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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)
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