its non-spatial version. In this paper, we show that this
model on a homogeneous tree captures features from both the non-spatial version
and the $\Z^d$ version. We also obtain comparison results between this model
and the basic contact process on general graphs.
http://arXiv.org/abs/math/0612564
http://front.math.ucdavis.edu/math.PR/0612564
(alternate)
Author(s): Walid Hachem (LTCI) and Oleksiy Khorunzhiy and Philippe Loubaton (IGM-LabInfo), Jamal Najim (LTCI), Leonid Pastur
Abstract: This paper adresses the behaviour of the mutual information of correlated
MIMO Rayleigh channels when the numbers of transmit and receive antennas
converge to infinity at the same rate. Using a new and simple approach based on
Poincar\'{e}-Nash inequality and on an integration by parts formula, it is
rigorously established that the mutual information converges to a Gaussian
random variable whose mean and variance are evaluated. These results confirm
previous evaluations based on the powerful but non rigorous replica method. It
is believed that the tools that are used in this paper are simple, robust, and
of interest for the communications engineering community.
http://arXiv.org/abs/cs/0612076
http://front.math.ucdavis.edu/cs.IT/0612076
(alternate)
Author(s): Richard G. Clegg
Abstract: This paper gives an exact closed form solution for the expected queue length
at equilibrium of a G/D/1 discrete time queuing system in which the arrival
process is a specific Markov-modulated process. A system of equations is given
which can calculate the probability that the queue has a given length. The
results are tested in simulation.
http://arXiv.org/abs/math/0612476
http://front.math.ucdavis.edu/math.PR/0612476
(alternate)
Author(s): Vladislav Vysotsky
Abstract: Let S_i be a random walk with standard exponential increments. We call
\sum_{i=1}^k S_i its k-step area. The random variable V = \inf_{k \ge 1}
\frac{2}{k(k+1)} \sum_{i=1}^k S_i plays important role in the study of
so-called one-dimensional sticky particles model. We find the distribution of V
and prove that P(V > t) = \sqrt{1-t} exp(-t/2) for t in [0,1]. We also show
that the variables \min_{1 \le k \le n} \frac{2n}{k(k+1)} \sum_{i=1}^k U_{i, n}
converge in distribution to V. Here U_{i, n} are the order statistics of n
i.i.d. random variables uniformly distributed on [0,1].
http://arXiv.org/abs/math/0612490
http://front.math.ucdavis.edu/math.PR/0612490
(alternate)
Author(s): Gilles Pag\`{e}s (PMA)
Abstract: We propose a multi-step Richardson-Romberg extrapolation method for the
computation of expectations $E f(X_{_T})$ of a diffusion $(X_t)_{t\in [0,T]}$
when the weak time discretization error induced by the Euler scheme admits an
expansion at an order $R\ge 2$. The complexity of the estimator grows as $R^2$
(instead of $2^R$) and its variance is asymptotically controlled by considering
some consistent Brownian increments in the underlying Euler schemes. Some Monte
carlo simulations carried with path-dependent options (lookback, barriers)
which support the conjecture that their weak time discretization error also
admits an expansion (in a different scale). Then an appropriate
Richardson-Romberg extrapolation seems to outperform the Euler scheme with
Brownian bridge.
http://arXiv.org/abs/math/0612523
http://front.math.ucdavis.edu/math.PR/0612523
(alternate)
Author(s): Dirk Tasche
Abstract: Determining contributions by sub-portfolios or single exposures to
portfolio-wide economic capital for credit risk is an important risk
measurement task. Often economic capital is measured as Value-at-Risk (VaR) of
the portfolio loss distribution. For many of the credit portfolio risk models
used in practice, then the VaR contributions have to be estimated from Monte
Carlo samples. In the context of a partly continuous loss distribution (i.e.
continuous except for a positive point mass on zero), we investigate how to
combine kernel estimation methods with importance sampling to achieve more
efficient (i.e. less volatile) estimation of VaR contributions.
http://arXiv.org/abs/math/0612470
http://front.math.ucdavis.edu/math.ST/0612470
(alternate)
Author(s): Bo'az Klartag and Gady Kozma
Abstract: Let N > n, and denote by K the convex hull of N independent standard gaussian
random vectors in an n-dimensional Euclidean space. We prove that with high
probability, the isotropic constant of K is bounded by a universal constant.
Thus we verify the hyperplane conjecture for the class of gaussian random
polytopes.
http://arXiv.org/abs/math/0612517
http://front.math.ucdavis.edu/math.MG/0612517
(alternate)
Author(s): Gerard Ben Arous and Veronique Gayrard and Alexey Kuptsov
Abstract: We introduce here a new universality conjecture for levels of random
Hamiltonians, in the same spirit as the local REM conjecture made by S. Mertens
and H. Bauke. We establish our conjecture for a wide class of Gaussian and
non-Gaussian Hamiltonians, which include the $p$-spin models, the
Sherrington-Kirkpatrick model and the number partitioning problem. We prove
that our universality result is optimal for the last two models by showing when
this universality breaks down.
http://arXiv.org/abs/math/0612373
http://front.math.ucdavis.edu/math.PR/0612373
(alternate)
Author(s): Maury Bramson and Ofer Zeitouni
Abstract: In this paper, we study the tightness of solutions for a family of recursive
equations. These equations arise naturally in the study of random walks on
tree-like structures. Examples include the maximal displacement of branching
random walk in one dimension, and the cover time of symmetric simple random
walk on regular binary trees. Recursion equations associated with the
distribution functions of these quantities have been used to establish weak
laws of large numbers. Here, we use these recursion equations to establish the
tightness of the corresponding sequences of distribution functions after
appropriate centering. We phrase our results in a fairly general context, which
we hope will facilitate their application in other settings.
http://arXiv.org/abs/math/0612382
http://front.math.ucdavis.edu/math.PR/0612382
(alternate)
Author(s): Mark Adler & Pierre van Moerbeke
Abstract: This paper is a step in the direction of understanding the behavior of
non-intersecting Brownian motions on the real line, when the number of
particles becomes large.
Consider 2k non-intersecting Brownian motions, all starting at the origin,
such that the k left paths end up at -a and the k right paths end up at +a at
time t=1. The Karlin-McGregor formula enables one to express the transition
probability in terms of a matrix model, consisting of Gaussian Hermitian random
matrices in a chain with external source. It is shown that the log of the
probability for this model satisfies a fourth order PDE with a quartic
non-linearity, obtained by means of the 3-component KP hierarchy and Virasoro
constraints.
When the number of particles grows very large, the particles will be
concentrated on two intervals near t=0 and on one interval near t=1. The
Pearcey process is the infinite-dimensional diffusion, near the critical
transition from two to one interval. An appropriate scaling limit of the PDE
for the finite model leads to a non-linear PDE for the multi-time transition
probabilities of the Pearcey process.
We conjecture that each of the Markov clouds (like the Pearcey process)
arising near phase transitions is related to some integrable system. Moreover,
there is an intimate connection between the integrable system and the
associated Riemann-Hilbert problem.
http://arXiv.org/abs/math/0612393
http://front.math.ucdavis.edu/math.PR/0612393
(alternate)
Author(s): V. P. Maslov
Abstract: The notion of density of a finite set is discussed. We proof a general
theorem of set theory which refines Bose-Einstein distribution.
http://arXiv.org/abs/math/0612394
http://front.math.ucdavis.edu/math.PR/0612394
(alternate)
Author(s): Hamed Hatami
Abstract: A celebrated theorem of Friedgut says that every function $f:\{0,1\}^n \to
\{0,1\}$ can be approximated by a function $g:\{0,1\}^n \to \{0,1\}$ with
$\|f-g\|_2^2 \le \epsilon$ which depends only on $e^{O(I_f/\epsilon)}$
variables where $I_f$ is the sum of the influences of the variables of $f$.
Dinur and Friedgut later showed that this statement also holds if we replace
the discrete domain $\{0,1\}^n$ with the continuous domain $[0,1]^n$, under the
extra assumption that $f$ is monotone. They conjectured that the condition of
monotonicity is unnecessary and can be removed.
We show that certain constant-depth decision trees provide counter-examples
to Dinur-Friedgut conjecture. This suggests a reformulation of the conjecture
in which the function $g:[0,1]^n \to \{0,1\}$ instead of depending on a small
number of variables has a decision tree of small depth. In fact we prove this
reformulation by showing that the depth of the decision tree of $g$ can be
bounded by $e^{O(I_f/\epsilon^2)}$.
http://arXiv.org/abs/math/0612405
http://front.math.ucdavis.edu/math.PR/0612405
(alternate)
Author(s): S\'{e}bastien Darses (PMA) and Ivan Nourdin (PMA)
Abstract: We study the dynamical properties of the Brownian diffusions having $\sigma
{\rm Id}$ as diffusion coefficient matrix and $b=\nabla U$ as drift vector. We
characterize this class through the equality $D^2_+=D^2_-$, where $D_{+}$
(resp. $D_-$) denotes the forward (resp. backward) stochastic derivative of
Nelson's type. Our proof is based on a remarkable identity for $D_+^2-D_-^2$
and on the use of the martingale problem. We also give a new formulation of a
famous theorem of Kolmogorov concerning reversible diffusions. We finally
relate our characterization to some questions about the complex stochastic
embedding of the Newton equation which initially motivated of this work.
http://arXiv.org/abs/math/0612413
http://front.math.ucdavis.edu/math.PR/0612413
(alternate)
Author(s): K.D. Elworthy and Xue-Mei Li
Abstract: An L2 theory of differential forms is proposed for the Banach manifold of
continuous paths on Riemannian manifolds M furnished with its Brownian motion
measure. Differentiation must be restricted to certain Hilbert space
directions, the H-tangent vectors. To obtain a closed exterior differential
operator the relevant spaces of differential forms, the H-forms, are perturbed
by the curvature of M. A Hodge decomposition is given for L2 H-one-forms, and
the structure of H-two -forms is described. The dual operator d* is analysed in
terms of a natural connection on the H-tangent spaces. Malliavin calculus is a
basic tool.
http://arXiv.org/abs/math/0612416
http://front.math.ucdavis.edu/math.PR/0612416
(alternate)
Author(s): Jean-Philippe Anker (MAPMO) and Bruno Schapira (MAPMO and PMA) and Bartosz Trojan (MAPMO)
Abstract: We obtain a global estimate of the transition density $p^n(0,x)$ associated
to a nearest neighbor random walk, called here "simple", on affine buildings of
type $\widetilde{A}_r$. Then we deduce a global estimate of the Green function.
This is the analogue of a result on Riemannian symmetric spaces of the
noncompact type.
http://arXiv.org/abs/math/0612385
http://front.math.ucdavis.edu/math.CA/0612385
(alternate)
Author(s): Hamed Hatami and Michael Molloy
Abstract: We determine under which conditions certain natural models of random
constraint satisfaction problems have sharp thresholds of satisfiability. These
models include graph and hypergraph homomorphism, the $(d,k,t)$-model, and
binary constraint satisfaction problems with domain size 3.
http://arXiv.org/abs/math/0612391
http://front.math.ucdavis.edu/math.CO/0612391
(alternate)
Author(s): V. P. Maslov
Abstract: We give a risk-minimizing formula for government investments taking into
account the zero intelligence law for financial markets.
http://arXiv.org/abs/math/0612395
http://front.math.ucdavis.edu/math.GM/0612395
(alternate)
Author(s): Liqun Wang and Klaus P\"otzelberger
Abstract: We propose an approach to compute the boundary crossing probabilities for a
class of diffusion processes which can be expressed as piecewise monotone (not
necessarily one-to-one) functionals of a standard Brownian motion. This class
includes many interesting processes in real applications, e.g.,
Ornstein-Uhlenbeck, growth processes and geometric Brownian motion with time
dependent drift. This method applies to both one-sided and two-sided general
nonlinear boundaries, which may be discontinuous. Using this approach explicit
formulas for boundary crossing probabilities for certain nonlinear boundaries
are obtained, which are useful in evaluation and comparison of various
omputational algorithms. Moreover, numerical computation can be easily done by
Monte Carlo integration and the approximation errors for general boundaries are
automatically calculated. Some numerical examples are presented.
http://arXiv.org/abs/math/0612337
http://front.math.ucdavis.edu/math.PR/0612337
(alternate)
Author(s): Jir\^o Akahori and Takahiro Tsuchiya
Abstract: This paper gives examples of explicit arbitrage-free term structure models
with L\'evy jumps via state price density approach. By generalizing quadratic
Gaussian models, it is found that the probability density function of a L\'evy
process is a "natural" scale for the process to be the state variable of a
market.
http://arXiv.org/abs/math/0612341
http://front.math.ucdavis.edu/math.PR/0612341
(alternate)
Author(s): Christian houdr\'e and Trevis J. Litherland
Abstract: Let $X_1, X_2, ..., X_n, ... $ be a sequence of iid random variables with
values in a finite alphabet $\{1,...,m\}$. Let $LI_n$ be the length of the
longest increasing subsequence of $X_1, X_2, ..., X_n.$ We express the limiting
distribution of $LI_n$ as functionals of $m$ and $(m-1)$-dimensional Brownian
motions. These expressions are then related to similar functionals appearing in
queueing theory, allowing us to further establish asymptotic behaviors as $m$
grows. The finite alphabet results are then used to treat the countable
(infinite) alphabet.
http://arXiv.org/abs/math/0612364
http://front.math.ucdavis.edu/math.PR/0612364
(alternate)
Author(s): Zongxia Liang and Tusheng Zhang
Abstract: In this paper, we establish the existence of the solutions $ (X, L)$ of
reflected stochastic differential equations with possible anticipating initial
random variables. The key is to obtain some substitution formula for
Stratonovich integrals via a uniform convergence of the corresponding Riemann
sums.
http://arXiv.org/abs/math/0612294
http://front.math.ucdavis.edu/math.PR/0612294
(alternate)
Author(s): Marc Peign\'e and Wolfgang Woess
Abstract: Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected
random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n -
Y_{n+1}|$. In this note, we study recurrence of this process, extending a
previous criterion. This is obtained by determining an invariant measure of the
embedded process of reflections.
http://arXiv.org/abs/math/0612306
http://front.math.ucdavis.edu/math.PR/0612306
(alternate)
Author(s): Vincent Bansaye (PMA)
Abstract: We consider a generalized version in continuous time of the parking problem
of Knuth. Files arrive following a Poisson point process and are stored on a
hardware identified with the real line, at the right of their arrival point. We
study here the evolution of the extremities of the data block straddling 0,
which is empty at time 0 and is equal to $\RRR$ at a deterministic time.
http://arXiv.org/abs/math/0612312
http://front.math.ucdavis.edu/math.PR/0612312
(alternate)
Author(s): J. Creutzig and T. Mueller-Gronbach and K. Ritter
Abstract: We study optimal approximation of stochastic processes by polynomial splines
with free knots. The number of free knots is either a priori fixed or may
depend on the particular trajectory. For the $s$-fold integrated Wiener process
as well as for scalar diffusion processes we determine the asymptotic behavior
of the average $L_p$-distance to the splines spaces, as the (expected) number
$k$ of free knots tends to infinity.
http://arXiv.org/abs/math/0612313
http://front.math.ucdavis.edu/math.PR/0612313
(alternate)
Author(s): J.F. Le Gall and F. Paulin
Abstract: We prove that scaling limits of random planar maps which are uniformly
distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to
the two-dimensional sphere. Our methods rely on the study of certain random
geodesic laminations of the disk.
http://arXiv.org/abs/math/0612315
http://front.math.ucdavis.edu/math.PR/0612315
(alternate)
Author(s): Boris Tsirelson
Abstract: Motions of the plane (shifts and rotations) correspond to automorphisms of
the type I Arveson system of white noise. I prove that automorphisms
corresponding to rotations cannot be extended to the type II Arveson system of
Warren's noise.
http://arXiv.org/abs/math/0612303
http://front.math.ucdavis.edu/math.OA/0612303
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: A paper studies a closed queueing network containing a server station and $k$
client stations. The server station is an infinite server queueing system, and
client stations are single server queueing systems with autonomous service,
i.e. every client station serves customers (units) only at random instants
generated by strictly stationary and ergodic sequence of random variables. The
total number of units in the network is $N$. The expected times between
departures in client stations are $(N\mu_j)^{-1}$. After service completion in
the server station a unit is transmitted to the $j$th client station with
probability $p_{j}$ $(j=1,2,...,k)$, and being processed in the $j$th client
station the unit returns to server station. The network is assumed to be in
Markov environment. The Markov environment is defined by initial state, and
phase space of dimension $d$. Then the routing matrix $p_{j}$ as well as
transmission rates (which are expressed via parameters of the network) depend
on the Markov state of the environment. The paper studies the queue-length
processes in client stations of this network, and is aimed to analysis of
performance measures associated with this network. The questions risen in this
paper have immediate relation to quality control of complex telecommunication
networks.
http://arXiv.org/abs/math/0612224
http://front.math.ucdavis.edu/math.PR/0612224
(alternate)
Author(s): Vladislav Vysotsky
Abstract: Consider a particle moving through a random medium, which consists of
spherical obstacles, randomly distributed in R^d. The particle is accelerated
by a constant external field; when colliding with an obstacle, the particle
inelastically reflects. We study the asymptotics of X(t), which denotes the
position of the particle at time t, as t tends to infinity. The result is a
functional limit theorem for X(t).
http://arXiv.org/abs/math/0612253
http://front.math.ucdavis.edu/math.PR/0612253
(alternate)
Author(s): Bruno Schapira (MAPMO and PMA)
Abstract: The aim of this note is to describe the Poisson boundary of the group of
invertible triangular matrices with coefficients in a number field. It
generalizes to any dimension and to any number field a result of Brofferio
\cite{Bro} concerning the Poisson boundary of random rational affinities.
http://arXiv.org/abs/math/0612272
http://front.math.ucdavis.edu/math.PR/0612272
(alternate)
Author(s): Nicolas Bouleau (CERMICS) and Christophe Chorro (CERMICS and CERMSEM)
Abstract: This article proposes a link between statistics and the theory of Dirichlet
forms used to compute errors. The error calculus based on Dirichlet forms is an
extension of classical Gauss' approach to error propagation. The aim of this
paper is to derive error structures from measurements. The links with Fisher's
information lay the foundations of a strong connection with experiment. We show
that this connection behaves well towards changes of variables and is related
to the theory of asymptotic statistics.
http://arXiv.org/abs/math/0612258
http://front.math.ucdavis.edu/math.ST/0612258
(alternate)
Author(s): M. Anoussis and D. Gatzouras
Abstract: Let $G$ be a semi-direct product $G=A\times_\phi K$ with $A$ Abelian and $K$
compact. We characterize spread-out probability measures on $G$ that are mixing
by convolutions by means of their Fourier transforms. A key tool is a spectral
radius formula for the Fourier transform of a regular Borel measure on $G$ that
we develop, and which is analogous to the well-known Beurling--Gelfand spectral
radius formula. For spread-out probability measures on $G$, we also
characterize ergodicity by means of the Fourier transform of the measure.
Finally, we show that spread-out probability measures on such groups are mixing
if and only if they are weakly mixing.
http://arXiv.org/abs/math/0612262
http://front.math.ucdavis.edu/math.FA/0612262
(alternate)
Author(s): Hari Bercovici and Jiun-Chau Wang
Abstract: We determine the distributional behavior for products of free random
variables in a general infinitesimal triangular array. In the case of positive
variables, the main theorem extends a result proved earlier for arrays with
identically distributed rows. The case of unitary variables is considered as
well.
http://arXiv.org/abs/math/0612278
http://front.math.ucdavis.edu/math.OA/0612278
(alternate)
Author(s): Ming Li (Univ. of Waterloo and BioInformatics Solutions Inc.) and Xin Chen (Univ. California, Santa Barbara), Xin Li (Univ. Western Ontario), Bin
Ma (Univ. Western Ontario), Paul Vitanyi (CWI and Univ. of Amsterdam)
Abstract: A new class of distances appropriate for measuring similarity relations
between sequences, say one type of similarity per distance, is studied. We
propose a new ``normalized information distance'', based on the noncomputable
notion of Kolmogorov complexity, and show that it is in this class and it
minorizes every computable distance in the class (that is, it is universal in
that it discovers all computable similarities). We demonstrate that it is a
metric and call it the {\em similarity metric}. This theory forms the
foundation for a new practical tool. To evidence generality and robustness we
give two distinctive applications in widely divergent areas using standard
compression programs like gzip and GenCompress. First, we compare whole
mitochondrial genomes and infer their evolutionary history. This results in a
first completely automatic computed whole mitochondrial phylogeny tree.
Secondly, we fully automatically compute the language tree of 52 different
languages.
http://arXiv.org/abs/cs/0111054
http://front.math.ucdavis.edu/cs.CC/0111054
(alternate)
Author(s): Rudi Cilibrasi and Paul M.B. Vitanyi
Abstract: We consider the problem of constructing an an optimal-weight tree from the
3*(n choose 4) weighted quartet topologies on n objects, where optimality means
that the summed weight of the embedded quartet topologiesis optimal (so it can
be the case that the optimal tree embeds all quartets as non-optimal
topologies). We present a heuristic for reconstructing the optimal-weight tree,
and a canonical manner to derive the quartet-topology weights from a given
distance matrix. The method repeatedly transforms a bifurcating tree, with all
objects involved as leaves, achieving a monotonic approximation to the exact
single globally optimal tree. This contrasts to other heuristic search methods
from biological phylogeny, like DNAML or quartet puzzling, which, repeatedly,
incrementally construct a solution from a random order of objects, and
subsequently add agreement values.
http://arXiv.org/abs/cs/0606048
http://front.math.ucdavis.edu/cs.DS/0606048
(alternate)
Author(s): James Allen Fill and David B. Wilson
Abstract: We analyze the two-player game of Knock 'em Down, asymptotically as the
number of tokens to be knocked down becomes large. Optimal play requires mixed
strategies with deviations of order sqrt(n) from the naive law-of-large numbers
allocation. Upon rescaling by sqrt(n) and sending n to infinity, we show that
optimal play's random deviations always have bounded support and have marginal
distributions that are absolutely continuous with respect to Lebesgue measure.
http://arXiv.org/abs/math/0612205
http://front.math.ucdavis.edu/math.PR/0612205
(alternate)
Author(s): Pierre Andreoletti (MAPMO)
Abstract: We consider Sinai's random walk in random environment. We prove that the
logarithm of the local time is a good estimator of the random potential
associated to the random environment. We give a constructive method allowing us
to built the random environment from a single trajectory of the random walk.
http://arXiv.org/abs/math/0612209
http://front.math.ucdavis.edu/math.PR/0612209
(alternate)
Author(s): Jak\v{s}a Cvitani\'{c} and Robert Liptser and Boris Rozovskii
Abstract: This paper is concerned with nonlinear filtering of the coefficients in asset
price models with stochastic volatility. More specifically, we assume that the
asset price process $S=(S_{t})_{t\geq0}$ is given by \[
dS_{t}=m(\theta_{t})S_{t} dt+v(\theta_{t})S_{t} dB_{t}, \] where
$B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function and
$\theta=(\theta_{t})_{t\geq0}$ is a c\'{a}dl\'{a}g strong Markov process. The
random process $\theta$ is unobservable. We assume also that the asset price
$S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<....$ This is an
appropriate assumption when modeling high frequency financial data (e.g.,
tick-by-tick stock prices).
In the above setting the problem of estimation of $\theta$ can be approached
as a special nonlinear filtering problem with measurements generated by a
multivariate point process $(\tau_{k},\log S_{\tau_{k}})$. While quite natural,
this problem does not fit into the ``standard'' diffusion or simple point
process filtering frameworks and requires more technical tools. We derive a
closed form optimal recursive Bayesian filter for $\theta_{t}$, based on the
observations of $(\tau_{k},\log S_{\tau_{k}})_{k\geq1}$. It turns out that the
filter is given by a recursive system that involves only deterministic
Kolmogorov-type equations, which should make the numerical implementation
relatively easy.
http://arXiv.org/abs/math/0612212
http://front.math.ucdavis.edu/math.PR/0612212
(alternate)
Author(s): Arvind Singh (PMA)
Abstract: We consider a diffusion process $X$ in a random potential $\V$ of the form
$\V_x = \S_x -\delta x$ where $\delta$ is a positive drift and $\S$ is a
strictly stable process of index $\alpha\in (1,2)$ with positive jumps. Then
the diffusion is transient and $X_t / \log^\alpha t$ converges in law towards
an exponential distribution. This behaviour contrasts with the case where $\V$
is a drifted Brownian motion and provides an example of a transient diffusion
in a random potential which is as "slow" as in the recurrent setting.
http://arXiv.org/abs/math/0612220
http://front.math.ucdavis.edu/math.PR/0612220
(alternate)
Author(s): C. Giardin\'a and J. Kurchan and F. Redig
Abstract: We study a model of heat conduction with stochastic diffusion of energy. We
obtain a dual particle process which describes the evolution of all the
correlation functions. An exact expression for the covariance of the energy
exhibits long-range correlations in the presence of a current. We discuss the
formal connection of this model with the simple symmetric exclusion process.
http://arXiv.org/abs/cond-mat/0612198
http://front.math.ucdavis.edu/cond-mat/0612198
(alternate)
Author(s): T. Byczkowski and M. Ryznar and J. Malecki
Abstract: The purpose of this paper is to find explicit formulas for basic objects
pertaining the local potential theory of the operator $(I-\Delta)^{\alpha/2}$,
$0<\alpha<2$. The potential theory of this operator is based on Bessel
potentials $J_{\alpha}=(I-\Delta)^{-\alpha/2}$. We compute the {\it harmonic
measure} of the half-space and write a concise form of the corresponding {\it
Green function} for the operator $(I-\Delta)^{\alpha/2}$. To achieve this we
analyze the so-called {\it relativistic $\alpha$-stable process} on $\R^d$
space, killed when exiting the half-space. In terms of this process we are
dealing here with the 1-{\it potential theory} or, equivalently, potential
theory of Schr{\"o}dinger operator based on the generator of the process with
Kato's potential $q=-1$.
http://arXiv.org/abs/math/0612176
http://front.math.ucdavis.edu/math.PR/0612176
(alternate)
Author(s): Tamer Oraby
Abstract: To analyze the limiting spectral distribution of some random block-matrices,
Girko [Girko, 2000] uses a system of canonical equations from [Girko, 98]. In
this paper, we use the method of moments to give an integral form for the
almost sure limiting spectral distribution of such matrices.
http://arXiv.org/abs/math/0612177
http://front.math.ucdavis.edu/math.PR/0612177
(alternate)
Author(s): Marie-Amelie Morlais
Abstract: In this paper, we consider the classical problem of utility maximization in a
financial market allowing jumps. Assuming that the constraint set is a compact
set, rather than a convex one, we use a dynamic method from which we derive a
specific BSDE. We then aim at showing existence and uniqueness results for the
introduced BSDE. This allows us to give an explicit expression of the value
function and characterize optimal strategies for our problem.
http://arXiv.org/abs/math/0612181
http://front.math.ucdavis.edu/math.PR/0612181
(alternate)
Author(s): C. P. Hughes and A. Nikeghbali and M. Yor
Abstract: We prove a multidimensional extension of Selberg's central limit theorem for
the logarithm of the Riemann zeta function on the critical line. The limit is a
totally disordered process, whose coordinates are all independent and Gaussian.
http://arXiv.org/abs/math/0612195
http://front.math.ucdavis.edu/math.PR/0612195
(alternate)
Author(s): Victor Goodman (Indiana University)
Abstract: We introduce a transform on the class of stochastic exponentials for
d-dimensional Brownian motions. Each stochastic exponential generates another
stochastic exponential under the transform. The new exponential process is
often merely a supermartingale even in cases where the original process is a
martingale. We determine a necessary and sufficient condition for the transform
to be a martingale process. The condition links expected values of the
transformed stochastic exponential to the distribution function of certain
time-integrals.
http://arXiv.org/abs/math/0612160
http://front.math.ucdavis.edu/math.PR/0612160
(alternate)
Author(s): Michel Fliess (INRIA Futurs)
Abstract: This note is answering an old questioning about the F\'{e}nyes-Nelson
stochastic mechanics. The Brownian nature of the quantum fluctuations, which
are associated to this mechanics, is deduced from Feynman's interpretation of
the Heisenberg uncertainty principle via infinitesimal random walks stemming
from nonstandard analysis. It is therefore no more necessary to combine those
fluctuations with a background field, which has never been well understood.
Most of the technical details are contained in an extended english abstract.
http://arXiv.org/abs/quant-ph/0612033
http://front.math.ucdavis.edu/quant-ph/0612033
(alternate)
Author(s): Yves Le Jan (LM-Orsay)
Abstract: The purpose of this note is to explore some simple relations between loop
measures, determinants, and Gaussian Markov fields.
http://arXiv.org/abs/math/0612112
http://front.math.ucdavis.edu/math.PR/0612112
(alternate)
Author(s): Anders Johansson and Anders \"Oberg
Abstract: In this paper we study the one-sided shift operator on a state space defined
by a finite alphabet. Using a scheme developed by Walters [13], we prove that
the sequence of iterates of the transfer operator converges under square
summability of variations of the g-function, a condition which gave uniqueness
of a g-measure in [7]. We also prove uniqueness of so-called G-measures,
introduced by Brown and Dooley [2], under square summability of variations.
http://arXiv.org/abs/math/0612131
http://front.math.ucdavis.edu/math.DS/0612131
(alternate)
Author(s): I. Kontoyiannis and S.P. Meyn
Abstract: Suppose the expectation E(F(X)) is to be estimated by the empirical averages
of the values of F on independent and identically distributed samples {X_i}. A
sampling rule called the ``screened'' estimator is introduced, and its
performance is studied. When the mean E(U(X)) of a different function U is
known, the estimates are ``screened,'' in that we only consider those which
correspond to times when the empirical average of the {U(X_i)} is sufficiently
close to its known mean. As long as U dominates F appropriately, the screened
estimates admit exponential error bounds, even when F(X) is heavy-tailed. The
main results are several nonasymptotic, explicit exponential bounds for the
screened estimates. A geometric interpretation, in the spirit of Sanov's
theorem, is given for the fact that the screened estimates always admit
exponential error bounds, even if the standard estimates do not. And when they
do, the screened estimates' error probability has a significantly better
exponent. This implies that screening can be interpreted as a variance
reduction technique. Our main mathematical tools come from large deviations
techniques. The results are illustrated by a detailed simulation example.
http://arXiv.org/abs/math/0612040
http://front.math.ucdavis.edu/math.PR/0612040
(alternate)
Author(s): Elchanan Mossel and Sebastien Roch
Abstract: We prove and extend a conjecture of Kempe, Kleinberg, and Tardos (KKT) on the
spread of influence in social networks. A social network can be represented by
a directed graph where the nodes are individuals and the edges indicate a form
of social relationship. A simple way to model the diffusion of ideas,
innovative behavior, or ``word-of-mouth'' effects on such a graph is to
consider an increasing process of ``infected'' (or active) nodes: each node
becomes infected once an activation function of the set of its infected
neighbors crosses a certain threshold value. Such a model was introduced by KKT
in \cite{KeKlTa:03,KeKlTa:05} where the authors also impose several natural
assumptions: the threshold values are (uniformly) random; and the activation
functions are monotone and submodular. For an initial set of active nodes $S$,
let $\sigma(S)$ denote the expected number of active nodes at termination. Here
we prove a conjecture of KKT: we show that the function $\sigma(S)$ is
submodular under the assumptions above. We prove the same result for the
expected value of any monotone, submodular function of the set of active nodes
at termination.
http://arXiv.org/abs/math/0612046
http://front.math.ucdavis.edu/math.PR/0612046
(alternate)
Author(s): C. Cotar and V. Limic
Abstract: We consider a class of strongly edge reinforced random walks, where the
corresponding reinforcement weight function is non-decreasing. It is known by
Limic and Tarr\`es (2006) that the attracting edge emerges with probability 1,
whenever the underlying graph is locally bounded. We study the asymptotic
behavior of the tail distribution of the (random) time of attraction. In
particular, we obtain exact (up to multiplicative constant) asymptotics if the
underlying graph has two edges. Next we show some extensions in the setting of
finite and bounded degree infinite graphs. A nice corollary is that if the
reinforcement weight has the form $W(k) = k^\rho$, $\rho>1$, then (universally
over finite graphs) the expected time to attraction is infinite if and only if
$\rho \leq 1+ \frac{1+\sqrt{5}}{2}$.
http://arXiv.org/abs/math/0612048
http://front.math.ucdavis.edu/math.PR/0612048
(alternate)
Author(s): Andreas Eberle and Carlo Marinelli
Abstract: Sequential Monte Carlo Samplers are a class of stochastic algorithms for
Monte Carlo integral estimation w.r.t. probability distributions, which combine
elements of Markov chain Monte Carlo methods and importance sampling/resampling
schemes. We develop a stability analysis by functional inequalities for a
nonlinear flow of probability measures describing the limit behavior of the
algorithms as the number of particles tends to infinity. Stability results are
derived both under global and local assumptions on the generator of the
underlying Metropolis dynamics. This allows us to prove that the combined
methods sometimes have good asymptotic stability properties in multimodal
setups where traditional MCMC methods mix extremely slowly. For example, this
holds for the mean field Ising model at all temperatures.
http://arXiv.org/abs/math/0612074
http://front.math.ucdavis.edu/math.PR/0612074
(alternate)
Author(s): Dimitris Bertsimas and Natasha Bushueva
Abstract: Employing probabilistic techniques we compute best possible upper and lower
bounds on the price of an option on one or two assets with continuous piecewise
linear payoff function based on prices of simple call options of possibly
distinct maturities and the no-arbitrage condition, but without any assumption
on the price dynamics of underlying assets. We show that the problem reduces to
solving linear optimization problems that we explicitly characterize. We report
numerical results that illustrate the effectiveness of the algorithms we
develop.
http://arXiv.org/abs/math/0612075
http://front.math.ucdavis.edu/math.PR/0612075
(alternate)
Author(s): Konstantin Avrachenkov and Nelly Litvak and Kim Son Pham
Abstract: The choice of the PageRank damping factor is not evident. The Google's choice
for the value c=0.85 was a compromise between the true reflection of the Web
structure and numerical efficiency. However, the Markov random walk on the
original Web Graph does not reflect the importance of the pages because it
absorbs in dead ends. Thus, the damping factor is needed not only for speeding
up the computations but also for establishing a fair ranking of pages. In this
paper, we propose new criteria for choosing the damping factor, based on the
ergodic structure of the Web Graph and probability flows. Specifically, we
require that the core component receives a fair share of the PageRank mass.
Using singular perturbation approach we conclude that the value c=0.85 is too
high and suggest that the damping factor should be chosen around 1/2. As a
by-product, we describe the ergodic structure of the OUT component of the Web
Graph in detail. Our analytical results are confirmed by experiments on two
large samples of the Web Graph.
http://arXiv.org/abs/math/0612079
http://front.math.ucdavis.edu/math.PR/0612079
(alternate)
Author(s): Christophe Bahadoran
Abstract: We consider asymmetric attractive particle systems with product invariant
measures in any space dimension. We show that, in the presence of open
boundaries, the hydrodynamic limit is a scalar conservation law with boundary
conditions in the sense defined by Bardos, Leroux and N\'{e}d\'{e}lec. When the
boundaries are parallel hyperplanes, we establish a large-time convergence
result for the entropy solution and derive the stationary profile for the
particle system. Models include current-density relations with arbitrarily many
maxima and minima.
http://arXiv.org/abs/math/0612094
http://front.math.ucdavis.edu/math.PR/0612094
(alternate)
Author(s): Joel A. Tropp
Abstract: This note presents a new proof of an important result due to Bourgain and
Tzafriri that provides a partial solution to the Kadison--Singer problem. The
result shows that every unit-norm matrix whose entries are relatively small in
comparison with its dimension can be paved by a partition of constant size.
That is, the coordinates can be partitioned into a constant number of blocks so
that the restriction of the matrix to each block of coordinates has norm less
than one half. The original proof of Bourgain and Tzafriri involves a long,
delicate calculation. The new proof relies on the systematic use of
symmetrization and Khintchine inequalities to estimate the norm of some random
matrices.
http://arXiv.org/abs/math/0612070
http://front.math.ucdavis.edu/math.MG/0612070
(alternate)
Author(s): Jacques Franchi
Abstract: K. G\"odel [G] discovered his celebrated solution to Einstein equations in
1949. Additional contributions were made by Kundt [K] and Hawking-Ellis
([H-E],5.7). On the other hand, a general Lorentz invariant operator,
associated to the so-called "relativistic diffusion'', and making sense in any
Lorentz manifold, was introduced by Franchi-Le Jan in [F-LJ]. Here is purposed
a first study of the relativistic diffusion in the framework of G\"odel's
universe, which contains matter.
http://arXiv.org/abs/math/0612020
http://front.math.ucdavis.edu/math.PR/0612020
(alternate)
Author(s): Franck Barthe (LSProba) and Cyril Roberto (LAMA)
Abstract: We provide a sufficient condition for a measure on the real line to satisfy a
modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov
and G\"{o}tze. Under mild assumptions the condition is also necessary.
Concentration inequalities are derived. This completes the picture given in
recent contributions by Gentil, Guillin and Miclo.
http://arXiv.org/abs/math/0612026
http://front.math.ucdavis.edu/math.PR/0612026
(alternate)
Author(s): Victor Goodman and Kyounghee Kim
Abstract: We find a simple expression for the probability density of $\int \exp (B_s -
s/2) ds$ in terms of its distribution function and the distribution function
for the time integral of $\exp (B_s + s/2)$. The relation is obtained with a
change of measure argument where expectations over events determined by the
time integral are replaced by expectations over the entire probability space.
We develop precise information concerning the lower tail probabilities for
these random variables as well as for time integrals of geometric Brownian
motion with arbitrary constant drift. In particular, $E[ \exp\big(\theta / \int
\exp (B_s)ds\big) ]$ is finite iff $\theta < 2$. We present a new formula for
the price of an Asian call option.
http://arXiv.org/abs/math/0612034
http://front.math.ucdavis.edu/math.PR/0612034
(alternate)
Author(s): Victor Goodman and Kyounghee Kim
Abstract: We construct a no-arbitrage model of bond prices where the long bond is used
as a numeraire. We develop bond prices and their dynamics without developing
any model for the spot rate or forward rates. The model is arbitrage free and
all nominal interest rates remain positive in the model. We give examples where
our model does not have a spot rate; other examples include both spot and
forward rates.
http://arXiv.org/abs/math/0612035
http://front.math.ucdavis.edu/math.PR/0612035
(alternate)
Author(s): Jesper Lykke Jacobsen (LPTMS and SPhT) and Hubert Saleur (SPhT)
Abstract: We study a model of densely packed self-avoiding loops on the annulus,
related to the Temperley Lieb algebra with an extra idempotent boundary
generator. Four different weights are given to the loops, depending on their
homotopy class and whether they touch the outer rim of the annulus. When the
weight of a contractible bulk loop x = q + 1/q satisfies -2 < x <= 2, this
model is conformally invariant for any real weight of the remaining three
parameters. We classify the conformal boundary conditions and give exact
expressions for the corresponding boundary scaling dimensions. The amplitudes
with which the sectors with any prescribed number and types of non contractible
loops appear in the full partition function Z are computed rigorously. Based on
this, we write a number of identities involving Z which hold true for any
finite size. When the weight of a contractible boundary loop y takes certain
discrete values, y_r = [r+1]_q / [r]_q with r integer, other identities
involving the standard characters K_{r,s} of the Virasoro algebra are
established. The connection with Dirichlet and Neumann boundary conditions in
the O(n) model is discussed in detail, and new scaling dimensions are derived.
When q is a root of unity and y = y_r, exact connections with the A_m type RSOS
model are made. These involve precise relations between the spectra of the loop
and RSOS model transfer matrices, valid in finite size. Finally, the results
where y=y_r are related to the theory of Temperley Lieb cabling.
http://arXiv.org/abs/math-ph/0611078
http://front.math.ucdavis.edu/math-ph/0611078
(alternate)
Author(s): R. Husseini and M. Kassmann
Abstract: Markov chain approximations of symmetric jump processes are investigated.
Tightness results and a central limit theorem are established. Moreover, given
the generator of a symmetric jump process with state space $\mathbbm{R}^d$ the
approximating Markov chains are constructed explicitly. As a byproduct we
obtain a definition of the Sobolev space $H^{\alpha/2}(\mathbbm{R}^d)$, $\alpha
\in (0,2)$, that is equivalent to the standard one.
http://arXiv.org/abs/math/0611934
http://front.math.ucdavis.edu/math.PR/0611934
(alternate)
Author(s): Jean-Fran\c{c}ois Delmas (CERMICS) and Benjamin Jourdain (CERMICS)
Abstract: The waste-recycling Monte Carlo (WR) algorithm, introduced by Frenkel, is a
modification of the Metropolis-Hastings algorithm, which makes use of all the
proposals, whereas the standard Metropolis-Hastings algorithm only uses the
accepted proposals. We prove the convergence of the WR algorithm and its
asymptotic normality. We give an example which shows that in general the WR
algorithm is not asymptotically better than the Metropolis-Hastings algorithm :
the WR algorithm can have an asymptotic variance larger than the one of the
Metropolis-Hastings algorithm. However, in the particular case of the
Metropolis-Hastings algorithm called Boltzmann algorithm, we prove that the WR
algorithm is asymptotically better than the Metropolis-Hastings algorithm.
http://arXiv.org/abs/math/0611949
http://front.math.ucdavis.edu/math.PR/0611949
(alternate)
Author(s): Emmanuel Candes and Justin Romberg
Abstract: We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from a
limited number of linear measurements. Given $m$ randomly selected samples of
$U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization
recovers $x^0$ exactly when the number of measurements exceeds \[ m\geq
\mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n, \] where $S$ is the number of
nonzero components in $x^0$, and $\mu$ is the largest entry in $U$ properly
normalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $\mu$,
the fewer samples needed.
The result holds for ``most'' sparse signals $x^0$ supported on a fixed (but
arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on
$T$ and the observed values of $Ux^0$ are drawn at random, the signal is
recovered with overwhelming probability. Moreover, there is a sense in which
this is nearly optimal since any method succeeding with the same probability
would require just about this many samples.
http://arXiv.org/abs/math/0611957
http://front.math.ucdavis.edu/math.ST/0611957
(alternate)
Author(s): Sourav Chatterjee and Ron Peled and Yuval Peres and Dan Romik
Abstract: For d>=3, we construct a non-randomized, fair and translation-equivariant
allocation of Lebesgue measure to the points of a standard Poisson point
process in R^d, defined by allocating to each of the Poisson points its basin
of attraction with respect to the flow induced by a gravitational force field
exerted by the points of the Poisson process. We prove that this allocation
rule is economical in the sense that the "allocation diameter", defined as the
diameter X of the basin of attraction containing the origin, is a random
variable with a rapidly decaying tail. Specifically, we have the tail bound:
P(X > R) < C exp[ -c R(log R)^(alpha_d) ], for all R>2, where: alpha_d =
(d-2)/d for d>=4; alpha_3 can be taken as any number <-4/3; and C,c are
positive constants that depend on d and alpha_d. This is the first construction
of an allocation rule of Lebesgue measure to a Poisson point process with
subpolynomial decay of the tail P(X>R).
http://arXiv.org/abs/math/0611886
http://front.math.ucdavis.edu/math.PR/0611886
(alternate)
Author(s): Yulia Mishura and Esko Valkeila
Abstract: We extend the classical Levy characterization of Brownian motion to
fractional Brownian motion.
http://arXiv.org/abs/math/0611913
http://front.math.ucdavis.edu/math.PR/0611913
(alternate)
Author(s): Brice Franke
Abstract: We prove a functional non-central limit theorem for jump-diffusions with
periodic coefficients driven by strictly stable Levy-processes with stability
index bigger than one. The limit process turns out to be a strictly stable Levy
process with an averaged jump-measure. Unlike in the situation where the
diffusion is driven by Brownian motion, there is no drift related enhancement
of diffusivity.
http://arXiv.org/abs/math/0611852
http://front.math.ucdavis.edu/math.PR/0611852
(alternate)
Author(s): Elise Janvresse (LMRS) and Beno\^{i}t Rittaud (IG) and Thierry De La Rue (LMRS)
Abstract: We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and
for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} =
|F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and
either + with probability $p$ or - with probability $1-p$ ($0
http://arXiv.org/abs/math/0611860
http://front.math.ucdavis.edu/math.PR/0611860
(alternate)
Author(s): Nizar Demni (PMA)
Abstract: In this paper, we study complex Wishart processes or the so-called Laguerre
processes. We give some interest to the behaviour of the eigenvalues process,
derive some useful SDE and compute both infinitesimal generator and semi-group.
We also give absolute-continuity relations between different indices.Then, we
compute the density function of the generalised Hartman-Watson law as well as
the law of the first hitting time of 0 when the size m=2.
http://arXiv.org/abs/math/0611863
http://front.math.ucdavis.edu/math.PR/0611863
(alternate)
Author(s): Shige Peng and Mingyu Xu
Abstract: Here we study a new numerical method for BSDE. Then we present a package of
our numerical algorithms of BSDE with convenient user--machine interface. This
package permit us to solve BSDE, reflected BSDE with one or two barriers as
well as BSDE with constraints. One of significant advantages of this package is
that users have a very convenient interface. Any users who know the ABC of BSDE
can use this package very easily. The interface of the input-output was also
carefully designed.
http://arXiv.org/abs/math/0611864
http://front.math.ucdavis.edu/math.PR/0611864
(alternate)
Author(s): Jacques Franchi
Abstract: Consider $ G:= PSL_2(\R)\equiv T^1\H^2$, a modular group $ \Gamma$, and the
homogeneous space $ \Gamma\sm G \equiv T^1(\Gamma\sm\H^2)$. Endow $ G $, and
then $ \Gamma\sm G $, with a canonical left-invariant metric, thereby equipping
it with a quasi hyperbolic geometry. Windings around handles and cusps of $
\Gamma\sm G $ are calculated by integrals of closed 1-forms of $ \Gamma\sm G $.
The main results express, in both Brownian and geodesic cases, the joint
convergence of the law of these integrals, with a stress on the asymptotic
independence between slow and fast windings. The non-hyperbolicity of $
\Gamma\sm G $ is responsible for a difference between the Brownian and geodesic
asymptotic behaviours, difference which does not exist at the level of the
Riemann surface $\Gamma\sm\H^2$ (and generally in hyperbolic cases).
Identification of the cohomology classes of closed 1-forms and with harmonic
1-forms, and equidistribution of large geodesic spheres, are also addressed.
http://arXiv.org/abs/math/0611866
http://front.math.ucdavis.edu/math.PR/0611866
(alternate)
Author(s): F. L. Toninelli (ENS Lyon and Umr--CNRS 5672)
Abstract: We consider models of directed polymers interacting with a one-dimensional
defect line on which random charges are placed. More abstractly, one starts
from renewal sequence on Z and gives a random (site-dependent) reward or
penalty to the occurrence of a renewal at any given point of Z. These models
are known to undergo a delocalization-localization transition, and the free
energy $\tf$ vanishes when the critical point is approached from the localized
region. We prove that the quenched correlation length $\xi$, defined as the
inverse of the rate of exponential decay of the two-point function, does not
diverge faster than $ 1/F$. We prove a lower bound also for the rate of
exponential decay of the disorder-averaged two-point function. We discuss how,
in the particular case where disorder is absent, this result can be seen as a
refinement of the classical renewal theorem, for a specific class of renewal
sequences.
http://arXiv.org/abs/math/0611868
http://front.math.ucdavis.edu/math.PR/0611868
(alternate)
Author(s): Shige Peng and Mingyu Xu
Abstract: In this paper, we study a type of reflected BSDE with a constraint and
introduce a new kind of nonlinear expectation via BSDE with a constraint and
prove the Doob-Meyer decomposition with respect to the super(sub)martingale
introduced by this nonlinear expectation. We then apply the results to the
pricing of American options in incomplete market.
http://arXiv.org/abs/math/0611869
http://front.math.ucdavis.edu/math.PR/0611869
(alternate)
Author(s): Mingyu Xu
Abstract: In this paper, we study the reflected BSDE with one continuous barrier, under
the monotonicity and general increasing condition on $y$ and non Lipschitz
condition on $z$. We prove the existence and uniqueness of the solution to
these equation by approximation method.
http://arXiv.org/abs/math/0611870
http://front.math.ucdavis.edu/math.PR/0611870
(alternate)
Author(s): S Satheesh and E Sandhya and K E Rajasekharan
Abstract: Generalizations and extensions of a first order autoregressive model of
Lawrance and Lewis (1981) are considered and characterized here.
http://arXiv.org/abs/math/0611878
http://front.math.ucdavis.edu/math.PR/0611878
(alternate)
Author(s): Javiera Barrera and Joaquin Fontbona
Abstract: We prove a law of large numbers for certain finite random partitions of
$[0,1]$, when the number of fragments go to $\infty$. Then, we apply it to
compute the limiting distribution of the transient search-cost of the
move-to-front rule for general classes of random and deterministic request
probabilities, when the list size goes to $\infty$.
http://arXiv.org/abs/math/0611882
http://front.math.ucdavis.edu/math.PR/0611882
(alternate)
Author(s): Nizar Demni (PMA) and Marguerite Zani (LAMA)
Abstract: In this paper, we derive a handable expression for the Jacobi process semi
group which is given by a bilinear series involving Jacobi polynomials. Our
attempt uses a subordination of the considered process by means of a suitable
random change. Once we did, we will be able, in the ultraspheric case, to
derive a LDP for a family of estimators based on a single trajectory of the
process.
http://arXiv.org/abs/math/0611884
http://front.math.ucdavis.edu/math.PR/0611884
(alternate)
Author(s): Julien Tailleur and Jorge Kurchan
Abstract: The transition from order to chaos has been a major subject of research since
the work of Poincare, as it is relevant in areas ranging from the foundations
of statistical physics to the stability of the solar system. Along this
transition, atypical structures like the first chaotic regions to appear, or
the last regular islands to survive, play a crucial role in many physical
situations. For instance, resonances and separatrices determine the fate of
planetary systems, and localised objects like solitons and breathers provide
mechanisms of energy transport in nonlinear systems such as Bose-Einstein
condensates and biological molecules. Unfortunately, despite the fundamental
progress made in the last years, most of the numerical methods to locate these
'rare' trajectories are confined to low-dimensional or toy models, while the
realms of statistical physics, chemical reactions, or astronomy are still hard
to reach. Here we implement an efficient method that allows one to work in
higher dimensions by selecting trajectories with unusual chaoticity. As an
example, we study the Fermi-Pasta-Ulam nonlinear chain in equilibrium and show
that the algorithm rapidly singles out the soliton solutions when searching for
trajectories with low level of chaoticity, and chaotic-breathers in the
opposite situation. We expect the scheme to have natural applications in
celestial mechanics and turbulence, where it can readily be combined with
existing numerical methods
http://arXiv.org/abs/cond-mat/0611672
http://front.math.ucdavis.edu/cond-mat/0611672
(alternate)
Author(s): David Pollard
Abstract: The paper reexamines an argument by Talagrand that leads to a remarkable
exponential tail bound for the concentration of probability near a set. The
main novelty is the replacement of a mysterious Calculus inequality by an
application of Jensen's inequality.
http://arXiv.org/abs/math/0611770
http://front.math.ucdavis.edu/math.PR/0611770
(alternate)
Author(s): Jomy Alappattu and Jim Pitman
Abstract: Starting from a sequence regarded as a walk through some set of values, we
consider the associated loop-erased walk as a sequence of directed edges, with
an edge from $i$ to $j$ if the loop erased walk makes a step from $i$ to $j$.
We introduce a coloring of these edges by painting edges with a fixed color as
long as the walk does not loop back on itself, then switching to a new color
whenever a loop is erased, with each new color distinct from all previous
colors. The pattern of colors along the edges of the loop-erased walk then
displays stretches of consecutive steps of the walk left untouched by the
loop-erasure process. Assuming that the underlying sequence generating the
loop-erased walk is a sequence of independent random variables, each uniform on
$[N]:=\{1, 2, ..., N\}$, we condition the walk to start at $N$ and stop the
walk when it first reaches the subset $[k]$, for some $1 \leq k \leq N-1$. We
relate the distribution of the random length of this loop-erased walk to the
distribution of the length of the first loop of the walk, via Cayley's
enumerations of trees, and via Wilson's algorithm. For fixed $N$ and $k$, and
$i = 1,2, ...$, let $B_i$ denote the event that the loop-erased walk from $N$
to $[k]$ has $i +1$ or more edges, and the $i^{th}$ and $(i+1)^{th}$ of these
edges are colored differently. We show that given that the loop-erased random
walk has $j$ edges for some $1\leq j \leq N-k$, the events $B_i$ for $1 \leq i
\leq j-1$ are independent, with the probability of $B_i$ equal to $1/(k+i+1)$.
This determines the distribution of the sequence of random lengths of
differently colored segments of the loop-erased walk, and yields asymptotic
descriptions of these random lengths as $N \to \infty$.
http://arXiv.org/abs/math/0611775
http://front.math.ucdavis.edu/math.PR/0611775
(alternate)
Author(s): Maurice J. Dupre and Frank J. Tipler
Abstract: We give a proof of Cox's Theorem on the product rule and sum rule for
conditional plausibility without assuming continuity or differentiablity of
plausibility. Instead, we extend the notion of plausibility to apply to
unknowns giving them plausible values. Our proof is enormously simpler than
others that have recently appeared in the literature, yet completely rigorous.
For example, we do not need to investigate the 11 possibilities for conditional
plausibilities as described on page 25 of Jaynes' recent book Probability
Theory.
http://arXiv.org/abs/math/0611795
http://front.math.ucdavis.edu/math.PR/0611795
(alternate)
Author(s): Peter Caithamer and Anna Karczewska
Abstract: We consider convolution-type stochastic Volterra equations with additive
Hilbert-valued fractional Brownian motion, $0
http://arXiv.org/abs/math/0611832
http://front.math.ucdavis.edu/math.PR/0611832
(alternate)
Author(s): M. D. Jara
Abstract: We obtain the equilibrium fluctuations for the empirical density of particles
for the zero-range process in the Sierpinski gasket. The limiting process is a
generalized Ornstein-Uhlenbeck process generated by the Neumann Laplacian and
its corresponding Dirichlet form on the gasket.
http://arXiv.org/abs/math/0611836
http://front.math.ucdavis.edu/math.PR/0611836
(alternate)
Author(s): Stefano Iacus and Masayuki Uchida and Nakahiro Yoshida
Abstract: A one dimensional diffusion process $X=\{X_t, 0\leq t \leq T\}$ is observed
only when its path lies over some threshold $\tau$. On the basis of the
observable part of the trajectory, the problem is to estimate finite
dimensional parameter in both drift and diffusion coefficient under a discrete
sampling scheme. It is assumed that the sampling occurs at regularly spaced
times intervals of length $h_n$ such that $h_n\cdot n =T$. The asymptotic is
considered as $T\to\infty$, $n\to\infty$, $n h_n^2\to 0$. Consistency and
asymptotic normality for estimators of parameters in both drift and diffusion
coefficient is proved.
http://arXiv.org/abs/math/0611781
http://front.math.ucdavis.edu/math.ST/0611781
(alternate)
Author(s): L. Bertini and S. Brassesco and P. Butt\`a
Abstract: We consider the van der Waals free energy functional in a bounded interval
with inhomogeneous Dirichlet boundary conditions imposing the two stable phases
at the endpoints. We compute the asymptotic free energy cost, as the length of
the interval diverges, of shifting the interface from the midpoint. We then
discuss the effect of thermal fluctuations by analyzing the \phi^4_1-measure
with Dobrushin boundary conditions. In particular, we obtain a nontrivial limit
in a suitable scaling in which the length of the interval diverges and the
temperature vanishes. The limiting state is not translation invariant and
describes a localized interface. This result can be seen as the probabilistic
counterpart of the variational convergence of the associated excess free
energy.
http://arXiv.org/abs/math-ph/0611077
http://front.math.ucdavis.edu/math-ph/0611077
(alternate)
Author(s): Daniela Bertacchi and Gustavo Posta and Fabio Zucca
Abstract: We study a generalized branching random walk where particles breed at a rate
which depends on the number of neighbouring particles. Under general
assumptions on the breeding rates we prove the existence of a phase where the
population survives without exploding. We construct a non trivial invariant
measure for this case.
http://arXiv.org/abs/math/0611720
http://front.math.ucdavis.edu/math.PR/0611720
(alternate)
Author(s): J. Beltr\'an and C. Landim
Abstract: We recover the Navier-Stokes equation as the incompressible limit of a
stochastic lattice gas in which particles are allowed to jump over a mesoscopic
scale. The result holds in any dimension assuming the existence of a smooth
solution of the Navier-Stokes equation in a fixed time interval. The proof does
not use non-gradient methods or the multi-scale analysis due to the long range
jumps.
http://arXiv.org/abs/math/0611721
http://front.math.ucdavis.edu/math.PR/0611721
(alternate)
Author(s): L. Mytnik and J. Villa
Abstract: The existence of self-intersection local time (SILT), when the time diagonal
is intersected, of the $(\alpha,d,\beta)$-superprocess is proved for
$d/2<\alpha $ and for a renormalized SILT when $d/(2+(1+\beta)^{-1})<\alpha
\leq d/2$. We also establish Tanaka-like formula for SILT.
http://arXiv.org/abs/math/0611727
http://front.math.ucdavis.edu/math.PR/0611727
(alternate)
Author(s): Majid Hosseini and Krishnamurthi Ravishankar
Abstract: We introduce a new self-interacting random walk on the integers in a dynamic
random environment and show that it converges to a pure diffusion in the
scaling limit. We also find a lower bound on the diffusion coefficient in some
special cases. With minor changes the same argument can be used to prove the
scaling limit of the corresponding walk in Z^d.
http://arXiv.org/abs/math/0611734
http://front.math.ucdavis.edu/math.PR/0611734
(alternate)
Author(s): Bruno Nietlispach
Abstract: We prove an analogue of the Kubilius main theorem for quasi-logarithmic
structures. This result extends the corresponding theorem of Arratia, Barbour
and Tavar\'{e} (2003) in the context of logarithmic structures, and of Zhang
(1996) in the context of additive arithmetic semigroups. In particular, our
theorem is valid for additive arithmetic semigroups where non-classical
``Beurling type'' prime number theorems hold true.
http://arXiv.org/abs/math/0611747
http://front.math.ucdavis.edu/math.PR/0611747
(alternate)
Author(s): Andrey A Dorogovtsev
Abstract: The article contains description of the functionals from the family of
coalescing Brownian particles. New type of the stochastic integral is
introduced and used.
http://arXiv.org/abs/math/0611748
http://front.math.ucdavis.edu/math.PR/0611748
(alternate)
Author(s): Andrey A Dorogovtsev
Abstract: This article is devoted to the stochastic anticipating equations with the
extended stochastic integral with respect to the Gaussian processes of a
special type and its application to the smoothing problem in the case when
noise is represented by the two jointly Gaussian Wiener processes, which can
have not a semimartingale property with respect to the joint filtration.
http://arXiv.org/abs/math/0611749
http://front.math.ucdavis.edu/math.PR/0611749
(alternate)
Author(s): Andrey A Dorogovtsev
Abstract: The weak limits of the measure-valued processes organized as a mass carried
by the interacting Brownian particles are described. As a limiting flow the
Arrattia flow is obtained.
http://arXiv.org/abs/math/0611750
http://front.math.ucdavis.edu/math.PR/0611750
(alternate)
Author(s): Andrey A Dorogovtsev
Abstract: This article is devoted to the stochastic anticipating equations with the
extended stochastic integral with respect to the Gaussian processes of a
special type. In the particular cases the solutions of such an equations are
the well-known Wiener functionals after the second quantization. As an
application the stochastic Kolmogorov equation for the conditional
distributions of the diffusion process is obtained. Also we will consider the
conditional variant of the Feynman--Kac formula. The two last sections of the
article are devoted to the smoothing problem in the case when noise is
represented by the two jointly Gaussian Wiener processes, which can have not a
semimartingale property with respect to the joint filtration.
http://arXiv.org/abs/math/0611751
http://front.math.ucdavis.edu/math.PR/0611751
(alternate)
Author(s): Arnaud De La Fortelle (CAOR) and Jean-Marc Lasgouttes (INRIA Rocquencourt), Cyril Furtlehner (INRIA Rocquencourt)
Abstract: Concepts and techniques from statistical physics inspired a new method for
traffic prediction. This method is particularly suitable in settings where the
only information available is floating car data. We propose a system, based on
the Ising model of statistical physics, which both reconstructs and predicts
the traffic in real time using a message-passing algorithm.
http://arXiv.org/abs/math/0611757
http://front.math.ucdavis.edu/math.PR/0611757
(alternate)
Author(s): Pieter C. Allaart
Abstract: Suppose $X_1,X_2,...$ are i.i.d. nonnegative random variables with finite
expectation, and for each $k$, $X_k$ is observed at the $k$-th arrival time
$S_k$ of a Poisson process with unit rate which is independent of the sequence
$\{X_k\}$. For $t>0$, comparisons are made between the expected maximum
$M(t):=\rE[\max_{k\geq 1} X_k \sI(S_k\leq t)]$ and the optimal stopping value
$V(t):=\sup_{\tau\in\TT}\sE[X_\tau \sI(S_\tau\leq t)]$, where $\TT$ is the set
of all $\NN$-valued random variables $\tau$ such that $\{\tau=i\}$ is
measurable with respect to the $\sigma$-algebra generated by
$(X_1,S_1),...,(X_i,S_i)$. For instance, it is shown that $M(t)/V(t)\leq
1+\alpha_0$, where $\alpha_0\doteq 0.34149$ satisfies $\int_0^1(y-y\ln
y+\alpha_0)^{-1} dy=1$; and this bound is asymptotically sharp as $t\to\infty$.
Another result is that $M(t)/V(t)<2-(1-e^{-t})/t$, and this bound is
asymptotically sharp as $t\downarrow 0$. Upper bounds for the difference
$M(t)-V(t)$ are also given, under the additional assumption that the $X_k$ are
bounded.
http://arXiv.org/abs/math/0611664
http://front.math.ucdavis.edu/math.PR/0611664
(alternate)
Author(s): Noam Berger and Marek Biskup and Christopher E. Hoffman and Gady Kozma
Abstract: We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$,
driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$. The
conductance law is i.i.d. subject to the condition that the probability of
$\omega_{xy}>0$ exceeds the percolation threshold on $\Z^d$. For environments
in which the origin is connected to infinity by bonds with positive
conductances, we prove that the $n$-step return probability $\cmss
P_\omega^n(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$,
while in $d\ge5$ it is bounded by a constant times $n^{-2}$. In $d=4$ we get an
upper bound proportional to $n^{-2}\log n$. The leading-order $1/n^2$ anomalous
decay in $d\ge5$ may be achieved in suitably chosen (i.i.d.) environments.
http://arXiv.org/abs/math/0611666
http://front.math.ucdavis.edu/math.PR/0611666
(alternate)
Author(s): Iva Kozakova
Abstract: In this article we study percolation on the Cayley graph of a free product of
groups. Such a graph has a tree-like structure which allows us to evaluate the
critical values of the phase transition, mean cluster size and the critical
exponent in bond percolation.
http://arXiv.org/abs/math/0611668
http://front.math.ucdavis.edu/math.PR/0611668
(alternate)
Author(s): B. Heller and M. Wang
Abstract: One may consider three types of statistical inference: Bayesian, frequentist,
and group invariance-based. The focus here is on the last method. We consider
the Poisson and binomial distributions in detail to illustrate a group
invariance method for constructing inferred distributions on parameter spaces
given observed results. These inferred distributions are obtained without using
Bayes' method and in particular without using a joint distribution of random
variable and parameter. In the Poisson and binomial cases, the final formulas
for inferred distributions coincide with the formulas for Bayes posteriors with
uniform priors.
http://arXiv.org/abs/math/0611675
http://front.math.ucdavis.edu/math.PR/0611675
(alternate)
Author(s): Jyotirmoy Sarkar
Abstract: A particle moves among the vertices of an $(m+1)$-gon which are labeled
clockwise as $0,1,...,m$. The particle starts at 0 and thereafter at each step
it moves to the adjacent vertex, going clockwise with a known probability $p$,
or counterclockwise with probability $1-p$. The directions of successive
movements are independent. What is the expected number of moves needed to visit
all vertices? This and other related questions are answered using recursive
relations.
http://arXiv.org/abs/math/0611676
http://front.math.ucdavis.edu/math.PR/0611676
(alternate)
Author(s): Oded Schramm and Scott Sheffield and David B. Wilson
Abstract: The conformal loop ensembles CLE(k), defined for k in [8/3, 8], are random
collections of loops in a planar domain which are conjectured scaling limits of
the O(n) loop models. We calculate the distribution of the conformal radii of
the nested loops surrounding a deterministic point. Our results agree with
predictions made by Cardy and Ziff and by Kenyon and Wilson for the O(n) model.
We also compute the expectation dimension of the CLE(k) gasket, which consists
of points not surrounded by any loop, to be 2-(8-k)(3k-8)/32k, which agrees
with the fractal dimension given by Duplantier for the O(n) model gasket.
http://arXiv.org/abs/math/0611687
http://front.math.ucdavis.edu/math.PR/0611687
(alternate)
Author(s): Enza Orlandi and Pierre Picco (CPT)
Abstract: We prove a quenched weak large deviations principle for the Gibbs measures of
a Random Field Kac Model (RFKM) in one dimension. The external random magnetic
field is given by symmetrically distributed Bernoulli random variables. The
results are valid for values of the temperature, $\beta^{-1}$, and magnitude,
$\theta$, of the field in the region where the free energy of the corresponding
random Curie Weiss model has only two absolute minima $m_\beta$ and $Tm_\beta$.
We give an explicit representation of the rate functional which is a positive
random functional determined by two distinct contributions. One is related to
the free energy cost ${\cal F}^*$ to undergo a phase change (the surface
tension). The ${\cal F}^*$ is the cost of one single phase change and depends
on the temperature and magnitude of the field. The other is a bulk contribution
due to the presence of the random magnetic field. We characterize the
minimizers of this random functional. We show that they are step functions
taking values $m_\beta$ and $Tm_\beta$. The points of discontinuity are
described by a stationary renewal process related to the $h-$extrema for a
bilateral Brownian motion studied by Neveu and Pitman, where $h$ in our context
is a suitable constant depending on the temperature and on magnitude of the
random field. As an outcome we have a complete characterization of the typical
profiles of RFKM (the ground states) which was initiated in [14] and extended
in [16].
http://arXiv.org/abs/math/0611688
http://front.math.ucdavis.edu/math.PR/0611688
(alternate)
Author(s): Philippe Carmona (LMJL)
Abstract: In this note we consider a chain of $N$ oscillators, whose ends are in
contact with two heat baths at different temperatures. Our main result is the
exponential convergence to the unique invariant probability measure (the
stationary state). We use the Lyapunov's function technique of Rey-Bellet and
coauthors with different model of heat baths, and adapt these techniques to two
new case recently considered in the literature by respectively Bernardin and
Olla, Lefevere and Schenkel
http://arXiv.org/abs/math/0611689
http://front.math.ucdavis.edu/math.PR/0611689
(alternate)
Author(s): Shui Feng
Abstract: Large deviation principles are established for the two-parameter
Poisson-Dirichlet distribution and two-parameter Dirichlet process when
parameter $\theta$ approaches infinity. The motivation for these results is to
understand the differences in terms of large deviations between the
two-parameter models and their one-parameter counterparts. New insight is
obtained about the role of the second parameter $\alpha$ through a comparison
with the corresponding results for the one-parameter Poisson-Dirichlet
distribution and Dirichlet process.
http://arXiv.org/abs/math/0611706
http://front.math.ucdavis.edu/math.PR/0611706
(alternate)
Author(s): Beno\^{i}t Collins and Michael Stolz
Abstract: We study random vectors of the form $({\rm Tr}(A^{(1)}V), ..., {\rm
Tr}(A^{(r)}V))$, where $V$ is a uniformly distributed element of a matrix
version of a classical compact symmetric space, and the $A^{(\nu)}$ are
deterministic parameter matrices. We show that for increasing matrix sizes
these random vectors converge to a joint Gaussian limit, and compute its
covariances. This generalizes previous work of Diaconis et al. for Haar
distributed matrices from the classical compact groups. The proof uses
integration formulae, due to Collins and \'{S}niady, for polynomial functions
on the classical compact groups.
http://arXiv.org/abs/math/0611708
http://front.math.ucdavis.edu/math.PR/0611708
(alternate)
Author(s): Liming Wu
Abstract: In in this paper we establish an explicit and sharp estimate of the spectral
gap (Poincar\'{e} inequality) and the transportation inequality for Gibbs
measures, under the Dobrushin uniqueness condition. Moreover, we give a
generalization of the Liggett's $M-\epsilon$ theorem for interacting particle
systems.
http://arXiv.org/abs/math/0611635
http://front.math.ucdavis.edu/math.PR/0611635
(alternate)
Author(s): B. Ferrario and F. Flandoli
Abstract: A stochastic version of a modified Navier-Stokes equation (introduced by
Prouse) is considered in a 3-dimensional torus. We prove existence and
uniqueness of martingale solutions. A different model with the non linearity
given by a power 5 of the velocity is analyzed; for the structure function of
this model, some insights towards an expression similar to that obtained by the
Kolmogorov 1941 theory of turbulence are presented.
http://arXiv.org/abs/math/0611637
http://front.math.ucdavis.edu/math.PR/0611637
(alternate)
Author(s): Cyril Roberto (LAMA) and Boguslaw Zegarlinski
Abstract: We study coercive inequalities in Orlicz spaces associated to the probability
measures on finite and infinite dimensional spaces which tails decay slower
than the Gaussian ones. We provide necessary and sufficient criteria for such
inequalities to hold and discuss relations between various classes of
inequalities.
http://arXiv.org/abs/math/0611638
http://front.math.ucdavis.edu/math.PR/0611638
(alternate)
Author(s): Guibao Liu
Abstract: We look at the semigroup generated by a system of heat equations.
Applications to testing normality and option pricing are addressed.
http://arXiv.org/abs/math/0611644
http://front.math.ucdavis.edu/math.PR/0611644
(alternate)
Author(s): Nils Berglund (CPT) and Bastien Fernandez (CPT) and Barbara Gentz
Abstract: We consider the dynamics of a periodic chain of N coupled overdamped
particles under the influence of noise. Each particle is subjected to a
bistable local potential, to a linear coupling with its nearest neighbours, and
to an independent source of white noise. We show that as the coupling strength
increases, the number of equilibrium points of the system changes from 3^N to
3. While for weak coupling, the system behaves like an Ising model with
spin-flip dynamics, for strong coupling (of the order N^2), it synchronises, in
the sense that all oscillators assume almost the same position in their
respective local potential most of the time. We derive the exponential
asymptotics for the transition times, and describe the most probable transition
paths between synchronised states, in particular for coupling intensities below
the synchronisation threshold. Our techniques involve a centre-manifold
analysis of the desynchronisation bifurcation, with a precise control of the
stability of bifurcating solutions, allowing us to give a detailed description
of the system's potential landscape, in which the metastable behaviour is
encoded.
http://arXiv.org/abs/math/0611647
http://front.math.ucdavis.edu/math.PR/0611647
(alternate)
Author(s): Nils Berglund (CPT) and Bastien Fernandez (CPT) and Barbara Gentz
Abstract: We consider the dynamics of a periodic chain of N coupled overdamped
particles under the influence of noise, in the limit of large N. Each particle
is subjected to a bistable local potential, to a linear coupling with its
nearest neighbours, and to an independent source of white noise. For strong
coupling (of the order N^2), the system synchronises, in the sense that all
oscillators assume almost the same position in their respective local potential
most of the time. In a previous paper, we showed that the transition from
strong to weak coupling involves a sequence of symmetry-breaking bifurcations
of the system's stationary configurations, and analysed in particular the
behaviour for coupling intensities slightly below the synchronisation
threshold, for arbitrary N. Here we describe the behaviour for any positive
coupling intensity \gamma of order N^2, provided the particle number N is
sufficiently large (as a function of \gamma/N^2). In particular, we determine
the transition time between synchronised states, as well as the shape of the
"critical droplet", to leading order in 1/N. Our techniques involve the control
of the exact number of periodic orbits of a near-integrable twist map, allowing
us to give a detailed description of the system's potential landscape, in which
the metastable behaviour is encoded.
http://arXiv.org/abs/math/0611648
http://front.math.ucdavis.edu/math.PR/0611648
(alternate)
Author(s): Giovanni Peccati (LSTA) and Igor Pr\"{u}nster (UNIVERSITY of Turin)
Abstract: A popular Bayesian nonparametric approach to survival analysis consists in
modeling hazard rates as kernel mixtures driven by a completely random measure.
In this paper we derive asymptotic results for linear and quadratic functionals
of such random hazard rates. In particular, we prove central limit theorems for
the cumulative hazard function and for the path-second moment and path-variance
of the hazard rate. Our techniques are based on recently established criteria
for the weak convergence of single and double stochastic integrals with respect
to Poisson random measures. We illustrate our results by considering specific
models involving kernels and random measures commonly exploited in practice.
http://arXiv.org/abs/math/0611652
http://front.math.ucdavis.edu/math.PR/0611652
(alternate)
Author(s): John T. Workman
Abstract: The following is a near complete set of notes of Bourgain's 1988 paper
"Almost Sure Convergence and Bounded Entropy." Both entropy results are
treated, as is one application. The proofs here are essentially those of
Bourgain's.
http://arXiv.org/abs/math/0611621
http://front.math.ucdavis.edu/math.CA/0611621
(alternate)
Author(s): Jose A. Ramirez and Brian Rider
Abstract: Consider Hill's operator Q = -D^2 + q(x) in which the potential q(x) is an
almost surely continuous and rotation invariant Gaussian process on the circle
of perimeter one. We prove a universality result for the shape of the
probability density function of the ground state energy
http://arXiv.org/abs/math/0611555
http://front.math.ucdavis.edu/math.PR/0611555
(alternate)
Author(s): Chunlin Wang
Abstract: Suppose that $\alpha \in (0,2)$ and that $X$ is an $\alpha$-stable-like
process on $\R^d$. Let $F$ be a function on $\R^d$ belonging to the class
$\bf{J_{d,\alpha}}$ (see Introduction) and $A_{t}^{F}$ be $\sum_{s \le
t}F(X_{s-},X_{s}), t> 0$, a discontinuous additive functional of $X$. With
neither $F$ nor $X$ being symmetric, under certain conditions, we show that the
Feynman-Kac semigroup $\{S_{t}^{F}:t \ge 0\}$ defined by $$
S_{t}^{F}f(x)=\mathbb{E}_{x}(e^{-A_{t}^{F}}f(X_{t}))$$ has a density $q$ and
that there exist positive constants $C_1,C_2,C_3$ and $C_4$ such that
$$C_{1}e^{-C_{2}t}t^{-\frac{d}{\alpha}}(1 \wedge
\frac{t^{\frac{1}{\alpha}}}{|x-y|})^{d+\alpha} \leq q(t,x,y) \leq
C_{3}e^{C_{4}t}t^{-\frac{d}{\alpha}}(1 \wedge
\frac{t^{\frac{1}{\alpha}}}{|x-y|})^{d+\alpha}$$ for all $(t,x,y)\in (0,\infty)
\times \R^d \times \R^d$.
http://arXiv.org/abs/math/0611565
http://front.math.ucdavis.edu/math.PR/0611565
(alternate)
Author(s): Chunlin Wang
Abstract: In this paper, we study the purely discontinuous Girsanov transforms which
were discussed in Chen and Song \cite{CS2} and Song \cite{S3}. We show that the
transition density of any purely discontinuous Girsanov transform of a
$\alpha$-stable-like process, which can be nonsymmetric, is comparable to the
transition density of the $\alpha$-stable-like process.
http://arXiv.org/abs/math/0611566
http://front.math.ucdavis.edu/math.PR/0611566
(alternate)
Author(s): Anne-Laure Basdevant (PMA) and Arvind Singh (PMA)
Abstract: We consider the model of the one-dimensional cookie random walk when the
initial cookie distribution is spatially uniform and the number of cookies per
site is finite. We give a criterion to decide whether the limiting speed of the
walk is non-zero. In particular, we show that a positive speed may be obtained
for just 3 cookies per site. We also prove a result on the continuity of the
speed with respect to the initial cookie distribution.
http://arXiv.org/abs/math/0611580
http://front.math.ucdavis.edu/math.PR/0611580
(alternate)
Author(s): Vladislav Kargin
Abstract: Let $X_i$ denote free identically-distributed random variables. This paper
investigates how the norm of products $\Pi_n=X_1 X_2 ... X_n$ behaves as $n$
approaches infinity. In addition, for positive $X_i$ it studies the asymptotic
behavior of the norm of $Y_n=X_1 \circ X_2 \circ ...\circ X_n$, where $\circ$
denotes the symmetric product of two positive operators: $A \circ
B=:A^{1/2}BA^{1/2}$.
It is proved that if the expectation of $X_i$ is 1, then the norm of the
symmetric product $Y_{n}$ is between $c_1 n^{1/2}$ and $c_2 n$ for certain
constant $c_1$ and $c_2$. That is, the growth in the norm is at most linear.
For the norm of the usual product $Pi_n$, it is proved that the limit of
$n^{-1}\log Norm(Pi_n)$ exists and equals $\log \sqrt{E(X_i^{\ast}X_{i})}.$ In
other words, the growth in the norm of the product is exponential and the rate
equals the logarithm of the Hilbert-Schmidt norm of operator X.
Finally, if $\pi $ is a cyclic representation of the algebra generated by
$X_i$, and if $\xi$ is a cyclic vector, then $n^{-1}\log Norm(\pi (\Pi_{n})
\xi)=\log \sqrt{E(X_{i}^{\ast}X_{i})}$ for all $n.$ In other words, the growth
in the length of the cyclic vector is exponential and the rate coincides with
the rate in the growth of the norm of the product.
These results are significantly different from analogous results for
commuting random variables and generalize results for random matrices derived
by Kesten and Furstenberg.
http://arXiv.org/abs/math/0611593
http://front.math.ucdavis.edu/math.PR/0611593
(alternate)
Author(s): R\'{e}mi Rhodes (LATP)
Abstract: We study a diffusion process with random space-time dependent coefficients.
Moreover the diffusion matrix is allowed to degenerate. An invariance principle
is proved provided that the diffusion coefficient is controlled by a time
independent one.
http://arXiv.org/abs/math/0611598
http://front.math.ucdavis.edu/math.PR/0611598
(alternate)
Author(s): Anders Karlsson and Fran\c{c}ois Ledrappier
Abstract: We prove a general noncommutative law of large numbers. This applies in
particular to random walks on any locally finite homogeneous graph, as well as
to Brownian motion on Riemannian manifolds which admit a compact quotient. It
also generalizes Oseledec's multiplicative ergodic theorem. In addition, we
show that $\epsilon$-shadows of any ballistic random walk with finite moment on
any group eventually intersect. Some related results concerning Coxeter groups
and mapping class groups are recorded in the last section.
http://arXiv.org/abs/math/0611607
http://front.math.ucdavis.edu/math.PR/0611607
(alternate)
Author(s): Eric Carlen and R. Vilela Mendes
Abstract: For the space of functions that can be approximated by linear chirps, we
prove a reconstruction theorem by random sampling at arbitrary rates.
http://arXiv.org/abs/math/0611608
http://front.math.ucdavis.edu/math.PR/0611608
(alternate)
Author(s): P. Mathieu
Abstract: We prove an almost sure invariance principle for a random walker among i.i.d.
conductances in $\Z^d$, $d\geq 2$. We assume conductances are bounded from
above but we dot require they are bounded from below.
http://arXiv.org/abs/math/0611613
http://front.math.ucdavis.edu/math.PR/0611613
(alternate)
Author(s): Chunlin Wang
Abstract: In this paper, we propose a method based on GMM (the generalized method of
moments) to estimate the parameters of stable distributions with $0<\alpha<2$.
We don't assume symmetry for stable distributions.
http://arXiv.org/abs/math/0611567
http://front.math.ucdavis.edu/math.ST/0611567
(alternate)
Author(s): Iain M. Johnstone
Abstract: Multivariate statistical analysis is concerned with observations on several
variables which are thought to possess some degree of inter-dependence. Driven
by problems in genetics and the social sciences, it first flowered in the
earlier half of the last century. Subsequently, random matrix theory (RMT)
developed, initially within physics, and more recently widely in mathematics.
While some of the central objects of study in RMT are identical to those of
multivariate statistics, statistical theory was slow to exploit the connection.
However, with vast data collection ever more common, data sets now often have
as many or more variables than the number of individuals observed. In such
contexts, the techniques and results of RMT have much to offer multivariate
statistics. The paper reviews some of the progress to date.
http://arXiv.org/abs/math/0611589
http://front.math.ucdavis.edu/math.ST/0611589
(alternate)
Author(s): Ravi Montenegro
Abstract: Previous authors have shown bounds on mixing time of random walks on finite
undirected Cayley graphs, both with and without self-loops. We extend this to
the most general case by showing that a similar bound holds for walks on all
finite directed Cayley graphs. These are shown by use of two new canonical path
theorems for mixing time of non-reversible Markov chains. The first result is
related to the traditional canonical path mixing result but holds for general
walks with small holding probability. The second theorem holds for all finite
Markov chains, even non-reversible walks with no holding probability.
Curiously, these results are shown by use of Evolving sets, whereas previous
path results were shown via Spectral gap.
http://arXiv.org/abs/math/0611585
http://front.math.ucdavis.edu/math.CO/0611585
(alternate)
Author(s): B. Klartag
Abstract: We investigate the rate of convergence in the central limit theorem for
convex sets. We obtain bounds with a power-law dependence on the dimension.
These bounds are asymptotically better than the logarithmic estimates which
follow from the original proof of the central limit theorem for convex sets.
http://arXiv.org/abs/math/0611577
http://front.math.ucdavis.edu/math.MG/0611577
(alternate)
Author(s): Ravi Montenegro
Abstract: We show how to apply the canonical path method to a non-reversible Markov
chain with no holding probability: a random walk used in Pollard's Rho
algorithm for discrete logarithm. This is used to show that the Pollard Rho
method for finding the discrete logarithm on a cyclic group $G$ requires
$O(\sqrt{|G|} (\log |G|)^{3/2})$ steps until a collision occurs and discrete
logarithm is possibly found, not far from the widely conjectured value of
$\Theta(\sqrt{|G|})$. Conversely, we find that arguments based on spectral gap,
spectral profile or log-Sobolev cannot be used to show the correct mixing bound
of the Pollard Rho walk, while coupling can give at best a small improvement on
our current bound for collision time.
http://arXiv.org/abs/math/0611586
http://front.math.ucdavis.edu/math.NT/0611586
(alternate)
Author(s): Anne-Laure Basdevant (PMA)
Abstract: In this paper, we study additive coalescents. Using their representation as
fragmentation processes, we prove that the law of a large class of eternal
additive coalescents is absolutely continuous with respect to the law of the
standard additive coalescent on any bounded time interval.
http://arXiv.org/abs/math/0611523
http://front.math.ucdavis.edu/math.PR/0611523
(alternate)
Author(s): David Brydges and Remco van der Hofstad and Wolfgang K\"onig
Abstract: We investigate the local times of a continuous-time Markov chain on an
arbitrary discrete state space. For fixed finite range of the Markov chain, we
derive an explicit formula for the joint density of all local times on the
range, at any fixed time. We use standard tools from the theory of stochastic
processes and finite-dimensional complex calculus.
We apply this formula in the following directions: (1) we derive large
deviation upper estimates for the normalized local times beyond the exponential
scale, (2) we derive the upper bound in Varadhan's lemma for any measurable
functional of the local times, \ch{and} (3) we derive large deviation upper
bounds for continuous-time simple random walk on large subboxes of $\Z^d$
tending to $\Z^d$ as time diverges. We finally discuss the relation of our
density formula to the Ray-Knight theorem for continuous-time simple random
walk on $\Z$, which is analogous to the well-known Ray-Knight description of
Brownian local times.
http://arXiv.org/abs/math/0611525
http://front.math.ucdavis.edu/math.PR/0611525
(alternate)
Author(s): Stan Zachary
Abstract: We give a simple and direct treatment of insensitivity in stochastic networks
which is quite general and which provides probabilistic insight into the
phenomenon. In the case of multi-class networks, the results generalise those
of Bonald and Proutiere (2002, 2003).
http://arXiv.org/abs/math/0611526
http://front.math.ucdavis.edu/math.PR/0611526
(alternate)
Author(s): Bruno Schapira (MAPMO and PMA)
Abstract: In this paper we study a random walk on an affine building, whose radial
part, when suitably normalized, converges to the Brownian motion of the Weyl
chamber (for the type $A$). This gives a new discrete approximation of this
process, which is different from the one of Biane \cite{Bia2}. The main
ingredients of the proof are a combinatorial formula on the building and the
estimate of the transition density proved in \cite{AST}. Moreover our result
extends in higher rank the correspondence at the probabilistic level between
Riemannian symmetric spaces of the noncompact type and their discrete
counterpart, which had been previously obtained by Bougerol and Jeulin in rank
one \cite{BJ}.
http://arXiv.org/abs/math/0611529
http://front.math.ucdavis.edu/math.PR/0611529
(alternate)
Author(s): D. Bloemker and G.A. Pavliotis and M. Hairer
Abstract: In this article we derive rigorously amplitude equations for stochastic PDEs
with quadratic nonlinearities, under the assumption that the noise acts only on
the stable modes and for an appropriate scaling between the distance from
bifurcation and the strength of the noise. We show that, due to the presence of
two distinct timescales in our system, the noise (which acts only on the fast
modes) gets transmitted to the slow modes and, as a result, the amplitude
equation contains both additive and multiplicative noise.
As an application we study the case of the one dimensional Burgers equation
forced by additive noise in the orthogonal subspace to its dominant modes. The
theory developed in the present article thus allows to explain theoretically
some recent numerical observations from [Rob03].
http://arXiv.org/abs/math/0611537
http://front.math.ucdavis.edu/math.PR/0611537
(alternate)
Author(s): Alexander Gnedin
Abstract: Random permutations with distribution conditionally uniform given the set of
record values can be generated in a unified way, coherently for all values of
$n$. Our central example is a two-parameter family of random permutations that
are conditionally uniform given the counts of upper and lower records. This
family interpolates between two versions of Ewens' distribution. We discuss
characterisations of the conditionally uniform permutations, their asymptotic
properties, constructions and relations to random partitions.
http://arXiv.org/abs/math/0611538
http://front.math.ucdavis.edu/math.PR/0611538
(alternate)
Author(s): Florent Benaych-Georges (PMA)
Abstract: In this text, extending results of A.Nica and M. Neagu, we study the
asymptotics of the number of cycles of a given length of a word in several
independent random permutations with restricted cycle lengths. Specifically,
for $A_1$,..., $A_k$ non empty sets of positive integers and for $w$ word in
the letters $g_1,g_1^{-1}$,..., $g_k,g_k^{-1}$, we consider, for all $n$ such
that it is possible, an independent family $s_1(n)$,..., $s_k(n)$ of random
permutations chosen uniformly among the permutations of $n$ objects which have
all their cycle lengths in respectively $A_1$,..., $A_k$, and for $l$ positive
integer, we are going to give asymptotics (as $n$ goes to infinity) on the
number $N_l(n)$ of cycles of length $l$ of the permutation obtained by changing
any letter $g_i$ in $w$ by $s_i(n)$. In many cases, we prove that the
distribution of $N_l(n)$ converges to a Poisson law with parameter $1/l$ and
that the family of random variables $(N_1(n), N_2(n),...)$ is asymptotically
independent. We notice the pretty surprising fact that from this point of view,
many things happen like if we considered the number of cycles of given lengths
of a single permutation with uniform distribution on the $n$-th symmetric
group.
http://arXiv.org/abs/math/0611500
http://front.math.ucdavis.edu/math.PR/0611500
(alternate)
Author(s): Patricia Goncalves
Abstract: We prove a Functional Central Limit Theorem for the position of a Tagged
Particle in the one-dimensional Asymmetric Simple Exclusion Process in the
hyperbolic scaling, starting from a Bernoulli product measure conditioned to
have a particle at the origin. We also prove that the position of the Tagged
Particle at time $t$ depends on the initial configuration, by the number of
empty sites in the interval $[0,(p-q)\alpha t]$ divided by $\alpha$ in the
hyperbolic and in a longer time scale, namely $N^{4/3}$.
http://arXiv.org/abs/math/0611505
http://front.math.ucdavis.edu/math.PR/0611505
(alternate)
Author(s): J.Martin Lindsay and Adam Skalski
Abstract: The theory of quantum stochastic convolution cocycles is extended to the
topological context of compact quantum groups. Stochastic convolution cocycles
on a C*-hyperbialgebra, which are Markov-regular, completely positive and
contractive, are shown to satisfy coalgebraic quantum stochastic differential
equations with completely bounded coefficients, and the structure of their
stochastic generators is obtained. Specialising to *-homomorphic convolution
cocycles on a C*-bialgebra the stochastic generators are shown to have
Schuermann form. Tentative definitions of quantum Levy process on a compact
quantum group, for which a reconstruction theorem is valid, are proposed. In
the examples given, connection to the algebraic theory is emphasised by a focus
on the case of full compact quantum groups.
http://arXiv.org/abs/math/0611497
http://front.math.ucdavis.edu/math.OA/0611497
(alternate)
Author(s): Boris Tsirelson
Abstract: The coordinate of a harmonic oscillator is measured at a time chosen at
random among three equiprobable instants: now, after one third of the period,
or after two thirds. The (total) probability that the outcome is positive
depends on the state of the oscillator. In the classical case the probability
varies between 1/3 and 2/3, but in the quantum case -- between 0.29 and 0.71.
http://arXiv.org/abs/quant-ph/0611147
http://front.math.ucdavis.edu/quant-ph/0611147
(alternate)
Author(s): Itai Benjamini and Ariel Yadin and Amir Yehudayoff
Abstract: A graph homomorphism between two graphs is a map from the vertex set of one
graph to the vertex set of the other graph, that maps edges to edges. In this
note we study the range of a uniformly chosen homomorphism from a graph G to
the infinite line Z. It is shown that if the maximal degree of G is
`sub-logarithmic', then the range of such a homomorphism is super-constant.
Furthermore, some examples are provided, suggesting that perhaps for graphs
with super-logarithmic degree, the range of a typical homomorphism is bounded.
In particular, a sharp transition is shown for a specific family of graphs
C_{n,k} (which is the tensor product of the n-cycle and a complete graph, with
self-loops, of size k). That is, given any function psi(n) tending to infinity,
the range of a typical homomorphism of C_{n,k} is super-constant for k = 2
log(n) - psi(n), and is 3 for k = 2 log(n) + psi(n).
http://arXiv.org/abs/math/0611416
http://front.math.ucdavis.edu/math.PR/0611416
(alternate)
Author(s): Werner Kirsch
Abstract: In certain bodies, like the Council of the EU, the member states have a
voting weight which depends on the population of the re- spective state. In
this article we ask the question which voting weight guarantees a `fair'
representation of the citizens in the union. The tra- ditional answer, the
square-root law by Penrose, is that the weight of a state (more precisely: the
voting power) should be proportional to the square-root of the population of
this state. The square root law is based on the assumption that the voters in
every state cast their vote inde- pendently of each other. In this paper we
concentrate on cases where the independence assumption is not valid.
http://arXiv.org/abs/math/0611418
http://front.math.ucdavis.edu/math.PR/0611418
(alternate)
Author(s): Vincent Bansaye (PMA)
Abstract: We consider a generalized version in continuous time of the parking problem
of Knuth. Files arrive following a Poisson point process and are stored on a
hardware identified with the real line. We specify the distribution of the
space of unoccupied locations at a fixed time and give its asymptotics when the
hardware is becoming full.
http://arXiv.org/abs/math/0611432
http://front.math.ucdavis.edu/math.PR/0611432
(alternate)
Author(s): Jean-Baptiste Gou\'er\'e (MAPMO)
Abstract: We consider the so-called Poisson Boolean model of continuum percolation. At
each point of an homogeneous Poisson point process on the Euclidean space
$\R^d$, we center a ball with random radius. We assume that the radii of the
balls are independent, identically distributed and independent of the point
process. We denote by $\Sigma$ the union of the balls and by $S$ the connected
component of $\Sigma$ that contains the origin. We show that $S$ is almost
surely bounded for small enough density $\lambda$ of the point process if and
only if the mean volume of the balls is finite. Let us denote by $D$ the
diameter of $S$ and by $R$ one of the random radii. We also show that, for all
positive real number $s$, $D^s$ is integrable for small enough $\lambda$ if and
only if $R^{d+s}$ is integrable.
http://arXiv.org/abs/math/0611369
http://front.math.ucdavis.edu/math.PR/0611369
(alternate)
Author(s): Philippe Soulier (MODAL'X) and Randal Douc (CMAP) and Fran\c{c}ois Roueff (LTCI)
Abstract: A new sufficient condition for the existence of a stationary causal solution
of an ARCH($\infty$) equation is provided. This condition allows to consider
polynomially decaying coefficients, so that it can be applied to the so-called
FIGARCH processes, whose existence is thus proved.
http://arXiv.org/abs/math/0611339
http://front.math.ucdavis.edu/math.ST/0611339
(alternate)
Author(s): Andreas Neuenkirch and Ivan Nourdin (PMA) and Andreas Roessler and Samy Tindel (IECN)
Abstract: In this paper we consider a n-dimensional stochastic differential equation
driven by a fractional Brownian motion with Hurst parameter H>1/3. After
solving this equation in a rather elementary way, following the approach of
Gubinelli, we show how to obtain an expansion for E[f(X\_t)] in terms of t,
where X denotes the solution to the SDE and f:R^n->R is a regular function.
With respect to the work by Baudoin and Coutin, where the same kind of problem
is considered, we try an improvement in three different directions: we are able
to take a drift into account in the equation, we parametrize our expansion with
trees (which makes it easier to use), and we obtain a sharp control of the
remainder.
http://arXiv.org/abs/math/0611306
http://front.math.ucdavis.edu/math.PR/0611306
(alternate)
Author(s): A. Gillett and M. Nuyens
Abstract: We introduce a continuum percolation model defined on the points of a
d-dimensional homogeneous Poisson process. Each Poisson point is connected to
all points within its connection range, which depends on the distances to the
other Poisson points. We show that the new model exhibits a phase transition,
and obtain results about the critical values in low and high dimensions.
http://arXiv.org/abs/math/0611315
http://front.math.ucdavis.edu/math.PR/0611315
(alternate)
Author(s): V. Vu
Abstract: In this survey, we discuss some basic problems concerning random matrices
with discrete distributions. Several new results, tools and conjectures will be
presented.
http://arXiv.org/abs/math/0611321
http://front.math.ucdavis.edu/math.CO/0611321
(alternate)
Author(s): P. Pfaffelhuber and A. Studeny
Abstract: Consider a genetic locus carrying a strongly beneficial allele which has
recently fixed in a large population. As strongly beneficial alleles fix
quickly, sequence diversity at partially linked neutral loci is reduced. This
phenomenon is known as a selective sweep. The fixation of the beneficial allele
not only affects sequence diversity at single neutral loci but also the joint
allele distribution of several partially linked neutral loci. This distribution
can be studied using the ancestral recombination graph for samples of partially
linked neutral loci during the selective sweep. To approximate this graph, we
extend recent work by Schweinsberg & Durrett 2005 and Etheridge, Pfaffelhuber &
Wakolbinger 2006 using a marked Yule tree for the genealogy at a single neutral
locus linked to a strongly beneficial one. We focus on joint genealogies at two
partially linked neutral loci in the case of large selection coefficients
\alpha and recombination rates \rho = O(\alpha/\log\alpha) between loci. Our
approach leads to a full description of the genealogy with accuracy of O((\log
\alpha)^{-2}) in probability. As an application, we derive the expectation of
Lewontin's D as a measure for non-random association of alleles.
http://arXiv.org/abs/q-bio/0611029
http://front.math.ucdavis.edu/q-bio.PE/0611029
(alternate)
Author(s): E. Porcu and J. Mateu and and G. Christakos
Abstract: The theory of quasi-arithmetic means is a powerful tool in the study of
covariance functions across space-time. In the present study we use
quasi-arithmetic functionals to make inferences about the permissibility of
averages of functions that are not, in general, permissible covariance
functions. This is the case, e.g., of the geometric and harmonic averages, for
which we obtain permissibility criteria. Also, some important inequalities
involving covariance functions and preference relations as well as algebraic
properties can be derived by means of the proposed approach. In particular, we
show that quasi-arithmetic covariances allow for ordering and preference
relations, for a Jensen-type inequality and for a minimal and maximal element
of their class. The general results shown in this paper are then applied to
study of spatial and spatiotemporal random fields. In particular, we discuss
the representation and smoothness properties of a weakly stationary random
field with a quasi-arithmetic covariance function. Also, we show that the
generator of the quasi-arithmetic means can be used as a link function in order
to build a space-time nonseparable structure starting from the spatial and
temporal margins, a procedure that is technically sound for those working with
copulas. Several examples of new families of stationary covariances obtainable
with this procedure are shown. Finally, we use quasi-arithmetic functionals to
generalise existing results concerning the construction of nonstationary
spatial covariances and discuss the applicability and limits of this
generalisation.
http://arXiv.org/abs/math/0611275
http://front.math.ucdavis.edu/math.PR/0611275
(alternate)
Author(s): Chris Howitt and Jon Warren
Abstract: Consider the following mechanism for the random evolution of a distribution
of mass on the integer lattice. At unit rate, independently for each site, the
mass at the site is split into two parts by choosing a random proportion
distributed according to some specified probability measure on [0,1] and
dividing the mass in that proportion. One part then moves to each of the two
adjacent sites. This paper considers a continuous analogue of this evolution,
which may be described by means of a stochastic flow of kernels, the theory of
which was developed by Le Jan and Raimond. One of their results is that such a
flow is characterized by specifying its N point motions, which form a
consistent family of Brownian motions. This means for each N we have a
diffusion in N dimensional Euclidean space, whose N co-ordinates are all
Brownian motions. Any M co-ordinates taken from the N-dimensional process are
distributed as the M-dimensional process in the family. Moreover, in this
setting, the only interactions between co-ordinates are local: when coordinates
differ in value they evolve independently of each other. In this paper we
explain how such multidimensional diffusions may be constructed and
characterized via martingale problems.
http://arXiv.org/abs/math/0611292
http://front.math.ucdavis.edu/math.PR/0611292
(alternate)
Author(s): Peter Mathe and Erich Novak
Abstract: We study the integration of functions with respect to an unknown density. We
compare the simple Monte Carlo method (which is almost optimal for a certain
large class of inputs) and compare it with the Metropolis algorithm (based on a
suitable ball walk).
Using MCMC we prove (for certain classes of inputs) that adaptive methods are
much better than nonadaptive ones. Actually, the curse of dimension (for
nonadaptive methods) can be broken by adaption.
http://arXiv.org/abs/math/0611285
http://front.math.ucdavis.edu/math.NA/0611285
(alternate)
Author(s): Adam Skalski
Abstract: Stochastic generators of completely positive and contractive quantum
stochastic convolution cocycles on a C*-hyperbialgebra are characterised. The
characterisation is used to obtain dilations and stochastic forms of
Stinespring decomposition for completely positive convolution cocycles on a
C*-bialgebra.
http://arXiv.org/abs/math/0611271
http://front.math.ucdavis.edu/math.OA/0611271
(alternate)
Author(s): Sourav Chatterjee
Abstract: We introduce a new version of Stein's method that reduces a large class of
normal approximation problems to variance bounding exercises, thus making a
connection between central limit theorems and concentration of measure. Unlike
Skorokhod embeddings, the object whose variance has to be bounded has an
explicit formula that makes it possible to carry out the program more easily.
As an application, we derive a general CLT for functions that are obtained as
combinations of many local contributions, where the definition of `local'
itself depends on the data. Several examples are given, including the solution
to a nearest-neighbor CLT problem posed by Peter Bickel.
http://arXiv.org/abs/math/0611213
http://front.math.ucdavis.edu/math.PR/0611213
(alternate)
Author(s): Jiri Cerny and Veronique Gayrard
Abstract: We study the simple random walk on the $n$-dimensional hypercube, in
particular its hitting times of large (possibly random) sets. We give simple
conditions on these sets ensuring that the properly-rescaled hitting time is
asymptotically exponentially distributed, uniformly in the starting position of
the walk. These conditions are then verified for percolation clouds with
densities that are much smaller than $(n \log n)^{-1}$. A main motivation
behind this paper is the study of the so-called aging phenomenon in the Random
Energy Model (REM), the simplest model of a mean-field spin glass. Our results
allow us to prove aging in the REM for all temperatures, thereby extending
earlier results to their optimal temperature domain.
http://arXiv.org/abs/math/0611242
http://front.math.ucdavis.edu/math.PR/0611242
(alternate)
Author(s): Yuri Bakhtin and Matilde Martinez
Abstract: We prove that a probability measure on a compact non-singular lamination by
hyperbolic Riemann surfaces is harmonic if and only if it is the projection of
a measure on the unit tangent bundle such that it is invariant under both the
geodesic and the horocycle flows.
http://arXiv.org/abs/math/0611235
http://front.math.ucdavis.edu/math.DS/0611235
(alternate)
Author(s): Jason Schweinsberg
Abstract: Let $(G_n)_{n=1}^{\infty}$ be a sequence of finite graphs, and let $Y_t$ be
the length of a loop-erased random walk on $G_n$ after $t$ steps. We show that
for a large family of sequences of finite graphs, which includes the case in
which $G_n$ is the $d$-dimensional torus of size-length $n$ for $d \geq 4$, the
process $(Y_t)_{t=0}^{\infty}$, suitably normalized, converges to the Rayleigh
process introduced by Evans, Pitman, and Winter. Our proof relies heavily on
ideas of Peres and Revelle, who used loop-erased random walks to show that the
uniform spanning tree on large finite graphs converges to the Brownian
continuum random tree of Aldous.
http://arXiv.org/abs/math/0611155
http://front.math.ucdavis.edu/math.PR/0611155
(alternate)
Author(s): Jean-Fran\c{c}ois Delmas (CERMICS)
Abstract: We define the height process for super-critical continuous state branching
processes with quadratic branching mechanism. It appears as a projective limit
of Brownian motions with positive drift reflected at 0 and a>0 as a goes to
infinity. Then we extend the pruning procedure of branching processes to the
super-critical case. This give a complete duality picture between pruning and
size proportional immigration for quadratic continuous state branching
processes.
http://arXiv.org/abs/math/0611172
http://front.math.ucdavis.edu/math.PR/0611172
(alternate)
Author(s): Gerard Ben Arous and Veronique Gayrard
Abstract: This work addresses potential theoretic questions for the standard nearest
neighbor random walk on the hypercube $\{-1,+1\}^N$. For a large class of
subsets $A\subset\{-1,+1\}^N$ we give precise estimates for the harmonic
measure of $A$, the mean hitting time of $A$, and the Laplace transform of this
hitting time. In particular, we give precise sufficient conditions for the
harmonic measure to be asymptotically uniform, and for the hitting time to be
asymptotically exponentially distributed, as $N\to\infty$. Our approach relies
on a $d$-dimensional extension of the Ehrenfest urn scheme called lumping and
covers the case where $d$ is allowed to diverge with $N$ as long as
$d\leq\alpha_0\frac{N}{\log N}$ for some constant $0<\alpha_0<1$.
http://arXiv.org/abs/math/0611178
http://front.math.ucdavis.edu/math.PR/0611178
(alternate)
Author(s): Yvan Le Borgne (LaBRI) and Jean-Fran\c{c}ois Marckert (LaBRI)
Abstract: In this paper, we revisit the enumeration of directed animals using gas
models. We show that there exists a natural construction of random directed
animals on any directed graph together with a particle system that explains at
the level of objects the formal link known between the density of the gas model
and the generating function of directed animals counted according to the area.
This provides some new methods to compute the generating function of directed
animals counted according to area, and leads in the particular case of the
square lattice to new combinatorial results and questions. A model of gas
related to directed animals counted according to area and perimeter on any
directed graph is also exhibited.
http://arXiv.org/abs/math/0611194
http://front.math.ucdavis.edu/math.PR/0611194
(alternate)
Author(s): Jeremy Quastel and Benedek Valko
Abstract: It is shown that white noise is an invariant measure for the Korteweg-deVries
equation on $\mathbb T$. This is a consequence of recent results of Kappeler
and Topalov establishing the well-posedness of the equation on appropriate
negative Sobolev spaces, together with a result of
Cambronero and McKean that white noise is the image under the Miura transform
(Ricatti map) of the (weighted) Gibbs measure for the modified KdV equation,
proven to be invariant for that equation by Bourgain.
http://arXiv.org/abs/math/0611152
http://front.math.ucdavis.edu/math.AP/0611152
(alternate)
Author(s): Ellen Baake and Hans-Otto Georgii
Abstract: We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.
http://arXiv.org/abs/q-bio/0611018
http://front.math.ucdavis.edu/q-bio.PE/0611018
(alternate)
Author(s): Krzysztof Burdzy and Zhen-Qing Chen
Abstract: In this paper we investigate three discrete or semi-discrete approximation
schemes for reflected Brownian motion on bounded Euclidean domains.
http://arXiv.org/abs/math/0611114
http://front.math.ucdavis.edu/math.PR/0611114
(alternate)
Author(s): Federico Camia and Charles M. Newman
Abstract: We review some of the recent progress on the scaling limit of two-dimensional
critical percolation; in particular, the convergence of the exploration path to
chordal SLE(6) and the "full" scaling limit of cluster interface loops. The
results given here on the full scaling limit and its conformal invariance
extend those presented previously. For site percolation on the triangular
lattice, the results are fully rigorous. We explain some of the main ideas,
skipping most technical details.
http://arXiv.org/abs/math/0611116
http://front.math.ucdavis.edu/math.PR/0611116
(alternate)
Author(s): Christian Benes
Abstract: The purpose of this note is to collect in one place a few results about
simple random walk and Brownian motion which are often useful. These include
standard results such as Beurling estimates, large deviation estimates, and a
method for coupling the two processes, as well as solutions to the discrete
Dirichlet problem in various domains which, to the author's knowledge, have not
been published anywhere. The main focus is on the two-dimensional processes.
http://arXiv.org/abs/math/0611127
http://front.math.ucdavis.edu/math.PR/0611127
(alternate)
Author(s): A. C. D. van Enter and C. Kuelske
Abstract: We consider statistical mechanics models of continuous spins in a disordered
environment. These models have a natural interpretation as effective interface
models. It is well-known that without disorder there are no interface Gibbs
measures in infinite volume in dimension d=2, while there are ``gradient Gibbs
measures'' describing an infinite-volume distribution for the increments of the
field, as was shown by Funaki and Spohn. In the present paper we show that
adding a disorder term prohibits the existence of such gradient Gibbs measures
for general interaction potentials in d=2. This non-existence result
generalizes the simple case of Gaussian fields where it follows from an
explicit computation. In d=3 where random gradient Gibbs measures are expected
to exist, our method provides a lower bound of the order of the inverse of the
distance on the decay of correlations of Gibbs expectations w.r.t. the
distribution of the random environment.
http://arXiv.org/abs/math/0611140
http://front.math.ucdavis.edu/math.PR/0611140
(alternate)
Author(s): A. Greven and F. den Hollander
Abstract: This paper considers a system of interacting diffusions labelled by the
d-dimensional integer lattice. The diffusions interact linearly, according to a
random walk transition kernel $a(\cdot,\cdot)$, and have an autonomous
quadratic diffusion function with diffusion parameter $b$. The ergodic
behaviour of such systems shows a delicate interplay between $a(cdot,\cdot)$
and $b$, which is described in detail. For instance, when $a(\cdot,\cdot)$ is
transient, there is a sequence of critical values $b_*>b_2>b_3>...$ at which
the equilibrium experiences a crossover: at $b_*$ from extinction to survival,
and at $b_m$ from infinite to finite $m$-th moment. Proofs are based on
$L^2$-theory, large deviations, and Palm theory.
http://arXiv.org/abs/math/0611141
http://front.math.ucdavis.edu/math.PR/0611141
(alternate)
Author(s): Christian Benes
Abstract: We study two variants of the notion of {\it holes} formed by planar simple
random walk of time duration $2n$ and the areas associated with them. We prove
in both cases that the number of holes of area greater than A(n), where
$\{A(n)\}$ is an increasing sequence, is, up to a logarithmic correction term,
asymptotic to $n\cdot A(n)^{-1}$ for a range of large holes, thus confirming an
observation by Mandelbrot. A consequence is that the largest hole has an area
which is logarithmically asymptotic to $n$. We also discuss the different
exponent of 5/3 observed by Mandelbrot for small holes.
http://arXiv.org/abs/math/0611144
http://front.math.ucdavis.edu/math.PR/0611144
(alternate)
Author(s): Argyn Kuketayev
Abstract: There is a sequence of random numbers x1,x2, ..., xn and so on. Numbers are
independent of each other, but all numbers are from the same continuous
distribution. If x1 < x2 > x3, then x2 is a local maximum. Here, we show that
the probability mass function (PMF) of idstribution of distances between local
maxima is non-parametric and the same for any probability distribution of
random numbers in the sequence, and that the average distance is exactly 3. We
present a method of computation of this PMF and its table for distances betwen
2 and 29. This PMF is confirmed to match distance distributions of sample
random number sequences, which were created by pseudo-random number generators
or obtained from "true" random number sources.
http://arXiv.org/abs/math/0611130
http://front.math.ucdavis.edu/math.ST/0611130
(alternate)
Author(s): Istvan Gy\"{o}ngy and Annie Millet (PMA and MATISSE and SAMOS)
Abstract: Stochastic evolutional equations with monotone operators are considered in
Banach spaces. Explicit and implicit numerical schemes are presented. The
convergence of the approximations to the solution of the equations is proved.
http://arXiv.org/abs/math/0611069
http://front.math.ucdavis.edu/math.PR/0611069
(alternate)
Author(s): Fabien Panloup (PMA)
Abstract: We study the rate of convergence of some recursive procedures based on some
"exact" or "approximate" Euler schemes which converge to the invariant measure
of an ergodic SDE driven by a L\'{e}vy process. The main interest of this work
is to compare the rates induced by exact and approximate Euler schemes. In our
main result, we show that replacing the small jumps by a Brownian component in
the approximate case preserves the rate induced by the exact Euler scheme for a
large class of L\'{e}vy processes.
http://arXiv.org/abs/math/0611072
http://front.math.ucdavis.edu/math.PR/0611072
(alternate)
Author(s): Annie Millet (MATISSE and Pma and Samos) and Pierre-Luc Morien (MODAL'x)
Abstract: We study the speed of convergence of the explicit and implicit space-time
discretization schemes of the solution $u(t,x)$ to a parabolic partial
differential equation in any dimension perturbed by a space-correlated Gaussian
noise. The coefficients only depend on $u(t,x)$ and the influence of the
correlation on the speed is observed.
http://arXiv.org/abs/math/0611073
http://front.math.ucdavis.edu/math.PR/0611073
(alternate)
Author(s): Caoline Cardon-Weber (PMA) and Annie Millet (PMA and MATISSE and SAMOS)
Abstract: In this paper we show that the Cahn-Hilliard stochastic SPDE has a function
valued solution in dimension 4 and 5 when the perturbation is driven by a
space-correlated Gaussian noise. This is done proving general results on SPDEs
with globally Lipschitz coefficients associated with operators on smooth
domains of $\mathbb{R}^d$ which are parabolic in the sense of Petrovskii}, and
do not necessarily define a semi-group of operators. We study the regularity of
the trajectories of the solutions and the absolute continuity of the law at
some given time and position.
http://arXiv.org/abs/math/0611090
http://front.math.ucdavis.edu/math.PR/0611090
(alternate)
Author(s): Takahiro Miyazaki and Makoto Katori and Norio Konno
Abstract: Quantization of a random-walk model is performed by giving a multi-component
qubit to a walker at site and by introducing a quantum coin, which is
represented by a unitary matrix. In quantum walks, the qubit of walker is mixed
according to the quantum coin at each time step, when the walker hops to other
sites. The standard (discrete) quantum-walk model in one-dimension is defined
by using a $2 \times 2$ unitary matrix for a walker with two-component qubit.
In this paper we use Wigner's $(2j+1)$-dimensional unitary representations of
rotations as quantum coins, where $j$ is a half-integer, and introduce a family
of one-dimensional quantum walks with $(2j+1)$-component qubits. For any value
of half-integer $j$, convergence of all moments of walker's pseudovelocity in
the long-time limit is proved. It is generally shown for the present models
that, if $(2j+1)$ is even, the limit distribution is given by a superposition
of $(2j+1)/2$ terms of scaled Konno's density functions, and if $(2j+1)$ is
odd, it is a superposition of $j$ terms of scaled Konno's density functions and
a Dirac's delta function at the origin. For the two-, three-, and
four-component models, the limit distribution functions are explicitly
calculated and their dependence on the parameters of quantum coins and on the
initial qubit of walker is completely determined. Comparison with computer
simulation results is also shown.
http://arXiv.org/abs/quant-ph/0611022
http://front.math.ucdavis.edu/quant-ph/0611022
(alternate)
Author(s): K.A. Borovkov and D.A. Odell
Abstract: We introduce a new class of dynamic point process models with simple and
intuitive dynamics that are based on the Voronoi tessellations generated by the
processes. Under broad conditions, these processes prove to be ergodic and
produce, on stabilisation, a wide range of clustering patterns. In the paper,
we present results of simulation studies of three statistical measures (Thiel's
redundancy, van Lieshout and Baddeley's $J$-function and the empirical
distribution of the Voronoi nearest neighbours' numbers) for inference on these
models from the clustering behaviour in the stationary regime. In particular,
we make comparisons with the area-interaction processes of Baddeley and van
Lieshout.
http://arXiv.org/abs/math/0611031
http://front.math.ucdavis.edu/math.PR/0611031
(alternate)
Author(s): Adrian R\"ollin
Abstract: We show by a surprisingly simple argument that the exchangeability condition,
which is key to the exchangeable pair approach in Stein's method for
distributional approximation, can be omitted in many standard settings. This is
achieved by replacing the usual antisymmetric function by a simpler one, for
which only equality in distribution is required. In the case of approximations
by continuous distributions we also slightly improve the constants appearing in
previous results. For Poisson approximation, Chatterjee et al. (2005) use a
different antisymmetric function, and additional error terms are needed if
their bound is to be extended beyond the exchangeable setting. There is a
strong connection between this new approach and Barbour's generator
interpretation of Stein's method.
http://arXiv.org/abs/math/0611050
http://front.math.ucdavis.edu/math.PR/0611050
(alternate)
Author(s): Robert Hardy and Simon C. Harris
Abstract: We present a formalization of the spine change of measure approach for
branching diffusions that improves on the scheme laid out for branching
Brownian motion in Kyprianou (2004) ["Travelling wave solutions to the KPP
equation, Ann. Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] which
itself made use of earlier works of Lyons et al (1997) ["A conceptual proof of
the Kesten-Stigum theorem for multi-type branching processes", Classical and
modern branching processes, IMA Vol. Math. Appl., vol.84, Springer, New York,
pp181-185]. We use our new formulation to interpret certain `Gibbs-Boltzmann'
weightings of particles and use this to give a new, intuitive and proof of a
more general `Many-to-One' result which enables expectations of sums over
particles in the branching diffusion to be calculated purely in terms of an
expectation of one particle. Significantly, our formalization has provided the
foundations that facilitate a variety of new, greatly simplified and more
intuitive proofs in branching diffusions: see, for example, the L^p convergence
of additive martingales in Hardy and Harris (2006) ["Spine proofs for
L^p-convergence of branching-diffusion martingales", arXiv:math.PR/0611056],
the path large deviation results for branching Brownian motion in Hardy and
Harris (2006) ["A conceptual approach to a path result for branching Brownian
motion", Stochastic Processes and their Applications,
doi:10.1016/j.spa.2006.05.010] and the large deviations for a continuous-typed
branching diffusion in Git et al (2006) ["Exponential growth rates in a typed
branching diffusion", Annals Applied Prob., (under revision)] and Hardy and
Harris (2004) ["A spine proof of a lower-bound for a typed branching
diffusion", no.0408, Mathematics Preprint, University of Bath].
http://arXiv.org/abs/math/0611054
http://front.math.ucdavis.edu/math.PR/0611054
(alternate)
Author(s): Robert Hardy and Simon C. Harris
Abstract: Using the foundations laid down in Hardy and Harris (2006) ["A new
formulation of the spine approach in branching diffusions",
arXiv:math.PR/0611054], we present new spine proofs of the L^p-convergence
p>=1) of some key `additive' martingales for three distinct models of branching
diffusions, including new results for a multi-type branching Brownian motion
and discussion of left-most particle speeds. The spine techniques we develop
give clear and simple arguments in the spirit of the conceptual spine proofs
found in Kyprianou (2004) ["Travelling wave solutions to the KPP equation, Ann.
Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] and Lyons et al (1997)
["A conceptual proof of the Kesten-Stigum theorem for multi-type branching
processes", Classical and modern branching processes, IMA Vol. Math. Appl.,
vol.84, Springer, New York, pp181-185], and they should also extend to more
general classes of branching diffusions. Importantly, the techniques in this
paper also pave the way for the path large-deviation results for branching
diffusions found in Hardy and Harris (2006) ["A conceptual approach to a path
result for branching Brownian motion", Stochastic Processes and their
Applications, doi:10.1016/j.spa.2006.05.010].
http://arXiv.org/abs/math/0611056
http://front.math.ucdavis.edu/math.PR/0611056
(alternate)
Author(s): D. Bloemker and F. Flandoli and M. Romito
Abstract: The paper analyses a model in surface growth, where uniqueness of weak
solutions seems to be out of reach. We provide the existence of a weak
martingale solution satisfying energy inequalities and having the Markov
property. Furthermore, under non-degeneracy conditions on the noise, we
establish that any such solution is strong Feller and has a unique invariant
measure.
http://arXiv.org/abs/math/0611021
http://front.math.ucdavis.edu/math.PR/0611021
(alternate)
Author(s): Y. Davydov (University Lille 1) and A. Illig (University of Versailles Saint Quentin)
Abstract: We are interested here in a birth-and-growth process where germs are born
according to a Poisson point process with invariant under translation in space
intensity measure. The germs can be born in free space and then start growing
until occupying the available space. In order to consider various way of
growing, we describe the crystals at each time through their geometrical
properties. In this general framework, the crystallization process can be
caracterized by the random field giving for a point in the space state the
first time this point is reached by a crystal. We prove under general
conditions that this random field is mixing in the sens of ergodic theory and
obtain estimates for the coefficient of absolute regularity.
http://arXiv.org/abs/math/0610966
http://front.math.ucdavis.edu/math.PR/0610966
(alternate)
Author(s): Eben Kenah and James Robins
Abstract: In an important paper, M.E.J. Newman claimed that a large class of
network-based stochastic SIR epidemic models are isomorphic to bond percolation
models, where the bonds are the edges of the contact network and the bond
occupation probability is equal to the marginal probability of transmission
from an infected node to a susceptible neighbor. In this paper, we show that
this isomorphism is incorrect and define a percolation model on a semi-directed
network that we call the percolation network that is exactly isomorphic to the
SIR epidemic model. We show that the percolation network model predicts the
same mean outbreak size (below the epidemic threshold), epidemic threshold, and
final size of an epidemic as the bond percolation model. However, we also show
that the bond percolation model fails to predict the correct outbreak size
distribution and probability of an epidemic for any SIR epidemic model with a
non-degenerate distribution of infectiousness. In a series of simulations, we
show that the percolation network model accurately predicts the probability of
an outbreak of size one and the probability of an epidemic, whereas the bond
percolation model underestimates the first and overestimates the latter. In an
appendix, we show that a percolation network model can be defined for any
time-homogeneous SIR epidemic model. We conclude that percolation networks are
a very general method of analyzing stochastic SIR epidemic models.
http://arXiv.org/abs/q-bio/0610057
http://front.math.ucdavis.edu/q-bio.QM/0610057
(alternate)
Author(s): Michael B. Marcus and Jay Rosen
Abstract: Let G=\{G(x),x\in R^1\} be a mean zero Gaussian processes with stationary
increments and set \si ^2(|x-y|)= E(G(x)-G(y))^2. Let f be a symmetric function
with Ef(\eta)<\ff, where \eta=N(0,1). When \si^2(s) is concave or when
\si^2(s)=s^r$, $1
http://arXiv.org/abs/math/0610894
http://front.math.ucdavis.edu/math.PR/0610894
(alternate)
Author(s): H. Gottschalk and B. Smii
Abstract: We consider a stochastic partial differential equation (SPDE) on a lattice
\partial_t X=(\Delta-m^2)X-\lambda X^p+\eta where $\eta$ is a space-time
L\'evy noise. A perturbative (in the sense of formal power series) strong
solution is given by a tree expansion, whereas the correlation functions of the
solution are given by a perturbative expansion with coefficients that are
represented as sums over a certain class of graphs, called Parisi-Wu graphs.
The perturbative expansion of the truncated (connected) correlation functions
is obtained via a Linked Cluster Theorem as a sums over connected graphs only.
The moments of the stationary solution can be calculated as well. In all these
solutions the cumulants of the single site distribution of the noise enter as
multiplicative constants. To determine them, e.g. by comparison with a
empirical correlation function, one can fit these constants (e.g. by the
methods of least squares) and thereby one (approximately) determines law of the
noise.
http://arXiv.org/abs/math/0610906
http://front.math.ucdavis.edu/math.PR/0610906
(alternate)
Author(s): David Coupier
Abstract: A $d$-dimensional ferromagnetic Ising model on a lattice torus is considered.
As the size of the lattice tends to infinity, two conditions ensuring a Poisson
approximation for the distribution of the number of occurrences in the lattice
of any given local configuration are suggested. The proof builds on the
Stein-Chen method. The rate of the Poisson approximation and the speed of
convergence to it are precised and make sense for the model. Thus, the two
sufficient conditions are traduced in terms of the magnetic field and the pair
potential. In particular, the Poisson approximation holds even if both
potentials diverge.
http://arXiv.org/abs/math/0610939
http://front.math.ucdavis.edu/math.PR/0610939
(alternate)
Author(s): Tanja Gernhard
Abstract: In a phylogenetic tree, we often don't have information about the time a
speciation event (inner node) occured. Under a neutral model for speciation, I
develop fast algorithms for calculating the probability that an inner node i is
the k-th speciation event. For the Yule and the coalescent model, I develop an
edge length estimation as well. Various properties of the Yule model are
discussed throughout the thesis.
http://arXiv.org/abs/math/0610919
http://front.math.ucdavis.edu/math.CO/0610919
(alternate)
Author(s): Peter Eichelsbacher and Michael Stolz
Abstract: In his seminal 1962 paper on the ``threefold way'', Freeman Dyson classified
the spaces of matrices that support the random matrix ensembles deemed relevant
from the point of view of classical quantum mechanics. Recently, Heinzner,
Huckleberry and Zirnbauer have obtained a similar classification based on less
restrictive assumptions, thus taking care of the needs of modern mesoscopic
physics. Their list is in one-to-one correspondence with the infinite families
of Riemannian symmetric spaces as classified by Cartan. The present paper
develops the corresponding random matrix theories, with a special emphasis on
large deviation principles.
http://arXiv.org/abs/math/0610811
http://front.math.ucdavis.edu/math.PR/0610811
(alternate)
Author(s): Ilwoo Cho
Abstract: In this paper, we define several measures induced by a finite directed graph.
The study themselves is interesting ont only in the noncommutative probability
point of view but also in the algebraic structure point of view, since to
define graph measures we defined several rough algebraic structures induced by
the given graph.
http://arXiv.org/abs/math/0610817
http://front.math.ucdavis.edu/math.PR/0610817
(alternate)
Author(s): Victor de la Pena and Henryk Gzyl and Patrick McDonald
Abstract: Let $G$ be a finite tree with root $r$ and associate to the internal vertices
of $G$ a collection of transition probabilities for a simple nondegenerate
Markov chain. Embedd $G$ into a graph $G^\prime$ constructed by gluing finite
linear chains of length at least 2 to the terminal vertices of $G.$ Then
$G^\prime$ admits distinguished boundary layers and the transition
probabilities associated to the internal vertices of $G$ can be augmented to
define a simple nondegenerate Markov chain $X$ on the vertices of $G^\prime.$
We show that the transition probabilities of $X$ can be recovered from the
joint distribution of first hitting time and first hitting place of $X$ started
at the root $r$ for the distinguished boundary layers of $G^\prime.$
http://arXiv.org/abs/math/0610821
http://front.math.ucdavis.edu/math.PR/0610821
(alternate)
Author(s): M. Grendar
Abstract: Posterior distribution over a countable set M of continuous data-sampling
distributions piles up at L-projection of the true distribution r on M,
provided that the L-projection is unique. If there are several L-projections of
r on M, then the posterior probability splits among them equally.
http://arXiv.org/abs/math/0610824
http://front.math.ucdavis.edu/math.PR/0610824
(alternate)
Author(s): Ionel Popescu
Abstract: In this paper we discuss general tridiagonal matrix models which are natural
extensions of the ones given by Dumitriu and Edelman. We prove here the
convergence of the distribution of the eigenvalues and compute the limiting
distributions in some particular cases. We also discuss the limit of
fluctuations, which, in a general context, turn out to be Gaussian. For the
case of several random matrices, we prove the convergence of the joint moments
and the convergence of the fluctuations to a Gaussian family.
http://arXiv.org/abs/math/0610827
http://front.math.ucdavis.edu/math.PR/0610827
(alternate)
Author(s): Peter Eichelsbacher and Wolfgang Konig
Abstract: We construct the conditional version of $k$ independent and identically
distributed random walks on $\R$ given that they stay in strict order at all
times. This is a generalisation of so-called non-colliding or non-intersecting
random walks, the discrete variant of Dyson's Brownian motions, which have been
considered yet only for nearest-neighbor walks on the lattice. Our only
assumptions are moment conditions on the steps and the validity of the local
central limit theorem. The conditional process is constructed as a Doob
$h$-transform with some positive regular function $V$ that is strongly related
with the Vandermonde determinant and reduces to that function for simple random
walk. Furthermore, we prove an invariance principle, i.e., a functional limit
theorem towards Dyson's Brownian motions, the continuous analogue.
http://arXiv.org/abs/math/0610850
http://front.math.ucdavis.edu/math.PR/0610850
(alternate)
Author(s): Geoffrey Grimmett and Dominic Welsh
Abstract: In writing this biographical memoir of John Hammersley, we have tried to
communicate something of the character of the person, and of the impact of his
scientific achievements across lattice models (for example, percolation,
self-avoiding walks, first-passage percolation, dimer models), stochastic
processes (including subadditive ergodic theory), Monte Carlo methods, applied
probability, statistics, and other areas to which he contributed. There is also
an extended account of his earlier life, taken from autobiographical notes
written around 1994, together with a list of his published work.
http://arXiv.org/abs/math/0610862
http://front.math.ucdavis.edu/math.PR/0610862
(alternate)
Author(s): Ionel Popescu
Abstract: We give a short proof of the free analogue of the Talagrand inequality for
the transportation cost to the semicircular which was originally proved by
Biane and Voiculescu. The proof is based on a convexity argument and is in the
spirit of the original Talagrand's proof. We also discuss the convergence,
fluctuations and large deviations of the energy of the eigenvalues of beta
ensembles, which gives also yet another proof of the convergence of the
eigenvalue distribution to the semicircle law.
http://arXiv.org/abs/math/0610826
http://front.math.ucdavis.edu/math.CA/0610826
(alternate)
Author(s): Tanja Gernhard and Daniel Ford and Rutger Vos and Mike Steel
Abstract: The reconstruction of large phylogenetic trees from data that violates
clocklike evolution (or as a supertree constructed from any m input trees)
raises a difficult question for biologists - how can one assign relative dates
to the vertices of the tree? In this paper we investigate this problem,
assuming a uniform distribution on the order of the inner vertices of the tree
(which includes, but is more general than, the popular Yule distribution on
trees). We derive fast algorithms for computing the probability that (i) any
given vertex in the tree was the j--th speciation event (for each j), and (ii)
any one given vertex is earlier in the tree than a second given vertex. We show
how the first algorithm can be used to calculate the expected length of any
given interior edge in any given tree that has been generated under either a
constant-rate speciation model, or the coalescent model.
http://arXiv.org/abs/math/0610840
http://front.math.ucdavis.edu/math.CO/0610840
(alternate)
Author(s): Lahcen Boulanba and M'hamed Eddahbi and Mohamed Mellouk
Abstract: In this paper we study a class of stochastic partial differential equations
in the whole space $\mathbb{R}^{d}$, with arbitrary dimension $d\geq 1$, driven
by a Gaussian noise white in time and correlated in space. The differential
operator is a fractional derivative operator. We show the existence, uniqueness
and H\"{o}lder's regularity of the solution. Then by means of Malliavin
calculus, we prove that the law of the solution has a smooth density with
respect to the Lebesgue measure.
http://arXiv.org/abs/math/0610769
http://front.math.ucdavis.edu/math.PR/0610769
(alternate)
Author(s): Heng-Qing Ye
Abstract: In this paper, we present counter-intuitive examples for the multiclass
queueing network system. In the system, each station may serve more than one
job class with differentiated service priority, and each job may require
service sequentially by more than one service station. In our examples, the
network performance is improved even when more workloads are admitted for
service.
http://arXiv.org/abs/math/0610784
http://front.math.ucdavis.edu/math.PR/0610784
(alternate)
Author(s): Itai Benjamini and Alain-Sol Sznitman
Abstract: We consider random walk on a discrete torus E of side-length N, in
sufficiently high dimension d. We investigate the percolative properties of the
vacant set corresponding to the collection of sites which have not been visited
by the walk up to time uN^d. We show that when u is chosen small, as N tends to
infinity, there is with overwhelming probability a unique connected component
in the vacant set which contains segments of length const log N. Moreover, this
connected component occupies a non-degenerate fraction of the total number of
sites N^d of E, and any point of E lies within distance an arbitrary fractional
power of N from this component.
http://arXiv.org/abs/math/0610802
http://front.math.ucdavis.edu/math.PR/0610802
(alternate)
Author(s): Robert O. Bauer
Abstract: We show that for the conformal restriction measure with exponent $b$ in the
unit disk on hulls $\gamma$ connecting $e^{ix}$ to 1 the probability of the
event that $\gamma$ avoids the disk of radius $q$ centered at zero decays like
$\exp(-b\pi x/(1-q))$ if either $b\in[5/8,1]\cup[5/4,\infty)$ and
$x\in(0,\pi]$, or if $b\in(1,5/4)$, $x\in(0,\pi)$, and $bx\le\pi$.
http://arXiv.org/abs/math/0610805
http://front.math.ucdavis.edu/math.PR/0610805
(alternate)
Author(s): Gr\'{e}gory Marc Miermont (LM-Orsay)
Abstract: We prove that critical multitype Galton-Watson trees converge after rescaling
to the Brownian continuum random tree, under the hypothesis that the offspring
distribution has finite covariance matrices. Our study relies on an ancestral
decomposition for marked multitype trees. We then couple the genealogical
structure with a spatial motion, whose step distribution may depend on the
structure of the tree in a local way, and show that the resulting discrete
spatial trees converge once suitably rescaled to the Brownian snake, under some
suitable moment assumptions.
http://arXiv.org/abs/math/0610807
http://front.math.ucdavis.edu/math.PR/0610807
(alternate)
Author(s): Benjamin Jourdain (CERMICS)
Abstract: The probabilistic equivalent formulation of Dupire's PDE is the Put-Call
duality equality. In local volatility models including exponential L\'{e}vy
jumps, we give a direct probabilistic proof for this result based on stochastic
flows arguments. This approach also enables us to check the probabilistic
equivalent formulation of various generalizations of Dupire's PDE recently
obtained by Pironneau by the adjoint equation technique in the case of complex
options.
http://arXiv.org/abs/math/0610809
http://front.math.ucdavis.edu/math.PR/0610809
(alternate)
Author(s): Nobuaki Sugimine and Masato Takei
Abstract: We investigate a limit theorem on traversable length inside semi-cylinder in
the 2-dimensional supercritical Bernoulli bond percolation, which gives an
extension of Theorem 2 in Grimmett(1981). This type of limit theorems was
originally studied for the extinction time for the 1-dimensional contact
process on a finite interval in Wagner and Anantharam(2005). Actually, our main
result Theorem 2.1 is stated under a rather general 2-dimensional bond
percolation setting.
http://arXiv.org/abs/math/0610744
http://front.math.ucdavis.edu/math.PR/0610744
(alternate)
Author(s): Marie-Amelie Morlais
Abstract: In this paper, we study a class of quadratic Backward Stochastic Differential
Equations (BSDEs) which arises naturally when studying the problem of utility
maximization with portfolio constraints. We first establish existence and
uniqueness results for such BSDEs and then, we give an application to the
utility maximization problem. Three cases of utility functions will be
discussed: the exponential, power and logarithmic ones.
http://arXiv.org/abs/math/0610749
http://front.math.ucdavis.edu/math.PR/0610749
(alternate)
Author(s): Martin Bender
Abstract: We consider a system of diffusing particles on the real line in a quadratic
external potential and with repulsive electrostatic interaction. The empirical
measure process is known to converge weakly to a deterministic measure-valued
process as the number of particles tends to infinity. Provided the initial
fluctuations are small, the rescaled linear statistics of the empirical measure
process converge in distribution to a Gaussian limit for sufficiently smooth
test functions. We derive explicit general formulae for the mean and covariance
in this central limit theorem by analyzing a partial differential equation
characterizing the limiting fluctuations.
http://arXiv.org/abs/math/0610750
http://front.math.ucdavis.edu/math.PR/0610750
(alternate)
Author(s): Zhenning Kong and Edmund M. Yeh
Abstract: Percolation theory has become a useful tool for the analysis of large scale
wireless networks. We investigate the fundamental problem of characterizing the
critical density for Poisson random geometric graphs in continuum percolation
theory. By using a probabilistic analysis which incorporates the clustering
effect in random geometric graphs, we develop a new class of lower bounds for
the critical density. In particular, the lower bound is substantially improved
to 0.833. This graph theoretical viewpoint provides a new approach and a deep
insight for the problem.
http://arXiv.org/abs/math/0610751
http://front.math.ucdavis.edu/math.PR/0610751
(alternate)
Author(s): Jorge A. Leon (CINVESTAV-Ipn) and Samy Tindel (IECN)
Abstract: In this paper we introduce a stochastic integral with respect to the solution
X of the fractional heat equation on [0,1], interpreted as a divergence
operator. This allows to use the techniques of the Malliavin calculus in order
to establish an It\^{o}-type formula for the process X.
http://arXiv.org/abs/math/0610753
http://front.math.ucdavis.edu/math.PR/0610753
(alternate)
Author(s): Yuri Bakhtin and Jonathan C. Mattingly
Abstract: We consider an infinite-dimensional dynamical system with polynomial
nonlinearity and additive noise given by a finite number of Wiener processes.
By studying how randomness is spread by the system we develop a counterpart of
Hormander's classical theory in this setting. We study the distributions of
finite-dimensional projections of the solutions and give conditions that
provide existence and smoothness of densities of these distributions with
respect to the Lebesgue measure. We also apply our results to concrete SPDEs
such as Stochastic Reaction Diffusion Equation and Stochastic 2D Navier--Stokes
System.
http://arXiv.org/abs/math/0610754
http://front.math.ucdavis.edu/math.PR/0610754
(alternate)
Author(s): Jean-Marc Derrien (LM-Brest)
Abstract: In this note, we prove without using Fourier analysis that the symmetric
square integrable random walks in $\Z^{2}$ are recurrent.
http://arXiv.org/abs/math/0610763
http://front.math.ucdavis.edu/math.PR/0610763
(alternate)
Author(s): Richard G. Clegg
Abstract: This paper describes, in detail, techniques for measuring the Hurst
parameter. Measurements are given on artificial data both in a raw form and
corrupted in various ways to check the robustness of the tools in question.
Measurements are also given on real data, both new data sets and well-studied
data sets. All data and tools used are freely available for download along with
simple ``recipes'' which any researcher can follow to replicate these
measurements.
http://arXiv.org/abs/math/0610756
http://front.math.ucdavis.edu/math.ST/0610756
(alternate)
Author(s): Alexander E. Holroyd and Terry Soo
Abstract: In this expository note, we construct a non-measurable set in the probability
space of coin flips indexed by the integers.
http://arXiv.org/abs/math/0610705
http://front.math.ucdavis.edu/math.PR/0610705
(alternate)
Author(s): Bela Bollobas and Oliver Riordan
Abstract: We make use of the recent proof that the critical probability for percolation
on random Voronoi tessellations is 1/2 to prove the corresponding result for
random Johnson-Mehl tessellations, as well as for two-dimensional slices of
higher dimensional Voronoi tessellations. Surprisingly, the proof is a little
simpler for these more complicated models.
http://arXiv.org/abs/math/0610716
http://front.math.ucdavis.edu/math.PR/0610716
(alternate)
Author(s): Nicolas Bouleau (LAMM)
Abstract: Because the stochastic calculus yields rarely random variables with laws
defined by explicit closed formulas, probabilistic numerical computations are
done most often by simulation. The simulation by the shift, whose field of
application is as wide as that of Monte Carlo method, is particularly relevant
when the simulations use, for each sample, a large number of calls to the
random function. We give here the theoretical features, the implementation and
the specific advantages of this method.
http://arXiv.org/abs/math/0610729
http://front.math.ucdavis.edu/math.PR/0610729
(alternate)
Author(s): Leonid Kontorovich
Abstract: We prove an elementary yet useful inequality bounding the maximal value of
certain linear programs. This leads directly to a bound on the martingale
difference for arbitrarily dependent random variables, providing a
generalization of some recent concentration of measure results. The linear
programming inequality may be of independent interest.
http://arXiv.org/abs/math/0610712
http://front.math.ucdavis.edu/math.FA/0610712
(alternate)
Author(s): Mario Annunziato
Abstract: In this paper the numerical approximation of solutions of Liouville-Master
Equations for time-dependent distribution functions of Piecewise Deterministic
Processes with memory is considered. These equations are linear hyperbolic PDEs
with non-constant coefficients, and boundary conditions that depend on
integrals over the interior of the integration domain. We construct a finite
difference method of the first order, by a combination of the upwind method,
for PDEs, and by a direct quadrature, for the boundary condition. We analyse
convergence of the numerical solution for distribution functions evolving
towards an equilibrium. Numerical results for two problems, whose analytical
solutions are known in closed form, illustrate the theoretical finding.
http://arXiv.org/abs/math/0610725
http://front.math.ucdavis.edu/math.NA/0610725
(alternate)
Author(s): R. J. Messikh
Abstract: For the two dimensional Ising model, we construct the adequate surface
tension near criticality. The latter quantity has been shown to play a central
role in the study of phase coexistence in a joint limit where the temperature
approaches the critical point from below and simultaneously the size of the
system increases fast enough.
http://arXiv.org/abs/math/0610636
http://front.math.ucdavis.edu/math.PR/0610636
(alternate)
Author(s): Don A. Dawson and Andreas Greven and Frank den Hollander and Rongfeng Sun and Jan M. Swart
Abstract: This paper studies countable systems of linearly and hierarchically
interacting diffusions taking values in the positive quadrant. These systems
arise in population dynamics for two types of individuals migrating between and
interacting within colonies. Their large-scale space-time behavior can be
studied by means of a renormalization program. This program, which has been
carried out successfully in a number of other cases (mostly one-dimensional),
is based on the construction and the analysis of a nonlinear renormalization
transformation, acting on the diffusion function for the components of the
system and connecting the evolution of successive block averages on successive
time scales. We identify a general class of diffusion functions on the positive
quadrant for which this renormalization transformation is well-defined and,
subject to a conjecture on its boundary behavior, can be iterated. Within
certain subclasses, we identify the fixed points for the transformation and
investigate their domains of attraction. These domains of attraction constitute
the universality classes of the system under space-time scaling.
http://arXiv.org/abs/math/0610645
http://front.math.ucdavis.edu/math.PR/0610645
(alternate)
Author(s): George P. Yanev
Abstract: We present a progress report for studies on maxima related to offspring in
branching processes. We summarize and discuss the findings on the subject that
appeared in the last ten years. Some of the results are refined and illustrated
with new examples.
http://arXiv.org/abs/math/0610647
http://front.math.ucdavis.edu/math.PR/0610647
(alternate)
Author(s): M. Baldini
Abstract: Let $X_t$ be a reversible and positive recurrent diffusion in $R^d$ described
by \begin{equation}\nonumber X_t=x+\sigma b(t)+\int_0^tm(X_s)\dif s,
\end{equation} where the diffusion coefficient $\sigma$ is a positive-definite
matrix and the drift $m$ is a smooth function. Let $X_t(A)$ denote the image of
a compact set $A\subset R^d$ under the stochastic flow generated by $X_t$. If
the divergence of the drift is strictly negative, there exists a set of
functions $u$ such that \[\lim_{t\to\infty} \int_{X_t(A)}u(x)\dif
x=0\quad{a.s.} \] A characterization of the functions $u$ is provided, as well
as lower and upper bounds for the exponential rate of convergence.
http://arXiv.org/abs/math/0610665
http://front.math.ucdavis.edu/math.PR/0610665
(alternate)
Author(s): T. Schreiber and Mathew D. Penrose and J. E. Yukich
Abstract: Consider the random sequential packing model with infinite input and in any
dimension. When the input consists of non-zero volume convex solids we show
that the total number of solids accepted over cubes of volume $\lambda$ is
asymptotically normal as $\lambda \to \infty$. We provide a rate of
approximation to the normal and show that the finite dimensional distributions
of the packing measures converge to those of a mean zero generalized Gaussian
field. The method of proof involves showing that the collection of accepted
solids satisfies the weak spatial dependence condition known as stabilization.
http://arXiv.org/abs/math/0610680
http://front.math.ucdavis.edu/math.PR/0610680
(alternate)
Author(s): Pierre Nolin (DMA and LM-Orsay)
Abstract: We study gradient percolation for site percolation on the triangular lattice.
This is a percolation model where the percolation probability depends linearly
on the location of the site. We prove the results predicted by physicists for
this model. More precisely, we describe the fluctuations of the interfaces
around their (straight) scaling limits, the expected and typical lengths of
these interfaces. These results build on the recent results for critical
percolation on this lattice by Smirnov, Lawler, Schramm and Werner, and on the
hyperscaling ideas developed by Kesten.
http://arXiv.org/abs/math/0610682
http://front.math.ucdavis.edu/math.PR/0610682
(alternate)
Author(s): Yefim I. Leifman
Abstract: A quantum-classical isomorphism is used in order to develop a Monte Carlo
simulation with controlled deviation from detailed balance, that is, in
proposed notions, with generalized detailed balance and known relative entropy
with respect to the reference process at each point. In order to apply this
method to molecular simulations a new partial chirotope realization algorithm,
based on linear programming methods, a new distance geometry algorithm and a
new all-atom off-lattice Monte Carlo method are proposed.
http://arXiv.org/abs/math/0610696
http://front.math.ucdavis.edu/math.PR/0610696
(alternate)
Author(s): I. Rushkin and E. Bettelheim and I. A. Gruzberg and P. Wiegmann
Abstract: We consider critical curves -- conformally invariant curves that appear at
critical points of two-dimensional statistical mechanical systems. We show how
to describe these curves in terms of the Coulomb gas formalism of conformal
field theory (CFT). We also provide links between this description and the
stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the
long-time limit of stochastic evolution of various SLE observables related to
CFT primary fields. We show how the multifractal spectrum of harmonic measure
and other fractal characteristics of critical curves can be obtained.
http://arXiv.org/abs/cond-mat/0610550
http://front.math.ucdavis.edu/cond-mat/0610550
(alternate)
Author(s): Scott Zrebiec
Abstract: We show that for Gaussian random SU(2)polynomials of a large degree $N$ the
probability that there are no zeros in the disk of radius $r$ is less than
$e^{-c_{1,r} N^2}$, and is also greater than $e^{-c_{2,r} N^2}$. Enroute to
this result, we also derive a more general result: probability estimates for
the event that the number of complex zeros of a random polynomial of high
degree deviates significantly from its mean.
http://arXiv.org/abs/math/0610686
http://front.math.ucdavis.edu/math.CV/0610686
(alternate)
Author(s): Konstantin Borovkov and David Odell
Abstract: We introduce a new class of spatial-temporal point processes based on Voronoi
tessellations. At each step of such a process, a point is chosen at random
according to a distribution determined by the associated Voronoi cells. The
point is then removed, and a new random point is added to the configuration.
The dynamics are simple and intuitive and could be applied to modeling natural
phenomena. We prove ergodicity of these processes under wide conditions.
http://arXiv.org/abs/math/0610606
http://front.math.ucdavis.edu/math.PR/0610606
(alternate)
Author(s): Jan van Neerven and Mark Veraar and Lutz Weis
Abstract: In this paper we construct a theory of stochastic integration of processes
with values in $\calL(H,E)$, where $H$ is a separable Hilbert space and $E$ is
a UMD Banach space. The integrator is an $H$-cylindrical Brownian motion. Our
approach is based on a two-sided $L^p$-decoupling inequality for UMD spaces due
to Garling, which is combined with the theory of stochastic integration of
$\calL(H,E)$-valued functions introduced recently by two of the authors. We
obtain various characterizations of the stochastic integral and prove versions
of the It\^o isometry, the Burkholder-Davis-Gundy inequalities, and the
representation theorem for Brownian martingales.
http://arXiv.org/abs/math/0610619
http://front.math.ucdavis.edu/math.PR/0610619
(alternate)
Author(s): Fabio Gobbi and Cecilia Mancini
Abstract: In this paper we consider two processes driven by diffusions and jumps. We
consider both finite activity and infinite activity jump components. Given
discrete observations we disentangle the covariation between the two diffusion
parts from the co-jumps. A commonly used approach to estimate the diffusion
covariation part is to take the sum of the cross products of the two processes
increments; however this estimator can be highly biased in the presence of jump
components, since it approaches the quadratic covariation containing also the
co-jumps. Our estimator is based on a threshold principle allowing to isolate
the jumps. %detect the presence of jumps. As a consequence we find an estimator
which is consistent. In the case of finite activity jump components the
estimator is also asymptotically Gaussian. We assess the performance of our
estimator for finite samples on four different simulated models.
http://arXiv.org/abs/math/0610621
http://front.math.ucdavis.edu/math.PR/0610621
(alternate)
Author(s): Rongfeng Sun and Jan M. Swart
Abstract: The (standard) Brownian web is a collection of coalescing one-dimensional
Brownian motions, starting from each point in space and time. It arises as the
diffusive scaling limit of a collection of coalescing random walks. We show
that it is possible to obtain a nontrivial limiting object if the random walks
in addition branch with a small probability. We call the limiting object the
Brownian net, and study some of its elementary properties.
http://arXiv.org/abs/math/0610625
http://front.math.ucdavis.edu/math.PR/0610625
(alternate)
Author(s): N. Tzvetkov
Abstract: We define a finite Borel measure of Gibbs type, supported by the Sobolev
spaces of negative indexes on the circle. The measure can be seen as a limit of
finite dimensional measures. These finite dimensional measures are invariant by
the ODE's which correspond to the projection of the Benjamin-Ono equation,
posed on the circle, on the first N>>1 modes in the trigonometric bases.
http://arXiv.org/abs/math/0610626
http://front.math.ucdavis.edu/math.AP/0610626
(alternate)
Author(s): Yasunari Higuchi and Xian-Yuan Wu
Abstract: We prove that the critical probability for the Sierpinski carpet lattice in
two dimensions is uniquely determined. The transition is sharp. This extends
the Kumagai's result to the original Sierpinski carpet lattice.
http://arXiv.org/abs/math/0610583
http://front.math.ucdavis.edu/math.PR/0610583
(alternate)
Author(s): Omar El-Dakkak (LSTA) and Giovanni Peccati (LSTA)
Abstract: Let X be a non-deterministic infinite exchangeable sequence with values in
{0,1}. We show that X is Hoeffding-decomposable if, and only if, X is either an
i.i.d. sequence or a Polya sequence. This completes the results established in
Peccati [2004]. The proof uses several combinatorial implications of the
correspondence between Hoeffding decomposability and weak independence. Our
results must be compared with previous characterizations of i.i.d. and Polya
sequences given by Hill et al. [1987] and Diaconis and Yilvisaker [1979].
http://arXiv.org/abs/math/0610590
http://front.math.ucdavis.edu/math.PR/0610590
(alternate)
Author(s): Nathalie Eisenbaum and Haya Kaspi
Abstract: Permanental processes can be viewed as a generalisation of squared centered
Gaussian processes. We develop in this paper two main subjects. The first one
analyses the connections of these processes with the local times of general
Markov processes. The second deals with Bosonian point processes and the
Bose-Einstein condensation. The obtained results in both directions are related
and based on the notion of infinite divisibility.
http://arXiv.org/abs/math/0610600
http://front.math.ucdavis.edu/math.PR/0610600
(alternate)
Author(s): Noga Alon and Itai Benjamini and Eyal Lubetzky and Sasha Sodin
Abstract: We compute the mixing rate of a non-backtracking random walk on a regular
expander. Using some properties of Chebyshev polynomials of the second kind, we
show that this rate may be up to twice as fast as the mixing rate of the simple
random walk. The closer the expander is to a Ramanujan graph, the higher the
ratio between the above two mixing rates is.
As an application, we show that if $G$ is a high-girth regular expander on
$n$ vertices, then a typical non-backtracking random walk of length $n$ on $G$
does not visit a vertex more than $(1+o(1))\frac{\log n}{\log\log n}$ times,
and this result is tight. In this sense, the multi-set of visited vertices is
analogous to the result of throwing $n$ balls to $n$ bins uniformly, in
contrast to the simple random walk on $G$, which almost surely visits some
vertex $\Omega(\log n)$ times.
http://arXiv.org/abs/math/0610550
http://front.math.ucdavis.edu/math.PR/0610550
(alternate)
Author(s): Serge Cohen and Renaud Marty
Abstract: This paper is devoted to establish an invariance principle where the limit
process is a multifractional Gaussian process with a multifractional function
which takes its values in (1/2, 1). Some properties, as regularity and local
self-similarity, of this process are studied. Moreover the limit process is
compared to the multifractional Brownian motion.
http://arXiv.org/abs/math/0610551
http://front.math.ucdavis.edu/math.PR/0610551
(alternate)
Author(s): Nicolas Bouleau (LAMM)
Abstract: The aim of this study is to clarify the consequences of recent theoretical
results for the numerical computation of expectation by the shift method, and
in particular to yield sufficient criteria for the existence of speed of
convergence of the type `iterated logarithm' in several situations. We deepen
the case of the Wiener space because it contains many situations useful in
applications.
http://arXiv.org/abs/math/0610560
http://front.math.ucdavis.edu/math.PR/0610560
(alternate)
Author(s): Joseph Najnudel
Abstract: In this paper, we construct a family of probability measures, by
penalizations of a Walsh Brownian motion with a weight dependent on its value
and its local time at a time t. We prove that this family converges to a
probability measure as t tends to infinity, and we study the behaviour of this
limit measure.
http://arXiv.org/abs/math/0610564
http://front.math.ucdavis.edu/math.PR/0610564
(alternate)
Author(s): Yves Le Jan (LM-Orsay)
Abstract: The purpose of this note is to extend Dynkin's isomorphim involving
functionals of the occupation field of a symmetric Markov processes and of the
associated Gaussian field to a suitable class of non symmetric Markov
processes.
http://arXiv.org/abs/math/0610571
http://front.math.ucdavis.edu/math.PR/0610571
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: Under the assumption that the initial population size of a Galton-Watson
branching process increases to infinity, the paper studies asymptotic behavior
of the population size before extinction. More specifically, we establish
asymptotic properties of the conditional moments (which are exactly defined in
the paper).
http://arXiv.org/abs/math/0610506
http://front.math.ucdavis.edu/math.PR/0610506
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: We show that the linear viscoelastic materials, and more generally the
physical phenomena to which Biot's relaxation theory is relevant, can be put in
correspondance with the laws of processes with independent increments. In the
one dimensional case this correspondence is one to one with subordinators and
gives rise naturally to a conjugation relation on subordinators.
http://arXiv.org/abs/math/0610507
http://front.math.ucdavis.edu/math.PR/0610507
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: The arbitrary functions principle says that the fractional part of $nX$
converges stably to an independent random variable uniformly distributed on the
unit interval, as soon as the random variable $X$ possesses a density or a
characteristic function vanishing at infinity. We prove a similar property for
random variables defined on the Wiener space when the stochastic measure
$dB\_s$ is crumpled on itself.
http://arXiv.org/abs/math/0610509
http://front.math.ucdavis.edu/math.PR/0610509
(alternate)
Author(s): Li Xin Zhang
Abstract: The exponential inequality of the maximum partial sums is a key to establish
the law of the iterated logarithm of negatively associated random variables. In
the one-indexed random sequence case, such inequalities for negatively
associated random variables are established by Shao (2000) by using his
comparison theorem between negatively associated and independent random
variables. In the multi-indexed random field case, the comparison theorem
fails. The purpose of this paper is to establish the Kolmogorov exponential
inequality as well a moment inequality of the maximum partial sums of a
negatively associated random field via a different method. By using these
inequalities, the sufficient and necessary condition for the law of the
iterated logarithm of a negatively associated random field to hold is obtained.
http://arXiv.org/abs/math/0610511
http://front.math.ucdavis.edu/math.PR/0610511
(alternate)
Author(s): Li-Xin Zhang and Wei Huang
Abstract: In literature, the central limit theorems for the product of sums of various
random variables have studied. The purpose of this note is to show that this
kind of results are corollary of the invariance principle.
http://arXiv.org/abs/math/0610515
http://front.math.ucdavis.edu/math.PR/0610515
(alternate)
Author(s): Li-Xin Zhang
Abstract: Let $X$, $X_1$, $X_2$, $...$ be i.i.d. random variables, and let $S_n=X_1+...
+ X_n$ be the partial sums and $M_n=\max_{k\le n}|S_k|$ be the maximum partial
sums. We give the sufficient and necessary conditions for a kind of limit
theorems to hold on the convergence rate of the tail probabilities of both
$S_n$ and $M_n$. These results are related to the law of the iterated
logarithm. The results of Gut and Spataru (2000) are special cases of ours.
http://arXiv.org/abs/math/0610519
http://front.math.ucdavis.edu/math.PR/0610519
(alternate)
Author(s): Li-Xin Zhang
Abstract: This paper gives sufficent and necessary conditions on a kind of limit
results to hold on the precise convergent rate of an infinite series of
probabilities on the Chung type law of the iterated logarithm.
http://arXiv.org/abs/math/0610520
http://front.math.ucdavis.edu/math.PR/0610520
(alternate)
Author(s): Li-Xin Zhang
Abstract: By using the strong approximation, this paper establishes several limit
results on the convergent rate of a infinite series of probabilities on the
other law of iterated logarithm.
http://arXiv.org/abs/math/0610521
http://front.math.ucdavis.edu/math.PR/0610521
(alternate)
Author(s): Jean Bertoin (PMA and DMA)
Abstract: It was shown recently that a Langevin process can be reflected at an energy
absorbing boundary. Here, we establish that the law of this reflecting process
can be characterized as the unique weak solution to a certain second order
stochastic differential equation with constraints, which is in sharp contrast
with a deterministic analog.
http://arXiv.org/abs/math/0610442
http://front.math.ucdavis.edu/math.PR/0610442
(alternate)
Author(s): H. Abels M. Kassmann
Abstract: We consider the linear integro-differential operator $L$ defined by \[ Lu(x)
=\int_\Rn (u(x+y) - u(x) - 1_{[1,2]}(\alpha) 1_{\{|y|\leq 2\}}(y)y \cdot \nabla
u(x)) k(x,y) \sd y . \] Here the kernel $k(x,y)$ behaves like
$|y|^{-d-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is H\"older-continuous
in the first variable, precise definitions are given below. The aim of this
work is twofold. On one hand, we study the unique solvability of the Cauchy
problem corresponding to $L$. On the other hand, we study the martingale
problem for $L$. The analytic results obtained for the deterministic parabolic
equation guarantee that the martingale problem is well-posed. Our strategy
follows the classical path of Stroock-Varadhan. The assumptions allow for cases
that have not been dealt with so far.
http://arXiv.org/abs/math/0610445
http://front.math.ucdavis.edu/math.PR/0610445
(alternate)
Author(s): Kevin Ford
Abstract: We give sharp, uniform estimates for the probability that a random walk of n
steps on the reals avoids a half-line [y,infinity) given that it ends at the
point x. The estimates hold for general continuous or lattice distributions
provided the 4th moment is finite.
http://arXiv.org/abs/math/0610450
http://front.math.ucdavis.edu/math.PR/0610450
(alternate)
Author(s): Itai Benjamini and Gady Kozma and Nicholas Wormald
Abstract: We show that the total variation mixing time of the simple random walk on the
giant component of supercritical Erdos-Renyi graphs is log^2 n. This statement
was only recently proved, independently, by Fountoulakis and Reed. Our proof
follows from a structure result for these graphs which is interesting in its
own right. We show that these graphs are "decorated expanders" - an expander
glued to graphs whose size has constant expectation and exponential tail, and
such that each vertex in the expander is glued to no more than a constant
number of decorations.
http://arXiv.org/abs/math/0610459
http://front.math.ucdavis.edu/math.PR/0610459
(alternate)
Author(s): Asaf Nachmias and Yuval Peres
Abstract: We provide simple proofs describing the behavior of the largest component of
the Erdos-Renyi random graph $G(n,p)$ outside of the scaling window,
$p={1+\eps(n) \over n}$ where $\eps(n) \to 0$ but $\eps(n)n^{1/3} \to \infty$.
http://arXiv.org/abs/math/0610466
http://front.math.ucdavis.edu/math.PR/0610466
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: We present recent advances on Dirichlet forms methods either to extend
financial models beyond the usual stochastic calculus or to study stochastic
models with less classical tools. In this spirit, we interpret the asymptotic
error on the solution of an sde due to the Euler scheme in terms of a Dirichlet
form on the Wiener space, what allows to propagate this error thanks to
functional calculus.
http://arXiv.org/abs/math/0610475
http://front.math.ucdavis.edu/math.PR/0610475
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: In order to develop a differential calculus for error propagation we study
local Dirichlet forms on probability spaces with square field operator $\Gamma$
-- i.e. error structures -- and we are looking for an object related to
$\Gamma$ which is linear and with a good behaviour by images. For this we
introduce a new notion called the measure valued gradient which is a randomized
square root of $\Gamma$. The exposition begins with inspecting some natural
notions candidate to solve the problem before proposing the measure-valued
gradient and proving its satisfactory properties.
http://arXiv.org/abs/math/0610485
http://front.math.ucdavis.edu/math.PR/0610485
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: Equipping the probability space with a local Dirichlet form with square field
operator $\Gamma$ and generator $A$ allows to improve Monte Carlo computations
of expectations, densities, and conditional expectations, as soon as we are
able to simulate a random variable $X$ together with $\Gamma[X]$ and $A[X]$. We
give examples on the Wiener space, on the Poisson space and on the Monte Carlo
space. When $X$ is real-valued we give an explicit formula yielding the density
at the speed of the law of large numbers.
http://arXiv.org/abs/math/0610486
http://front.math.ucdavis.edu/math.PR/0610486
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: In the framework of risk management, for the study of the sensitivity of
pricing and hedging in stochastic financial models to changes of parameters and
to perturbations of the stock prices, we propose an error calculus which is an
extension of the Malliavin calculus based on Dirichlet forms. Although useful
also in physics, this error calculus is well adapted to stochastic analysis and
seems to be the best practicable in finance. This tool is explained here
intuitively and with some simple examples.
http://arXiv.org/abs/math/0610489
http://front.math.ucdavis.edu/math.PR/0610489
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: We study the error calculus from a mathematical point of view, in particular
for the infinite dimensional models met in stochastic analysis. Gauss was the
first to propose an error calculus. It can be reinforced by an extension
principle based on Dirichlet forms which gives more strength to the coherence
property. One gets a Lipschitzian complete error calculus which behaves well by
images and by products and allows a quick and easy construction of the basic
mathematical tools of Malliavin calculus. This allows also to revisit the
delicate question of error permanency that Poincar\'{e} emphasized. This error
calculus is connected with statistics by mean of the notion of Fisher
information.
http://arXiv.org/abs/math/0610491
http://front.math.ucdavis.edu/math.PR/0610491
(alternate)
Author(s): J.A.D. Appleby and G. Berkolaiko and A. Rodkina
Abstract: We consider stochastic difference equation
$$ x_{n+1} = x_n (1 - h f(x_n) + \sqrt{h} g(x_n) \xi_{n+1}), $$ where
functions f and g are nonlinear and bounded, random variables \xi_i are
independent and h>0 is a nonrandom parameter. We establish results on
asymptotic stability and instability of the trivial solution x_n=0. We also
show, that for some natural choices of the nonlinearities f and g, the rate of
decay of x_n is approximately polynomial: we find \alpha>0 such that x_n decay
faster than n^{-\alpha+\epsilon} but slower than n^{-\alpha-\epsilon} for any
\epsilon>0. It also turns out that if g(x) decays faster than f(x) as x->0, the
polynomial rate of decay can be established exactly, x_n n^\alpha -> const. On
the other hand, if the coefficient by the noise does not decay fast enough, the
approximate decay rate is the best possible result.
http://arXiv.org/abs/math/0610425
http://front.math.ucdavis.edu/math.PR/0610425
(alternate)
Author(s): Leonid Kontorovich
Abstract: We derive sufficient conditions for a family $(X^n,\rho_n,P_n)$ of metric
probability spaces to have the measure concentration property. Specifically, if
the sequence $\{P_n\}$ of probability measures satisfies a strong mixing
condition (which we call $\eta$-mixing) and the sequence of metrics
$\{\rho_n\}$ is what we call $\Psi$-dominated, we show that $(X^n,\rho_n,P_n)$
is a normal Levy family. We establish these properties for some metric
probability spaces, including the possibly novel $X=[0,1]$, $\rho_n=\ell_1$
case.
http://arXiv.org/abs/math/0610427
http://front.math.ucdavis.edu/math.PR/0610427
(alternate)
Author(s): Beniamin Goldys and Bohdan Maslowski
Abstract: For an arbitrary Hilbert space-valued Ornstein-Uhlenbeck process we construct
the Ornstein-Uhlenbeck Bridge connecting a starting point $x$ and an endpoint
$y$ that belongs to a certain linear subspace of full measure. We derive also a
stochastic evolution equation satisfied by the OU Bridge and study its basic
properties. The OU Bridge is then used to investigate the Markov transition
semigroup associated to a nonlinear stochastic evolution equation with additive
noise. We provide an explicit formula for the transition density and study its
regularity. Given the Strong Feller property and the existence of an invariant
measure we show that the transition semigroup maps $L^p$ functions into
continuous functions. We also show that transition operators are $q$-summing
for some $q>p>1$, in particular of Hilbert-Schmidt type.
http://arXiv.org/abs/math/0610386
http://front.math.ucdavis.edu/math.PR/0610386
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: We use the language of errors to handle local Dirichlet forms with square
field operator (cf [2]). Let us consider, under the hypotheses of Donsker
theorem, a random walk converging weakly to a Brownian motion. If in addition
the random walk is supposed to be erroneous, the convergence occurs in the
sense of Dirichlet forms and induces the Ornstein-Uhlenbeck structure on the
Wiener space. This quite natural result uses an extension of Donsker theorem to
functions with quadratic growth. As an application we prove an invariance
principle for the gradient of the maximum of the Brownian path computed by
Nualart and Vives.
http://arXiv.org/abs/math/0610392
http://front.math.ucdavis.edu/math.PR/0610392
(alternate)
Author(s): Itai Benjamini and Raphael Rossignol
Abstract: We study a model of random electric networks with Bernoulli resistances. In
the case of the lattice Z^2, we show that the point-to-point effective
resistance has a small variance compared to its expected value, whereas for
Z^d, with d different from 2, expectation and variance are of the same order.
Similar results are obtained in the context of p-resistance. The proofs rely on
a modified Poincare inequality due to Falik and Samorodnitsky.
http://arXiv.org/abs/math/0610393
http://front.math.ucdavis.edu/math.PR/0610393
(alternate)
Author(s): Clement Dombry and Nadine Guillotin-Plantard
Abstract: We study a Curie-Weiss model with a random external field generated by a
dynamical system. Probabilistic limit theorems (weak law of large numbers,
central limit theorems) are proven for the corresponding magnetization.
http://arXiv.org/abs/math/0610394
http://front.math.ucdavis.edu/math.PR/0610394
(alternate)
Author(s): Clive G. Wells
Abstract: We prove the existence of a solution to an equation governing the number
density within a compact domain of a discrete particle system for a prescribed
class of particle interactions taking into account the effects of the diffusion
and drift of the set of particles. Each particle carries a number of internal
coordinates which may evolve continuously in time, determined by what we will
refer to as the internal drift, or discretely via the interaction kernels.
Perfectly reflecting boundary conditions are imposed on the system and all the
processes may be spatially and temporally inhomogeneous. We use a relative
compactness argument to construct a sequence of measures that converge weakly
to a solution of the governing equation. Since the proof of existence is a
constructive one, it provides a stochastic approximation scheme that can be
used for the numerical study of molecular dynamics.
http://arXiv.org/abs/math/0610412
http://front.math.ucdavis.edu/math.PR/0610412
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: We consider a random variable $Y$ and approximations $Y\_n$, defined on the
same probability space with values in the same measurable space as $Y$. We are
interested in situations where the approximations $Y\_n$ allow to define a
Dirichlet form in the space $L^2(P\_Y)$ where $P\_Y$ is the law of $Y$. Our
approach consists in studying both biases and variances. The article attempts
to propose a general theoretical framework. It is illustrated by several
examples.
http://arXiv.org/abs/math/0610389
http://front.math.ucdavis.edu/math.FA/0610389
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: A comparison of the "theory of random sequences" developed during the
twentieth century and the axiomatic approach of probability theory proposed by
Kolmogorov shows the importance of sigma-additivity as extension tool.
Similarly, the Cauchy criterion appears to be an extension tool for
mathematical analysis. The Dirichlet forms theory possesses also such an
extension tool. They are the source of the fruitfulness of these languages and
the condition of their creativity. A connection is given with the so-called
Richard paradox.
http://arXiv.org/abs/math/0610390
http://front.math.ucdavis.edu/math.HO/0610390
(alternate)
Author(s): Murali K. Ganapathy and Prasad Tetali
Abstract: We settle an open problem, raised by Y. Peres and D. Revelle, concerning the
$L^2$ mixing time of the random walk on the lamplighter graph. We also provide
general bounds relating the entropy decay of a Markov chain to the separation
distance of the chain, and show that the lamplighter graphs once again provide
examples of tightness of our results.
http://arXiv.org/abs/math/0610345
http://front.math.ucdavis.edu/math.PR/0610345
(alternate)
Author(s): J. Michael Harrison
Abstract: Due to a printing error the above mentioned article [Annals of Applied
Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous
equations appearing incorrectly in the print version of this paper. The entire
article follows as it should have appeared. IMS apologizes to the author and
the readers for this error. A recent paper by Harrison and Van Mieghem
explained in general mathematical terms how one forms an ``equivalent workload
formulation'' of a Brownian network model. Denoting by $Z(t)$ the state vector
of the original Brownian network, one has a lower dimensional state descriptor
$W(t)=MZ(t)$ in the equivalent workload formulation, where $M$ can be chosen as
any basis matrix for a particular linear space. This paper considers Brownian
models for a very general class of open processing networks, and in that
context develops a more extensive interpretation of the equivalent workload
formulation, thus extending earlier work by Laws on alternate routing problems.
A linear program called the static planning problem is introduced to articulate
the notion of ``heavy traffic'' for a general open network, and the dual of
that linear program is used to define a canonical choice of the basis matrix
$M$. To be specific, rows of the canonical $M$ are alternative basic optimal
solutions of the dual linear program. If the network data satisfy a natural
monotonicity condition, the canonical matrix $M$ is shown to be nonnegative,
and another natural condition is identified which ensures that $M$ admits a
factorization related to the notion of resource pooling.
http://arXiv.org/abs/math/0610352
http://front.math.ucdavis.edu/math.PR/0610352
(alternate)
Author(s): Nicolas Bouleau (CERMICS)
Abstract: The error on a real quantity Y due to the graduation of the measuring
instrument may be represented, when the graduation is regular and fines down,
by a Dirichlet form on R whose square field operator do not depend on the
probability law of Y as soon as this law possesses a continuous density. This
feature is related to the "arbitrary functions principle" (Poincar\'{e}, Hopf).
We give extensions of this property to multivariate case and infinite
dimensional case for approximations of the Brownian motion. We use a Girsanov
theorem for Dirichlet forms which has its own interest. Connections are given
with discretization of stochastic differential equations.
http://arXiv.org/abs/math/0610355
http://front.math.ucdavis.edu/math.PR/0610355
(alternate)
Author(s): Marek Biskup
Abstract: Reflection positivity (RP) is a property of Gibbs measures exhibited by a
class of lattice spin systems that includes the Ising, Potts and Heisenberg
models. The RP property is useful because of its two basic consequences:
infrared bound and chessboard estimates. These are one of basic (and rather
efficient) tools for proving phase transitions in many models of physical
interest. The content of the notes presented hereby are the lectures on
reflection positivity and its consequences that the author delivered at the
Prague Summer School on Mathematical Statistical Mechanics in September 2006.
The notes summarize both the classical material on the subject from the late
1970s as well as some of the more recent developments.
http://arXiv.org/abs/math-ph/0610025
http://front.math.ucdavis.edu/math-ph/0610025
(alternate)
Author(s): Stefan Adams and Tony Dorlas
Abstract: We study large deviations principles for $ N $ random processes on the
lattice $ \Z^d $ with finite time horizon $ [0,\beta] $ under a symmetrised
measure where all initial and terminal points are uniformly given by a random
permutation. That is, given a permutation $ \sigma $ of $ N $ elements and a
vector $ (x_1,...,x_N) $ of $ N $ initial points we let the random processes
terminate in the points $ (x_{\sigma(1)},...,x_{\sigma(N)}) $ and then sum over
all possible permutations and initial points, weighted with an initial
distribution. There is a two-level random mechanism and we prove two-level
large deviations principles for the mean of empirical path measures, for the
mean of paths and for the mean of occupation local times under this symmetrised
measure. The symmetrised measure cannot be written as any product of single
random process distributions. We show a couple of important applications of
these results in quantum statistical mechanics using the Feynman-Kac formulae
representing traces of certain trace class operators. In particular we prove a
non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein
statistics and mean field interactions.
A special case of our large deviations principle for the mean of occupation
local times of $ N $ simple random walks has the Donsker-Varadhan rate function
as the rate function for the limit $ N\to\infty $ but for finite time $ \beta
$. We give an interpretation in quantum statistical mechanics for this
surprising result.
http://arXiv.org/abs/math-ph/0610026
http://front.math.ucdavis.edu/math-ph/0610026
(alternate)
Author(s): Dapeng Zhan
Abstract: We define a family of SLE-type processes in finitely connected domains, which
are called continuous LERW (loop-erased random walk). A continuous LERW
describes a random curve in a finitely connected domain that starts from a
prime end and ends at a certain target set, which could be an interior point,
or a prime end, or a side arc. It is defined using the usual chordal Loewner
equation with the driving function being $\sqrt 2 B(t)$ plus a drift term. The
distributions of continuous LERW are conformally invariant. A continuous LERW
preserves a family of local martingales, which are composed of generalized
Poisson kernels, normalized by their behaviors near the target set. These local
martingales resemble the discrete martingales preserved by the corresponding
LERW on the discrete approximation of the domain. For all kinds of targets, if
the domain satisfies certain boundary conditions, we use these martingales to
prove that when the mesh of the discrete approximation is small enough, the
continuous LERW and the corresponding discrete LERW can be coupled together,
such that after a suitable parametrization, with probability close to 1, the
two curves are uniformly close to each other.
http://arXiv.org/abs/math/0610304
http://front.math.ucdavis.edu/math.PR/0610304
(alternate)
Author(s): Christophe Andrieu and \'{E}ric Moulines
Abstract: In this paper we study the ergodicity properties of some adaptive Markov
chain Monte Carlo algorithms (MCMC) that have been recently proposed in the
literature. We prove that under a set of verifiable conditions, ergodic
averages calculated from the output of a so-called adaptive MCMC sampler
converge to the required value and can even, under more stringent assumptions,
satisfy a central limit theorem. We prove that the conditions required are
satisfied for the independent Metropolis--Hastings algorithm and the random
walk Metropolis algorithm with symmetric increments. Finally, we propose an
application of these results to the case where the proposal distribution of the
Metropolis--Hastings update is a mixture of distributions from a curved
exponential family.
http://arXiv.org/abs/math/0610317
http://front.math.ucdavis.edu/math.PR/0610317
(alternate)
Author(s): Brad Luen and Kavita Ramanan and Ilze Ziedins
Abstract: We consider a symmetric tree loss network that supports single-link (unicast)
and multi-link (multicast) calls to nearest neighbors and has capacity $C$ on
each link. The network operates a control so that the number of multicast calls
centered at any node cannot exceed $C_V$ and the number of unicast calls at a
link cannot exceed $C_E$, where $C_E$, $C_V\leq C$. We show that uniqueness of
Gibbs measures on the infinite tree is equivalent to the convergence of certain
recursions of a related map. For the case $C_V=1$ and $C_E=C$, we precisely
characterize the phase transition surface and show that the phase transition is
always nonmonotone in the arrival rate of the multicast calls. This model is an
example of a system with hard constraints that has weights attached to both the
edges and nodes of the network and can be viewed as a generalization of the
hard core model that arises in statistical mechanics and combinatorics. Some of
the results obtained also hold for more general models than just the loss
network. The proofs rely on a combination of techniques from probability theory
and dynamical systems.
http://arXiv.org/abs/math/0610321
http://front.math.ucdavis.edu/math.PR/0610321
(alternate)
Author(s): M. A. Steel and L. A. Sz\'{e}kely
Abstract: A widely studied model for generating sequences is to ``evolve'' them on a
tree according to a symmetric Markov process. We prove that model trees tend to
be maximally ``far apart'' in terms of variational distance.
http://arXiv.org/abs/math/0610323
http://front.math.ucdavis.edu/math.PR/0610323
(alternate)
Author(s): Erik Ekstr\"{o}m and Stephane Villeneuve
Abstract: We show, under weaker assumptions than in the previous literature, that a
perpetual optimal stopping game always has a value. We also show that there
exists an optimal stopping time for the seller, but not necessarily for the
buyer. Moreover, conditions are provided under which the existence of an
optimal stopping time for the buyer is guaranteed. The results are illustrated
explicitly in two examples.
http://arXiv.org/abs/math/0610324
http://front.math.ucdavis.edu/math.PR/0610324
(alternate)
Author(s): Olivier Zindy (PMA)
Abstract: We consider Sinai's walk in i.i.d. random scenery and focus our attention on
a conjecture of R\'ev\'esz \cite{r05} concerning the upper limits of Sinai's
walk in random scenery when the scenery is bounded from above. A close study of
the competition between the concentration property for Sinai's walk and
negative values for the scenery enables us to prove that the conjecture is true
if the scenery has "thin" negative tails and is false otherwise.
http://arXiv.org/abs/math/0610326
http://front.math.ucdavis.edu/math.PR/0610326
(alternate)
Author(s): Mario V. W\"{u}thrich
Abstract: We define a heteropolymer in a medium with random droplets. We prove that for
this model we have two regimes: a delocalized one and a localized one. In the
localized regime we prove tightness to the droplets, whereas in the delocalized
regime we prove diffusive path behavior.
http://arXiv.org/abs/math/0610328
http://front.math.ucdavis.edu/math.PR/0610328
(alternate)
Author(s): Abdelkader Mokkadem and Mariane Pelletier
Abstract: The first aim of this paper is to establish the weak convergence rate of
nonlinear two-time-scale stochastic approximation algorithms. Its second aim is
to introduce the averaging principle in the context of two-time-scale
stochastic approximation algorithms. We first define the notion of asymptotic
efficiency in this framework, then introduce the averaged two-time-scale
stochastic approximation algorithm, and finally establish its weak convergence
rate. We show, in particular, that both components of the averaged
two-time-scale stochastic approximation algorithm simultaneously converge at
the optimal rate $\sqrt{n}$.
http://arXiv.org/abs/math/0610329
http://front.math.ucdavis.edu/math.PR/0610329
(alternate)
Author(s): Terry Soo
Abstract: Consider independent fair coin flips at each site of the lattice Z^d. We
study non-randomized translation-invariant perfect matching rules of occupied
sites to unoccupied sites and determine bounds on the distance from a site to
its partner. In particular, in d=2, if Z is the distance from the origin to its
partner then we obtain that if 0 < p < 2/3, then the p-th moment of Z is
finite. This is related to an open problem of Holroyd and Peres.
http://arXiv.org/abs/math/0610334
http://front.math.ucdavis.edu/math.PR/0610334
(alternate)
Author(s): Alexei Borodin and Grigori Olshanski
Abstract: We present a solution to a problem suggested by Philippe Biane: We prove that
a certain Plancherel-type probability distribution on partitions converges, as
partitions get large, to a new determinantal random point process on the set
{0,1,2,...} of nonnegative integers. This can be viewed as an edge limit
ransition. The limit process is determined by a correlation kernel on
{0,1,2,...} which is expressed through the Hermite polynomials, we call it the
discrete Hermite kernel. The proof is based on a simple argument which derives
convergence of correlation kernels from convergence of unbounded self-adjoint
difference operators.
Our approach can also be applied to a number of other probabilistic models.
As an example, we discuss a bulk limit for one more Plancherel-type model of
random partitions.
http://arXiv.org/abs/math/0610240
http://front.math.ucdavis.edu/math.PR/0610240
(alternate)
Author(s): Anna Karczewska and Carlos Lizama
Abstract: In this paper, stochastic Volterra equations driven by cylindrical and
genuine Wiener process in Hilbert space are investigated. Sufficient conditions
for existence of strong solutions are given. The key role is played by
convergence of $\alpha$-times resolvent families.
http://arXiv.org/abs/math/0610241
http://front.math.ucdavis.edu/math.PR/0610241
(alternate)
Author(s): Irina Ignatiouk-Robert
Abstract: The complete representation of the Martin compactification for reflected
random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the
full Martin compactification is in general not homeomorphic to the ``radial''
compactification obtained by Ney and Spitzer for the homogeneous random walks
in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times\N$ to a
point of on the Martin boundary does not imply convergence of the sequence
$z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large
deviation properties of the scaled processes and uses Pascal's method combined
with the ratio limit theorem. The existence of non-radial limits is related to
non-linear optimal large deviation trajectories.
http://arXiv.org/abs/math/0610242
http://front.math.ucdavis.edu/math.PR/0610242
(alternate)
Author(s): Andre Goldman
Abstract: We prove that the Palm measure of the Ginibre process is obtained by removing
a gaussian distributed point from the process and adding the origin. We obtain
also precise formulas describing the law of the typical cell of the
Ginibre-Voronoi tessellation. We show that near the cell's germs a more
important part of the area is captured in the Ginibre-Voronoi tessellation than
in the case of the Poisson-Voronoi tessellation. Moments areas of corresponding
subdomains of cells are explicitly evaluated.
http://arXiv.org/abs/math/0610243
http://front.math.ucdavis.edu/math.PR/0610243
(alternate)
Author(s): Anna Karczewska and Carlos Lizama
Abstract: In this paper stochastic Volterra equations admitting exponentially bounded
resolvents are studied. After obtaining convergence of resolvents, some
properties of stochastic convolutions are given. The paper provides a
sufficient condition for a stochastic convolution to be a strong solution to a
stochastic Volterra equation.
http://arXiv.org/abs/math/0610244
http://front.math.ucdavis.edu/math.PR/0610244
(alternate)
Author(s): Victor F. Araman and Peter W. Glynn
Abstract: Consider a random walk $S=(S_n:n\geq 0)$ that is ``perturbed'' by a
stationary sequence $(\xi_n:n\geq 0)$ to produce the process
$(S_n+\xi_n:n\geq0)$. This paper is concerned with computing the distribution
of the all-time maximum $M_{\infty}=\max \{S_k+\xi_k:k\geq0\}$ of perturbed
random walk with a negative drift. Such a maximum arises in several different
applications settings, including production systems, communications networks
and insurance risk. Our main results describe asymptotics for
$\mathbb{P}(M_{\infty}>x)$ as $x\to\infty$. The tail asymptotics depend greatly
on whether the $\xi_n$'s are light-tailed or heavy-tailed. In the light-tailed
setting, the tail asymptotic is closely related to the Cram\'{e}r--Lundberg
asymptotic for standard random walk.
http://arXiv.org/abs/math/0610271
http://front.math.ucdavis.edu/math.PR/0610271
(alternate)
Author(s): Serge Cohen and Gennady Samorodnitsky
Abstract: We describe a new class of self-similar symmetric $\alpha$-stable processes
with stationary increments arising as a large time scale limit in a situation
where many users are earning random rewards or incurring random costs. The
resulting models are different from the ones studied earlier both in their
memory properties and smoothness of the sample paths.
http://arXiv.org/abs/math/0610272
http://front.math.ucdavis.edu/math.PR/0610272
(alternate)
Author(s): Benoit Collins and Piotr Sniady
Abstract: We study the asymptotics of representations of a fixed compact Lie group. We
prove that the limit behavior of a sequence of such representations can be
described in terms of certain random matrices; in particular operations on
representations (for example: tensor product, restriction to a subgroup)
correspond to some natural operations on random matrices (respectively: sum of
independent random matrices, taking the corners of a random matrix). Our method
of proof is to treat the canonical block matrix associated to a representation
as a random matrix with non-commutative entries.
http://arXiv.org/abs/math/0610285
http://front.math.ucdavis.edu/math.PR/0610285
(alternate)
Author(s): Abdelmalek Abdesselam
Abstract: We give a rigorous nonperturbative construction of a massless discrete
trajectory for Wilson's exact renormalization group. The model is a three
dimensional Euclidean field theory with a modified free propagator. The
trajectory realizes the mean field to critical crossover from the ultraviolet
Gaussian fixed point to an analog recently constructed by Brydges, Mitter and
Scoppola of the Wilson-Fisher nontrivial fixed point.
http://arXiv.org/abs/math-ph/0610018
http://front.math.ucdavis.edu/math-ph/0610018
(alternate)
Author(s): Hjalmar Rosengren
Abstract: We study multivariable Christoffel-Darboux kernels, which may be viewed as
reproducing kernels for antisymmetric orthogonal polynomials, and also as
correlation functions for products of characteristic polynomials of random
hermitian matrices. Using their interpretation as reproducing kernels, we
obtain simple proofs of pfaffian and determinant formulas, as well as Schur
polynomial expansions, for such kernels. In subsequent work, these results are
applied in combinatorics (enumeration of marked shifted tableaux) and number
theory (representation of integers as sums of squares).
http://arXiv.org/abs/math/0606391
http://front.math.ucdavis.edu/math.CA/0606391
(alternate)
Author(s): Bartlomiej Dyda and Tadeusz Kulczycki
Abstract: We study the semigroup of the symmetric $\alpha$-stable process in bounded
domains in $\R^2$. We obtain a variational formula for the spectral gap, i.e.
the difference between two first eigenvalues of the generator of this
semigroup. This variational formula allows us to obtain lower bound estimates
of the spectral gap for convex planar domains which are symmetric with respect
to both coordinate axes. For rectangles, using "midconcavity" of the first
eigenfunction, we obtain sharp upper and lower bound estimates of the spectral
gap.
http://arXiv.org/abs/math/0610283
http://front.math.ucdavis.edu/math.SP/0610283
(alternate)
Author(s): Johan Tykesson
Abstract: We consider the Poisson-Boolean continuum percolation model on the hyperbolic
disc. We show that there are intensities for the underlying Poisson point
process for which there are infinitely many unbounded connected components in
the covered and vacant regions of the hyperolic disc.
http://arXiv.org/abs/math/0610202
http://front.math.ucdavis.edu/math.PR/0610202
(alternate)
Author(s): Lancelot F. James
Abstract: This paper develops some general calculus for GGC and Dirichlet process means
functionals. It then proceeds via an investigation of positive Linnik random
variables, and more generally random variables derived from compositions of a
stable subordinator with GGC subordinators, to establish various distributional
equivalences between these models and phenomena connected to local times and
occupation times of what are defined as randomly skewed Bessel processes and
bridges. This yields a host of interesting identities and explicit density
formula for these models. Randomly skewed Bessel processes and bridges may be
seen as a randomization of their p-skewed counterparts developed in Barlow,
Pitman and Yor~(1989) and Pitman and Yor~(1997), and are shown to naturally
arise via exponential tilting. As a special result it is shown that the
occupation time of a p-skewed random Bessel process or (generalized) bridge is
equivalent in distribution to the the occupation time of a non-trivial randomly
skewed process.
http://arXiv.org/abs/math/0610218
http://front.math.ucdavis.edu/math.PR/0610218
(alternate)
Author(s): Thorsten Rheinl\"{a}nder and Gallus Steiger
Abstract: We determine the minimal entropy martingale measure for a general class of
stochastic volatility models where both price process and volatility process
contain jump terms which are correlated. This generalizes previous studies
which have treated either the geometric L\'{e}vy case or continuous price
processes with an orthogonal volatility process. We proceed by linking the
entropy measure to a certain semi-linear integro-PDE for which we prove the
existence of a classical solution.
http://arXiv.org/abs/math/0610219
http://front.math.ucdavis.edu/math.PR/0610219
(alternate)
Author(s): Dmitry Kramkov and Mihai S\^{{\i}}rbu
Abstract: We study the two-times differentiability of the value functions of the primal
and dual optimization problems that appear in the setting of expected utility
maximization in incomplete markets. We also study the differentiability of the
solutions to these problems with respect to their initial values. We show that
the key conditions for the results to hold true are that the relative risk
aversion coefficient of the utility function is uniformly bounded away from
zero and infinity, and that the prices of traded securities are sigma-bounded
under the num\'{e}raire given by the optimal wealth process.
http://arXiv.org/abs/math/0610224
http://front.math.ucdavis.edu/math.PR/0610224
(alternate)
Author(s): N. Lanchier and C. Neuhauser
Abstract: Competition is a major force in structuring ecological communities. The
strength of competition can be measured using the concept of a niche. A niche
comprises the set of requirements of an organism in terms of habitat,
environment and functional role. The more niches overlap, the stronger
competition is. The niche breadth is a measure of specialization: the smaller
the niche space of an organism, the more specialized the organism is. It
follows that, everything else being equal, generalists tend to be more
competitive than specialists. In this paper, we compare the outcome of
competition among generalists and specialists in a spatial versus a nonspatial
habitat in a heterogeneous environment. Generalists can utilize the entire
habitat, whereas specialists are restricted to their preferred habitat type. We
find that although competitiveness decreases with specialization, specialists
are more competitive in a spatial than in a nonspatial habitat as patchiness
increases.
http://arXiv.org/abs/math/0610227
http://front.math.ucdavis.edu/math.PR/0610227
(alternate)
Author(s): E. D. Andjel and M. V. Menshikov and V. V. Sisko
Abstract: We show that certain Markov jump processes associated to crystal growth
models are positive recurrent when the parameters satisfy a rather natural
condition.
http://arXiv.org/abs/math/0610172
http://front.math.ucdavis.edu/math.PR/0610172
(alternate)
Author(s): K. Bahlali and M. Eddahbi and M. Mellouk
Abstract: We consider stochastic partial differential equations on $\mathbb{R}^{d},
d\geq 1$, driven by a Gaussian noise white in time and colored in space, for
which the pathwise uniqueness holds. By using the Skorokhod representation
theorem we establish various strong stability results. Then, we give an
application to the convergence of the Picard successive approximation. Finally,
we show that in the sense of Baire category, almost all stochastic partial
differential equations with continuous and bounded coefficients have the
properties of existence and uniqueness of solutions as well as the continuous
dependence on the coefficients.
http://arXiv.org/abs/math/0610174
http://front.math.ucdavis.edu/math.PR/0610174
(alternate)
Author(s): Emmanuelle Cl\'{e}ment and Arturo Kohatsu-Higa and Damien Lamberton
Abstract: In this article we develop a new methodology to prove weak approximation
results for general stochastic differential equations. Instead of using a
partial differential equation approach as is usually done for diffusions, the
approach considered here uses the properties of the linear equation satisfied
by the error process. This methodology seems to apply to a large class of
processes and we present as an example the weak approximation of stochastic
delay equations.
http://arXiv.org/abs/math/0610178
http://front.math.ucdavis.edu/math.PR/0610178
(alternate)
Author(s): Benjamin Chan and Richard Durrett
Abstract: Neuhauser [Probab. Theory Related Fields 91 (1992) 467--506] considered the
two-type contact process and showed that on $\mathbb{Z}^2$ coexistence is not
possible if the death rates are equal and the particles use the same dispersal
neighborhood. Here, we show that it is possible for a species with a long-, but
finite, range dispersal kernel to coexist with a superior competitor with
nearest-neighbor dispersal in a model that includes deaths of blocks due to
``forest fires.''
http://arXiv.org/abs/math/0610179
http://front.math.ucdavis.edu/math.PR/0610179
(alternate)
Author(s): Peter Neal
Abstract: We consider a multitype epidemic model which is a natural extension of the
randomized Reed--Frost epidemic model. The main result is the derivation of an
asymptotic Gaussian limit theorem for the final size of the epidemic. The
method of proof is simpler, and more direct, than is used for similar results
elsewhere in the epidemics literature. In particular, the results are
specialized to epidemics upon extensions of the Bernoulli random graph.
http://arXiv.org/abs/math/0610180
http://front.math.ucdavis.edu/math.PR/0610180
(alternate)
Author(s): Fabien Campillo (IRISA / INRIA Rennes) and Vivien Rossi (IURC)
Abstract: In many situations it is important to be able to propose $N$ independent
realizations of a given distribution law. We propose a strategy for making $N$
parallel Monte Carlo Markov Chains (MCMC) interact in order to get an
approximation of an independent $N$-sample of a given target law. In this
method each individual chain proposes candidates for all other chains. We prove
that the set of interacting chains is itself a MCMC method for the product of
$N$ target measures. Compared to independent parallel chains this method is
more time consuming, but we show through concrete examples that it possesses
many advantages: it can speed up convergence toward the target law as well as
handle the multi-modal case.
http://arXiv.org/abs/math/0610181
http://front.math.ucdavis.edu/math.PR/0610181
(alternate)
Author(s): Erhan Bayraktar and Savas Dayanik and Ioannis Karatzas
Abstract: We study the quickest detection problem of a sudden change in the arrival
rate of a Poisson process from a known value to an unknown and unobservable
value at an unknown and unobservable disorder time. Our objective is to design
an alarm time which is adapted to the history of the arrival process and
detects the disorder time as soon as possible. In previous solvable versions of
the Poisson disorder problem, the arrival rate after the disorder has been
assumed a known constant. In reality, however, we may at most have some prior
information about the likely values of the new arrival rate before the disorder
actually happens, and insufficient estimates of the new rate after the disorder
happens. Consequently, we assume in this paper that the new arrival rate after
the disorder is a random variable. The detection problem is shown to admit a
finite-dimensional Markovian sufficient statistic, if the new rate has a
discrete distribution with finitely many atoms. Furthermore, the detection
problem is cast as a discounted optimal stopping problem with running cost for
a finite-dimensional piecewise-deterministic Markov process. This optimal
stopping problem is studied in detail in the special case where the new arrival
rate has Bernoulli distribution. This is a nontrivial optimal stopping problem
for a two-dimensional piecewise-deterministic Markov process driven by the same
point process. Using a suitable single-jump operator, we solve it fully,
describe the analytic properties of the value function and the stopping region,
and present methods for their numerical calculation. We provide a concrete
example where the value function does not satisfy the smooth-fit principle on a
proper subset of the connected, continuously differentiable optimal stopping
boundary, whereas it does on the complement of this set.
http://arXiv.org/abs/math/0610184
http://front.math.ucdavis.edu/math.PR/0610184
(alternate)
Author(s): Niels Richard Hansen
Abstract: We consider local alignments without gaps of two independent Markov chains
from a finite alphabet, and we derive sufficient conditions for the number of
essentially different local alignments with a score exceeding a high threshold
to be asymptotically Poisson distributed. From the Poisson approximation a
Gumbel approximation of the maximal local alignment score is obtained. The
results extend those obtained by Dembo, Karlin and Zeitouni [Ann. Probab. 22
(1994) 2022--2039] for independent sequences of i.i.d. variables.
http://arXiv.org/abs/math/0610187
http://front.math.ucdavis.edu/math.PR/0610187
(alternate)
Author(s): Thomas P. Hayes and Eric Vigoda
Abstract: We present an improved coupling technique for analyzing the mixing time of
Markov chains. Using our technique, we simplify and extend previous results for
sampling colorings and independent sets. Our approach uses properties of the
stationary distribution to avoid worst-case configurations which arise in the
traditional approach. As an application, we show that for $k/\Delta >1.764$,
the Glauber dynamics on $k$-colorings of a graph on $n$ vertices with maximum
degree $\Delta$ converges in $O(n\log n)$ steps, assuming $\Delta =\Omega(\log
n)$ and that the graph is triangle-free. Previously, girth $\ge 5$ was needed.
As a second application, we give a polynomial-time algorithm for sampling
weighted independent sets from the Gibbs distribution of the hard-core lattice
gas model at fugacity $\lambda <(1-\epsilon)e/\Delta$, on a regular graph $G$
on $n$ vertices of degree $\Delta =\Omega(\log n)$ and girth $\ge 6$. The best
known algorithm for general graphs currently assumes $\lambda <2/(\Delta -2)$.
http://arXiv.org/abs/math/0610188
http://front.math.ucdavis.edu/math.PR/0610188
(alternate)
Author(s): I. Barany and V. H. Vu
Abstract: Choose $n$ random, independent points in $\R^d$ according to the standard
normal distribution. Their convex hull $K_n$ is the {\sl Gaussian random
polytope}. We prove that the volume and the number of faces of $K_n$ satisfy
the central limit theorem, settling a well known conjecture in the field.
http://arXiv.org/abs/math/0610192
http://front.math.ucdavis.edu/math.CO/0610192
(alternate)
Author(s): A. Ayache and N. Tzvetkov
Abstract: We study L^p properties of Gaussian random series with particular attention
to the case of radial functions.
http://arXiv.org/abs/math/0610139
http://front.math.ucdavis.edu/math.PR/0610139
(alternate)
Author(s): Friedrich G\"otze and Mikhail Gordin
Abstract: Universal limits for the eigenvalue correlation functions in the bulk of the
spectrum are shown for a class of nondeterminantal random matrices known as the
fixed trace ensemble.
http://arXiv.org/abs/math/0610149
http://front.math.ucdavis.edu/math.PR/0610149
(alternate)
Author(s): Jan Fricke
Abstract: We will show that for any $n\ge N$ points on the $N$-dimensional sphere $S^N$
there is a closed hemisphere which contains at least
$\lfloor\frac{n+N+1}{2}\rfloor$ of these points. This bound is sharp and we
will calculate the amount of sets which realize this value.
If we change to open hemispheres things will be easier. For any $n$ points on
the sphere there is an open hemisphere which contains at least
$\lfloor\frac{n+1}{2}\rfloor$ of these points, independent of the dimension.
This bound is sharp.
http://arXiv.org/abs/math/0610140
http://front.math.ucdavis.edu/math.MG/0610140
(alternate)
Author(s): Steven N. Evans and David Steinsaltz and Kenneth W. Wachter
Abstract: A probability model is presented for the dynamics of mutation-selection
balance in a infinite-population infinite-sites setting sufficiently general to
cover mutation-driven changes in full age-specific demographic schedules. An
earlier work by the same authors presented a haploid model -- without genetic
recombination -- of similar scope. This work complements that model, adding
genetic recombination, based on a well-known general discrete-population
genetic model of N. Barton and M. Turelli. The model with recombination is a
flow on Poisson intensities, substantially different from the haploid model. It
is shown that the new model arises from the haploid model when recombination is
added, in the limit as generations per unit time go to infinity, and selection
strength and mutation per generation go to 0.
http://arXiv.org/abs/q-bio/0609046
http://front.math.ucdavis.edu/q-bio.PE/0609046
(alternate)
Author(s): Massimo Campanino and Dmitry Ioffe and Yvan Velenik
Abstract: We develop a fluctuation theory of connectivities for subcritical random
cluster models. The theory is based on a non-perturbative description of long
connected clusters in terms of essentially one-dimensional chains of
irreducible objects. Our construction leads to an effective random walk
representation of percolation clusters. The results include a derivation of a
sharp Ornstein-Zernike type asymptotic formula for 2-point functions, a proof
of analyticity and strict convexity of inverse correlation length and a proof
of an invariance principle for connected clusters under diffusive scaling. In
two dimensions, duality considerations enable a reformulation of these results
for supercritical nearest-neighbour random cluster measures, in particular for
nearest-neighbour Potts models in the phase transition regime. Accordingly, we
prove that equilibrium crystal shapes are always analytic and strictly convex
and that the interfaces between different phases are always diffusive. Thus, no
roughening transition is possible in the whole regime where our results apply.
Our results hold under an assumption of exponential decay of finite volume
wired connectivities in rectangular domains that is conjectured to hold in the
whole subcritical regime; the latter is known to be true, in any dimensions,
when q=1, q=2, and when q is sufficiently large. In two dimensions the
assumption holds whenever there is an exponential decay of connectivities in
the infinite volume measure. By duality this includes all supercritical
nearest-neighbour Potts models with positive surface tension between ordered
phases.
http://arXiv.org/abs/math/0610100
http://front.math.ucdavis.edu/math.PR/0610100
(alternate)
Author(s): K.Ramanan
Abstract: This work introduces the extended Skorokhod problem (ESP) and associated
extended Skorokhod map (ESM) that enable a pathwise construction of reflected
diffusions that are not necessarily semimartingales. Roughly speaking, given
the closure G of an open connected set in R^J, a non-empty convex cone d(x) in
R^J, specified at each point x on the boundary of G, and a cadlag trajectory
\psi taking values in R^J, the ESM defines a constrained version \phi of \psi
that takes values in G and is such that the increments of \phi - \psi on any
interval [s,t] lie in the closed convex hull of the directions d(\phi(u)), u in
(s,t]. General deterministic properties of the ESP are first established under
the only assumption that the graph of d(.) is closed. Next, for a class of
multi-dimensional ESPs on polyhedral domains, pathwise uniqueness and existence
of strong solutions to the associated stochastic differential equations is
established. In addition, it is also proved that these reflected diffusions are
semimartingales on [0,\tau_0], where \tau_0 is the time to hit the set of
points x on the boundary for which d(x) contains a line. One motivation for the
study of this class of reflected diffusions is that they arise as
approximations of queueing networks in heavy traffic that use the so-called
generalised processor sharing discipline.
http://arXiv.org/abs/math/0610103
http://front.math.ucdavis.edu/math.PR/0610103
(alternate)
Author(s): Nicoletta Cancrini and Fabio Martinelli and Cyril Roberto (LAMA) and Cristina Toninelli (PMA)
Abstract: We analyze the density and size dependence of the relaxation time for
kinetically constrained spin models (KCSM) intensively studied in the physical
literature as simple models sharing some of the features of a glass transition.
KCSM are interacting particle systems on $\Z^d$ with Glauber-like dynamics,
reversible w.r.t. a simple product i.i.d Bernoulli($p$) measure. The essential
feature of a KCSM is that the creation/destruction of a particle at a given
site can occur only if the current configuration of empty sites around it
satisfies certain constraints which completely define each specific model. No
other interaction is present in the model. From the mathematical point of view,
the basic issues concerning positivity of the spectral gap inside the
ergodicity region and its scaling with the particle density $p$ remained open
for most KCSM (with the notably exception of the East model in $d=1$
\cite{Aldous-Diaconis}). Here for the first time we: i) identify the ergodicity
region by establishing a connection with an associated bootstrap percolation
model; ii) develop a novel multi-scale approach which proves positivity of the
spectral gap in the whole ergodic region; iii) establish, sometimes optimal,
bounds on the behavior of the spectral gap near the boundary of the ergodicity
region and iv) establish pure exponential decay for the persistence function.
Our techniques are flexible enough to allow a variety of constraints and our
findings disprove certain conjectures which appeared in the physical literature
on the basis of numerical simulations.
http://arXiv.org/abs/math/0610106
http://front.math.ucdavis.edu/math.PR/0610106
(alternate)
Author(s): Michael Blank and Sergey Pirogov
Abstract: The notion of a successful coupling of Markov processes, based on the idea
that both components of the coupled system ``intersect'' in finite time with
probability one, is extended to cover situations when the coupling is
unnecessarily Markovian and its components are only converging (in a certain
sense) to each other with time. Under these assumptions the unique ergodicity
of the original Markov process is proven. A price for this generalization is
the weak convergence to the unique invariant measure instead of the strong one.
Applying these ideas to infinite interacting particle systems we consider even
more involved situations when the unique ergodicity can be proven only for a
restriction of the original system to a certain class of initial distributions
(e.g. translational invariant ones). Questions about the existence of invariant
measures with a given particle density are discussed as well.
http://arXiv.org/abs/math/0610118
http://front.math.ucdavis.edu/math.PR/0610118
(alternate)
Author(s): Peter Balint (1) and Balint Toth (1) and Peter Toth (2) ((1) Institute of Mathematics, Technical University of Budapest, (2) Renyi Institute, Hungarian
Academy of Sciences)
Abstract: We consider the M -> 0 limit for tagged particle diffusion in a 1-dimensional
Rayleigh-gas, studied originaly by Sinai and Soloveichik (1986), respectively
by Szasz and Toth (1986). In this limit we derive a new type of model for
tagged paricle diffusion, with Calogero-Moser-Sutherland (i.e. inverse
quadratic) interaction potential between the two central particles. Computer
simulations on this new model reproduce exactly the numerical value of the
limiting variance obtained by Boldrighini, Frigio and Tognetti (2002).
http://arXiv.org/abs/math/0610125
http://front.math.ucdavis.edu/math.PR/0610125
(alternate)
Author(s): Richard D. Gill
Abstract: I explain quantum nonlocality experiments and discuss how to optimize them.
Statistical tools from missing data maximum likelihood are crucial. New results
are given on CGLMP, CH and ladder inequalities. Open problems are also
discussed.
http://arXiv.org/abs/math/0610115
http://front.math.ucdavis.edu/math.ST/0610115
(alternate)
Author(s): Alan Hammond and Fraydoun Rezakhanlou
Abstract: We prove uniform bounds on moments X_a = \sum_{m}{m^a f_m(x,t)} of the
Smoluchowski coagulation equations with diffusion, valid in any dimension. If
the collision propensities \alpha(n,m) of mass n and mass m particles grow more
slowly than (n+m)(d(n) + d(m)), and the diffusion rate d(\cdot) is
non-increasing and satisfies m^{-b_1} \leq d(m) \leq m^{-b_2} for some b_1 and
b_2 satisfying 0 \leq b_2 < b_1 < \infty, then any weak solution satisfies X_a
\in L^{\infty}(\mathbb{R}^d \times [0,T]) \cap L^1(\mathbb{R}^d \times [0,T])
for every a \in \mathbb{N} and T \in (0,\infty), (provided that certain moments
of the initial data are finite). As a consequence, we infer that these
conditions are sufficient to ensure uniqueness of a weak solution and its
conservation of mass.
http://arXiv.org/abs/math/0610090
http://front.math.ucdavis.edu/math.AP/0610090
(alternate)
Author(s): Itai Benjamini and Carlos Hoppen and Eran ofek and Pawel Pralat and Nick Wormald
Abstract: A geodesic in a graph G is a shortest path between two vertices of G. For a
specific function e(n) of n, we define an almost geodesic cycle C in G to be a
cycle in which for every two vertices u and v in C, the distance d_G(u,v) is at
least d_C(u,v)-e(n). Let f(n) be any function tending to infinity with n. We
consider a random d-regular graph on n vertices. We show that almost all pairs
of vertices belong to an almost geodesic cycle C with e(n)= \log_{d-1}
\log_{d-1} n +f(n) and |C|=2\log_{d-1}n+O(f(n)). Along the way, we obtain
results on near-geodesic paths. We also give the limiting distribution of the
number of geodesics between two random vertices in this random graph.
http://arXiv.org/abs/math/0610089
http://front.math.ucdavis.edu/math.MG/0610089
(alternate)
Author(s): Kenneth S. Alexander
Abstract: We consider a polymer, with monomer locations modeled by the trajectory of a
Markov chain, in the presence of a potential that interacts with the polymer
when it visits a particular site 0. We assume that probability of an excursion
of length $n$ is given by $n^{-c}\phi(n)$ for some $13/2$, at high temperature,
the quenched and annealed curves differ significantly only in a very small
neighborhood of the critical point--the size of this neighborhood scales as
$\beta^{1/(2c-3)}$ where $\beta$ is the inverse temperature. For $c<3/2$, given
$\epsilon>0$, for sufficiently high temperature the quenched and annealed
curves are within a factor of $1-\epsilon$ for all $u$ near the critical point;
in particular the quenched and annealed critical points are equal. For $c=3/2$
the regime depends on the slowly varying function $\phi$.
http://arXiv.org/abs/math/0610008
http://front.math.ucdavis.edu/math.PR/0610008
(alternate)
Author(s): Eddy Mayer-Wolf and Moshe Zakai
Abstract: The model considered is that of "signal plus white noise". Known connections
between the non-causal filtering error and mutual information are combined with
new ones involving the causal estimation error, in a general abstract setup.
The results are shown to be invariant under a wide class of causality patterns;
they are applied to the derivation of the causal estimation error of a Gaussian
non-stationary filtering problem and to a multidimensional extension of the
Yovits-Jackson formula.
http://arXiv.org/abs/math/0610024
http://front.math.ucdavis.edu/math.PR/0610024
(alternate)
Author(s): Irina Ignatiouk-Robert
Abstract: It is proved that the weak large deviation principle of the scaled processes
$Z^\eps(t) = \eps Z(t/\eps)$ implies the weak large deviation principle for the
scaled Green's measures of the Markov process $Z(t)$.
http://arXiv.org/abs/math/0610040
http://front.math.ucdavis.edu/math.PR/0610040
(alternate)
Author(s): Dimitrios Cheliotis
Abstract: For any recurrent random walk (S_n)_{n>0} on R, there are increasing
sequences (g_n)_{n>0} converging to infinity for which (g_n S_n)_{n>0} has at
least one finite accumulation point. For one class of random walks, we give a
criterion on (g_n)_{n>0} and the distribution of S_1 determining the set of
accumulation points for (g_n S_n)_{n>0}. This extends, with a simpler proof, a
result of K.L. Chung and P. Erdos. Finally, for recurrent, symmetric random
walks, we give a criterion characterizing the increasing sequences (g_n)_{n>0}
of positive numbers for which liminf g_n|S_n|=0.
http://arXiv.org/abs/math/0610056
http://front.math.ucdavis.edu/math.PR/0610056
(alternate)
Author(s): Dimitrios Cheliotis
Abstract: According to a theorem of S. Schumacher, for a diffusion X in an environment
determined by a stable process that belongs to an appropriate class and has
index a, it holds that X_t/(log t)^a converges in distribution, as t goes to
infinity, to a random variable having an explicit description in terms of the
environment. We compute the density of this random variable in the case the
stable process is spectrally one-sided. This computation extends a result of H.
Kesten and quantifies the bias that the asymmetry of the environment causes to
the behavior of the diffusion.
http://arXiv.org/abs/math/0610057
http://front.math.ucdavis.edu/math.PR/0610057
(alternate)
Author(s): Vladislav Kargin
Abstract: An analogue of the Berry-Esseen inequality is proved for the speed of
convergence of free additive convolutions of bounded probability measures. The
obtained rate of convergence is of the order n^{-1/2}, the same as in the
classical case. An example with binomial measures shows that this estimate
cannot be improved without imposing further restrictions on convolved measures.
http://arXiv.org/abs/math/0610072
http://front.math.ucdavis.edu/math.PR/0610072
(alternate)
Author(s): Vladislav Kargin
Abstract: This paper derives sufficient conditions for supeconvergence of sums of
bounded free random variables, and provides an estimate on the rate of
superconvergence.
http://arXiv.org/abs/math/0610075
http://front.math.ucdavis.edu/math.PR/0610075
(alternate)
Author(s): Robin Pemantle
Abstract: The models surveyed include generalized Polya urns, reinforced random walks,
interacting urn models, and continuous reinforced processes. Emphasis is on
methods and results, with sketches provided of some proofs. Applications are
discussed in statistics, biology, economics and a number of other areas.
http://arXiv.org/abs/math/0610076
http://front.math.ucdavis.edu/math.PR/0610076
(alternate)
Author(s): Denis S. Grebenkov
Abstract: Transport phenomena are ubiquitous in nature and known to be important for
various scientific domains. Examples can be found in physics, electrochemistry,
heterogeneous catalysis, physiology, etc. To obtain new information about
diffusive or Laplacian transport towards a semi-permeable or resistive
interface, one can study the random trajectories of diffusing particles
modeled, in a first approximation, by the partially reflected Brownian motion.
This stochastic process turns out to be a convenient mathematical foundation
for discrete, semi-continuous and continuous theoretical descriptions of
diffusive transport.
This paper presents an overview of these topics with a special emphasis on
the close relation between stochastic processes with partial reflections and
Laplacian transport phenomena. We give selected examples of these phenomena
followed by a brief introduction to the partially reflected Brownian motion and
related probabilistic topics (e.g., local time process and spread harmonic
measure). A particular attention is paid to the use of the Dirichlet-to-Neumann
operator. Some practical consequences and further perspectives are discussed.
http://arXiv.org/abs/math/0610080
http://front.math.ucdavis.edu/math.PR/0610080
(alternate)
Author(s): M. Gubinelli
Abstract: The one dimensional Korteweg-de Vries equation on a periodic domain and with
initial condition in negative Sobolev spaces is studied using ideas from the
theory of rough paths. We discuss convergence of Galerkin approximations and
the presence of a random force of white-noise type in time.
http://arXiv.org/abs/math/0610006
http://front.math.ucdavis.edu/math.AP/0610006
(alternate)
Author(s): Emmanuel Kowalski
Abstract: We describe a very general abstract form of sieve based on a large-sieve
inequality which generalizes both the classical sieve inequality of Montgomery
(and its higher-dimensional variants), and our recent sieve for Frobenius over
function fields. The general framework suggests new applications. We get some
first results on the number of prime divisors of ``most'' elements of an
elliptic divisibility sequence, and we develop in some detail ``probabilistic''
sieves for random walks on arithmetic groups, e.g., estimating the probability
of finding a reducible characteristic polynomial at some step of a random walk
on SL(n,Z). In addition to the sieve principle, the applications depend on
bounds for a large sieve constant. To prove such bounds involves a variety of
deep results, including Property (T) or expanding properties of Cayley graphs,
and the Riemann Hypothesis over finite fields. It seems likely that this sieve
can have further applications.
http://arXiv.org/abs/math/0610021
http://front.math.ucdavis.edu/math.NT/0610021
(alternate)
Author(s): Mathieu Merle
Abstract: The goal of this work is to find the asymptotics of the hitting probability
of a distant point for the voter model on the integer lattice started from a
single 1 at the origin. In dimensions 2 or 3, we obtain the precise asymptotic
behavior of this probability. We use the scaling limit of the voter model
started from a single 1 at the origin in terms of super-Brownian motion under
its excursion measure. This invariance principle was stated by Bramson, Cox and
Le Gall, as a consequence of a theorem of Cox, Durrett and Perkins. Less
precise estimates are derived in dimensions greater than 4.
http://arXiv.org/abs/math/0609826
http://front.math.ucdavis.edu/math.PR/0609826
(alternate)
Author(s): Leonid Kontorovich and Kavita Ramanan
Abstract: We use the martingale method to establish concentration inequalities for a
class of dependent random sequences on a countable state space, with the
constants in the inequalities expressed in terms of certain mixing
coefficients. Along the way, we obtain bounds on certain martingale differences
associated with the random sequences, which may be of independent interest. As
an application of our result, we also derive a concentration inequality for
inhomogeneous Markov chains, and establish an extremal property associated with
their martingale difference bounds. This work complements certain concentration
inequalities obtained by Marton and Samson, while also providing a different
proof of some known results.
http://arXiv.org/abs/math/0609835
http://front.math.ucdavis.edu/math.PR/0609835
(alternate)
Author(s): M.T. Barlow and R.F. Bass and Z.-Q. Chen. and M. Kassmann
Abstract: We consider the symmetric non-local Dirichlet form $(E, F)$ given by \[ E
(f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$ the closure
of the set of $C^1$ functions on $R^d$ with compact support with respect to
$E_1$, where $E_1 (f, f):=E (f, f)+\int_{R^d} f(x)^2 dx$, and where the jump
kernel $J$ satisfies \[ \kappa_1|y-x|^{-d-\alpha} \leq J(x,y) \leq
\kappa_2|y-x|^{-d-\beta} \] for $0<\alpha< \beta <2, |x-y|<1$. This assumption
allows the corresponding jump process to have jump intensities whose size
depends on the position of the process and the direction of the jump. We prove
upper and lower estimates on the heat kernel. We construct a strong Markov
process corresponding to $(E, F)$. We prove a parabolic Harnack inequality for
nonnegative functions that solve the heat equation with respect to $E$. Finally
we construct an example where the corresponding harmonic functions need not be
continuous.
http://arXiv.org/abs/math/0609842
http://front.math.ucdavis.edu/math.PR/0609842
(alternate)
Author(s): G.Oshanin (1 and 2) and R.Voituriez (1) and S.Nechaev (3) and O.Vasilyev (2) and F.Hivert (4)((1) LPTMC, Universite Paris 6, France; (2) Inhomogeneous
Condensed Matter Department, Max-Planck-Institute Stuttgart, Germany; (3)
LPTMS, Universite Paris-Sud, France; (4) LITIS/LIFAR, Universite de Rouen,
France)
Abstract: We survey recent results on some one- and two-dimensional patterns generated
by random permutations of natural numbers. In the first part, we discuss
properties of random walks, evolving on a one-dimensional regular lattice in
discrete time $n$, whose moves to the right or to the left are induced by the
rise-and-descent sequence associated with a given random permutation. We
determine exactly the probability of finding the trajectory of such a
permutation-generated random walk at site $X$ at time $n$, obtain the
probability measure of different excursions and define the asymptotic
distribution of the number of "U-turns" of the trajectories - permutation
"peaks" and "through". In the second part, we focus on some statistical
properties of surfaces obtained by randomly placing natural numbers $1,2,3,
>...,L$ on sites of a 1d or 2d square lattices containing $L$ sites. We
calculate the distribution function of the number of local "peaks" - sites the
number at which is larger than the numbers appearing at nearest-neighboring
sites - and discuss some surprising collective behavior emerging in this model.
http://arXiv.org/abs/cond-mat/0609718
http://front.math.ucdavis.edu/cond-mat/0609718
(alternate)
|
|
|
|
|
|
