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Probability Abstracts 94
This document contains abstracts 4514-4721 from
Aug-1-2006 to Set-30-2006.
They have been mailed on Oct 1st, 2006.
Author(s): Sasha Sodin
Abstract: The normalised volume measure on the $ell_p^n$ unit ball (for p between 1 and
2) satisfies the following isoperimetric inequality: the boundary measure of a
set of measure $a$ is at least $c n^1/p a' log^{1-1/p} (1/a')$, where $a' =
min(a, 1 - a)$.
http://arXiv.org/abs/math/0607398
http://front.math.ucdavis.edu/math.PR/0607398
(alternate)
Author(s): Jason Fulman
Abstract: Random walk on the set of irreducible representations of a finite group is
investigated. For the symmetric and general linear groups, a sharp convergence
rate bound is obtained and a cutoff phenomenon is proved. As a related result,
an asymptotic description of Plancherel measure of the finite general linear
groups is given.
http://arXiv.org/abs/math/0607399
http://front.math.ucdavis.edu/math.PR/0607399
(alternate)
Author(s): Rami Atar and Krzysztof Burdzy
Abstract: We analyze a pair of reflected Brownian motions in a planar domain $D$, for
which the increments of both processes form mirror images of each other when
the processes are not on the boundary. We show that for $D$ in a class of
smooth convex planar domains, the two processes remain ordered forever,
according to a certain partial order.
This is used to prove that the second eigenvalue is simple for the Laplacian
with Neumann boundary conditions for the same class of domains.
http://arXiv.org/abs/math/0607400
http://front.math.ucdavis.edu/math.PR/0607400
(alternate)
Author(s): Glenn Merlet (IRMAR)
Abstract: We analyze the asymptotic behavior of random variables $x(n,x\_0)$ defined by
$x(0,x\_0)=x\_0$ and $x(n+1,x\_0)=A(n)x(n,x\_0)$, where $\sAn$ is a stationary
and ergodic sequence of random matrices with entries in the semi-ring
\mbox{$\R\cup\{-\infty\}$} whose addition is the $\max$ and whose
multiplication is $+$. Such sequences modelize a large class of discrete event
systems, among which timed event graphs, 1-bounded Petri nets, some queuing
networks, train or computer networks. We give necessary conditions for
$(\frac{1}{n}x(n,x\_0))\_{n\in\N}$ to converge almost surely. Then, we prove a
general scheme to give partial converse theorems. When $\max\_{A\_{ij}(0)\neq
-\infty}|A\_{ij}(0)|$ is integrable, it allows us: - to give a necessary and
sufficient condition for the convergence of $(\frac{1}{n}x(n,0))\_{n\in\N}$
when the sequence $(A(n))\_{n\in\N}$ is i.i.d., - to prove that, if $(A(n)
)\_{n\in\N}$ satisfy a condition of reinforced ergodicity and a condition of
fixed structure (i.e. $\P(A\_{ij}(0)=-\infty)\in\{0,1\}$), then
$(\frac{1}{n}x(n,0))\_{n\in\N}$ converges almost-surely, - and to reprove the
convergence of $(\frac{1}{n}x(n,0))\_{n\in\N}$ if the diagonal entries are
never $-\infty$.
http://arXiv.org/abs/math/0607406
http://front.math.ucdavis.edu/math.PR/0607406
(alternate)
Author(s): Jinqiao Duan and Andrei V. Fursikov
Abstract: The authors consider stochastic aspects of the stabilization problem for two
and three-dimensional Oseen equations with help of feedback control defined on
a part of the fluid boundary. Stochastic issues arise when inevitable
unpredictable fluctuations in numerical realization of stabilization procedures
are taken into account and they are supposed to be independent identically
distributed random variables. Under this assumption the solution to the
stabilization problem obtained via boundary feedback control can be described
by a Markov chain or a discrete random dynamical system. It is shown that this
random dynamical system possesses a unique, exponentially attracting, invariant
measure, namely, this random dynamical system is ergodic. This gives adequate
statistical description of the stabilization process on the stage when
stabilized solution has to be retained near zero (i.e. near unstable state of
equilibrium).
http://arXiv.org/abs/math/0607429
http://front.math.ucdavis.edu/math.AP/0607429
(alternate)
Author(s): Vitor Araujo
Abstract: Considering random noise in finite dimensional parameterized families of
diffeomorphisms of a compact finite dimensional boundaryless manifold M, we
show the existence of time averages for almost every orbit of each point of M,
imposing mild conditions on the families. Moreover these averages are given by
a finite number of physical absolutely continuous stationary probability
measures.
We use this result to deduce that situations with infinitely many sinks and
Henon-like attractors are not stable under random perturbations, e.g.,
Newhouse's and Colli's phenomena in the generic unfolding of a quadratic
homoclinic tangency by a one-parameter family of diffeomorphisms.
http://arXiv.org/abs/math/0607433
http://front.math.ucdavis.edu/math.DS/0607433
(alternate)
Author(s): Vitor Araujo
Abstract: Let f be a diffeomorphism of a compact finite dimensional boundaryless
manifold M exhibiting infinitely many coexisting attractors. Assume that each
attractor supports a stochastically stable probability measure and that the
union of the basins of attraction of each attractor covers Lebesgue almost all
points of M. We prove that the time averages of almost all orbits under random
perturbations are given by a finite number of probability measures. Moreover
these probability measures are close to the probability measures supported by
the attractors when the perturbations are close to the original map f.
http://arXiv.org/abs/math/0607434
http://front.math.ucdavis.edu/math.DS/0607434
(alternate)
Author(s): Krzysztof Bogdan and Tadeusz Kulczycki and Mateusz Kwa\'snicki
Abstract: We prove a uniform boundary Harnack inequality for nonnegative harmonic
functions of the fractional Laplacian on arbitrary open set $D$. This yields a
unique representation of such functions as integrals against measures on
$D^c\cup \{\infty\}$ satisfying an integrability condition. The corresponding
Martin boundary of $D$ is a subset of the Euclidean boundary determined by an
integral test.
http://arXiv.org/abs/math/0607561
http://front.math.ucdavis.edu/math.PR/0607561
(alternate)
Author(s): Jean-Francois Le Gall
Abstract: We discuss scaling limits of large bipartite planar maps. If p is a fixed
integer strictly greater than 1, we consider a random planar map M(n) which is
uniformly distributed over the set of all 2p-angulations with n faces. Then, at
least along a suitable subsequence, the metric space M(n) equipped with the
graph distance rescaled by the factor n to the power -1/4 converges in
distribution as n tends to infinity towards a limiting random compact metric
space, in the sense of the Gromov-Hausdorff distance. We prove that the
topology of the limiting space is uniquely determined independently of p, and
that this space can be obtained as the quotient of the Continuum Random Tree
for an equivalence relation which is defined from Brownian labels attached to
the vertices. We also verify that the Hausdorff dimension of the limit is
almost surely equal to 4.
http://arXiv.org/abs/math/0607567
http://front.math.ucdavis.edu/math.PR/0607567
(alternate)
Author(s): Erick Herbin and Ely Merzbach
Abstract: We prove that a set-indexed process is a set-indexed fractional Brownian
motion if and only if its projections on all the increasing paths are
one-parameter time changed fractional Brownian motions. As an application, we
present an integral representation for such processes.
http://arXiv.org/abs/math/0607575
http://front.math.ucdavis.edu/math.PR/0607575
(alternate)
Author(s): Parthanil Roy and Gennady Samorodnitsky
Abstract: We establish a connection between the structure of a stationary symmetric
alpha-stable random field (0 < alpha < 2) and ergodic theory of non-singular
group actions, elaborating on a previous work by Rosinski (2000). With the help
of this connection, we study the extreme values of the field over increasing
boxes. Depending on the ergodic theoretical and group theoretical structures of
the underlying action, we observe different kinds of asymptotic behavior of
this sequence of extreme values.
http://arXiv.org/abs/math/0607587
http://front.math.ucdavis.edu/math.PR/0607587
(alternate)
Author(s): Ilya A. Gruzberg
Abstract: Conformally-invariant curves that appear at critical points in
two-dimensional statistical mechanics systems, and their fractal geometry have
received a lot of attention in recent years. On the one hand, Schramm has
invented a new rigorous as well as practical calculational approach to critical
curves, based on a beautiful unification of conformal maps and stochastic
processes, and by now known as Schramm-Loewner evolution (SLE). On the other
hand, Duplantier has applied boundary quantum gravity methods to calculate
exact multifractal exponents associated with critical curves.
In the first part of this paper I provide a pedagogical introduction to SLE.
I present mathematical facts from the theory of conformal maps and stochastic
processes related to SLE. Then I review basic properties of SLE and provide
practical derivation of various interesting quantities related to critical
curves, including fractal dimensions and crossing probabilities.
The second part of the paper is devoted to a way of describing critical
curves using boundary conformal field theory (CFT) in the so-called Coulomb gas
formalism. This description provides an alternative (to quantum gravity) way of
obtaining the multifractal spectrum of critical curves using only traditional
methods of CFT based on free bosonic fields.
http://arXiv.org/abs/math-ph/0607046
http://front.math.ucdavis.edu/math-ph/0607046
(alternate)
Author(s): Christian L\'{e}onard (MODAL'X and CMAP)
Abstract: We present a general method, based on conjugate duality, for solving a convex
minimization problem without assuming unnecessary topological restrictions on
the constraint set. It leads to dual equalities and characterizations of the
minimizers without constraint qualification. As an example of application, the
Monge-Kantorovich optimal transport problem is solved in great detail. In
particular, the optimal transport plans are characterized without restriction.
This characterization improves the already existing literature on the subject.
http://arXiv.org/abs/math/0607604
http://front.math.ucdavis.edu/math.OC/0607604
(alternate)
Author(s): Alain Rouault (LM-Versailles)
Abstract: This is a companion paper of arxiv math.PR/050921. It concentrates on
asymptotic properties of determinants of random matrices in the Jacobi
ensemble. Let $M \in {\cal M}\_{n\_1 + n\_2,r}(`R)$ (with $r \leq n\_1 + n\_2$)
be a matrix whose entries are standard i.i.d. Gaussian. If $M^T = (M\_1^T,
M\_2^T)$ with $M\_1 \in {\cal M}\_{n\_1,r}$ and $M\_2 \in {\cal M}\_{n\_2,r}$,
then, $W\_1 := M\_1^T M\_1$ and $W\_2 := M\_2^T M\_2$ are independent $r\times
r$ Wishart matrices with parameters $n\_1$ and $n\_2$ and $M^T M = W\_1 + W\_2$
is Wishart with parameter $n\_1+ n\_2$. Then ${\cal Z} := (W\_1 + W\_2)^{-1/2}
W\_1 (W\_1 + W\_2)^{-1/2}$ has a Beta matrix variate distribution with
parameters $n\_1/2, n\_2/2$ (sometimes called the Jacobi distribution). We set
$n\_1 = \lfloor n\tau\_1 \rfloor$, $n\_2 = \lfloor n\tau\_2 \rfloor$, $r=
\lfloor nt\rfloor$ $t\in [0, \tau\_1)$ and let $n \to \infty$; we define ${\cal
Z}\_n (t)$ as the corresponding matrix and $\Theta\_n (t) := |{\cal Z}\_n(t)|$
as its determinant. In the Jacobi ensemble, the Kshirsagar's theorem decomposes
$\Theta\_n (t)$ into a product of independent beta distributed variables. This
allows us to study the process $\frac{1}{n} (n^{-1} \log \Theta\_n (t), t \in
[0,\tau\_1))$ and the asymptotic behavior of the sequence $\{\frac{1}{n}
n^{-1}\log \Theta\_n \}\_n$ as $n\to \infty$ with $\tau\_1$ and $\tau\_2$ fixed
: a.s. convergence, fluctuations, large deviations. We connect the results for
marginals (fixed $t$) with those obtained by the study of the empirical
spectral distribution. In the whole paper, we consider the problem of general
$\beta$, where the particular cases $\beta = 1,2,4$ correspond to real,
complex, and quaternionic matrices.
http://arXiv.org/abs/math/0607767
http://front.math.ucdavis.edu/math.PR/0607767
(alternate)
Author(s): Jianming Xia
Abstract: The results on the mean-variance hedging problem in Gouri\'eroux, Laurent and
Pham (1998), Rheinl\"ander and Schweizer (1997) and Arai (2005) are extended to
discontinuous semimartingale models. When the num\'eraire method is used, we
only assume the Radon-Nikodym derivative of the variance-optimal signed
martingale measure (VSMM) is non-zero almost surely (but may be strictly
negative). When discussing the relation between the solutions and the
Galtchouk-Kunita-Watanabe decompositions under the VSMM, we only assume the
VSMM is equivalent to the reference probability.
http://arXiv.org/abs/math/0607775
http://front.math.ucdavis.edu/math.PR/0607775
(alternate)
Author(s): Adrian R\"ollin
Abstract: It is shown that the method of exchangeable pairs introduced by Stein (1986)
for normal approximation can effectively be used for translated Poisson
approximation. Introducing an additional smoothness condition, one can obtain
approximation results in total variation and also in a local limit metric. The
result is applied in particular to the anti-voter model on finite graphs as
analysed by Rinott and Rotar (1997), obtaining the same rate of convergence,
but now for a stronger metric.
http://arXiv.org/abs/math/0607781
http://front.math.ucdavis.edu/math.PR/0607781
(alternate)
Author(s): Kalvis M. Jansons
Abstract: We consider the stochastic Stokes' drift of a flexible dumbbell. The dumbbell
consists of two isotropic Brownian particles connected by a linear string with
zero natural length, and is advected by a sinusoidal wave. We find an
asymptotic approximation for the Stokes' drift in the limit of a weak wave, and
find good agreement with the results of a Monte Carlo simulation.
Interestingly, the dependence of the Stokes' drift on the strength of the
spring is not monotonic.
http://arXiv.org/abs/math/0607797
http://front.math.ucdavis.edu/math.PR/0607797
(alternate)
Author(s): Gersende Fort (TSI) and Sean Meyn and Eric Moulines (TSI) and Pierre Priouret (PMA)
Abstract: Fluid limit techniques have become a central tool to analyze queueing
networks over the last decade, with applications to performance analysis,
simulation, and optimization. In this paper some of these techniques are
extended to a general class of skip-free Markov chains. As in the case of
queueing models, a fluid approximation is obtained by scaling time, space, and
the initial condition by a large constant. The resulting fluid limit is the
solution of an ordinary differential equation (ODE) in ``most'' of the state
space. Stability and finer ergodic properties for the stochastic model then
follow from stability of the set of fluid limits. Moreover, similar to the
queueing context where fluid models are routinely used to design control
policies, the structure of the limiting ODE in this general setting provides an
understanding of the dynamics of the Markov chain. These results are
illustrated through application to Markov Chain Monte Carlo.
http://arXiv.org/abs/math/0607800
http://front.math.ucdavis.edu/math.PR/0607800
(alternate)
Author(s): P. Caputo and A. Faggionato
Abstract: We consider the random walk on a simple point process on R^d, d>1, whose jump
rates decay exponentially in the A-power of jump length. The case A=1
corresponds to the phonon-induced variable-range hopping in disordered solids
in the regime of strong Anderson localization. Under mild assumptions on the
point process, we show for A in (0,d) that the random walk confined to a cubic
box of side L has a.s. Cheeger constant of order at least L^{-1} and mixing
time of order L^2. For the Poisson point process we prove that at A=d there is
a transition from diffusive to subdiffusive behavior of the random walk.
http://arXiv.org/abs/math/0607805
http://front.math.ucdavis.edu/math.PR/0607805
(alternate)
Author(s): Yuval Peres and Scott Sheffield
Abstract: Fix a bounded domain Omega in R^d, a continuous function F on the boundary of
Omega, and constants epsilon>0, p>1, and q>1 with p^{-1} + q^{-1} = 1. For each
x in Omega, let u^epsilon(x) be the value for player I of the following
two-player, zero-sum game. The initial game position is x. At each stage, a
fair coin is tossed and the player who wins the toss chooses a vector v of
length at most epsilon to add to the game position, after which a random
``noise vector'' with mean zero and variance (q/p)|v|^2 in each orthogonal
direction is also added. The game ends when the game position reaches some y on
the boundary of Omega, and player I's payoff is F(y).
We show that (for sufficiently regular Omega) as epsilon tends to zero the
functions u^epsilon converge uniformly to the unique p-harmonic extension of F.
Using a modified game (in which epsilon gets smaller as the game position
approaches the boundary), we prove similar statements for general bounded
domains Omega and resolutive functions F.
These games and their variants interpolate between the tug of war games
studied by Peres, Schramm, Sheffield, and Wilson (p=infinity) and the
motion-by-curvature games introduced by Spencer and studied by Kohn and Serfaty
(p=1). They generalize the relationship between Brownian motion and the
ordinary Laplacian and yield new results about p-capacity and p-harmonic
measure.
http://arXiv.org/abs/math/0607761
http://front.math.ucdavis.edu/math.AP/0607761
(alternate)
Author(s): Vitor Araujo
Abstract: We obtain a large deviation bound for continuous observables on suspension
semiflows over a non-uniformly expanding base transformation with non-flat
singularities or criticalities, where the roof function defining the suspension
behaves like the logarithm of the distance to the singular/critical set of the
base map. That is, given a continuous function we consider its space average
with respect to a physical measure and compare this with the time averages
along orbits of the semiflow, showing that the Lebesgue measure of the set of
points whose time averages stay away from the space average tends to zero
exponentially fast as time goes to infinity. Suspension semiflows model the
dynamics of flows admitting cross-sections, where the dynamics of the base is
given by the Poincar\'e return map and the roof function is the return time to
the cross-section. The results are applicable in particular to semiflows
modeling the geometric Lorenz attractors and the Lorenz flow, as well as other
semiflows with multidimensional non-uniformly expanding base with non-flat
singularities and/or criticalities under slow recurrence rate conditions to
this singular/critical set. We are also able to obtain exponentially fast
escape rates from subsets without full measure.
http://arXiv.org/abs/math/0607771
http://front.math.ucdavis.edu/math.DS/0607771
(alternate)
Author(s): Witold Bednorz
Abstract: We improve constants in the Rademacher-Menchov inequality.
http://arXiv.org/abs/math/0608023
http://front.math.ucdavis.edu/math.PR/0608023
(alternate)
Author(s): Yueyun Hu (LAGA) and Zhan Shi (PMA)
Abstract: We consider a recurrent random walk in random environment on a regular tree.
Under suitable general assumptions upon the distribution of the environment, we
show that the walk exhibits an unusual slow movement: the order of magnitude of
the walk in the first $n$ steps is $(\log n)^3$.
http://arXiv.org/abs/math/0608036
http://front.math.ucdavis.edu/math.PR/0608036
(alternate)
Author(s): Dmitri L. Finkelshtein and Yuri G. Kondratiev and Eugene W. Lytvynov
Abstract: A Kawasaki dynamics in continuum is a dynamics of an infinite system of
interacting particles in $\mathbb{R}^d$ which randomly hop over the space. In
this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs
measure $mu$ as invariant measure. We study a scaling limit of such a dynamics,
derived through a scaling of the jump rate. Informally, we expect that, in the
limit, only jumps of ``infinite length'' will survive, i.e., we expect to
arrive at a Glauber dynamics in continuum (a birth-and-death process in
$\mathbb{R}^d$). We prove that, in the low activity-high temperature regime,
the generators of the Kawasaki dynamics converge to the generator of a Glauber
dynamics. The convergence is on the set of exponential functions, in the
$L^2(\mu)$-norm. Furthermore, additionally assuming that the potential of pair
interaction is positive, we prove the weak convergence of the
finite-dimensional distributions of the processes.
http://arXiv.org/abs/math/0608051
http://front.math.ucdavis.edu/math.PR/0608051
(alternate)
Author(s): Leonid Kontorovich
Abstract: We prove what appears to be the first concentration of measure result for
hidden Markov processes. Our bound is stated in terms of the contraction
coefficients of the underlying Markov process, and strictly generalizes the
Markov process concentration results of Marton (1996) and Samson (2000).
Somewhat surprisingly, the bound turns out to be the same as for ordinary
Markov processes; this property, however, fails for general hidden/observed
process pairs.
http://arXiv.org/abs/math/0608064
http://front.math.ucdavis.edu/math.PR/0608064
(alternate)
Author(s): Gideon Amir and Eyal Lubetzky
Abstract: In [Amir et al.], the authors consider the generalization $\Gor$ of the
Erd\H{o}s-R\'enyi random graph process $G$, where instead of adding new edges
uniformly, $\Gor$ gives a weight of size 1 to missing edges between pairs of
isolated vertices, and a weight of size $K\in[0,\infty)$ otherwise. This can
correspond to the linking of settlements or the spreading of an epidemic. The
authors investigate $\tgor(K)$, the critical time for the appearance of a giant
component as a function of $K$, and prove that
$\tgor=(1+o(1))\frac{4}{\sqrt{3K}}$, using a proper timescale.
In this work, we show that a natural variation of the model $\Gor$ has
interesting properties. Define the process $\Gand$, where a weight of size $K$
is assigned to edges between pairs of non-isolated vertices, and a weight of
size 1 otherwise. We prove that the asymptotical behavior of the giant
component threshold is essentially the same for $\Gand$, and namely $\tgand /
\tgor$ tends to $\frac{64\sqrt{6}}{\pi(24+\pi^2)}\approx 1.47$ as $K\to\infty$.
However, the corresponding thresholds for connectivity satisfy $\tcand /
\tcor=\max\{{1/2},K\}$ for every $K>0$. Following the methods of [Amir et al.],
$\tgand$ is characterized as the singularity point to a system of differential
equations, and computer simulations of both models agree with the analytical
results as well as with the asymptotic analysis. In the process, we answer the
following question: when does a giant component emerge in a graph process where
edges are chosen uniformly out of all edges incident to isolated vertices,
while such exist, and otherwise uniformly? This corresponds to the value of
$\tgand(0)$, which we show to be ${3/2}+\frac{4}{3\mathrm{e}^2-1}$.
http://arXiv.org/abs/math/0608097
http://front.math.ucdavis.edu/math.CO/0608097
(alternate)
Author(s): Marek Biskup and Lincoln Chayes and Steven A. Kivelson
Abstract: We consider the Ising systems in $d$ dimensions with nearest-neighbor
ferromagnetic interactions and long-range repulsive (antiferromagnetic)
interactions which decay with a power, $s$, of the distance. The physical
context of such models is discussed; primarily this is $d=2$ and $s=3$ where,
at long distances, genuine magnetic interactions between genuine magnetic
dipoles are of this form. We prove that when the power of decay lies above $d$
and does not exceed $d+1$, then for all temperatures, the spontaneous
magnetization is zero. In contrast, we also show that for powers exceeding
$d+1$ (with $d\ge2$) magnetic order can occur.
http://arXiv.org/abs/math-ph/0608009
http://front.math.ucdavis.edu/math-ph/0608009
(alternate)
Author(s): Omer Angel and Jesse Goodman and Frank den Hollander and Gordon Slade
Abstract: We consider invasion percolation on a rooted regular tree. For the infinite
cluster invaded from the root, we identify the scaling behaviour of its
$r$-point function for any $r \ge 2$ and of its volume both at a given height
and below a given height. In addition, we derive scaling estimates for simple
random walk on the cluster starting from the root. We find that while the power
laws of the scaling are the same as for the incipient infinite cluster for
ordinary percolation, the scaling functions differ. Thus, somewhat
surprisingly, the two clusters behave differently. We show that the invasion
percolation cluster is stochastically dominated by the incipient infinite
cluster. Far above the root, the two clusters have the same law locally, but
not globally. A key ingredient in the proofs is an analysis of the forward
maximal weights along the backbone of the invasion percolation cluster. These
weights decay towards the critical value for ordinary percolation, but only
slowly, and this slow decay causes an anomalous scaling behaviour.
http://arXiv.org/abs/math/0608132
http://front.math.ucdavis.edu/math.PR/0608132
(alternate)
Author(s): Adrian R\"ollin
Abstract: Stein's method is used to approximate sums of discrete and locally dependent
random variables by a centered and symmetric Binomial distribution. Under
appropriate smoothness properties of the summands, the same order of accuracy
as in the Berry-Essen Theorem is achieved. The approximation of the total
number of points of a point processes is also considered. The results are
applied to the exceedances of the $r$-scans process and to the Mat\'ern
hardcore point process type I.
http://arXiv.org/abs/math/0608138
http://front.math.ucdavis.edu/math.PR/0608138
(alternate)
Author(s): Glauco Valle
Abstract: We investigate the evolution of the random interfaces in a two dimensional
Potts model at zero temperature under Glauber dynamics for some particular
initial conditions. We prove that under space-time diffusive scaling the shape
of the interfaces converges in probability to the solution of a non-linear
parabolic equation. This Law of Large Numbers is obtained from the Hydrodynamic
limit of a coupling between an exclusion process and an inhomogeneous one
dimensional zero range process with asymmetry at the origin.
http://arXiv.org/abs/math/0608142
http://front.math.ucdavis.edu/math.PR/0608142
(alternate)
Author(s): Martin T. Barlow and Antal A. Jarai and Takashi Kumagai and Gordon Slade
Abstract: We consider simple random walk on the incipient infinite cluster for the
spread-out model of oriented percolation on $Z^d \times Z_+$. In dimensions
$d>6$, we obtain bounds on exit times, transition probabilities, and the range
of the random walk, which establish that the spectral dimension of the
incipient infinite cluster is 4/3, and thereby prove a version of the
Alexander--Orbach conjecture in this setting. The proof divides into two parts.
One part establishes general estimates for simple random walk on an arbitrary
infinite random graph, given suitable bounds on volume and effective resistance
for the random graph. A second part then provides these bounds on volume and
effective resistance for the incipient infinite cluster in dimensions $d>6$, by
extending results about critical oriented percolation obtained previously via
the lace expansion.
http://arXiv.org/abs/math/0608164
http://front.math.ucdavis.edu/math.PR/0608164
(alternate)
Author(s): Claudio Landim (LMRS) and Aniura Milan\'{e}s (ICEX) and Stefano Olla (CEREMADE)
Abstract: We prove nonequilibrium fluctuations for the boundary driven symmetric simple
exclusion process. We deduce from this result the stationary fluctuations.
http://arXiv.org/abs/math/0608165
http://front.math.ucdavis.edu/math.PR/0608165
(alternate)
Author(s): Desheng Yang and Jinqiao Duan
Abstract: Nonlinear systems are often subject to random influences. Sometimes the noise
enters the system through physical boundaries and this leads to stochastic
dynamic boundary conditions. A dynamic, as opposed to static, boundary
condition involves the time derivative as well as spatial derivatives for the
system state variables on the boundary. Although stochastic \emph{static}
(Neumann or Dirichet type) boundary conditions have been applied for stochastic
partial differential equations, not much is known about the dynamical impact of
stochastic \emph{dynamic} boundary conditions. The purpose of this paper is to
study possible impacts of stochastic dynamic boundary conditions on the long
term dynamics of the Cahn-Hilliard equation arising in the materials science.
We show that the dimension estimation of the random attractor increases as the
coefficient for the dynamic term in the stochastic dynamic boundary condition
decreases. However, the dimension of the random attractor is not affected by
the corresponding stochastic static boundary condition.
http://arXiv.org/abs/math/0608133
http://front.math.ucdavis.edu/math.DS/0608133
(alternate)
Author(s): Vitor Araujo
Abstract: The concept of random dynamical system is a comparatively recent development
combining ideas and methods from the well developed areas of probability theory
and dynamical systems.
Due to our inaccurate knowledge of the particular physical system or due to
computational or theoretical limitations (lack of sufficient computational
power, inefficient algorithms or insufficiently developed mathematical or
physical theory, for example), the mathematical models never correspond exactly
to the phenomenon they are meant to model. Moreover when considering practical
systems we cannot avoid either external noise or measurement or inaccuracy
errors, so every realistic mathematical model should allow for small errors
along orbits not to disturb too much the long term behavior. To be able to cope
with unavoidable uncertainty about the ``correct'' parameter values, observed
initial states and even the specific mathematical formulation involved, we let
randomness be embedded within the model to begin with.
We present the most basic classes of models in what follows, then define the
general concept and present some developments and examples of applications.
http://arXiv.org/abs/math/0608162
http://front.math.ucdavis.edu/math.DS/0608162
(alternate)
Author(s): Edouard Maurel-Segala
Abstract: Perturbation of the GUE are known in physics to be related to enumeration of
graphs on surfaces. We investigate this idea and show that for a small convex
perturbation, we can perform a genus expansion: the moments of the empirical
measure can be developed into a series whose g-th term is a generating function
of graphs on a surface of genus g.
http://arXiv.org/abs/math/0608192
http://front.math.ucdavis.edu/math.PR/0608192
(alternate)
Author(s): Benoit Collins and Alice Guionnet and Edouard Maurel-Segala
Abstract: We prove that the limit of various unitary matrix integrals, including the
Itzykson-Zuber integral, exists in a small parameters region and is analytic in
these parameters.
http://arXiv.org/abs/math/0608193
http://front.math.ucdavis.edu/math.PR/0608193
(alternate)
Author(s): Steven N. Evans and David Steinsaltz
Abstract: A fissioning organism may purge unrepairable damage by bequeathing it
preferentially to one of its daughters. We propose a superprocess model, and
show that when damage accumulates deterministically, optimal growth is achieved
by unequal division of damage between the daughters.
http://arXiv.org/abs/q-bio/0608008
http://front.math.ucdavis.edu/q-bio.PE/0608008
(alternate)
Author(s): Konstantin Borovkov and Vladimir Vatutin
Abstract: We consider growing random recursive trees in random environment, in which at
each step a new vertex is attached (by an edge of a random length) to an
existing tree vertex according to a probability distribution that assigns the
tree vertices masses proportional to their random weights. The main aim of the
paper is to study the asymptotic behaviour of the distance from the newly
inserted vertex to the tree's root and that of the mean numbers of outgoing
vertices as the number of steps tends to infinity. Most of the results are
obtained under the assumption that the random weights have a product form with
independent identically distributed factors.
http://arXiv.org/abs/math/0608211
http://front.math.ucdavis.edu/math.PR/0608211
(alternate)
Author(s): J. van den Berg and O. H\"{a}ggstr\"{o}m and J. Kahn
Abstract: Consider the one-dimensional contact process. About ten years ago, N. Konno
stated the conjecture that, for all positive integers $n,m$, the upper
invariant measure has the following property: Conditioned on the event that $O$
is infected, the events $\{$All sites $-n,...,-1$ are healthy$\}$ and $\{$All
sites $1,...,m$ are healthy$\}$ are negatively correlated. We prove (a stronger
version of) this conjecture, and explain that in some sense it is a dual
version of the planar case of one of our results in \citeBHK.
http://arXiv.org/abs/math/0608216
http://front.math.ucdavis.edu/math.PR/0608216
(alternate)
Author(s): Ronald Meester
Abstract: We describe infinite clusters which arise in nearest-neighbour percolation
for so-called cocycle measures on the square lattice. These measures arise
naturally in the study of random transformations. We show that infinite
clusters have a very specific form and direction. In concrete situations, this
leads to a quick decision whether or not a certain cocycle measure percolates.
We illustrate this with two examples which are interesting in their own right.
http://arXiv.org/abs/math/0608217
http://front.math.ucdavis.edu/math.PR/0608217
(alternate)
Author(s): Frank den Hollander and Jeffrey E. Steif
Abstract: . In this paper we give a survey of some recent results for random walk in
random scenery (RWRS). On $\mathbb {Z}^d$, $d\geq 1$, we are given a random
walk with i.i.d. increments and a random scenery with i.i.d. components. The
walk and the scenery are assumed to be independent. RWRS is the random process
where time is indexed by $\mathbb {Z}$, and at each unit of time both the step
taken by the walk and the scenery value at the site that is visited are
registered. We collect various results that classify the ergodic behavior of
RWRS in terms of the characteristics of the underlying random walk (and discuss
extensions to stationary walk increments and stationary scenery components as
well). We describe a number of results for scenery reconstruction and close by
listing some open questions.
http://arXiv.org/abs/math/0608219
http://front.math.ucdavis.edu/math.PR/0608219
(alternate)
Author(s): Franz Merkl and Silke W. W. Rolles
Abstract: We review results on linearly edge-reinforced random walks. On finite graphs,
the process has the same distribution as a mixture of reversible Markov chains.
This has applications in Bayesian statistics and it has been used in studying
the random walk on infinite graphs. On trees, one has a representation as a
random walk in an independent random environment. We review recent results for
the random walk on ladders: recurrence, a representation as a random walk in a
random environment, and estimates for the position of the random walker.
http://arXiv.org/abs/math/0608220
http://front.math.ucdavis.edu/math.PR/0608220
(alternate)
Author(s): Xiaofeng Shao and Wei Biao Wu
Abstract: We obtain invariance principles for a wide class of fractionally integrated
nonlinear processes. The limiting distributions are shown to be fractional
Brownian motions. Under very mild conditions, we extend earlier ones on long
memory linear processes to a more general setting. The invariance principles
are applied to the popular R/S and KPSS tests.
http://arXiv.org/abs/math/0608223
http://front.math.ucdavis.edu/math.PR/0608223
(alternate)
Author(s): Jolanta K. Misiewicz
Abstract: A random vector ${\bf X}$ is weakly stable iff for all $a,b\in \mathbb{R}$
there exists a random variable $\Theta$ such that $a{\bf X}+b{\bf
X}'\stackrel{d}{=}{\bf X}\Theta$. This is equivalent (see \cite{MOU}) with the
condition that for all random variables $Q_1,Q_2$ there exists a random
variable $\Theta$ such that $$ X Q_1 + X' Q_2 \stackrel{d}{=} X \Theta, $$
where ${\bf X},{\bf X}',Q_1,Q_2,\Theta$ are independent. In this paper we
define generalized convolution of measures defined by the formula $$ L(Q_1)
\oplus_{\mu} L(Q_2) = L(\Theta), $$ if the equation $(*)$ holds for ${\bf
X},Q_1,Q_2,\Theta$ and $\mu ={\cal L}(\Theta)$. We study here basic properties
of this convolution, basic properties of $\oplus_{\mu}$-infinitely divisible
distributions, $\oplus_{\mu}$-stable distributions and give a series of
examples.
http://arXiv.org/abs/math/0608225
http://front.math.ucdavis.edu/math.PR/0608225
(alternate)
Author(s): F. M. Dekking and P. Liardet
Abstract: This paper considers 1-dimensional generalized random walks in random
scenery. That is, the steps of the walk are generated by an arbitrary
stationary process, and also the scenery is a priori arbitrary stationary.
Under an ergodicity condition--which is satisfied in the classical case--a
simple proof of the distinguishability of periodic sceneries is given.
http://arXiv.org/abs/math/0608218
http://front.math.ucdavis.edu/math.DS/0608218
(alternate)
Author(s): Klaus Schmidt
Abstract: We survey distributional properties of $\mathbb{R}^d$-valued cocycles of
finite measure preserving ergodic transformations (or, equivalently, of
stationary random walks in $\mathbb{R}^d$) which determine recurrence or
transience.
http://arXiv.org/abs/math/0608221
http://front.math.ucdavis.edu/math.DS/0608221
(alternate)
Author(s): Alberto Fernandez and Sergio Gomez
Abstract: In agglomerative hierarchical clustering, pair-group methods suffer from a
problem of non-uniqueness when two or more distances between different clusters
coincide during the amalgamation process. The traditional approach for solving
this drawback has been to take any arbitrary criterion in order to break ties
between distances, which results in different hierarchical classifications
depending on the criterion followed. In this article we propose a
variable-group algorithm that consists in grouping more than two clusters at
the same time when ties occur. We give a tree representation for the results of
the algorithm, which we call a "multidendrogram", as well as a generalisation
of the Lance and Williams' formula which enables the implementation of the
algorithm in a recursive way.
http://arXiv.org/abs/cs/0608049
http://front.math.ucdavis.edu/cs.IR/0608049
(alternate)
Author(s): Fabrice Baudoin
Abstract: The purpose of this work is to provide a general formalism for the study in
small times of heat evolution semigroups associated to operators that can be
written as sum of squares. We give a representation of such heat kernels as the
averaging over the set of Brownian paths of the exponential of an infinite Lie
series. The method we develop is an alternative to It\^o's theory of stochastic
differential equations for small times problems and can be applied in a more
general setting. In order to illustrate the method, we apply this formalism to
give a new short proof of Atiyah-Singer local index theorem.
http://arXiv.org/abs/math/0608231
http://front.math.ucdavis.edu/math.PR/0608231
(alternate)
Author(s): Alexander Bulinski and Alexey Shashkin
Abstract: A strong invariance principle is established for random fields which satisfy
dependence conditions more general than positive or negative association. We
use the approach of Cs\"{o}rg\H{o} and R\'{e}v\'{e}sz applied recently by Balan
to associated random fields. The key step in our proof combines new moment and
maximal inequalities, established by the authors for partial sums of
multiindexed random variables, with the estimate of the convergence rate in the
CLT for random fields under consideration.
http://arXiv.org/abs/math/0608237
http://front.math.ucdavis.edu/math.PR/0608237
(alternate)
Author(s): Yves Guivarc'h
Abstract: We consider the following recurrence relation with random i.i.d. coefficients
$(a_n,b_n)$: $$ x_{n+1}=a_{n+1} x_n+b_{n+1} $$ where $a_n\in
GL(d,\mathbb{R}),b_n\in \mathbb{R}^d$. Under natural conditions on $(a_n,b_n)$
this equation has a unique stationary solution, and its support is non-compact.
We show that, in general, its law has a heavy tail behavior and we study the
corresponding directions. This provides a natural construction of laws with
heavy tails in great generality. Our main result extends to the general case
the results previously obtained by H. Kesten in [16] under positivity or
density assumptions, and the results recently developed in [17] in a special
framework.
http://arXiv.org/abs/math/0608239
http://front.math.ucdavis.edu/math.PR/0608239
(alternate)
Author(s): Nathael Gozlan (MODAL'X)
Abstract: In this paper, we give necessary and sufficient conditions for Talagrand's
like transportation cost inequalities on the real line. This brings a new wide
class of examples of probability measures enjoying a dimension-free
concentration of measure property. Another byproduct is the characterization of
modified Log-Sobolev inequalities for Log-concave probability measures on R.
http://arXiv.org/abs/math/0608241
http://front.math.ucdavis.edu/math.PR/0608241
(alternate)
Author(s): M. N. M. van Lieshout
Abstract: . Markov chains in time, such as simple random walks, are at the heart of
probability. In space, due to the absence of an obvious definition of past and
future, a range of definitions of Markovianity have been proposed. In this
paper, after a brief review, we introduce a new concept of Markovianity that
aims to combine spatial and temporal conditional independence.
http://arXiv.org/abs/math/0608242
http://front.math.ucdavis.edu/math.PR/0608242
(alternate)
Author(s): Laurent Decreusefond and Pascal Moyal
Abstract: In this paper, we present a functional fluid limit theorem and a functional
central limit theorem for a queue with an infinity of servers M/GI/$\infty$.
The system is represented by a point-measure valued process keeping track of
the remaining processing times of the customers in service. The convergence in
law of a sequence of such processes is proved by compactness-uniqueness
methods, and the deterministic fluid limit is the solution of an integrated
equation in the space $\S^{\prime}$ of tempered distributions. We then
establish the corresponding central limit theorem, i.e. the approximation of
the normalized error process by a $\S^{\prime}$-valued diffusion.
http://arXiv.org/abs/math/0608258
http://front.math.ucdavis.edu/math.PR/0608258
(alternate)
Author(s): Rahul Roy
Abstract: For a marked point process $\{(x_i,S_i)_{i\geq 1}\}$ with $\{x_i\in
\Lambda:i\geq 1\}$ being a point process on $\Lambda \subseteq \mathbb{R}^d$
and $\{S_i\subseteq R^d:i\geq 1\}$ being random sets consider the region
$C=\cup_{i\geq 1}(x_i+S_i)$. This is the covered region obtained from the
Boolean model $\{(x_i+S_i):i\geq 1\}$. The Boolean model is said to be
completely covered if $\Lambda \subseteq C$ almost surely. If $\Lambda$ is an
infinite set such that ${\bf s}+\Lambda \subseteq \Lambda$ for all ${\bf s}\in
\Lambda$ (e.g. the orthant), then the Boolean model is said to be eventually
covered if ${\bf t}+\Lambda \subseteq C$ for some ${\bf t}$ almost surely. We
discuss the issues of coverage when $\Lambda$ is $\mathbb{R}^d$ and when
$\Lambda$ is $[0,\infty)^d$.
http://arXiv.org/abs/math/0608238
http://front.math.ucdavis.edu/math.CO/0608238
(alternate)
Author(s): Romuald Lenczewski
Abstract: We introduce and study a new type of convolution of probability measures
called the orthogonal convolution, which is related to the monotone
convolution. Using this convolution, we derive alternating decompositions of
the free additive convolution of compactly supported probability measures in
free probability. These decompositions are directly related to alternating
decompositions of the associated subordination functions. In particular, they
allow us to compute free additive convolutions of compactly supported measures
without using free cumulants or $R$-transforms. In simple cases,
representations of the corresponding Cauchy transforms as continued fractions
are obtained in a natural way. Moreover, this approach establishes a clear
connection between convolutions and products associated with the main notions
of independence (free, monotone and boolean) in noncommutative probability.
Finally, our result leads to natural decompositions of the free product of
rooted graphs.
http://arXiv.org/abs/math/0608236
http://front.math.ucdavis.edu/math.OA/0608236
(alternate)
Author(s): Itai Benjamini and Ori Gurel-Gurevich and and Boris Solomyak
Abstract: We consider a Branching Random Walk on $\R$ whose step size decreases by a
fixed factor, $01/2$ the limit measure is almost surely (a.s.) absolutely continuous
with respect to the Lebesgue measure, but for Pisot $1/b$ it is a.s. singular;
(2) for all $b > (\sqrt{5}-1)/2$ the support of the measure is a.s. the closure
of its interior; (3) for Pisot $1/b$ the support of the measure is
``fractured'': it is a.s. disconnected and the components of the complement are
not isolated on both sides.
http://arXiv.org/abs/math/0608271
http://front.math.ucdavis.edu/math.PR/0608271
(alternate)
Author(s): Mike A. Steel and Laszlo A. Szekely
Abstract: This paper continues our earlier investigations into the inversion of random
functions in a general (abstract) setting. In Section 2 we investigate a
concept of invertibility and the invertibility of the composition of random
functions. In Section 3 we resolve some questions concerning the number of
samples required to ensure the accuracy of parametric maximum likelihood
estimation (MLE). A direct application to phylogeny reconstruction is given.
http://arXiv.org/abs/math/0608273
http://front.math.ucdavis.edu/math.PR/0608273
(alternate)
Author(s): Dee Denteneer and Frank den Hollander and Evgeny Verbitskiy
Abstract: The present volume is a Festschrift for Mike Keane, on the occasion of his
65th birthday on January 2, 2005. It contains 29 contributions by Mike's
closest colleagues and friends, covering a broad range of topics in Dynamics
and Stochastics. To celebrate Mike's scientific achievements, a conference
entitled ``Dynamical Systems, Probability Theory and Statistical Mechanics''
was organized in Eindhoven, The Netherlands, during the week of January 3--7,
2005. This conference was hosted jointly by EURANDOM and by Philips Research.
It drew over 80 participants from 5 continents, which is a sign of the warm
affection and high esteem for Mike felt by the international mathematics
community.
http://arXiv.org/abs/math/0608289
http://front.math.ucdavis.edu/math.PR/0608289
(alternate)
Author(s): Alexander Gnedin and Jim Pitman
Abstract: A simple explicit construction is provided of a partition-valued
fragmentation process whose distribution on partitions of $[n]=\{1,...,n\}$ at
time $\theta \ge 0$ is governed by the Ewens sampling formula with parameter
$\theta$. These partition-valued processes are exchangeable and consistent, as
$n$ varies. They can be derived by uniform sampling from a corresponding mass
fragmentation process defined by cutting a unit interval at the points of a
Poisson process with intensity $\theta x^{-1} \diff x$ on ${\mathbb R}_+$,
arranged to be intensifying as $\theta$ increases.
http://arXiv.org/abs/math/0608307
http://front.math.ucdavis.edu/math.PR/0608307
(alternate)
Author(s): Eugene Lytvynov and Lin Mei
Abstract: Let $X$ be a locally compact, second countable Hausdorff topological space.
We consider a family of commuting Hermitian operators $a(\Delta)$ indexed by
all measurable, relatively compact sets $\Delta$ in $X$ (a quantum stochastic
process over $X$). For such a family, we introduce the notion of a correlation
measure. We prove that, if the family of operators possesses a correlation
measure which satisfies some condition of growth, then there exists a point
process over $X$ having the same correlation measure. Furthermore, the
operators $a(\Delta)$ can be realized as multiplication operators in the
$L^2$-space with respect to this point process. In the proof, we utilize the
notion of $\star$-positive definiteness, proposed in [Y. G. Kondratiev and T.\
Kuna, {\it Infin. Dimens. Anal. Quantum Probab. Relat. Top.} {\bf 5} (2002),
201--233]. In particular, our result extends the criterion of existence of a
point process from that paper to the case of the topological space $X$, which
is a standard underlying space in the theory of point processes. As
applications, we discuss particle densities of the quasi-free representations
of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like
(e.g. para-fermions and para-bosons of order 2) point processes.
In particular, we prove that any fermion point process corresponding to a
Hermitian kernel may be derived in this way.
http://arXiv.org/abs/math/0608334
http://front.math.ucdavis.edu/math.PR/0608334
(alternate)
Author(s): Yurij M. Berezansky and Eugene W. Lytvynov and Artem D. Pulemyotov
Abstract: By definition, a Jacobi field $J=(J(\phi))_{\phi\in H_+}$ is a family of
commuting selfadjoint three-diagonal operators in the Fock space $\mathcal
F(H)$. The operators $J(\phi)$ are indexed by the vectors of a real Hilbert
space $H_+$. The spectral measure $\rho$ of the field $J$ is defined on the
space $H_-$ of functionals over $H_+$. The image of the measure $\rho$ under a
mapping $K^+:T_-\to H_-$ is a probability measure $\rho_K$ on $T_-$. We obtain
a family $J_K$ of operators whose spectral measure is equal to $\rho_K$. We
also obtain the chaotic decomposition for the space $L^2(T_-,d\rho_K)$.
http://arXiv.org/abs/math/0608335
http://front.math.ucdavis.edu/math.PR/0608335
(alternate)
Author(s): S. Albeverio and A. Daletskii and E. Lytvynov
Abstract: Spaces of differential forms over configuration spaces with Poisson measures
are constructed. The corresponding Laplacians (of Bochner and de Rham type) on
1-forms and associated semigroups are considered. Their probabilistic
interpretation is given.
http://arXiv.org/abs/math/0608337
http://front.math.ucdavis.edu/math.PR/0608337
(alternate)
Author(s): S. Albeverio and A. Daletskii and E. Lytvynov
Abstract: The space $\Gamma_X$ of all locally finite configurations in a
Riemannian manifold $X$ of infinite volume is considered. The deRham complex
of square-integrable differential forms over $\Gamma_X$, equipped with the
Poisson measure, and the corresponding deRham cohomology are studied. The
latter is shown to be unitarily isomorphic to a certain Hilbert tensor algebra
generated by the $L^2$-cohomology of the underlying manifold $X$.
http://arXiv.org/abs/math/0608338
http://front.math.ucdavis.edu/math.PR/0608338
(alternate)
Author(s): Yu. Kondratiev and E. Lytvynov
Abstract: The paper is devoted to the study of Gamma white noise analysis. We define an
extended Fock space $\Gama(\Ha)$ over $\Ha=L^2(\R^d, d\sigma)$, and show how to
include the usual Fock space ${\cal F} (\Ha)$ in it as a subspace. We introduce
in $\Gama(\Ha)$ operators $a(\xi)=\int_{\R^d} dx \xi(x)a(x)$, $\xi\in S$, with
$a(x)=\dig_x+2\dig_x\di_x+1+\di_x +\dig_x\di_x\di_x$, where $\dig_x$ and
$\di_x$ are the creation and annihilation operators at $x$. We show that
$(a(\xi))_{\xi\in S}$ is a family of commuting selfadjoint operators in
$\Gama(\Ha)$ and construct the Fourier transform in generalized joint
eigenvectors of this family. This transform is a unitary $I$ between
$\Gama(\Ha)$ and the $L^2$-space $L^2(S',d\mu_{\mathrm G})$, where
$\mu_{\mathrm G}$ is the measure of Gamma white noise with intensity $\sigma$.
The image of $a(\xi)$ under $I$ is the operator of multiplication by
$\la\cdot,\xi\ra$, so that $a(\xi)$'s are Gamma field operators. The Fock
structure of the Gamma space determined by $I$ coincides with that discovered
in {\bf [}{\it Infinite Dimensional Analysis,
Quantum Probability and Related Topics} {\bf 1} (1998), 91--117{\bf ]}. We
note that $I$ extends in a natural way the multiple stochastic integral (chaos)
decomposition of the ``chaotic'' subspace of the Gamma space. Next, we
introduce and study spaces of test and generalized functions of Gamma white
noise and derive explicit formulas for the action of the creation, neutral, and
Gamma annihilation operators on these spaces.
http://arXiv.org/abs/math/0608340
http://front.math.ucdavis.edu/math.PR/0608340
(alternate)
Author(s): Yu. M. Berezansky and Yu. G. Kondratiev and T. Kuna and E. Lytvynov
Abstract: The paper is devoted to the study of configuration space analysis by using
the projective spectral theorem. For a manifold $X$, let $\Gamma_X$, resp.\
$\Gamma_{X,0}$ denote the space of all, resp. finite configurations in $X$. The
so-called $K$-transform, introduced by A. Lenard, maps functions on
$\Gamma_{X,0}$ into functions on $\Gamma_{X}$ and its adjoint $K^*$ maps
probability measures on $\Gamma_X$ into $\sigma$-finite measures on
$\Gamma_{X,0}$. For a probability measure $\mu$ on $\Gamma_X$,
$\rho_\mu:=K^*\mu$ is called the correlation measure of $\mu$. We consider the
inverse problem of existence of a probability measure $\mu$ whose correlation
measure $\rho_\mu$ is equal to a given measure $\rho$. We introduce an
operation of $\star$-convolution of two functions on $\Gamma_{X,0}$ and suppose
that the measure $\rho$ is $\star$-positive definite, which enables us to
introduce the Hilbert space ${\cal H}_\rho$ of functions on $\Gamma_{X,0}$ with
the scalar product $(G^{(1)},G^{(2)})_{{\cal H}_{\rho}}=
\int_{\Gamma_{X,0}}(G^{(1)}\star\bar G{}^{(2)})(\eta) \rho(d\eta)$. Under a
condition on the growth of the measure $\rho$ on the $n$-point configuration
spaces, we construct the Fourier transform in generalized joint eigenvectors of
some special family $A=(A_\phi)_{\phi\in\D}$, $\D:=C_0^\infty(X)$, of commuting
selfadjoint operators in ${\cal H}_\rho$. We show that this Fourier transform
is a unitary between ${\cal H}_{\rho}$ and the $L^2$-space
$L^2(\Gamma_X,d\mu)$, where $\mu$ is the spectral measure of $A$. Moreover,
this unitary coincides with the $K$-transform, while the measure $\rho$ is the
correlation measure of $\mu$.
http://arXiv.org/abs/math/0608343
http://front.math.ucdavis.edu/math.PR/0608343
(alternate)
Author(s): S. Albeverio and Yu. G. Kondratiev and E. W. Lytvynov and g. F. Us
Abstract: We carry out analysis and geometry on a marked configuration space
$\Omega^M_X$ over a Riemannian manifold $X$ with marks from a space $M$. We
suppose that $M$ is a homogeneous space $M$ of a Lie group $G$. As a
transformation group $\frak A$ on $\Omega_X^M$ we take the ``lifting'' to
$\Omega_X^M$ of the action on $X\times M$ of the semidirect product of the
group $\operatorname{Diff}_0(X)$ of diffeomorphisms on $X$ with compact support
and the group $G^X$ of smooth currents, i.e., all $C^\infty$ mappings of $X$
into $G$ which are equal to the identity element outside of a compact set. The
marked Poisson measure $\pi_\sigma$ on $\Omega_X^M$ with L\'evy measure
$\sigma$ on $X\times M$ is proven to be quasiinvariant under the action of
$\frak A$. Then, we derive a geometry on $\Omega_X^M$ by a natural ``lifting''
of the corresponding geometry on $X\times M$. In particular, we construct a
gradient $\nabla^\Omega$ and a divergence $\operatorname{div}^\Omega$. The
associated volume elements, i.e., all probability measures $\mu$ on
$\Omega_X^M$ with respect to which $\nabla^\Omega$ and
$\operatorname{div}^\Omega$ become dual operators on $L^2(\Omega_X^M;\mu)$, are
identified as the mixed marked Poisson measures with mean measure equal to a
multiple of $\sigma$. As a direct consequence of our results, we obtain marked
Poisson space representations of the group $\frak A$ and its Lie algebra $\frak
a$. We investigate also Dirichlet forms and Dirichlet operators connected with
(mixed) marked Poisson measures.
http://arXiv.org/abs/math/0608344
http://front.math.ucdavis.edu/math.PR/0608344
(alternate)
Author(s): Yu. G. Kondratiev and E. W. Lytvynov and G. F. Us
Abstract: We carry out analysis and geometry on a marked configuration space
$\Omega_X^{R_+}$ over a Riemannian manifold $X$ with marks from the space $R_+$
as a natural generalization of the work {\bf [}{\it J. Func. Anal}. {\bf 154}
(1998),
444--500{\bf ]}. As a transformation group $\mathfrak G$ on this space, we
take the ``lifting'' to $\Omega_X^{R_+}$ of the action on $X\times R_+$ of the
semidirect product of the group Diff of diffeomorphisms on $X$ with compact
support and the group $R_+^X$ of smooth currents, i.e., all $C^\infty$ mappings
of $X$ into $R_+$ which are equal to one outside a compact set. The marked
Poisson measure $\pi$ on $\Omega_X^{R_+}$ with L\'evy measure $\sigma$ is
proven to be quasiinvariant under the action of $\mathfrak G$. Then, we derive
a geometry on $\Omega_X^{R_+}$ by a natural ``lifting'' of the corresponding
geometry on $X\times R_+$. In particular, we construct a gradient
$\nabla^\Omega$ and divergence $div^\Omega$. The associated volume elements,
i.e., all probability measures $\mu$ on $\Omega_X^{R_+}$ with respect to which
$\nabla^\Omega$ and $div^\Omega$ become dual operators on $L^2(\Omega_X^{R_+}
,\mu)$ are identified as the mixed Poisson measures with mean measure equal to
a multiple of $\sigma$. As a direct consequence of our results, we obtain
marked Poisson space representations of the group $\mathfrak G$ and its Lie
algebra $\mathfrak g$. We investigate also Dirichlet forms and Dirichlet
operators connected with (mixed) marked Poisson measures. In particular, we
obtain conditions of ergodicity of the semigroups generated by the Dirichlet
operators. A possible generalization of the results of the paper to the case
where the marks belong to a homogeneous space of a Lie group is noted.
http://arXiv.org/abs/math/0608347
http://front.math.ucdavis.edu/math.PR/0608347
(alternate)
Author(s): S. Albeverio and A. Daletskii and E. Lytvynov
Abstract: Spaces of differential forms over configuration spaces with Poisson measures
are constructed. The corresponding Laplacians (of Bochner and de Rham type) on
forms and associated semigroups are considered. Their probabilistic
interpretation is given.
http://arXiv.org/abs/math/0608349
http://front.math.ucdavis.edu/math.PR/0608349
(alternate)
Author(s): Biao Wu
Abstract: In this paper we study multiagent models with time-varying type change.
Assume that there exist a closed system of $N$ agents classified into $r$ types
according to their states of an internal system; each agent changes its type by
an internal dynamics of the internal states or by the relative frequency of
different internal states among the others, e.g., multinomial sampling. We
investigate the asymptotic behavior of the empirical distributions of the
agents' types as $N$ goes to infinity, by the weak convergence criteria for
time-inhomogeneous Markov processes and the theory of Volterra integral
equations of the second kind. We also prove convergence theorems of these
models evolving in random environment.
http://arXiv.org/abs/math/0608352
http://front.math.ucdavis.edu/math.PR/0608352
(alternate)
Author(s): Markus Flury
Abstract: We investigate the free energy of nearest-neighbor random walks on $\mathbb
Z^d$, endowed with a drift along the first axis, and evolving in a nonnegative
random potential given by i.i.d. random variables. Our main result concerns the
ballistic regime in dimensions $d\geq 4$, at what we show that quenched and
annealed Lyapunov exponents are equal, as soon as the strength of the potential
is small enough.
http://arXiv.org/abs/math/0608357
http://front.math.ucdavis.edu/math.PR/0608357
(alternate)
Author(s): A. Faggionato and P. Mathieu
Abstract: We consider a random walk on the support of an ergodic simple point process
on R^d, d>1, furnished with independent energy marks. The jump rates of the
random walk decay exponentially in the jump length and depend on the energy
marks via a Boltzmann-type factor. This is an effective model for the
phonon-induced hopping of electrons in disordered solids in the regime of
strong Anderson localization. Under mild assumptions on the point process we
prove an upper bound of the asymptotic diffusion matrix of the random walk in
agreement with Mott law. A lower bound in agreement with Mott law was proved in
\cite{FSS}.
http://arXiv.org/abs/math-ph/0608033
http://front.math.ucdavis.edu/math-ph/0608033
(alternate)
Author(s): Yonatan Gutman and Michael Hochman
Abstract: A function $J$ defined on a family $C$ of stationary processes is finitely
observable if there is a sequence of functions $s_n$ such that $s_n(x_1 ...
x_n)\to J(X)$ in probability for every process $X=(x_n)\in C$. Recently,
Ornstein and Weiss roved the striking result that if $C$ is the class of
aperiodic ergodic finite valued processes, then the only finitely observable
isomorphism invariant on $C$ is entropy. We sharpen this in several ways. Our
main theorem is that if $X \to Y$ is a zero-entropy extension of finite entropy
ergodic systems and $C$ is the family of processes arising from $X$ and $Y$,
then every finitely observable function on $C$ is constant. This implies
Ornstein and Weiss' result, and extends it to many other families of processes,
e.g. it shows that there are no nontrivial finitely observable isomorphism
invariants for processes arising from Kronecker systems, mild and strong mixing
zero entropy systems. It also implies that any finitely observable isomorphism
invariant defined on the family of processes arising from irrational rotations
must be constant for rotations belonging to a set of full Lebesgue measure.
http://arXiv.org/abs/math/0608310
http://front.math.ucdavis.edu/math.DS/0608310
(alternate)
Author(s): Michael Hochman
Abstract: An empirical statistic for a class $C$ of stationary processes is a function
$g$ which assigns to each process $(X_n)\in C$ with distribution $P$ and to
each sample $X_1,...,X_n$ of the process a real number $g_P(X_1,...,X_n)$. We
describe a condition on $g$ which implies that the sequence
$(g_P(X_1,...,X_n))_{n=1}^{\infty}$ obeys a (universal) upcrossing inequality,
that is, that the probability that this sequence fluctuates across some
interval $k$ times decays to zero with $k$. As applications we get upcrossing
inequalities for the ergodic theorem (recovering known results), and get
upcrossing inequalities for the Shannon-McMillan-Breiman theorem and for the
Kolmogorov complexity statistic.
http://arXiv.org/abs/math/0608311
http://front.math.ucdavis.edu/math.DS/0608311
(alternate)
Author(s): E. Lytvynov
Abstract: We review some recent developments in white noise analysis and quantum
probability. We pay a special attention to spaces of test and generalized
functionals of some L\'evy white noises, as well as as to the structure of
quantum white noise on these spaces.
http://arXiv.org/abs/math/0608380
http://front.math.ucdavis.edu/math.PR/0608380
(alternate)
Author(s): E. Lytvynov
Abstract: The paper is devoted to construction and investigation of some riggings of
the $L^2$-space of Poisson white noise. A particular attention is paid to the
existence of a continuous version of a function from a test space, and to the
property of an algebraic structure under pointwise multiplication of functions
from a test space.
http://arXiv.org/abs/math/0608383
http://front.math.ucdavis.edu/math.PR/0608383
(alternate)
Author(s): Marton Balazs and Timo Seppalainen
Abstract: We prove that the variance of the current across a characteristic is of order
t^{2/3} in a stationary asymmetric simple exclusion process, and that the
diffusivity has order t^{1/3}. The proof proceeds via couplings to show the
corresponding results for the expected deviations and variance of a second
class particle.
http://arXiv.org/abs/math/0608400
http://front.math.ucdavis.edu/math.PR/0608400
(alternate)
Author(s): Sakhnovich Lev
Abstract: For a broad class of the Levy processes the new form (convolution type) of
the infinitesimal generators is introduced. It leads to the new notions: a
truncated generator, a quasi-potential. The probability of the Levy process
remaining within the given domain is estimated.
http://arXiv.org/abs/math/0608402
http://front.math.ucdavis.edu/math.PR/0608402
(alternate)
Author(s): Stefan Grosskinsky
Abstract: We study the equivalence of ensembles for stationary measures of interacting
particle systems with two conserved quantities and unbounded local state space.
The main motivation is a condensation transition in the zero-range process
which has recently attracted attention. Establishing the equivalence of
ensembles via convergence in specific relative entropy, we derive the phase
diagram for the condensation transition, which can be understood in terms of
the domain of grand-canonical measures. Of particular interest, also from a
mathematical point of view, are the convergence properties of the Gibbs free
energy on the boundary of that domain, involving large deviations and
multivariate local limit theorems of subexponential distributions.
http://arXiv.org/abs/math-ph/0608029
http://front.math.ucdavis.edu/math-ph/0608029
(alternate)
Author(s): Viorel Barbu and Carlo Marinelli
Abstract: We study the existence theory for parabolic variational inequalities in
weighted $L^2$ spaces with respect to excessive measures associated with a
transition semigroup. We characterize the value function of optimal stopping
problems for finite and infinite dimensional diffusions as a generalized
solution of such a variational inequality. The weighted $L^2$ setting allows us
to cover some singular cases, such as optimal stopping for stochastic equations
with degenerate diffusion coefficient. As an application of the theory, we
consider the pricing of American-style contingent claims. Among others, we
treat the cases of assets with stochastic volatility, of path-dependent
payoffs, and of interest-rate derivatives.
http://arXiv.org/abs/math/0608379
http://front.math.ucdavis.edu/math.AP/0608379
(alternate)
Author(s): Manuel Lladser
Abstract: Given an integer m>=1, let || || be a norm in R^{m+1} and let S denote the
set of points with nonnegative coordinates in the unit sphere with respect to
this norm. Consider for each 1<= j<= m a function f_j(z) that is analytic in an
open neighborhood of the point z=0 in the complex plane and with possibly
negative Taylor coefficients. Given a vector n=(n_0,...,n_m) with nonnegative
integer coefficients, we develop a method to systematically associate a
parameter-varying integral to study the asymptotic behavior of the coefficient
of z^{n_0} of the Taylor series of (f_1(z))^{n_1}...(f_m(z))^{n_m}, as ||n||
tends to infinity. The associated parameter-varying integral has a phase term
with well specified properties that make the asymptotic analysis of the
integral amenable to saddle-point methods: for many directions d in S, these
methods ensure uniform asymptotic expansions for the Taylor coefficient of
z^{n_0} of (f_1(z))^{n_1}...(f_m(z))^{n_m}, provided that n/||n|| stays
sufficiently close to d as ||n|| blows up to infinity. Our method finds
applications in studying the asymptotic behavior of the coefficients of a
certain multivariable generating functions as well as in problems related to
the Lagrange inversion formula for instance in the context random planar maps.
http://arXiv.org/abs/math/0608398
http://front.math.ucdavis.edu/math.CO/0608398
(alternate)
Author(s): Kevin Ford and Gerald Tenenbaum
Abstract: We study large partial sums, localized with respect to the sums of variances,
of a sequence of centered random variables. An application is given to the
distribution of prime factors of typical integers.
http://arXiv.org/abs/math/0608411
http://front.math.ucdavis.edu/math.PR/0608411
(alternate)
Author(s): Richard W. Kenyon and David B. Wilson
Abstract: We study groves on planar graphs, which are forests in which every tree
contains one or more of a special set of vertices on the outer face, referred
to as nodes. Each grove partitions the set of nodes. When a random grove is
selected, we show how to compute the various partition probabilities as
functions of the electrical properties of the graph when viewed as a resistor
network. We prove that for any partition sigma, Pr[grove has type sigma] /
Pr[grove is a tree] is a dyadic-coefficient polynomial in the pairwise
resistances between the nodes, and Pr[grove has type sigma] / Pr[grove has
maximal number of trees] is an integer-coefficient polynomial in the entries of
the Dirichlet-to-Neumann matrix. We give analogous integer-coefficient
polynomial formulas for the pairings of chains in the double-dimer model. We
show that the distribution of pairings of contour lines in the Gaussian free
field with certain natural boundary conditions is identical to the distribution
of pairings in the scaling limit of the double-dimer model. These partition
probabilities are relevant to multichordal SLE_2, SLE_4, and SLE_8.
http://arXiv.org/abs/math/0608422
http://front.math.ucdavis.edu/math.PR/0608422
(alternate)
Author(s): Marton Balazs and Timo Seppalainen
Abstract: We consider a large class of nearest neighbor attractive stochastic
interacting systems that includes the asymmetric simple exclusion, zero range,
bricklayers' and the symmetric K-exclusion processes. We provide exact formulas
that connect particle flux (or surface growth) fluctuations to the two-point
function of the process and to the motion of the second class particle. Such
connections have only been available for simple exclusion where they were of
great use in particle current fluctuation investigations.
http://arXiv.org/abs/math/0608437
http://front.math.ucdavis.edu/math.PR/0608437
(alternate)
Author(s): Jafar Shaffaf
Abstract: The determination of the density functions for products of random elements
from specified classes of matrices is a basic problem in random matrix theory
and is also of interest in theoretical physics. For connected simple Lie groups
of $2\times 2$ matrices and conjugacy and spherical classes a complete solution
is given here. The problem/solution can be re-stated in terms of the structure
of certain Hecke algebras attached to groups of $2\times 2$ matrices.
http://arXiv.org/abs/math/0608440
http://front.math.ucdavis.edu/math.RT/0608440
(alternate)
Author(s): Teunis J. Ott and Jason Swanson
Abstract: The Transmission Control Protocol (TCP) is a Transport Protocol used in the
Internet. Ott has introduced a more general class of candidate Transport
Protocols called "protocols in the TCP Paradigm". The long run objective of
studying this larger class is to find protocols with promising performance
characteristics. This paper studies Markov chain models derived from protocols
in the TCP Paradigm. Protocols in the TCP Paradigm, as TCP, protect the network
from congestion by reducing the "Congestion Window" (the amount of data allowed
to be sent but not yet acknowledged) when there is packet loss or packet
marking, and increasing it when there is no loss. When loss of different
packets are assumed to be independent events and the probability p of loss is
assumed to be constant, the protocol gives rise to a Markov chain {W_n}, where
W_n is the size of the congestion window after the transmission of the n-th
packet. For a wide class of such Markov chains, we prove weak convergence
results, after appropriate rescaling of time and space, as p tends to 0. The
limiting processes are defined by stochastic differential equations. Depending
on certain parameter values, the stochastic differential equation can define an
Ornstein-Uhlenbeck process or can be driven by a Poisson process.
http://arXiv.org/abs/math/0608476
http://front.math.ucdavis.edu/math.PR/0608476
(alternate)
Author(s): Leonid Kontorovich
Abstract: We prove an apparently novel concentration of measure result for Markov tree
processes. The bound we derive reduces to the known bounds for Markov processes
when the tree is a chain, thus strictly generalizing the known Markov process
concentration results. We employ several techniques of potential independent
interest, especially for obtaining similar results for more general directed
acyclic graphical models.
http://arXiv.org/abs/math/0608511
http://front.math.ucdavis.edu/math.PR/0608511
(alternate)
Author(s): Tryphon T. Georgiou
Abstract: We present an intrinsic metric that quantifies distances between power
spectral density functions. The metric was derived by the author in a recent
arXiv-report (math.OC/0607026) as the geodesic distance between spectral
density functions with respect to a particular pseudo-Riemannian metric
motivated by a quadratic prediction problem. We provide an independent
verification of the metric inequality and discuss certain key properties of the
induced topology.
http://arXiv.org/abs/math/0608486
http://front.math.ucdavis.edu/math.OC/0608486
(alternate)
Author(s): Arni S. R. Srinivasa Rao
Abstract: We consider previously well-known models in epidemiology where the parameter
for incubation period is used as one of the important components to explain the
dynamics of the variables. Such models are extended here to explain the
dynamics with respect to a given therapy that prolongs the incubation period. A
deconvolution method is demonstrated for estimation of parameters in the
situations when no-therapy and multiple therapies are given to the infected
population. The models and deconvolution method are extended in order to study
the impact of therapy in age-structured populations. A generalisation for a
situation when n- types of therapies are available is given.
http://arXiv.org/abs/q-bio/0608028
http://front.math.ucdavis.edu/q-bio.QM/0608028
(alternate)
Author(s): P. J. Fitzsimmons and K. Yano
Abstract: It is proved that generalized excursion measures can be constructed via time
change of Ito's Brownian excursion measure. A tightness-like condition on
strings is introduced to prove a convergence theorem of generalized excursion
measures. The convergence theorem is applied to obtain a conditional limit
theorem, a kind of invariance principle where the limit is the Bessel meander.
http://arXiv.org/abs/math/0608530
http://front.math.ucdavis.edu/math.PR/0608530
(alternate)
Author(s): Mathew D. Penrose
Abstract: In ballistic deposition (BD), $(d+1)$-dimensional particles fall sequentially
at random towards an initially flat, large but bounded $d$-dimensional surface,
and each particle sticks to the first point of contact. For both lattice and
continuum BD, a law of large numbers in the thermodynamic limit establishes
convergence of the mean height and surface width of the interface to constants
$h(t)$ and $w(t)$, respectively, depending on time $t$. We show that $h(t)$ is
asymptotically linear in $t$, while $w(t)$ grows at least logarithmically in
$t$ when $d=1$. We also give duality results saying that the height above the
origin for deposition onto an initially flat surface is equidistributed with
the maximum height for deposition onto a surface growing from a single site.
http://arXiv.org/abs/math/0608540
http://front.math.ucdavis.edu/math.PR/0608540
(alternate)
Author(s): Wolfgang Konig and Peter Morters and Nadia Sidorova
Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation
$\partial_t u(t,z)=\Delta u(t,z)+\xi(z) u(t,z)$ on $(0,\infty)\times {\mathbb
Z}^d$ with random potential $(\xi(z) \colon z\in {\mathbb Z}^d)$. We consider
independent and identically distributed potential variables, such that
Prob$(\xi(z)>x)$ decays polynomially as $x\uparrow\infty$. If $u$ is initially
localised in the origin, i.e. if $u(0,x)=\one_0(x)$, we show that, at any large
time $t$, the solution is completely localised in a single point with high
probability. More precisely, we find a random process $(Z_t \colon t\ge 0)$
with values in $\Z^d$ such that $\lim_{t \uparrow\infty}
u(t,Z_t)/\sum_{z\in\Z^d} u(t,z) =1,$ in probability. We also identify the
asymptotic behaviour of $Z_t$ in terms of a weak limit theorem.
http://arXiv.org/abs/math/0608544
http://front.math.ucdavis.edu/math.PR/0608544
(alternate)
Author(s): Jonathan Rowe and Boris Mitavskiy
Abstract: Dynamical processes taking place on networks have received much attention in
recent years, especially on various models of random graphs (including small
world and scale free networks). They model a variety of phenomena, including
the spread of information on the Internet; the outbreak of epidemics in a
spatially structured population; and communication between randomly dispersed
processors in an ad hoc wireless network. Typically, research has concentrated
on the existence and size of a large connected component (representing, say,
the size of the epidemic) in a percolation model, or uses differential
equations to study the dynamics using a mean-field approximation in an infinite
graph. Here we investigate the time taken for information to propagate from a
single source through a finite network, as a function of the number of nodes
and the network topology. We assume that time is discrete, and that nodes
attempt to transmit to their neighbors in parallel, with a given probability of
success. We solve this problem exactly for several specific topologies, and use
a large-deviation theorem to derive general asymptotic bounds, which apply to
any family of networks where the diameter grows at least logarithmically in the
number of nodes. We use these bounds, for example, to show that a scale-free
network has propagation time logarithmic in the number of nodes, and inversely
proportional to the transmission probability.
http://arXiv.org/abs/math/0608561
http://front.math.ucdavis.edu/math.PR/0608561
(alternate)
Author(s): Andras Telcs
Abstract: In this paper characterizations of graphs satisfying heat kernel estimates
for a wide class of space-time scaling functions are given. The equivalence of
the two-sided heat kernel estimate and the parabolic Harnack inequality is also
shown via the equivalence of the upper (lower) heat kernel estimate to the
parabolic mean value (and super mean value) inequality.
http://arXiv.org/abs/math/0608594
http://front.math.ucdavis.edu/math.PR/0608594
(alternate)
Author(s): C\'edric Boutillier and B\'eatrice de Tili\`ere
Abstract: The dimer model on a graph embedded in the torus can be interpreted as a
collection of random self-avoiding loops. We prove that when the mesh of the
graph tends to zero, and the aspect of the torus is fixed, the winding number
of this collection of loops converges in law to a two-dimensional discrete
Gaussian distribution. This is the first mathematical proof of a result known
to physicists in the context of toroidal 2-D critical models, and their mapping
to the massless free field on the torus.
http://arXiv.org/abs/math/0608600
http://front.math.ucdavis.edu/math.PR/0608600
(alternate)
Author(s): Andras Telcs
Abstract: This paper presents estimates for the distribution of the exit time from
balls and short time asymptotics for measure metric Dirichlet spaces. The
estimates cover the classical Gaussian case, the sub-diffusive case which can
be observed on particular fractals and further less regular cases as well. The
proof is based on a new chaining argument and it is free of volume growth
assumptions.
http://arXiv.org/abs/math/0608615
http://front.math.ucdavis.edu/math.PR/0608615
(alternate)
Author(s): Shalom Benaim and Peter Friz
Abstract: In a recent article the authors obtained a formula which relates explicitly
the tail of risk neutral returns with the wing behavior of the Black Scholes
implied volatility smile. In situations where precise tail asymptotics are
unknown but a moment generating function is available we first establish, under
easy-to-check conditions, tail asymptoics on logarithmic scale as soft
applications of standard Tauberian theorems. Such asymptotics are enough to
make the tail-wing formula work and we so obtain a version of Lee's moment
formula with the novel guarantee that there is indeed a limiting slope when
plotting implied variance against log-strike. We apply these results to
time-changed Levy models and the Heston model. In particular, the
term-structure of the wings can be analytically understood.
http://arXiv.org/abs/math/0608619
http://front.math.ucdavis.edu/math.PR/0608619
(alternate)
Author(s): Alexander Gnedin
Abstract: For a class of random partitions of an infinite set a de Finetti-type
representation is derived, and in one special case a central limit theorem for
the number of blocks is shown.
http://arXiv.org/abs/math/0608621
http://front.math.ucdavis.edu/math.PR/0608621
(alternate)
Author(s): G.Molchan
Abstract: Let x(s), s in R^d be a Gaussian self-similar random process of index H. We
consider the problem of log-asymptotics for the probability p(T) that x(s),
x(0)=0 does not exceed a fixed level in a star-shaped increasing domain T*U as
T >> 1. General conditions are given to guarantee the existence of the limit of
(-log p(T))/L(T) as T >> 1 for a slowly increasing function L(T).
http://arXiv.org/abs/math/0608630
http://front.math.ucdavis.edu/math.PR/0608630
(alternate)
Author(s): Martin Forde
Abstract: Building on an insight in Carr&Lee\cite{CarrLee03}, we establish a simple
relationship between the prices of Eigenfunction contracts and the mgf of the
time-change, under a model where the Stock price is a diffusion process
evaluated at an independent stochastic clock. In particular, we characterize
the tail behaviour (Theorems \ref{thm:CEVtail}, \ref{thm:CEVstocvoltail}) and
the small-time behaviour (Theorem \ref{thm:CEVLargeDev}) of a CEV diffusion,
and a time-changed CEV diffusion. We describe the small-time behaviour of the
Heston subordinator (Theorem \ref{thm:HestonLDP}) using large deviations
theory, which shows that the previous three results are applicable to the
CEV-Heston stochastic volatility model discussed in Atlan&Leblanc\cite{Atlan}.
We also use a general result by Norris&Stroock\cite{NorrisStroock} to
characterize the tail behaviour of the transition densities for a general
Dupire local volatility model\cite{Dupire94}, in terms of an Energy functional
(Corollary \ref{cor:SN}). Finally, in section 3, we discuss calibration issues
for a time-changed diffusion model. Specifically, for the time-changed CEV
model, we show that if we wish to apply an extended version of the
Carr-Lee\cite{CarrLee03} methodology to infer the characteristic function of
the time-change from an observed single-maturity smile, then the tails of the
distribution of the time-change have to have sub-exponential behaviour, or else
we have to use \textit{analytic continuation}
http://arXiv.org/abs/math/0608634
http://front.math.ucdavis.edu/math.PR/0608634
(alternate)
Author(s): Michael C. Mackey and Marta Tyran-Kaminska
Abstract: We establish a new functional central limit theorem result for non-invertible
measure preserving maps that are not necessarily ergodic, using the
Perron-Frobenius operator. We apply the result to asymptotically periodic
transformations and give an extensive specific example of asymptotically
periodic transformations by using the tent map.
http://arXiv.org/abs/math/0608637
http://front.math.ucdavis.edu/math.PR/0608637
(alternate)
Author(s): Olivier Garet (MAPMO) and R\'{e}gine Marchand (IECN)
Abstract: Consider two epidemics whose expansions on $\mathbb{Z}^d$ are governed by two
families of passage times that are distinct and stochastically comparable. We
prove that when the weak infection survives, the space occupied by the strong
one is almost impossible to detect: for instance, it could not be observed by a
medium resolution satellite. We also recover the same fluctuations with respect
to the asymptotic shape as in the case where the weak infection evolves alone.
In dimension two, we prove that one species finally occupies a set with full
density, while the other one only occupies a set of null density. We also prove
that the H\"{a}ggstr\"{o}m-Pemantle non-coexistence result "except perhaps for
a denumerable set" can be extended to families of stochastically comparable
passage times indexed by a continuous parameter.
http://arXiv.org/abs/math/0608667
http://front.math.ucdavis.edu/math.PR/0608667
(alternate)
Author(s): Olivier Garet (MAPMO)
Abstract: This paper concerns maximal flows on $\mathbb{Z}^2$ traveling from a convex
set to infinity, the flows being restricted by a random capacity. For every
compact convex set $A$, we prove that the maximal flow $\Phi(nA)$ between $nA$
and infinity is such that $\Phi(nA)/n$ almost surely converges to the integral
of a deterministic function over the boundary of $A$. The limit can also be
interpreted as the optimum of a deterministic continuous max-flow problem. We
derive some properties of the infinite cluster in supercritical Bernoulli
percolation.
http://arXiv.org/abs/math/0608676
http://front.math.ucdavis.edu/math.PR/0608676
(alternate)
Author(s): Alexander V. Kolesnikov
Abstract: We find sufficient conditions for a probability measure $\mu$ to satisfy an
inequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_{\R^d} f^2 d
\mu} \Bigr) d \mu \le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|f|}
\Bigr) d \mu + A \int_{\R^d} f^2 d \mu, $$ where $F$ is concave and $c$ (a cost
function) is convex. In particular, for every convex $\mu$ satisfying
$\int_{\R^d} e^{\epsilon |x|^{\alpha}} d\mu < \infty$ for some $\epsilon>0$, $1
< \alpha \le 2$, we establish a family of tight inequalities interpolating
between the $F$-Sobolev and modified log-Sobolev inequalities.
http://arXiv.org/abs/math/0608681
http://front.math.ucdavis.edu/math.PR/0608681
(alternate)
Author(s): Uwe Einmahl and Deli Li
Abstract: In a recent paper by the authors a general result characterizing two-sided
LIL behavior for real valued random variables has been established. In this
paper, we show that there are analogous results in the Banach space setting.
One of our main new tools is an improved Fuk-Nagaev type inequality in Banach
space which should be of independent interest.
http://arXiv.org/abs/math/0608687
http://front.math.ucdavis.edu/math.PR/0608687
(alternate)
Author(s): Samir Belhaouari and Thomas Mountford and Glauco Valle
Abstract: We show that for the voter model on $\{0,1\}^{\mathbb{Z}}$ corresponding to a
random walk with kernel $p(\cdot)$ and starting from unanimity to the right and
opposing unanimity to the left, a tight interface between 0's and 1's exists if
$p(\cdot)$ has finite second moment but does not if $p(\cdot)$ fails to have
finite moment of order $\alpha$ for some $\alpha <2$.
http://arXiv.org/abs/math/0608690
http://front.math.ucdavis.edu/math.PR/0608690
(alternate)
Author(s): M. V. Menshikov and Andrew R. Wade
Abstract: We give criteria for ergodicity, transience and null recurrence for the
random walk in random environment on {0,1,2,...}, with reflection at the
origin, where the random environment is subject to a vanishing perturbation.
Our results complement existing criteria for random walks in random
environments and for Markov chains with asymptotically zero drift, and are
significantly different to these previously studied cases. Our method is based
on a martingale technique - the method of Lyapunov functions.
http://arXiv.org/abs/math/0608696
http://front.math.ucdavis.edu/math.PR/0608696
(alternate)
Author(s): M. V. Menshikov and Andrew R. Wade
Abstract: We study the random walk in random environment on {0,1,2,...}, where the
environment is subject to a vanishing (random) perturbation. The two particular
cases we consider are: (i) random walk in random environment perturbed from
Sinai's regime; (ii) simple random walk with random perturbation. We give
almost sure results on how far the random walker will be from the origin after
a long time t, for almost every environment. We give both upper and lower
almost sure bounds. These bounds are of order $(\log t)^\beta$, for $\beta \in
(1,\infty)$, depending on the perturbation. In addition, in the ergodic cases,
we give results on the rate of decay of the stationary distribution.
http://arXiv.org/abs/math/0608697
http://front.math.ucdavis.edu/math.PR/0608697
(alternate)
Author(s): Hyungsu Kim and Chul Ki Ko and Sungchul Lee
Abstract: When we use the entropy method to get the tail bounds, typically the left
tail bounds are not good comparing with the right ones. Up to now this
asymmetry has been observed many times. Surprisingly we find an entropy method
for the left tail that works in the exactly same way that it works for the
right tail.
http://arXiv.org/abs/math/0608706
http://front.math.ucdavis.edu/math.PR/0608706
(alternate)
Author(s): L. Pastur
Abstract: We study the variance and the Laplace transform of the probability law of
linear eigenvalue statistics of unitary invariant Matrix Models of
n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test
function of statistics is smooth enough and using the asymptotic formulas by
Deift et al for orthogonal polynomials with varying weights, we show first that
if the support of the Density of States of the model consists of two or more
intervals, then in the global regime the variance of statistics is a
quasiperiodic function of n generically in the potential, determining the
model. We show next that the exponent of the Laplace transform of the
probability law is not in general 1/2variance, as it should be if the Central
Limit Theorem would be valid, and we find the asymptotic form of the Laplace
transform of the probability law in certain cases.
http://arXiv.org/abs/math/0608719
http://front.math.ucdavis.edu/math.PR/0608719
(alternate)
Author(s): Alexei Borodin (1) and Patrik L. Ferrari (2) and Michael Pr\"ahofer (2) and Tomohiro Sasamoto (3) ((1) Caltech, (2) TU-Muenchen, (3) Chiba University)
Abstract: We consider the joint distributions of particle positions for the continuous
time totally asymmetric simple exclusion process (TASEP). They are expressed as
Fredholm determinants with a kernel defining a signed determinantal point
measure. We then consider certain periodic initial conditions and determine the
kernel in the scaling limit. This result has been announced first in a letter
by one of us and here we provide a self-contained derivation. Connections to
last passage directed percolation and random matrices are also briefly
discussed.
http://arXiv.org/abs/math-ph/0608056
http://front.math.ucdavis.edu/math-ph/0608056
(alternate)
Author(s): Roy Wagner
Abstract: We prove a tail estimate for the variable $\sum f(X_i)$, where $(X_i)_i$ is
the trajectory of a random walk on a graph (or a reversible Markov chain). The
estimate is in terms of the maximum of the function, its variance, and the
spectral gap of the graph. Our proof is more elementary than other proofs in
the literature, and for some parameter regimes our results are sharper. We
obtain Bernstein and Bennett-type inequalitis, as well as an inequality for
subgaussian variables.
http://arXiv.org/abs/math/0608740
http://front.math.ucdavis.edu/math.PR/0608740
(alternate)
Author(s): Sho Matsumoto
Abstract: Jack function theory is useful for the calculation of the moment of the
characteristic polynomials in Dyson's circular $\beta$-ensembles (C$\beta$E).
We define a $q$-analogue of the C$\beta$E and calculate moments of
characteristic polynomials via Macdonald function theory. By this
$q$-deformation, the asymptotics calculation of these moments becomes simple
and the ordinary C$\beta$E case is recovered as $q \to 1$. Further, by using a
hyperdeterminant which is a simple generalization of a determinant, we give a
Jacobi-Trudi type formula for Jack symmetric functions of rectangular shapes.
http://arXiv.org/abs/math/0608751
http://front.math.ucdavis.edu/math.PR/0608751
(alternate)
Author(s): Bernard Shiffman and Steve Zelditch
Abstract: Our main results are asymptotic formulas for the variance of the number
$\mathcal{N}^U_N$ of zeros of $m$ Gaussian random polynomials of degree $N$ in
an open set $U\subset C^m$ with smooth boundary as the degree $N\to\infty$, and
more generally for the zeros of $m$ random holomorphic sections of high powers
of any positive line bundle over any $m$-dimensional compact K\"ahler manifold.
Our result for number statistics states that the variance of the number
$\mathcal{N}^U_N$ of zeros in $U$ is asymptotic to $N^{m-1/2} \nu_{mm}
Vol(\partial U)$, where $\nu_{mm}$ is a universal constant depending only on
the dimension $m$. We also give variance results for $Vol(Z^k_N\cap U)$, where
$Z^k_N$ denotes the set of simultaneous zeros of $k
http://arXiv.org/abs/math/0608743
http://front.math.ucdavis.edu/math.CV/0608743
(alternate)
Author(s): Adam Skalski
Abstract: A concept of quantum stochastic convolution cocycle is introduced and studied
in two different contexts -- purely algebraic and operator space theoretic. A
quantum stochastic convolution cocycle is a quantum stochastic process on a
coalgebra satisfying the convolution cocycle relation and the initial condition
given by the counit. The notion generalises that of quantum Levy process, which
in turn is a noncommutative probability counterpart of classical Levy process
on a group.
Convolution cocycles arise as solutions of quantum stochastic differential
equations. In turn every sufficiently regular cocycle satisfies an equation of
that type. This is proved along with the corresponding existence and uniqueness
of solutions for coalgebraic quantum stochastic differential equations. The
stochastic generators of unital *-homomorphic cocycles are characterised in
terms of structure maps on a *-bialgebra. This yields a simple proof of the
Schurmann Reconstruction Theorem for a quantum Levy process; it also yields a
topological version for a quantum Levy process on a C*-bialgebra. Precise
characterisation of the stochastic generators of completely positive and
contractive quantum stochastic convolution cocycles in the C*-algebraic context
is given, leading to some dilation results. A few examples are presented and
some interpretations offered for quantum stochastic convolution cocycles and
their stochastic generators on different types of *-bialgebra.
http://arXiv.org/abs/math/0608756
http://front.math.ucdavis.edu/math.OA/0608756
(alternate)
Author(s): Peter Friz and Nicolas Victoir
Abstract: Multi-dimensional continuous local martingales, enhanced with their
stochastic area process, give rise to geometric rough paths with a.s. finite
homogenous p-variation, p>2. Here we go one step further and establish
quantitative bounds of the p-variation norm in the form of a BDG inequality.
Our proofs are based on old ideas by Lepingle. We also discuss geodesic and
piecewise linear approximations.
http://arXiv.org/abs/math/0608783
http://front.math.ucdavis.edu/math.PR/0608783
(alternate)
Author(s): Brian Rider and Balint Virag
Abstract: For the plane, sphere, and hyperbolic plane we consider the canonical
invariant determinantal point processes with intensity rho dnu, where nu is the
corresponding invariant measure. We show that as rho converges to infinity,
after centering, these processes converge to invariant H1 noise. More
precisely, for all functions f in the interesection of H1(nu) and L1(nu) the
distribution of sum f(z) - rho/pi integral f dnu converges to Gaussian with
mean 0 and variance given by ||f||_H1^2 / (4 pi).
http://arXiv.org/abs/math/0608785
http://front.math.ucdavis.edu/math.PR/0608785
(alternate)
Author(s): Peter Constantin and Gautam Iyer
Abstract: We discuss stochastic representations of advection diffusion equations with
variable diffusivity, stochastic integrals of motion and generalized relative
entropies.
http://arXiv.org/abs/math/0608797
http://front.math.ucdavis.edu/math.AP/0608797
(alternate)
Author(s): Peter Friz and Nicolas Victoir
Abstract: We consider uniformly subelliptic operators on certain unimodular Lie groups
of polynomial growth. It was shown by Saloff-Coste and Stroock that classical
results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then
observed by Sturm that many proofs extend naturally to the setting of locally
compact Dirichlet spaces. We relate these results to what is known as rough
path theory by showing that they provide a natural and powerful analytic
machinery for construction and study of (random) geometric Hoelder rough paths.
(In particular, we obtain a simple construction of the Lyons-Stoica stochastic
area for a diffusion process with uniformly elliptic generator in divergence
form.) Our approach then enables us to establish a number of far-reaching
generalizations of classical theorems in diffusion theory including Wong-Zakai
approximations, Freidlin-Wentzell sample path large deviations and the
Stroock-Varadhan support theorem. The latter was conjectured by T. Lyons in his
recent St. Flour lecture.
http://arXiv.org/abs/math/0609007
http://front.math.ucdavis.edu/math.PR/0609007
(alternate)
Author(s): H\'ector Cancela (INCO and UdelaR) and Ernesto Mordecki (CMAT and UdelR)
Abstract: We give an estimate of the number of geometrically distinct open tours $\G$
for a knight on a chessboard. We use a randomization of Warnsdorff rule to
implement importance sampling in a backtracking scheme, correcting the observed
bias of the original rule, according to the proposed principle that ``most
solutions follow Warnsdorff rule most of the time''. After some experiments in
order to test this principle, and to calibrate a parameter, interpreted as a
distance of a general solution from a Warnsdorff solution, we conjecture that
$\G=1.22\times 10^{15}$.
http://arXiv.org/abs/math/0609009
http://front.math.ucdavis.edu/math.PR/0609009
(alternate)
Author(s): Zhenxin Liu
Abstract: Conley index theory is a very powerful tool in the study of dynamical
systems, differential equations and bifurcation theory. In this paper, we make
an attempt to generalize the Conley index to discrete random dynamical systems.
And we mainly follow the Conley index for maps given by Franks and Richeson in
\cite{Fra}. Furthermore, we simply discuss the relations of isolated invariant
sets between time-continuous random dynamical systems and the corresponding
time-$h$ maps. For applications we give several examples to illustrate our
results.
http://arXiv.org/abs/math/0609011
http://front.math.ucdavis.edu/math.DS/0609011
(alternate)
Author(s): W. Jaworski and M. Neufang
Abstract: Let $G$ be a locally compact group and $\pi$ a representation of $G$ by
weakly^* continuous isometries acting in a dual Banach space $E$. Given a
probability measure $\mu$ on $G$ we study the Choquet-Deny equation
$\pi(\mu)x=x$, $x\in E$. We prove that the solutions of this equation form the
range of a projection of norm 1 and can be represented by means of a ``Poisson
formula'' on the same boundary space that is used to represent the bounded
harmonic functions of the random walk of law $\mu$. The relation between the
space of solutions of the Choquet-Deny equation in $E$ and the space of bounded
harmonic functions can be understood in terms of a construction resembling the
$W^*$-crossed product and coinciding precisely with the crossed product in the
special case of the Choquet-Deny equation in the space $E=B(L^2(G))$ of bounded
linear operators on $L^2(G)$. Other general properties of the Choquet-Deny
equation in a Banach space are also discussed.
http://arXiv.org/abs/math/0609035
http://front.math.ucdavis.edu/math.FA/0609035
(alternate)
Author(s): Tim D. Austin (UC and Los Angeles)
Abstract: The mechanism of transmission of an action potential along the axon of a
neuron has been heavily studied by biophysicists for over fifty years, and
several detailed models now exist to describe axonal behaviour. Older models
have been purely deterministic, predicting behaviour by various representative
quantities evolving according to differential equations. More recently,
however, stochastic elements have been included to represent more faithfully a
large number of unpredictable sub-processes at work at the scale of individual
protein molecules within the axon.
In this paper we consider the classical differential equations of Hodgkin and
Huxley and a natural refinement of them to include a layer of stochastic
behaviour, modelled by a large number of finite-state-space Markov processes
coupled to a simple modification of the original Hodgkin-Huxley PDE. We first
prove existence, uniqueness and some regularity for the stochastic process, and
then show that in a suitable limit as the number of stochastic components of
the stochastic model increases and their individual contributions decrease the
process that they determine converges to the trajectory predicted by the
deterministic PDE, uniformly up to finite time horizons in probability. In a
sense, this verifies the consistency of the deterministic and stochastic
processes.
http://arXiv.org/abs/math/0609068
http://front.math.ucdavis.edu/math.PR/0609068
(alternate)
Author(s): Greg Markowsky
Abstract: Let $B_t$ be a one dimensional Brownian motion, and let $\alpha'$ denote the
derivative of the intersection local time of $B_t$ as defined in Jay Rosen's
work (see references). The object of this paper is to prove the following
formula $(1/2)\alpha'_t(x) + (1/2)sgn(x)t = \int_0^t L_s^{B_s - x}dB_s -
\int_0^t sgn(B_t - B_u - x) du$ which was given as a formal identity by Rosen
without proof.
http://arXiv.org/abs/math/0609084
http://front.math.ucdavis.edu/math.PR/0609084
(alternate)
Author(s): Martin Forde
Abstract: Using a result by Doss\cite{Doss77} and the G\"{a}rtner-Ellis theorem, we
prove, by bounding the It\^{o} map, that under certain bounds on the diffusion
coefficients, the transition densities of a one-dimensional diffusion process
satisfy the \textit{large deviation principle} (Theorem \ref{thm:Tails}). We
prove a similar result for a diffusion proces on the line evaluated at an
independent stochastic clock, when the arithmetic average of the time-change
also satisfies the LDP (Theorem (\ref{thm:stocvoltail}), as it does for the
well know Cox-Ingersoll-Ross subordinator (Theorem \ref{thm:HestonLDP}).
http://arXiv.org/abs/math/0609117
http://front.math.ucdavis.edu/math.PR/0609117
(alternate)
Author(s): Bruno Nietlispach
Abstract: Quasi-logarithmic combinatorial structures are a class of decomposable
combinatorial structures which extend the logarithmic class. In order to obtain
asymptotic approximations to their component spectrum, it is necessary first to
establish an approximation to the sum of an associated sequence of independent
random variables in terms of the Dickman distribution. This in turn requires an
argument that refines the Mineka coupling by incorporating a blocking
construction, leading to exponentially sharper coupling rates for the sums in
question. Applications include distributional limit theorems for the size of
the largest component and for the vector of counts of the small components in a
quasi-logarithmic combinatorial structure.
http://arXiv.org/abs/math/0609129
http://front.math.ucdavis.edu/math.CO/0609129
(alternate)
Author(s): Bruno Nietlispach
Abstract: We show that in quasi-logarithmic additive number systems all partition sets
have asymptotic density, and we obtain a corresponding monadic second-order
limit law for adequate classes of relational structures.
http://arXiv.org/abs/math/0609143
http://front.math.ucdavis.edu/math.CO/0609143
(alternate)
Author(s): Scott Sheffield
Abstract: We construct and study the conformal loop ensembles CLE(kappa), defined for
all kappa between 8/3 and 8, using branching variants of SLE(kappa) called
exploration trees. The conformal loop ensembles are random collections of
countably many loops in a planar domain that are characterized by certain
conformal invariance and Markov properties. We conjecture that they are the
scaling limits of various random loop models from statistical physics,
including the O(n) loop models.
http://arXiv.org/abs/math/0609167
http://front.math.ucdavis.edu/math.PR/0609167
(alternate)
Author(s): Nayantara Bhatnagar and Pietro Caputo and Prasad Tetali and Eric Vigoda
Abstract: We study Markov chains which model genome rearrangements. These models are
useful for studying the equilibrium distribution of chromosomal lengths, and
are used in methods for estimating genomic distances. The primary Markov chain
studied in this paper is the top-swap Markov chain. The top-swap chain is a
card-shuffling process with n cards divided over k decks, where the cards are
ordered within each deck. A transition consists of choosing a random pair of
cards, and if the cards lie in different decks, we cut each deck at the chosen
card and exchange the tops of the two decks. We prove precise bounds on the
relaxation time (inverse spectral gap) of the top-swap chain. In particular, we
prove the relaxation time is of order n+k. This resolves an open question of
Durrett.
http://arXiv.org/abs/math/0609171
http://front.math.ucdavis.edu/math.PR/0609171
(alternate)
Author(s): Stilian A. Stoev and George Michailidis and Murad S. Taqqu
Abstract: In this paper, a novel approach to the problem of estimating the heavy-tail
exponent alpha>0 of a distribution is proposed. It is based on the fact that
block-maxima of size m of the independent and identically distributed data
scale at a rate of m^{1/alpha}. This scaling rate can be captured well by the
max-spectrum plot of the data that leads to regression based estimators.
Consistency and asymptotic normality of these estimators is established under
mild conditions on the behavior of the tail of the distribution. The results
are obtained by establishing bounds on the rate of convergence of moment-type
functionals of heavy-tailed maxima. Such bounds often yield exact rates of
convergence and are of independent interest. Practical issues on the automatic
selection of tuning parameters for the estimators and corresponding confidence
intervals are also addressed. Extensive numerical simulations show that the
proposed method proves competitive for both small and large sample sizes and
for a large range of tail exponents. The method is shown to be more robust than
the classical Hill plot and is illustrated on two data sets of insurance claims
and natural gas field sizes.
http://arXiv.org/abs/math/0609163
http://front.math.ucdavis.edu/math.ST/0609163
(alternate)
Author(s): Julien Reynier (INRIA Rocquencourt)
Abstract: Congestion on the Internet is an old problem but still a subject of intensive
research. The TCP protocol with its AIMD (Additive Increase and Multiplicative
Decrease) behavior hides very challenging problems; one of them is to
understand the interaction between a large number of users with delayed
feedback. This article will focus on two modeling issues of TCP which appeared
to be important to tackle concrete scenarios when implementing the model
proposed in [Baccelli McDonald Reynier 02] firstly the modeling of the maximum
TCP window size: this maximum can be reached quickly in many practical cases;
secondly the delay structure: the usual Little-like formula behaves really
poorly when queuing delays are variable, and may change dramatically the
evolution of the predicted queue size, which makes it useless to study
drop-tail or RED (Random Early Detection) mechanisms. Within proposed TCP
modeling improvements, we are enabled to look at a concrete example where RED
should be used in FIFO routers instead of letting the default drop-tail happen.
We study mathematically fixed points of the window size distribution and local
stability of RED. An interesting case is when RED operates at the limit when
the congestion starts, it avoids unwanted loss of bandwidth and delay
variations.
http://arXiv.org/abs/cs/0609014
http://front.math.ucdavis.edu/cs.NI/0609014
(alternate)
Author(s): E. Mordecki and A. Szepessy and R. Tempone and G. E. Zouraris
Abstract: This work develops Monte Carlo Euler adaptive time stepping methods for the
weak approximation problem of jump diffusion driven stochastic differential
equations. The main result is the derivation of a new expansion for the
omputational error, with computable leading order term in a posteriori form,
based on stochastic flows and discrete dual backward problems which extends the
results in [STZ]. These expansions lead to efficient and accurate computation
of error estimates. Adaptive algorithms for either stochastic time steps or
quasi-deterministic time steps are described. Numerical examples show the
performance of the proposed error approximation and of the described adaptive
time-stepping methods.
http://arXiv.org/abs/math/0609186
http://front.math.ucdavis.edu/math.NA/0609186
(alternate)
Author(s): Bhupendra Gupta
Abstract: Let $X_1,X_2,...$ be an infinite sequence of i.i.d. random vectors
distributed exponentially with parameter $\lam .$ For each $y$ and $n\geq 1,$
form a graph $G_n(y)$ with vertex set $V_n = \{X_1,...,X_n\},$ two vertices are
connected if and only if edge distance between them is greater then $y$, i.e,
$\|X_i-X_j\| \leq y.$ Almost-sure asymptotic rates of convergence/divergence
are obtained for the minimum and maximum vertex degree of the random geometric
graph, as the number of vertices becomes large $n,$ and the edge distance
varies with the number of vertices.
http://arXiv.org/abs/math/0609193
http://front.math.ucdavis.edu/math.PR/0609193
(alternate)
Author(s): Andrei Khrennikov
Abstract: We compare the classical Kolmogorov and quantum probability models. We show
that the gap between these model is not so huge as it was commonly believed.
The main structures of quantum theory (interference of probabilities, Born's
rule, complex probabilistic amplitudes, Hilbert state space, representation of
observables by operators) are present in a latent form in the Kolmogorov model.
In particular, we obtain ``interference of probabilities'' without to appeal to
the Hilbert space formalism. We interpret ``interference of probabilities'' as
a perturbation (by a $\cos$-term) of the conventional formula of total
probability. Our classical derivation of quantum probabilistic formalism can
stimulate applications of quantum methods outside of microworld : in
psychology, biology, economy,...
http://arXiv.org/abs/math/0609197
http://front.math.ucdavis.edu/math.PR/0609197
(alternate)
Author(s): Lutz Mattner
Abstract: The approach of Kleitman (1970) and Kanter (1976) to multivariate
concentration function inequalities is generalized in order to obtain for
deviation probabilities of sums of independent symmetric random variables a
lower bound depending only on deviation probabilities of the terms of the sum.
This bound is optimal up to discretization effects, improves on a result of
Nagaev (2001), and complements the comparison theorems of Birnbaum (1948) and
Pruss (1997). Birnbaum's theorem for unimodal random variables is extended to
the lattice case.
http://arXiv.org/abs/math/0609200
http://front.math.ucdavis.edu/math.PR/0609200
(alternate)
Author(s): Philippe Blanchard and Tyll Krueger and Madeleine Sirugue-Collin
Abstract: We study growth properties of the number of paths of lenght k for a variant
of Cameo graphs introduced in an earlier paper. Sharp results are obtained for
threshold for the k-path connectivity and the essential diameter.
http://arXiv.org/abs/math/0609202
http://front.math.ucdavis.edu/math.PR/0609202
(alternate)
Author(s): Jo\~{a}o Amaro de Matos and Rui Dil\~{a}o and Bruno Ferreira
Abstract: In the context of a Black-Scholes economy and with a no-arbitrage argument,
we derive arbitrarily accurate lower and upper bounds for the value of European
options on a stock paying a discrete dividend. Setting the option price error
below the smallest monetary unity, both bounds coincide, and we obtain the
exact value of the option.
http://arXiv.org/abs/math/0609212
http://front.math.ucdavis.edu/math.PR/0609212
(alternate)
Author(s): Sylvie Corteel and Lauren K. Williams
Abstract: The partially asymmetric exclusion process (PASEP) is an important model from
statistical mechanics which describes a system of interacting particles hopping
left and right on a one-dimensional lattice of N sites. It is partially
asymmetric in the sense that the probability of hopping left is q times the
probability of hopping right. Additionally, particles may enter from the left
with probability alpha and exit from the right with probability beta.
It has been observed that the (unique) stationary distribution of the PASEP
has remarkable connections to combinatorics -- see for example the papers of
Derrida, Duchi and Schaeffer, and Corteel. Most recently we proved that in fact
the (normalized) probability of being in a particular state of the PASEP can be
viewed as a certain weight generating function for permutation tableaux of a
fixed shape. (This result implies the previous combinatorial results.) However,
our proof relied on the matrix ansatz of Derrida et al, and hence did not give
an intuitive explanation of why one should expect the steady state distribution
of the PASEP to involve such nice combinatorics.
In this paper we define a Markov chain -- which we call the PT chain -- on
the set of permutation tableaux which projects to the PASEP in a very strong
sense. This gives a new proof of our previous result which bypasses the matrix
ansatz altogether. Furthermore, via the bijection from permutation tableaux to
permutations, the PT chain can also be viewed as a Markov chain on the
symmetric group. Another nice feature of the PT chain is that it possesses a
certain symmetry which extends the "particle-hole symmetry" of the PASEP. More
specifically, this is a graph-automorphism on the state diagram of the PT chain
which is an involution; this has a simple description in terms of permutations.
http://arXiv.org/abs/math/0609188
http://front.math.ucdavis.edu/math.CO/0609188
(alternate)
Author(s): Kevin Ford
Abstract: We give sharp, uniform estimates for the probability that the empirical
distribution function for n uniform-[0,1] random variables stays to one side of
a given line.
http://arXiv.org/abs/math/0609224
http://front.math.ucdavis.edu/math.PR/0609224
(alternate)
Author(s): Greg Markowsky
Abstract: In this paper we will examine the derivative of intersection local time of
Brownian motion and symmetric stable processes in $R^2$. These processes do not
exist when defined in the canonical way. The purpose of this paper is to
exhibit the correct rate for renormaliztion of these processes.
http://arXiv.org/abs/math/0609265
http://front.math.ucdavis.edu/math.PR/0609265
(alternate)
Author(s): Gideon Amir and Christopher Hoffman
Abstract: Benjamini, \olle, Peres and Steif introduced the model of dynamical random
walk on $\Z^d$ \cite{ds}. This is a continuum of random walks indexed by a
parameter $t$. They proved that for $d=3,4$ there almost surely exist $t$ such
that the random walk at time $t$ visits the origin infinitely often, but for $d
\geq 5$ there almost surely do not exist such $t$.
Hoffman showed that for $d=2$ there almost surely exists $t$ such that the
random walk at time $t$ visits the origin only finitely many times \cite{H1}.
We refine the results of \cite{H1} for dynamical random walk on $\z^2$, showing
that with probability one the are times when the origin is visited only a
finite number of times while other points are visited infinitely often.
http://arXiv.org/abs/math/0609267
http://front.math.ucdavis.edu/math.PR/0609267
(alternate)
Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
Abstract: We prove functional limits theorems for the occupation time process of a
system of particles moving independently in $R^d$ according to a symmetric
$\alpha$-stable L\'evy process, and starting off from an inhomogeneous Poisson
point measure with intensity measure
$\mu(dx)=(1+|x|^{\gamma})^{-1}dx,\gamma>0$, and other related measures. In
contrast to the homogeneous case $(\gamma=0)$, the system is not in equilibrium
and ultimately it vanishes, and there are more different types of occupation
time limit processes depending on arrangements of the parameters $\gamma, d$
and $\alpha$. The case $\gamma
http://arXiv.org/abs/math/0609290
http://front.math.ucdavis.edu/math.PR/0609290
(alternate)
Author(s): Ludmila L. Zaitseva
Abstract: The comparison theorem for skew Brownian motions is proved. As the corollary
we get the estimate on ${\Cal L}_1-$distance between two skew Brownian motions
started from different points. Using this result we prove the continuous
dependence on starting point of one class of generalized diffusion processes
constructed as the strong solution to an SDE.
http://arXiv.org/abs/math/0609305
http://front.math.ucdavis.edu/math.PR/0609305
(alternate)
Author(s): Ludmila L. Zaitseva
Abstract: We show the complete proof of the Markov property of the strong solution to a
multidimensional SDE whose coefficients involve local time on a hyperplane of
the unknown process.
http://arXiv.org/abs/math/0609307
http://front.math.ucdavis.edu/math.PR/0609307
(alternate)
Author(s): Ciprian A. Tudor and Frederi G. Viens
Abstract: We apply the techniques of stochastic integration with respect to the
fractional Brownian motion and the theory of regularity and supremum estimation
for stochastic processes to study the maximum likelihood estimator (MLE) for
the drift parameter of stochastic processes satisfying stochastic equations
driven by fractional Brownian motion with any level of Holder-regularity (any
Hurst parameter). We prove existence and strong consistency of the MLE for
linear and nonlinear equations. We also prove that a version of the MLE using
only discrete observations is still a strongly consistent estimator.
http://arXiv.org/abs/math/0609295
http://front.math.ucdavis.edu/math.ST/0609295
(alternate)
Author(s): Ravi Montenegro
Abstract: We show bounds on total variation and $L^{\infty}$ mixing times, spectral gap
and magnitudes of the complex valued eigenvalues of a general (non-reversible
non-lazy) Markov chain with a minor expansion property. This leads to the first
known bounds for the non-lazy simple and max-degree walks on a (directed)
graph, and even in the lazy case they are the first bounds of the optimal
order. In particular, it is found that within a factor of two or four, the
worst case of each of these mixing time and eigenvalue quantities is a walk on
a cycle with clockwise drift.
http://arXiv.org/abs/math/0609303
http://front.math.ucdavis.edu/math.CO/0609303
(alternate)
Author(s): F. Flandoli and M. Romito
Abstract: A 3D stochastic Navier-Stokes equation with a suitable non degenerate
additive noise is considered. The regularity in the initial conditions of every
Markov transition kernel associated to the equation is studied by a simple
direct approach. A by-product of the technique is the equivalence of all
transition probabilities associated to every Markov transition kernel.
http://arXiv.org/abs/math/0609317
http://front.math.ucdavis.edu/math.PR/0609317
(alternate)
Author(s): M. Romito
Abstract: The existence of suitable weak solutions of 3D Navier-Stokes equations,
driven by a random body force, is proved. These solutions satisfy a local
balance of energy. Moreover it is proved also the existence of a statistically
stationary solution.
http://arXiv.org/abs/math/0609318
http://front.math.ucdavis.edu/math.PR/0609318
(alternate)
Author(s): Alexander M.G. Cox and Jan Obloj (PMA)
Abstract: The Skorokhod Embedding problem is well understood when the underlying
process is a Brownian motion. We examine the problem when the underlying is the
simple symmetric random walk and when no external randomisation is allowed. We
prove that any measure on Z can be embedded by means of a minimal stopping
time. However, in sharp contrast to the Brownian setting, we show that the set
of measures which can be embedded in a uniformly integrable way is strictly
smaller then the set of centered probability measures: specifically it is a
fractal set which we characterise as an iterated function system. Finally, we
define the natural extension of several known constructions from the Brownian
setting and show that these constructions require us to further restrict the
sets of target laws.
http://arXiv.org/abs/math/0609330
http://front.math.ucdavis.edu/math.PR/0609330
(alternate)
Author(s): Mathilde Weill (DMA)
Abstract: We prove some asymptotic results for the radius and the profile of large
random bipartite planar maps. Using a bijection due to Bouttier, Di Francesco
and Guitter between rooted bipartite planar maps and certain two-type trees
with positive labels, we derive our results from a conditional limit theorem
for two-type spatial trees. Finally we apply our estimates to separating
vertices of bipartite planar maps: with probability close to one when $n$ goes
to infinity, a random $2\ka$-angulation with $n$ faces has a separating vertex
whose removal disconnects the map into two components each with size greater
that $n^{1/2-\vep}$.
http://arXiv.org/abs/math/0609334
http://front.math.ucdavis.edu/math.PR/0609334
(alternate)
Author(s): Bartlomiej Bartek Blaszczyszyn (INRIA Rocquencourt) and Bozidar Radunovic (INRIA Rocquencourt)
Abstract: We consider a hybrid wireless sensor network with regular and transmit-only
sensors. The transmit-only sensors do not have receiver circuit, hence are
cheaper and less energy consuming, but their transmissions cannot be
coordinated. Regular sensors, also called cluster-heads, are responsible for
receiving information from transmit-only sensors and forwarding it to sinks.
The main goal of such a hybrid network is to reduce the cost of deployment
while achieving some performance constraints (minimum coverage, sensing rate,
etc). In this paper we are interested in the communication between
transmit-only sensors and cluster-heads. We develop a detailed analytical model
of the physical and MAC layer using tools from queuing theory and stochastic
geometry. (The MAC model, that we call Erlang's loss model with interference,
might be of independent interest as adequate for any non-slotted; i.e.,
unsynchronized, wireless communication channel.) We give an explicit formula
for the frequency of successful packet reception by a cluster-head, given
sensors' locations. We further define packet admission policies at a
cluster-head, and we calculate the optimal policies for different performance
criteria. Finally we show that the proposed hybrid network, using the optimal
policies, can achieve substantial cost savings as compared to conventional
architectures.
http://arXiv.org/abs/cs/0609038
http://front.math.ucdavis.edu/cs.NI/0609038
(alternate)
Author(s): A. C. D. van Enter and W. M. Ruszel
Abstract: We present a class of examples of nearest-neighbour, boubded-spin models, in
which the low-temperature Gibbs measures do not converge as the temperature is
lowered to zero, in any dimension.
http://arXiv.org/abs/math-ph/0609024
http://front.math.ucdavis.edu/math-ph/0609024
(alternate)
Author(s): Ery Arias-Castro
Abstract: Let {(Z_i,W_i):i=1,...,n} be uniformly distributed in [0,1]^d * G(k,d), where
G(k,d) denotes the space of k-dimensional linear subspaces of R^d. For a
differentiable function f from [0,1]^k to [0,1]^d we say that f interpolates
(z,w) in [0,1]^d * G(k,d) if there exists x in [0,1]^k such that f(x) = z and
vec{f}(x) = w, where vec{f}(x) denotes the tangent space at x defined by f. For
a smoothness class F of H\"older type, we obtain probability bounds on the
maximum number of points a function f in F interpolates.
http://arXiv.org/abs/math/0609340
http://front.math.ucdavis.edu/math.PR/0609340
(alternate)
Author(s): Yong Liu and Huaizhong Zhao
Abstract: In this paper, we show the existence and uniqueness of the stationary
solution $u(t,\omega)$ and stationary point $Y(\omega)$ of the differentiable
random dynamical system $U:R\times L^2[0,1]\times \Omega\to L^2[0,1]$ generated
by the stochastic Burgers equation with $L^2[0,1]$-noise and large viscosity,
especially, $u(t,\omega)=U(t,Y(\omega),\omega)=Y(\theta(t,\omega))$, and
$Y(\omega) \in H^1[0,1]$ is the unique solution of the following equation in
$L^2[0,1]$
$$
Y(\omega)={1/2}\int_{-\infty}^0T_\nu(-s)\frac{\partial
(Y(\theta(s,\omega))^2}{\partial x}ds +\int_{-\infty}^0T_\nu(-s)dW_s(\omega),
$$ where $\theta$ is the group of $P$-preserving ergodic transformation on
the canonical probability pace $(\Omega, {\cal F}, P)$ such that
$\theta(t,\omega)(s)=W(t+s)-W(t)$.
http://arXiv.org/abs/math/0609344
http://front.math.ucdavis.edu/math.PR/0609344
(alternate)
Author(s): S. Janson and R. Neininger
Abstract: We study a random fragmentation process and its associated random tree. The
process has earlier been studied by Dean and Majumdar (J. Phys. A: Math. Gen.,
vol. 35, L501--L507), who found a phase transition: the number of
fragmentations is asymptotically normal in some cases but not in others,
depending on the position of roots of a certain characteristic equation. This
parallels the behaviour of discrete analogues with various random trees that
have been studied in computer science. We give rigorous proofs of this phase
transition, and add further details.
The proof uses the contraction method. We extend some previous results for
recursive sequences of random variables to families of random variables with a
continuous parameter; we believe that this extension has independent interest.
http://arXiv.org/abs/math/0609350
http://front.math.ucdavis.edu/math.PR/0609350
(alternate)
Author(s): Sylvain Rubenthaler (JAD) and Tobias Ryd\'{e}n (CENTRE for Mathematical Sciences), Magnus Wiktorsson (CENTRE for Mathematical Sciences)
Abstract: Using classical simulated annealing to maximise a function $\psi$ defined on
a subset of $\R^d$, the probability $\p(\psi(\theta\_n)\leq
\psi\_{\max}-\epsilon)$ tends to zero at a logarithmic rate as $n$ increases;
here $\theta\_n$ is the state in the $n$-th stage of the simulated annealing
algorithm and $\psi\_{\max}$ is the maximal value of $\psi$. We propose a
modified scheme for which this probability is of order $n^{-1/3}\log n$, and
hence vanishes at an algebraic rate. To obtain this faster rate, the
exponentially decaying acceptance probability of classical simulated annealing
is replaced by a more heavy-tailed function, and the system is cooled faster.
We also show how the algorithm may be applied to functions that cannot be
computed exactly but only approximated, and give an example of maximising the
log-likelihood function for a state-space model.
http://arXiv.org/abs/math/0609353
http://front.math.ucdavis.edu/math.PR/0609353
(alternate)
Author(s): Greg Anderson and Ofer Zeitouni
Abstract: We consider random hermitian matrices in which distant above-diagonal entries
are independent but nearby entries may be correlated. We find the limit of the
empirical distribution of eigenvalues by combinatorial methods. We also prove
that the limit has algebraic Stieltjes transform by an argument based on
dimension theory of noetherian local rings.
http://arXiv.org/abs/math/0609364
http://front.math.ucdavis.edu/math.PR/0609364
(alternate)
Author(s): Yooyoung Koo (Sungkyunkwan Univ) and Sungchul Lee (Yonsei Univ)
Abstract: Let $L$ be the Euclidean functional with $p$-th power-weighted edges.
Examples include the sum of the $p$-th power-weighted lengths of the edges in
minimal spanning trees, traveling salesman tours, and minimal matchings.
Motivated by the works of Steele, Redmond and Yukich (1994, 1996) have shown
that for $n$ i.i.d. sample points $\{X_1,...,X_n\}$ from $[0,1]^d$,
$L(\{X_1,...,X_n\})/n^{(d-p)/d}$ converges a.s. to a finite constant. Here we
bound the rate of convergence of $EL(\{X_1,...,X_n\})/n^{(d-p)/d}$.
http://arXiv.org/abs/math/0609382
http://front.math.ucdavis.edu/math.PR/0609382
(alternate)
Author(s): Michael Drmota and Svante Janson and Ralph Neininger
Abstract: We study the profile $X_{n,k}$ of random search trees including binary search
trees and $m$-ary search trees. Our main result is a functional limit theorem
of the normalized profile $X_{n,k}/\E X_{n,k}$ for $k = \lfloor \alpha \log
n\rfloor$ in a certain range of $\alpha$.
A central feature of the proof is the use of the contraction method to prove
convergence in distribution of certain random analytic functions in a complex
domain. This is based on a general theorem on the contraction method for random
variables in an infinite dimensional Hilbert space. As part of the proof, we
show that the Zolotarev metric is complete for a Hilbert space.
http://arXiv.org/abs/math/0609385
http://front.math.ucdavis.edu/math.PR/0609385
(alternate)
Author(s): Frank Oertel
Abstract: By investigating in detail discontinuities of the first kind of real-valued
functions and the analysis of unordered sums, where the summands are given by
values of a positive real-valued function, we develop a measure-theoretical
framework which in particular allows us to describe \textit{rigorously} the
representation and meaning of sums of jumps of type $\sum_{0 < s \leq t} \Phi
\circ | \Delta X_s |$, where $X : \Omega \times \R_+ \longrightarrow \R$ is a
stochastic process with regulated trajectories, $t \in \R_+$ and $\Phi : \R_+
\longrightarrow \R_+$ is a strictly increasing function which maps 0 to 0 (cf.
Proposition \ref{prop:sum of jumps on R+ with invertible function}). Moreover,
our approach enables a natural extension of the jump measure of c\`{a}dl\`{a}g
and adapted processes to an integer-valued random measure of optional processes
with regulated trajectories which need not necessarily to be right- or
left-continuous (cf. Theorem \ref{thm:optional random measures}). In doing so,
we provide a detailed and constructive proof of the fact that the set of all
discontinuities of the first kind of a given real-valued function on $\R$ is at
most countable (cf. Lemma \ref{lemma:right limits and left limits}, Theorem
\ref{thm:at most countably many jumps on compact intervals} and Theorem
\ref{thm:at most countably many jumps on R+}).
By using the powerful analysis of unordered sums, we hope that our
contributions fill an existing gap in the literature, since neither a detailed
proof of (the frequently used) Theorem \ref{thm:at most countably many jumps on
compact intervals} nor a precise definition of sums of jumps seems to be
available yet.
http://arXiv.org/abs/math/0609395
http://front.math.ucdavis.edu/math.PR/0609395
(alternate)
Author(s): Frank Oertel and Mark Owen
Abstract: Consider a financial market in which an agent trades with utility-induced
restrictions on wealth. We prove that the utility-based super-replication price
of an unbounded (but sufficiently integrable) contingent claim is equal to the
supremum of its discounted expectations under pricing measures with finite
entropy. Central to our proof is the representation of a cone $C_\V$ of
utility-based super-replicable contingent claims as the polar cone of the set
of finite entropy separating measures. $C_\V$ is shown to be the closure, under
a relevant weak topology, of the cone of all (sufficiently integrable)
contingent claims that can be dominated by a zero-financed terminal wealth. As
our approach shows, those terminal wealths need {\it not} necessarily stem from
{\it admissible} trading strategies only.
We investigate also the natural dual of this result, and show that the polar
cone of $C_\V$ is the cone generated by separating measures with {\it finite
loss-entropy}. For an agent whose utility function is unbounded from above, the
set of pricing measures with finite loss-entropy can be slightly larger than
the set of pricing measures with finite entropy. Indeed, we prove that the
former set is the closure of the latter under a suitable weak topology.
Finally, we show how our framework can be applied to another field of
mathematical economics and how it sheds a different light on Farkas' Lemma and
its infinite dimensional version there.
http://arXiv.org/abs/math/0609402
http://front.math.ucdavis.edu/math.PR/0609402
(alternate)
Author(s): Frank Oertel and Mark Owen
Abstract: Consider a financial market in which an agent trades with utility-induced
restrictions on wealth. For a utility function which satisfies the condition of
reasonable asymptotic elasticity at $-\infty$ we prove that the utility-based
super-replication price of an unbounded (but sufficiently integrable)
contingent claim is equal to the supremum of its discounted expectations under
pricing measures with finite {\it loss-entropy}. For an agent whose utility
function is unbounded from above, the set of pricing measures with finite
loss-entropy can be slightly larger than the set of pricing measures with
finite entropy. Indeed, the former set is the closure of the latter under a
suitable weak topology.
Central to our proof is the representation of a cone $C_U$ of utility-based
super-replicable contingent claims as the polar cone to the set of finite
loss-entropy pricing measures. The cone $C_U$ is defined as the closure, under
a relevant weak topology, of the cone of all (sufficiently integrable)
contingent claims that can be dominated by a zero-financed terminal wealth.
We investigate also the natural dual of this result and show that the polar
cone to $C_U$ is generated by those separating measures with finite
loss-entropy. The full two-sided polarity we achieve between measures and
contingent claims yields an economic justification for the use of the cone
$C_U$, and an open question.
http://arXiv.org/abs/math/0609403
http://front.math.ucdavis.edu/math.PR/0609403
(alternate)
Author(s): Dan Shiber
Abstract: In this paper we develop the geometry of information for a single matrix
random matrix model, with two goals: proving a Cramer-Rao theorem for
estimators on random matrices, and calculating the Legendre transform of
pressure and entropy with respect to a metric duality. In our development we
recover several quantities from free probability: Voiculescu's conjugate
variable is the tangent vector to the GUE pertrubation model, giving rise to a
metric which turns out to be Voiculescu's Free Fisher Information measure;
Hiai's Legendre transform of free pressure agrees with our Legendre transform
of pressure; and Speicher's covariance of fluctuations naturally arises as the
metric on perturbations of the random matrix model. Incidentally, we obtain a
new kind of convexity for the free entropy of the limit of a random matrix
model.
http://arXiv.org/abs/math/0609372
http://front.math.ucdavis.edu/math.OA/0609372
(alternate)
Author(s): Luigi Manca
Abstract: The dynamic programming approach for the control of a 3D flow governed by the
stochastic Navier-Stokes equations for incompressible fluid in a bounded domain
is studied. By a compactness argument, existence of solutions for the
associated Hamilton-Jacobi-Bellman equation is proved. Finally, existence of an
optimal control through the feedback formula and of an optimal state is
discussed.
http://arXiv.org/abs/math/0609389
http://front.math.ucdavis.edu/math.OC/0609389
(alternate)
Author(s): Eric Gautier
Abstract: We consider stochastic nonlinear Schrodinger equations driven by an additive
noise. The noise is fractional in time with Hurst parameter H in (0,1). It is
also colored in space and the space correlation operator is assumed to be
nuclear. We study the local well-posedness of the equation. Under adequate
assumptions on the initial data, the space correlations of the noise and for
some saturated nonlinearities, we prove a sample path large deviations
principle and a support result. These results are stated in a space of
exploding paths which are Holder continuous in time until blow-up. We treat the
case of Kerr nonlinearities when H > 1/2.
http://arXiv.org/abs/math/0609423
http://front.math.ucdavis.edu/math.PR/0609423
(alternate)
Author(s): Arnaud Debussche and Eric Gautier
Abstract: We consider random perturbations of the focusing cubic one dimensional
nonlinear Schrodinger equation. The noises, either additive or multiplicative,
are white in time and colored in space. In the additive case, a white noise
limit is considered. We study the small noise asymptotic of the tails of the
center and mass of a pulse at a fixed coordinate when the initial datum is null
or a soliton profile. Our main tools are large deviation results at the level
of paths. Upper and lower bounds are obtained from bounds for the optimal
control problems derived from the rate function of the large deviation
principles. Our results are in perfect agreement with several results from
physics. These results had been obtained with arguments which seem difficult to
fully justify mathematically. Some results are new.
http://arXiv.org/abs/math/0609424
http://front.math.ucdavis.edu/math.PR/0609424
(alternate)
Author(s): Arnaud Debussche (IRMAR) and Eric Gautier (IRMAR)
Abstract: We consider random perturbations of the focusing cubic one dimensional
nonlinear Schr\"{o}dinger equation. The noises, either additive or
multiplicative, are white in time and colored in space. In the additive case, a
"white noise limit" is considered. We study the small noise asymptotic of the
tails of the center and mass of a pulse at a fixed coordinate when the initial
datum is null or a soliton profile. Our main tools are large deviation results
at the level of paths. Upper and lower bounds are obtained from bounds for the
optimal control problems derived from the rate function of the large deviation
principles. Our results are in perfect agreement with several results from
physics.These results had been obtained with arguments which seem difficult to
fully justify mathematically. Some results are new.
http://arXiv.org/abs/math/0609434
http://front.math.ucdavis.edu/math.PR/0609434
(alternate)
Author(s): Christian Berg and Henrik L. Pedersen
Abstract: We show that the median $m(x)$ in the gamma distribution with parameter $x$
is a strictly convex function on the positive half-line
http://arXiv.org/abs/math/0609442
http://front.math.ucdavis.edu/math.PR/0609442
(alternate)
Author(s): P. Chigansky and R. Liptser
Abstract: Let $\xi(u)$, $u\in \Real$ be an ergodic stationary Markov chain, taking a
finite number of values, and consider the diffusion process generated by the
SDE $$ dX^\eps_t = b(X^\eps_t)dt +\eps^\kappa\xi\big(X^\eps_t/\eps\big)dB_t $$
with a small positive scaling parameter $\eps$, where $B=(B_t)_{t\in\Real_+}$
is a Brownian motion, independent of $\xi$, and $\kappa\ge 0$ is a fixed
constant. Such model describes evolution of a particle, perturbed by a small
white noise disturbance, whose intensity is switched by the random environment
$\xi$. We show that for $\kappa\in (0,1/6)$, the process $X^\eps$ satisfies the
same Large Deviations Principle (LDP) of the Freidlin-Wentzell type as the
process $\hat{X}^\epsilon$: $$ dX^\eps_t = b(\hat{X}^\eps_t)dt +
\eps^\kappa\sqrt{\mathbf{a}}dB_t, $$ with $\mathbf{a}=\dfrac{1}{\E
\xi^{-2}(0)}$. For $\kappa=0$, $X^\epsilon$ converges weakly to the the
solution of the SDE $dX_t=b(X_t)dt+\sqrt{\mathbf{a}}dB_t.$
http://arXiv.org/abs/math/0609443
http://front.math.ucdavis.edu/math.PR/0609443
(alternate)
Author(s): Markus Mueller
Abstract: Kolmogorov complexity and algorithmic probability quantify the randomness and
universal a priori probability of finite binary strings. Nevertheless, they
share the disadvantage of depending on the choice of the universal computer
which is used as a reference computer to count the program lengths. In this
paper, we propose an approach to algorithmic probability that tries to
eliminate this machine-dependence.
Elaborating on the idea that computers with ``atypical'' algorithmic
probability should be hard to emulate, we define the notion of emulation
complexity. This naturally leads to a Markov process of universal computers
that randomly emulate each other, yielding stationary probability distributions
on the computers and finite binary strings.
By proving symmetry relations with respect to input and output
transformations, we show that properties of individual computers are
successfully eliminated. Our approach is not limited to prefix-free computers,
but can be applied to more general sets of computers. The question for what
computer sets such stationary distributions exist remains open in general, but
is answered in some special cases and is shown to be closely related to the
aforementioned symmetry relations.
http://arXiv.org/abs/cs/0608095
http://front.math.ucdavis.edu/cs.IT/0608095
(alternate)
Author(s): Rodrigo Ba\~nuelos and Krzysztof Bogdan
Abstract: We study Fourier multipliers which result from modulating jumps of L\'evy
processes. Using the theory of martingale transforms we prove that these
operators are bounded in $L^p(\Rd)$ for $1
http://arXiv.org/abs/math/0609432
http://front.math.ucdavis.edu/math.FA/0609432
(alternate)
Author(s): Pablo A. Ferrari and Valentin V. Sisko
Abstract: We consider zero-range processes in Z^d with site dependent jump rates. The
rate for a particle jump from site x to y in Z^d is given by \lambda_x g(k)
p(y-x), where p(\cdot) is a probability in Z^d, g(k) is a bounded nondecreasing
function of the number k of particles in x and \lambda = \{\lambda_x\} is a
collection of i.i.d. random variables with values in (c,1], for some c>0. For
almost every realization of the environment \lambda the zero-range process has
product invariant measures \{\nu_{\lambda,v}: 0\le v \le c\} parametrized by v,
the average total jump rate from any given site. The density of a measure,
defined by the asymptotic average number of particles per site, is an
increasing function of v. There exists a product invariant measure
\nu_{\lambda,c}, with maximal density. Let \mu be a probability measure
concentrating mass on configurations whose number of particles at site x grows
less than exponentially with \|x\|. Denoting by S_{\lambda}(t) the semigroup of
the process, we prove that all weak limits of \{\mu S_{\lambda}(t), t\ge 0 \}
as t \to \infty are dominated, in the natural partial order, by
\nu_{\lambda,c}. In particular, if \mu dominates \nu_{\lambda,c}, then \mu
S_{\lambda}(t) converges to \nu_{\lambda,c}. The result is particularly
striking when the maximal density is finite and the initial measure has a
density above the maximal.
http://arXiv.org/abs/math/0609469
http://front.math.ucdavis.edu/math.PR/0609469
(alternate)
Author(s): L. F. James and A. Lijoi and I. Pruenster
Abstract: The present paper provides exact expression for the probability distribution
of linear functionals of the two--parameter Poisson-Dirichlet process
PD$(\alpha,\theta)$. Distributional results that follow from the application of
an inversion formula for a (generalized) Cauchy--Stieltjes transform are
achieved. Moreover, several interesting integral identities are obtained by
exploiting a correspondence between the mean functional of a Poisson--Dirichlet
process and the mean functional of a suitable Dirichlet process. Finally, some
distributional characterizations in terms of mixture representations are
illustrated. Our formulae are relevant to occupation time phenomena connected
with Brownian motion and more general Bessel processes, as well as to models
arising in Bayesian nonparametric statistics.
http://arXiv.org/abs/math/0609488
http://front.math.ucdavis.edu/math.PR/0609488
(alternate)
Author(s): L. V. Bogachev and A. V. Gnedin and Yu.V. Yakubovich
Abstract: We consider the occupancy problem where balls are thrown independently at
infinitely many boxes with fixed positive frequencies. It is well known that
the random number of boxes occupied by the first n balls is asymptotically
normal if its variance V_n tends to infinity. In this work, we mainly focus on
the opposite case where V_n is bounded, and derive a simple necessary and
sufficient condition for convergence of V_n to a finite limit, thus settling a
long-standing question raised by Karlin in the seminal paper of 1967. One
striking consequence of our result is that the possible limit may only be a
positive integer number. Some new conditions for other types of behavior of the
variance, like boundedness or convergence to infinity, are also obtained. The
proofs are based on the poissonization techniques.
http://arXiv.org/abs/math/0609498
http://front.math.ucdavis.edu/math.PR/0609498
(alternate)
Author(s): Alexander G. Tartakovsky
Abstract: In 1960s Shiryaev developed Bayesian theory of change detection in
independent and identically distributed (i.i.d.) sequences. In Shiryaev's
classical setting the goal is to minimize an average detection delay under the
constraint imposed on the average probability of false alarm. Recently,
Tartakovsky and Veeravalli (2005) developed a general Bayesian asymptotic
change-point detection theory (in the classical setting) that is not limited to
a restrictive i.i.d. assumption. It was proved that Shiryaev's detection
procedure is asymptotically optimal under traditional average false alarm
probability constraint, assuming that this probability is small. In the present
paper, we consider a less conventional approach where the constraint is imposed
on the global, supremum false alarm probability. An asymptotically optimal
Bayesian change detection procedure is proposed and thoroughly evaluated for
both i.i.d. and non-i.i.d. models when the global false alarm probability
approaches zero.
http://arXiv.org/abs/math/0609467
http://front.math.ucdavis.edu/math.ST/0609467
(alternate)
Author(s): Y. Kozitsky and T. Pasurek
Abstract: A rigorous description of the equilibrium thermodynamic properties of an
infinite system of interacting $\nu$-dimensional quantum anharmonic oscillators
is given. The oscillators are indexed by the elements of a countable set
$\mathbb{L}\subset \mathbb{R}^d$, possibly irregular; the anharmonic potentials
vary from site to site. The description is based on the representation of the
Gibbs states in terms of path measures -- the so called Euclidean Gibbs
measures. It is proven that: (a) the set of such measures $\mathcal{G}^{\rm t}$
is non-void and compact; (b) every $\mu \in \mathcal{G}^{\rm t}$ obeys an
exponential integrability estimate, the same for the whole set
$\mathcal{G}^{\rm t}$; (c) every $\mu \in \mathcal{G}^{\rm t}$ has a
Lebowitz-Presutti type support; (d) $\mathcal{G}^{\rm t}$ is a singleton at
high temperatures. In the case of attractive interaction and $\nu=1$ we prove
that $|\mathcal{G}^{\rm t}|>1$ at low temperatures. The uniqueness of Gibbs
measures due to quantum effects and at a nonzero external field are also proven
in this case. Thereby, a qualitative theory of phase transitions and quantum
effects, which interprets most important experimental data known for the
corresponding physical objects, is developed. The mathematical result of the
paper is a complete description of the set $\mathcal{G}^{\rm t}$, which refines
and extends the results known for models of this type.
http://arXiv.org/abs/math-ph/0609045
http://front.math.ucdavis.edu/math-ph/0609045
(alternate)
Author(s): Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS)
Abstract: We construct a continuous state branching process with immigration (CBI)
whose immigration depends on the CBI itself and we recover a continuous state
branching process (CB). This provides a dual construction of the pruning at
nodes of CB introduced by the authors in a previous paper. This construction is
a natural way to model neutral mutation. Using exponential formula, we compute
the probability of extinction of the original type population in a critical or
sub-critical quadratic branching, conditionally on the non extinction of the
total population.
http://arXiv.org/abs/math/0609518
http://front.math.ucdavis.edu/math.PR/0609518
(alternate)
Author(s): Mark Braverman and Omid Etesami and Elchanan Mossel
Abstract: In this paper we study a game called {\em mafia} in which different players
have different types of information, communication, and functionality. The
players communicate and function in a way that resembles some real life
situations. We consider two types of operations. First, there are operations
that follow an open democratic discussion. Second, some subgroups of the
players who may have different interests, make decisions based on their own
group interest. A key ingredient here is that the identity of each subgroup is
known only to the members of that group.
http://arXiv.org/abs/math/0609534
http://front.math.ucdavis.edu/math.PR/0609534
(alternate)
Author(s): Jean B. Lasserre
Abstract: We specialize Schm\"udgen's Positivstellensatz and its Putinar and Jacobi and
Prestel refinement, to the case of a polynomial $f\in R[X,Y]+R[Y,Z]$, positive
on a compact basic semi algebraic set $K$ described by polynomials in $R[X,Y]$
and $R[Y,Z]$ only, or in $R[X]$ and $R[Y,Z]$ only (i.e. $K$ is a cartesian
product). In particular, we show that the preordering $P(g,h)$ (resp. quadratic
module $Q(g,h)$) generated by the polynomials $\{g_j\}\subset R[X,Y]$ and
$\{h_k\}\subset R[Y,Z]$ that describe $K$, is replaced with $P(g)+P(h)$ (resp.
$Q(g)+Q(h)$), so that the absence of coupling between $X$ and $Z$ is also
preserved in the representation. A similar result applies with Krivine's
Positivstellensatz involving the cone generated by $\{g_j,h_k\}$.
http://arXiv.org/abs/math/0609529
http://front.math.ucdavis.edu/math.AC/0609529
(alternate)
Author(s): Omer Angel and Alexander E. Holroyd and Dan Romik and Balint Virag
Abstract: A sorting network is a shortest path from 12...n to n...21 in the Cayley
graph of S_n generated by nearest-neighbour swaps. We prove that for a uniform
random sorting network, as n->infinity the space-time process of swaps
converges to the product of semicircle law and Lebesgue measure. We conjecture
that the trajectories of individual particles converge to random sine curves,
while the permutation matrix at half-time converges to the projected surface
measure of the 2-sphere. We prove that, in the limit, the trajectories are
Holder-1/2 continuous, while the support of the permutation matrix lies within
a certain octagon. A key tool is a connection with random Young tableaux.
http://arXiv.org/abs/math/0609538
http://front.math.ucdavis.edu/math.PR/0609538
(alternate)
Author(s): Amir Dembo and Alice Guionnet and Christian Mazza
Abstract: We analyze the coupled non-linear integro-differential equations whose
solutions is the thermodynamical limit of the empirical correlation and
response functions in the Langevin dynamics for spherical p-spin disordered
mean-field models. We provide a mathematically rigorous derivation of their FDT
solution (for the high temperature regime) and of certain key properties of
this solution, which are in agreement with earlier derivations based on
physical grounds.
http://arXiv.org/abs/math/0609546
http://front.math.ucdavis.edu/math.PR/0609546
(alternate)
Author(s): David Aldous and Charles Bordenave and Marc Lelarge
Abstract: We study the relation between the minimal spanning tree (MST) on many random
points and the "near-minimal" tree which is optimal subject to the constraint
that a proportion $\delta$ of its edges must be different from those of the
MST. Heuristics suggest that, regardless of details of the probability model,
the ratio of lengths should scale as $1 + \Theta(\delta^2)$. We prove this
scaling result in the model of the lattice with random edge-lengths. In the
2-dimensional Euclidean model, by exploiting the well-known connection between
MSTs and continuum percolation we can prove the scaling result up to an Ansatz
that a known technical result for lattice percolation extends to continuum
percolation.
http://arXiv.org/abs/math/0609547
http://front.math.ucdavis.edu/math.PR/0609547
(alternate)
Author(s): Robert Burton and Yevgeniy Kovchegov
Abstract: We provide a coupling proof that the transposition shuffle on a deck of n
cards is mixing of rate $n\log(n)$ with a moderate constant. This has already
been shown by Diaconis and Shahshahani but no natural coupling proof has been
demonstrated to date. We also enlarge the methodology of coupling to include
intuitive but nonadapted coupling rules, for example, to take in account future
events and to prepare for their occurrence.
http://arXiv.org/abs/math/0609568
http://front.math.ucdavis.edu/math.PR/0609568
(alternate)
Author(s): Guangyu Yang and Yu Miao
Abstract: In this paper, we prove Strassen's strong invariance principle for a
vector-valued additive functionals of a Markov chain via the martingale
argument and the theory of fractional coboundaries. The hypothesis is a moment
bound on the resolvent.
http://arXiv.org/abs/math/0609593
http://front.math.ucdavis.edu/math.PR/0609593
(alternate)
Author(s): Tatyana S. Turova and Thomas Vallier
Abstract: We study a random graph model which combines properties of the edge
percolation model on Z^d and a classical random graph G(n,c/n). We show that
this model, being a homogeneous random graph, has a natural relation to the
so-called "rank 1 case" of inhomogeneous random graphs. This allows us to use
the newly developed theory of inhomogeneous random graphs to describe
completely the phase diagram in the case d=1. The phase transition is similar
to the classical random graph, it is of the second order. We also find the
scaled size of the largest connected component above the phase transition.
http://arXiv.org/abs/math/0609594
http://front.math.ucdavis.edu/math.PR/0609594
(alternate)
Author(s): C. Bahadoran and H. Guiol and K. Ravishankar and E. Saada
Abstract: We consider attractive irreducible conservative particle systems on
$\mathbb{Z}$, without necessarily nearest-neighbor jumps or explicit invariant
measures. We prove that for such systems, the hydrodynamic limit under Euler
time scaling exists and is given by the entropy solution to some scalar
conservation law with Lipschitz-continuous flux. Our approach is a
generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1--30],
from which we relax the assumption that the process has explicit invariant
measures.
http://arXiv.org/abs/math/0609605
http://front.math.ucdavis.edu/math.PR/0609605
(alternate)
Author(s): Gideon Amir and Ori Gurel-Gurevich
Abstract: Consider the Cayley graph of the cyclic group of prime order q with k
uniformly chosen generators. For k fixed, we prove that the diameter of said
graph is asymptotically (in q) of order q^(1/k).
http://arXiv.org/abs/math/0609620
http://front.math.ucdavis.edu/math.PR/0609620
(alternate)
Author(s): Paul Fendley
Abstract: Loop models have been widely studied in physics and mathematics, in problems
ranging from polymers to topological quantum computation to Schramm-Loewner
evolution. I present new loop models which have critical points described by
conformal field theories. Examples include both fully-packed and dilute loop
models with critical points described by the superconformal minimal models and
the SU(2)_2 WZW models. The dilute loop models are generalized to include
SU(2)_k models as well.
http://arXiv.org/abs/cond-mat/0609435
http://front.math.ucdavis.edu/cond-mat/0609435
(alternate)
Author(s): Siu-Ah Ng
Abstract: The classical notion of a Levy process is generalized to one that takes
values in an arbitrary model of a first order language. This is achieved by
defining a convolution product and the infinite divisibility with respect to
it.
http://arXiv.org/abs/math/0609608
http://front.math.ucdavis.edu/math.LO/0609608
(alternate)
Author(s): Rafal Kulik
Abstract: In this paper we characterize the limiting behavior of sums of extreme values
of long range dependent sequences defined as functionals of linear processes
with finite variance. The extremal sums behave completely different by compared
to the i.i.d case. In particular, though we still have asymptotic normality,
the scaling factor is relatively bigger than in the i.i.d case, meaning that
the maximal terms have relatively smaller contribution to the whole sum. Also,
the scaling need not depend on the tail index of the underlying marginal
distribution, as it is well-known to be so in the i.i.d. situation.
Furthermore, subordination may completely change the asymptotic properties of
sums of extremes.
http://arXiv.org/abs/math/0609625
http://front.math.ucdavis.edu/math.PR/0609625
(alternate)
Author(s): L\'{e}onard Gallardo and Marc Yor
Abstract: Dunkl processes are martingales as well as c\`{a}dl\`{a}g homogeneous Markov
processes taking values in $\mathbb{R}^d$ and they are naturally associated
with a root system. In this paper we study the jumps of these processes, we
describe precisely their martingale decompositions into continuous and purely
discontinuous parts and we obtain a Wiener chaos decomposition of the
corresponding $L^2$ spaces of these processes in terms of adequate mixed
multiple stochastic integrals.
http://arXiv.org/abs/math/0609679
http://front.math.ucdavis.edu/math.PR/0609679
(alternate)
Author(s): Marie F. Kratz and Jos\'{e} R. Le\'{o}n
Abstract: Cram\'{e}r and Leadbetter introduced in 1967 the sufficient condition
\[\frac{r''(s)-r''(0)}{s}\in L^1([0,\delta],dx),\qquad \delta>0,\] to have a
finite variance of the number of zeros of a centered stationary Gaussian
process with twice differentiable covariance function $r$. This condition is
known as the Geman condition, since Geman proved in 1972 that it was also a
necessary condition. Up to now no such criterion was known for counts of
crossings of a level other than the mean. This paper shows that the Geman
condition is still sufficient and necessary to have a finite variance of the
number of any fixed level crossings. For the generalization to the number of a
curve crossings, a condition on the curve has to be added to the Geman
condition.
http://arXiv.org/abs/math/0609682
http://front.math.ucdavis.edu/math.PR/0609682
(alternate)
Author(s): Serguei Dachian and Boris Nahapetian (IMNASA)
Abstract: The problem of characterization of Gibbs random fields is considered. Various
Gibbsianness criteria are obtained using the earlier developed one-point
framework which in particular allows to describe random fields by means of
either one-point conditional or one-point finite-conditional distributions. The
main outcome are the criteria in terms of one-point finite-conditional
distribution, on the basis of which a simple and comprehensible definition of
Gibbs random field is given.
http://arXiv.org/abs/math/0609688
http://front.math.ucdavis.edu/math.PR/0609688
(alternate)
Author(s): Martynas Manstavi\v{c}ius
Abstract: Khoshnevisan and Xiao showed in [Ann. Probab. 33 (2005) 841--878] that the
statement about almost surely vanishing Bessel--Riesz capacity of the image of
a Borel set $G\subset\mathbb{R}_+$ under a symmetric L\'{e}vy process $X$ in
$\mathbb{R}^d$ is equivalent to the vanishing of a deterministic $f$-capacity
for a particular function $f$ defined in terms of the characteristic exponent
of $X$. The authors conjectured that a similar statement is true for all
L\'{e}vy processes in $\mathbb{R}^d$. We show that the conjecture is true
provided we extend the definition of $f$ and require certain integrability
conditions which cannot be avoided in general.
http://arXiv.org/abs/math/0609696
http://front.math.ucdavis.edu/math.PR/0609696
(alternate)
Author(s): Blandine Berard Bergery (IECN) and Pierre Vallois (IECN)
Abstract: We give some approximations of the local time process $(L\_t^x)\_{t\geqslant
0}$ at level $x$ of the real Brownian motion $(X\_t)$. We prove that $
\frac{1}{\epsilon}\int\_0^t (\indi\_{\{x
http://arXiv.org/abs/math/0609701
http://front.math.ucdavis.edu/math.PR/0609701
(alternate)
Author(s): Jonathan C. Mattingly and Toufic Suidan and Eric Vanden-Eijnden
Abstract: We analyze a class of linear shell models subject to stochastic forcing in
finitely many degrees of freedom. The unforced systems considered formally
conserve energy. Despite being formally conservative, we show that these
dynamical systems support dissipative solutions (suitably defined) and, as a
result, may admit unique (statistical) steady states when the forcing term is
nonzero. This claim is demonstrated via the complete characterization of the
solutions of the system above for specific choices of the coupling
coefficients. The mechanism of anomalous dissipations is shown to arise via a
cascade of the energy towards the modes ($a_n$) with higher $n$; this is
responsible for solutions with interesting energy spectra, namely $\EE |a_n|^2$
scales as $n^{-\alpha}$ as $n\to\infty$. Here the exponents $\alpha$ depend on
the coupling coefficients $c_n$ and $\EE$ denotes expectation with respect to
the equilibrium measure. This is reminiscent of the conjectured properties of
the solutions of the Navier-Stokes equations in the inviscid limit and their
accepted relationship with fully developed turbulence. Hence, these simple
models illustrate some of the heuristic ideas that have been advanced to
characterize turbulence, similar in that respect to the random passive scalar
or random Burgers equation, but even simpler and fully solvable.
http://arXiv.org/abs/math-ph/0607047
http://front.math.ucdavis.edu/math-ph/0607047
(alternate)
Author(s): Haifeng Qian and Sachin S. Sapatnekar
Abstract: This paper presents a new stochastic preconditioning approach. For symmetric
diagonally-dominant M-matrices, we prove that an incomplete LDL factorization
can be obtained from random walks, and used as a preconditioner for an
iterative solver, e.g., conjugate gradient. It is argued that our factor
matrices have better quality, i.e., better accuracy-size tradeoffs, than
preconditioners produced by existing incomplete factorization methods.
Therefore the resulting preconditioned conjugate gradient (PCG) method requires
less computation than traditional PCG methods to solve a set of linear
equations with the same error tolerance, and the advantage increases for larger
and denser sets of linear equations. These claims are verified by numerical
tests, and we provide techniques that can potentially extend the theory to more
general types of matrices.
http://arXiv.org/abs/math/0609672
http://front.math.ucdavis.edu/math.NA/0609672
(alternate)
Author(s): Marzio Cassandro and Enza Orlandi and Pierre Picco
Abstract: Estimate (3.39) which appears in the proof of Proposition 3.4 in [Ann.
Probab. 27 (1999) 1414--1467, doi:10.1214/aop/1022677454] is wrong. We present
below a corrected proof which introduces an extra factor 2 in equations (3.34)
and (3.35). This has no consequence in the rest of the paper since Proposition
3.4 is used to estimate only ratios; see (3.23) and (3.25).
http://arXiv.org/abs/math/0609719
http://front.math.ucdavis.edu/math.PR/0609719
(alternate)
Author(s): Lo\"ic Herv\'e (IRMAR)
Abstract: Let $Q$ be a transition probability on a measurable space $E$, let
$(X\_n)\_n$ be a Markov chain associated to $Q$, and let $\xi$ be a real-valued
measurable function on $E$, and $S\_n = \sum\_{k=1}^{n} \xi(X\_k)$. Under
functional hypotheses on the action of $Q$ and its Fourier kernels $Q(t)$, we
investigate the rate of convergence in the central limit theorem for the
sequence $(\frac{S\_n}{\sqrt n})\_n$. According to the hypotheses, we prove
that the rate is, either $O(n^{-\frac{\tau}{2}})$ for all $\tau<1$, or
$O(n^{-{1/2}})$. We apply the spectral method of Nagaev which is improved by
using a perturbation theorem of Keller and Liverani and a method of martingale
difference reduction. When $E$ is not compact or $\xi$ is not bounded, the
conditions required here are weaker than the ones usually imposed when the
standard perturbation theorem is used. For example, in the case of
$V$-geometric ergodic chains or Lipschitz iterative models, the rate of
convergence in the c.l.t is $O(n^{-{1/2}})$ under a third moment condition on
$\xi$.
http://arXiv.org/abs/math/0609720
http://front.math.ucdavis.edu/math.PR/0609720
(alternate)
Author(s): Michel Benaim and Raphael Rossignol
Abstract: We provide a new exponential concentration inequality for
First Passage Percolation valid for a wide class of edge times distributions.
This improves and extends a result by Benjamini, Kalai and Schramm which gave a
variance bound for Bernoulli edge times. Our approach is based on some
functional inequalities extending the work of Rossignol and Falik and
Samorodnitsky.
http://arXiv.org/abs/math/0609730
http://front.math.ucdavis.edu/math.PR/0609730
(alternate)
Author(s): G.Oshanin (1) and H.S.Wio (2) and K.Lindenberg (3) and S.F.Burlatsky (4)((1) LPTMC, Universite Paris 6, France; (2) Instituto de Fisica de Cantabria,
Santander, Spain; (3) Department of Chemistry and Biochemistry, University of
California at San Diego, USA; (4) United Technologies Research Center, UT
Corp, USA)
Abstract: We study the search kinetics of an immobile target by a concentration of
randomly moving searchers. The object of the study is to optimize the
probability of detection within the constraints of our model. The target is
hidden on a one-dimensional lattice in the sense that searchers have no a
priori information about where it is, and may detect it only upon encounter.
The searchers perform random walks in discrete time n=0,1,2, ..., N, where N is
the maximal time the search process is allowed to run. With probability \alpha
the searchers step on a nearest-neighbour, and with probability (1-\alpha) they
leave the lattice and stay off until they land back on the lattice at a fixed
distance L away from the departure point. The random walk is thus intermittent.
We calculate the probability P_N that the target remains undetected up to the
maximal search time N, and seek to minimize this probability. We find that P_N
is a non-monotonic function of \alpha, and show that there is an optimal choice
\alpha_{opt}(N) of \alpha well within the intermittent regime, 0 <
\alpha_{opt}(N) < 1, whereby P_N can be orders of magnitude smaller compared to
the "pure" random walk cases \alpha =0 and \alpha = 1.
http://arXiv.org/abs/cond-mat/0609641
http://front.math.ucdavis.edu/cond-mat/0609641
(alternate)
Author(s): Myl\`ene Ma\"{\i}da
Abstract: We establish a large deviation principle for the largest eigenvalue of a rank
one deformation of a matrix from the GUE or GOE. As a corollary, we get another
proof of the phenomenon, well-known in learning theory and finance, that the
largest eigenvalue separates from the bulk if the perturbation is large enough.
A large part of the paper is devoted to an auxiliary result on the continuity
of spherical integrals, in the case when one of the matrix is of rank one, as
studied in a previous work.
http://arXiv.org/abs/math/0609738
http://front.math.ucdavis.edu/math.PR/0609738
(alternate)
Author(s): Sergio De Carvalho Bezerra (IECN) and Samy Tindel (IECN)
Abstract: In this note, we consider a SK (Sherrington--Kirkpatrick)-type model on Z^d
for d greater or equal to 1, weighted by a function allowing to any single spin
to interact with a small proportion of the other ones. In the thermodynamical
limit, we investigate the equivalence of this model with the usual SK spin
system, through the study of the fluctuations of the free energy.
http://arXiv.org/abs/math/0609754
http://front.math.ucdavis.edu/math.PR/0609754
(alternate)
Author(s): Evgeny Baklanov (Novosibirsk State University)
Abstract: The strong law of large numbers for linear combinations of functions of order
statistics ($L$-statistics) based on weakly dependent random variables is
proven. We also establish the Glivenko--Cantelli theorem for $\phi$-mixing
sequences of identically distributed random variables.
http://arXiv.org/abs/math/0609758
http://front.math.ucdavis.edu/math.PR/0609758
(alternate)
Author(s): Markus Flury
Abstract: We establish large deviation principles and phase transition results for both
quenched and annealed settings of nearest-neighbor random walks with constant
drift in random nonnegative potentials on $\mathbb Z^d$. We complement the
analysis of \cite{Zer}, where a shape theorem on the Lyapunov functions and a
large deviation principle in absence of the drift are achieved for the quenched
setting.
http://arXiv.org/abs/math/0609766
http://front.math.ucdavis.edu/math.PR/0609766
(alternate)
Author(s): Moshe Pollak and Alexander G. Tartakovsky
Abstract: We consider the first exit time of a nonnegative Harris-recurrent Markov
process from the interval $[0,A]$ as $A\to\infty$. We provide a method of proof
of asymptotic exponentiality of the first exit time (suitably standardized)
that does not rely on embedding a regeneration process. We provide examples for
which regeneration embedding fails to yield a proof, whereas our method
succeeds. We show that under certain conditions the moment generating function
of a suitably standardized version of the first exit time converges to that of
$\Exp(1)$. The results are applied to the evaluation of a distribution of run
length to false alarm in change-point detection problems.
http://arXiv.org/abs/math/0609780
http://front.math.ucdavis.edu/math.PR/0609780
(alternate)
Author(s): Andreas Greven and Peter Pfaffelhuber and Anita Winter
Abstract: We consider the space of complete and separable metric spaces which are
equipped with a probability measure. A notion of convergence is given based on
the philosophy that a sequence of metric measure spaces converges if and only
if all finite subspaces sampled from these spaces converge. This topology is
metrized following Gromov's idea of embedding two metric spaces isometrically
into a common metric space combined with the Prohorov metric between
probability measures on a fixed metric space. We show that for this topology
convergence in distribution follows - provided the sequence is tight - from
convergence of all randomly sampled finite subspaces. We give a
characterization of tightness based on quantities which are reasonably easy to
calculate. Subspaces of particular interest are the space of real trees and of
ultra-metric spaces equipped with a probability measure. As an example we
characterize convergence in distribution for the (ultra-)metric measure spaces
given by the random genealogies of the $\Lambda$-coalescents. We show that the
$\Lambda$-coalescent defines an infinite (random) metric measure space if and
only if the so-called ``dust-free''-property holds.
http://arXiv.org/abs/math/0609801
http://front.math.ucdavis.edu/math.PR/0609801
(alternate)
Author(s): Alexei Borodin and Grigori Olshanski
Abstract: The paper deals with a 3-parameter family of probability measures on the set
of partitions, called the z-measures. The z-measures first emerged in
connection with the problem of harmonic analysis on the infinite symmetric
group. They are a special and distinguished case of Okounkov's Schur measures.
It is known that any Schur measure determines a determinantal point process on
the 1-dimensional lattice. In the particular case of z-measures, the
correlation kernel of this process, called the discrete hypergeometric kernel,
has especially nice properties. The aim of the paper is to derive the discrete
hypergeometric kernel by a new method, based on a relationship between the
z-measures and the Meixner orthogonal polynomial ensemble. The present paper
can be viewed as an introduction to another our paper where the same approach
is applied to studying a dynamical model related to the z-measures (Markov
processes on partitions, Prob. Theory Rel. Fields 135 (2006), 84-152; arXiv:
math-ph/0409075).
http://arXiv.org/abs/math/0609806
http://front.math.ucdavis.edu/math.PR/0609806
(alternate)
Author(s): Nicolas Petrelis
Abstract: We consider a simple random walk of length N denoted by $(S_{i})_{i\in
\{1,...,N\}}$, and we define independently a double sequence
$(\gamma^{j}_{i})_{i\geq 1,j\geq 1}$ of i.i.d. random variables and
$(w_i)_{i\geq 1}$ a sequence of centered i.i.d. random variables. We set
$\beta\geq 0$, $\lambda\geq 0$, $h\geq 0$ and $K \in \mathbb{N}$ and transform
the measure of each random trajectory with the Hamiltonian $\lambda
\sum_{i=1}^{N} (w_i+h) \sign(S_i)+\beta \sum_{j=-K}^{K}\sum_{i=1}^{N}
\gamma_{i}^{j} \boldsymbol{1}_{\{S_{i}=j\}}$. This new path measure describes
an hydrophobic homopolymer interacting with a layer of width $2K$ around an
interface between oil and water.
In this article we prove the convergence at weak coupling (namely when $h$
and $\beta$ go to 0) of this discrete model towards its continuous counterpart.
To that aim we develop a technique of coarse graining introduced by Bolthausen
and den Hollander in \cite{BDH}. This result shows in particular that the
randomness of the pinning around the interface vanishes as the coupling becomes
weaker.
We also introduce a new model of polymer interacting with infinitely many
horizontal interfaces located at heights $(P_k)_{k\in\mathbb{Z}}$ through the
Hamiltoninan $\beta\sum_{i=1}^{N}\sum_{j\in\mathbb{Z}}\gamma_i^j\
\ind_{\{S_i=P_k\}}$ and we extend the former convergence result to a particular
case of this model, namely when the widths between successive interfaces are
equal.
http://arXiv.org/abs/math/0609814
http://front.math.ucdavis.edu/math.PR/0609814
(alternate)
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