Probability Abstracts 80
This document contains abstracts 2506-2615.
They have been mailed on April 30, 2004.
2506. ON THE CENTRAL AND LOCAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE
SEQUENCES
Mohamed El Machkouri, Dalibor Volny
Let $(\Omega, \A, \mu)$ be a Lebesgue space and $T$ an ergodic measure
preserving automorphism on $\Omega$ with positive entropy. We show that there
is a bounded and strictly stationary martingale difference sequence defined on
$\Omega$ with a common non-degenerate lattice distribution satisfying the
central limit theorem with an arbitrarily slow rate of convergence and not
satisfying the local limit theorem. A similar result is established for
martingale difference sequences with densities provided the entropy is
infinite. In addition, the martingale difference sequence may be chosen to be
strongly mixing.
2507. CONDITIONAL MOMENTS OF Q-MEIXNER PROCESSES
Wlodzimierz Bryc and Jacek Wesolowski
We show that stochastic processes with linear conditional expectations and
quadratic conditional variances are Markov, and their transition probabilities
are related to a three-parameter family of orthogonal polynomials which
generalize the Meixner polynomials. Special cases of these processes are known
to arise from the non-commutative generalizations of the Levy processes.
2508. PHASE TRANSITION OF THE LARGEST EIGENVALUE FOR NON-NULL COMPLEX SAMPLE
COVARIANCE MATRICES
Jinho Baik, Gerard Ben Arous and Sandrine Peche
We compute the limiting distributions of the largest eigenvalue of a complex
Gaussian sample covariance matrix when both the number of samples and the
number of variables of each sample become large. When all but finitely many,
say $r$, eigenvalues of the covariance matrix are equal to 1, the dependence of
the limiting distribution of the largest eigenvalue of the sample covariance
matrix on those distinguished $r$ eigenvalues of the covariance matrix is
completely characterized and we obtain an infinite sequence of new distribution
functions that generalize the Tracy-Widom distributions of the random matrix
theory. Especially a phase transition phenomena is observed. Our results also
apply to a last passage percolation model and a queuing model.
2509. ON METHOD OF STATISTICAL DIFFERENTIALS
Rohitha Goonatilake
The method of statistical differentials, which approximates the mean and
variance of transformations of random variables is used in many areas of
mathematics. This paper will discuss the conditions under which such an
approximation will be exact, and also explore their accuracy in terms of error
bounds under certain moment conditions.
2510. MULTI-EXCITED RANDOM WALKS ON INTEGERS
Martin P.W. Zerner
We introduce a class of nearest-neighbor integer random walks in random and
non-random media, which includes excited random walks considered in the
literature. At each site the random walker has a drift to the right, the
strength of which depends on the environment at that site and on how often the
walker has visited that site before. We give exact criteria for recurrence and
transience and consider the speed of the walk.
2511. PERPETUAL INTEGRAL FUNCTIONALS AS HITTING AND OCCUPATION TIMES
Paavo Salminen, Marc Yor
Let $X$ be a linear diffusion and $f$ a non-negative, Borel measurable
function. We are interested in finding conditions on $X$ and $f$ which imply
that the perpetual integral functional $$ I^X_\infty(f):=\int_0^\infty f(X_t)
dt $$ is identical in law with the first hitting time of a point for some other
diffusion. This phenomenon may often be explained using random time change.
Because of some potential applications in mathematical finance, we are
considering mainly the case when $X$ is a Brownian motion with drift $\mu>0,$
denoted $\{B^{(\mu)}_t: t\geq 0\},$ but it is obvious that the method presented
is more general. We also review the known examples and give new ones. In
particular, results concerning one-sided functionals $$ \int_0^\infty
f(B^{(\mu)}_t) {\bf 1}_{\{B^{(\mu)}_t<0\}} dt\quad {\rm and}\quad \int_0^\infty
f(B^{(\mu)}_t) {\bf 1}_{\{B^{(\mu)}_t>0\}} dt $$ are presented.
This approach generalizes the proof, based on the random time change
techniques, of the fact that the Dufresne functional (this corresponds to
$f(x)=\exp(-2x)),$ playing quite an important r\^ole in the study of geometric
Brownian motion, is identical in law with the first hitting time for a Bessel
process. Another functional arising naturally in this context is %associated to
the function $$ \int_0^\infty \big(a+\exp(B^{(\mu)}_t)\big)^{-2} dt, $$ which
is seen, in the case $\mu=1/2,$ to be identical in law with the first hitting
time for a Brownian motion with drift $\mu=a/2.$
The paper is concluded by discussing how the Feynman-Kac formula can be used
to find the distribution of a perpetual integral functional.
2512. CURVED WIENER SPACE ANALYSIS
Bruce K. Driver
These notes represent a much expanded and updated version of the
\textquotedblleft mini course\textquotedblright that the author gave at the ETH
(Z\"{u}rich) and the University of Z\"{u}rich in February of 1995. The purpose
of these notes is to first provide some basic background to Riemannian geometry
and stochastic calculus on manifolds and then to cover some of the more recent
developments pertaining to analysis on \textquotedblleft curved Wiener
spaces.\textquotedblright Essentially no differential geometry is assumed.
However, it is assumed that the reader is comfortable with stochastic calculus
and differential equations on Euclidean spaces.
2513. STOCHASTIC PROCESSES WITH VALUYES IN RIEMANNIAN ADMISSIBLE COMPLEX:
ISOTROPIC PROCESS, WIENER MEASURE AND BROWNIAN MOTION
Taoufik Bouziane
The purpose of this work was to construct a {\it Brownian motion} with values
in simplicial complexes with piecewise differential structure. After a {\it
martingale} theory attempt, we constructed a family of continuous Markov
processes with values in an admissible complex ; we named every process of this
family, {\it isotropic transport process}. We showed that the family of the
isotropic processes contains a subsequence, which converged weakly to a
measure; we named it the {\it Wiener measure}. Then, we constructed, thanks to
the finite dimensional distributions of the Wiener measure, a new continuous
Markov process with values in an admissible complex : the Brownian motion. We
finished by a geometric analysis of this Brownian motion, to determinate, under
hypothesis on the complex, the recurrent or transient behavior of such process.
2514. ORTHOGONAL POLYNOMIAL ENSEMBLES IN PROBABILITY THEORY
Wolfgang Koenig
We survey a number of models from physics, statistical mechanics, probability
theory and combinatorics, which are each described in terms of an {\it
orthogonal polynomial ensemble}. The most prominent example is apparently the
Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble.
In recent years, a number of further interesting models were found to lead to
orthogonal polynomial ensembles, among which the corner growth model, the PNG
droplet, non-colliding random processes, the length of the longest increasing
subsequence of a random permutation, and others.
Much attention has been paid to universal classes of asymptotic behaviors of
these models in the limit of large particle numbers, in particular the spacings
between the particles and the fluctuation behavior of the largest particle.
Computer simulations suggest that the connections go even farther and also
comprise the zeros of the Riemann zeta function. The existing proofs require a
substantial technical machinery and heavy tools from various parts of
mathematics, in particular complex analysis, combinatorics and variational
analysis.
We survey various models, explain the questions and problems, and make
remarks on the relations between the models. Furthermore, we concisely outline
some elements of the proofs of some of the most important results. This text is
aimed at the non-expert who wants to achieve a quick survey over the field.
2515. THE PARABOLIC ANDERSON MODEL
Juergen Gaertner, Wolfgang Koenig
This is a survey on the intermittent behavior of the parabolic {Anderson}
model, which is the Cauchy problem for the heat equation with random potential
on the lattice $\Z^d$. We first introduce the model and give heuristic
explanations of the long-time behavior of the solution, both in the annealed
and the quenched setting for time-independent potentials. We thereby consider
examples of potentials studied in the literature. In the particularly important
case of an i.i.d. potential with double-exponential tails we formulate the
asymptotic results in detail. Furthermore, we explain that, under mild
regularity assumptions, there are only four different universality classes of
asymptotic behaviors. Finally, we study the moment Lyapunov exponents for
space-time homogeneous catalytic potentials generated by a {Poisson} field of
random walks.
2516. ON FIRST-PASSAGE-TIME DENSITIES FOR CERTAIN SYMMETRIC MARKOV CHAINS
Antonio Di Crescenzo and Annapatrizia Nastro
The spatial symmetry property of truncated birth-death processes studied in
Di Crescenzo [6] is extended to a wider family of continuous-time Markov
chains. We show that it yields simple expressions for first-passage-time
densities and avoiding transition probabilities, and apply it to a bilateral
birth-death process with jumps. It is finally proved that this symmetry
property is preserved within the family of strongly similar Markov chains.
2517. ON SYMMETRIC RANDOM WALKS WITH RANDOM CONDUCTANCES ON $\Z^D$
L. R. G. Fontes, P. Mathieu
We study models of continuous time, symmetric, $\Z^d$-valued random walks in
random environments. One of our aims is to derive estimates on the decay of
transition probabilities in a case where a uniform ellipticity assumption is
absent. We consider the case of independent conductances with a polynomial tail
near 0, and obtain precise asymptotics for the annealed return probability and
convergence times for the random walk confined to a finite box.
2518. THE EXPLORATION PROCESS OF INHOMOGENEOUS CONTINUUM RANDOM TREES, AND AN
EXTENSION OF JEULIN'S LOCAL TIME IDENTITY
David J Aldous, Gregory Miermont, Jim Pitman
We study the inhomogeneous continuum random trees (ICRT) that arise as weak
limits of birthday trees. We give a description of the exploration process, a
function defined on [0,1] that encodes the structure of an ICRT, and also of
its width process, determining the size of layers in order of height. These
processes turn out to be transformations of bridges with exchangeable
increments, which have already appeared in other ICRT related topics such as
stochastic additive coalescence. The results rely on two different
constructions of birthday trees from processes with exchangeable increments, on
weak convergence arguments, and on general theory on continuum random trees.
2519. EXCHANGEABLE FRAGMENTATION-COALESCENCE PROCESSES AND THEIR EQUILIBRIUM
MEASURES
Julien Berestycki
We define and study a family of Markov processes with state space the compact
set of all partitions of N that we call exchangeable fragmentation-coalescence
processes. They can be viewed as a combination of exchangeable fragmentation as
defined by Bertoin and of homogenous coalescence as defined by Pitman and
Schweinsberg or Mohle and Sagitov. We show that they admit a unique invariant
probability measure and we study some properties of their paths and of their
equilibrium measure.
2520. MEASURE CONVOLUTION SEMIGROUPS AND NON-INFINITELY DIVISIBLE PROBABILITY
DISTRIBUTIONS
Aubrey Wulfsohn
Let $\mu$ be a probability measure (or corresponding random variable) such
that all moments $\mu_n$ exist. Knowledge of the moments is not sufficient to
determine infinite divisibility of the measure; we show also that infinitely
divisible, and in particular lognormal, distributions lose infinitely
divisibilty when censored in certain ways even if all moments are arbitrarily
close to those of the uncensored distribution. The moments of a composition of
k copies of $\mu$ are expressed as combinatorial compositions of the $\mu_n$.
We express the moments of the compositions in the context of occupancy
problems, arranging n balls in k cells; the classical convolution is described
by Maxwell-Boltzmann statistics and is multinomial. For certain non-infinitely
divisible measures with moments increasing fast enough the indexing of a k-cell
combinatorial composition is extended to indexing by non-negative real t and we
construct classical convolution measure semigroups from amongst the t-indexed
classes. We prove also that when a random variable with infinitely divisible
distribution is embedded in a L\'evy process $(Y_t)$ then the t-indexed
Maxwell-Boltzmann is the law of $Y_t$. In order to get moment-based multinomial
compositions indexed by a continuum we use random measures and random
distributions rather than random variables. An alternative approach to
embeddability of a non-infinitely divisible $\mu$ is by considering
non-classical convolution measure semigroups; for example embedding $\mu$ in a
Boolean convolution measure semigroup and retaining the multinomial character
of the moments.embedding $\mu$ in a Boolean convolution measure semigroup and
retaining the multinomial character of the moments. Embedding $\mu$ in an
Urbanik convolution measure semigroup loses their multinomial character .
2521. POISSON BOUNDARY FOR FINITELY GENERATED GROUPS OF RATIONAL AFFINITIES
Sara Brofferio
The group of affine transformations with rational coefficients, $aff(Q)$,
acts naturally on the real line, but also on the $p$-adic fields. The aim of
this note is to show that all these actions are necessary and sufficient to
represent bounded $\mu$-harmonic functions for a probability measure $\mu$ on
$aff(Q)$ that is supported by a finitely generated sub-group, that is to
describe the Poisson boundary.
2522. THE POISSON BOUNDARY OF RANDOM RATIONAL AFFINITIES
Sara Brofferio
The group of affine transformations with rational coefficients acts naturally
on the real line, but also on the $p$-adic fields. The aim of this note is to
show that, for random walks whose laws have a finite first moment, all these
actions are necessary and sufficient to describe the Poisson boundary, which is
in fact the product of all the fields that contract in mean.
2523. BOUNDARIES AND HARMONIC FUNCTIONS FOR RANDOM WALKS WITH RANDOM
TRANSITION PROBABILITIES
Vadim A. Kaimanovich, Yuri Kifer, Ben-Zion Rubshtein
The usual random walk on a group (homogeneous both in time and in space) is
determined by a probability measure on the group. In a random walk with random
transition probabilities this single measure is replaced with a stationary
sequence of measures, so that the resulting (random) Markov chains are still
space homogeneous, but no longer time homogeneous. We study various notions of
measure theoretical boundaries associated with this model and establish an
analogue of the Poisson formula for (random) bounded harmonic functions. Under
natural conditions on transition probabilities we identify these boundaries for
several classes of groups with hyperbolic properties and prove the boundary
triviality (i.e., the absence of non-constant random bounded harmonic
functions) for groups of subexponential growth, in particular, for nilpotent
groups.
2524. A PHASE TRANSITION IN THE RANDOM TRANSPOSITION RANDOM WALK
Nathanael Berestycki, Rick Durrett
Our work is motivated by Bourque and Pevzner's (2002) simulation study of the
effectiveness of the parsimony method in studying genome rearrangement, and
leads to a surprising result about the random transposition walk on the group
of permutations on $n$ elements.
Consider this walk in continuous time starting at the identity and let $D_t$
be the minimum number of transpositions needed to go back to the identity from
the location at time $t$.
$D_t$ undergoes a phase transition: the distance $D_{cn/2} \sim u(c)n$, where
$u$ is an explicit function satisfying $u(c)=c/2$ for $c \le 1$ and $u(c)<c/2$
for $c>1$. In other words, the distance to the identity is roughly linear
during the subcritical phase, and after critical time $n/2$ it becomes
sublinear.
In addition, we describe the fluctuations of $D_{cn/2}$ about its mean in
each of the three regimes (subcritical, critical and supercritical).
The techniques used involve viewing the cycles in the random permutation as a
coagulation-fragmentation process and relating the behavior to the
\Erd\H{o}s-Renyi random graph model.
2525. GREEN KERNEL ESTIMATES AND THE FULL MARTIN BOUNDARY FOR RANDOM WALKS ON
LAMPLIGHTER GROUPS AND DIESTEL-LEADER GRAPHS
Sara Brofferio and Wolfgang Woess
We determine the precise asymptotic behaviour (in space) of the Green kernel
of simple random walk with drift on the Diestel-Leader graph $DL(q,r)$, where
$q,r \ge 2$. The latter is the horocyclic product of two homogeneous trees with
respective degrees $q+1$ and $r+1$. When $q=r$, it is the Cayley graph of the
wreath product (lamplighter group) ${\mathbb Z}_q \wr {\mathbb Z}$ with respect
to a natural set of generators. We describe the full Martin compactification of
these random walks on $DL$-graphs and, in particular, lamplighter groups. This
completes and provides a better approach to previous results of Woess, who has
determined all minimal positive harmonic functions.
2526. A PICKANDS TYPE ESTIMATOR OF THE EXTREME VALUE INDEX
Laurent Gardes, Stephane Girard
One of the main goal of extreme value analysis is to estimate the probability
of rare events given a sample from an unknown distribution. The upper tail
behavior of this distribution is described by the extreme value index. We
present a new estimator of the extreme value index adapted to any domain of
attraction. Its construction is similar to the one of Pickands' estimator. its
weak consistency and its asymptotic distribution are established and a bias
reduction method is proposed. Our estimator is compared with classical extreme
value index estimators through a simulation study.
2527. THE BEURLING ESTIMATE FOR A CLASS OF RANDOM WALKS
Gregory F. Lawler and Vlada Limic
An estimate of Beurling states that if K is a curve from 0 to the unit circle
in the complex plane, then the probability that a Brownian motion starting at
-eps reaches the unit circle without hitting the curve is bounded above by c
eps^{1/2}. This estimate is very useful in analysis of boundary behavior of
conformal maps, especially for connected but rough boundaries. The
corresponding estimate for simple random walk was first proved by Kesten. In
this note we extend this estimate to random walks with zero mean, and finite
(3+delta) moment.
2528. THE EFFECT OF SERVICE TIME VARIABILITY ON MAXIMUM QUEUE LENGTHS IN
M^X/G/1 QUEUES
Misja Nuyens, Rhonda Righter and Ger Koole
We study the impact of service-time distributions on the distribution of the
maximum queue length during a busy period for the M^X/G/1 queue. The maximum
queue length is an important random variable to understand when designing the
buffer size for finite buffer (M/G/1/n) systems. We show the somewhat
surprising result that for non-preemptive queueing disciplines and for three
variations of the preemptive LCFS discipline, the maximum queue length during a
busy period is smaller when service times are more variable (in the convex
sense).
2529. LAMPLIGHTERS, DIESTEL-LEADER GRAPHS, RANDOM WALKS, AND HARMONIC
FUNCTIONS
Wolfgang Woess
The lamplighter group over $\mathbb Z$ is the wreath product $\mathbb Z_q \wr
\mathbb Z$. With respect to a natural generating set, its Cayley graph is the
Diestel-Leader graph $DL(q,q)$. We study harmonic functions for the "simple"
Laplacian on this graph, and more generally, for a class of random walks on
$DL(q,r)$, where $q,r \ge 2$. The DL-graphs are horocyclic products of two
trees, and we give a full description of all positive harmonic functions in
terms of the boundaries of these two trees. In particular, we determine the
minimal Martin boundary, that is, the set of minimal positive harmonic
functions.
2530. QUADRI-TILINGS OF THE PLANE
B. de Tiliere
Quadri-tilings of the plane are tilings by quadrilaterals made of adjacent
right triangles. Quadri-tilings are in bijection with dimer configurations on
graphs arising from rhombus tilings of the plane. Assigning "critical" weights
to the edges of such a graph, we construct a natural explicit Gibbs measure,
and prove that it is asymptotically independent of the structure of the graph.
We give an explicit expression for a measure on the set of dimer configurations
of all graphs arising from 60- degree rhombus tilings of the plane, whose
marginals are the above Gibbs measures. We construct two "height functions" on
60-degree rhombus quadri-tilings, and thereby interpret them as surfaces in a
4-dimensional space.
2531. HITTING TIMES FOR INDEPENDENT RANDOM WALKS
Amine Asselah and Pablo A. Ferrari
We consider a system of asymmetric independent random walks on $Z^d$, denoted
by $\{\eta_t,t\in R\}$, stationary under the product Poisson measure
$\nu_{\rho}$ of marginal density $\rho>0$. We fix a pattern $A$, an increasing
local event, and denote by $\tau$ the hitting time of $A$. By using a Loss
Network representation of our system, at small density, we obtain a coupling
between the laws of $\eta_t$ conditioned on $\{\tau>t\}$ for all times $t$.
When $d\ge 3$, this provides bounds on the rate of convergence of the law of
$\eta_t$ conditioned on $\{\tau>t\}$ towards its limiting probability measure
as $t$ tends to infinity. We also treat the case where the initial measure is
{\it close} to $\nu_{\rho}$ without being product.
2532. UNIVERSALITY IN SHERRINGTON-KIRKPATRICK'S SPIN GLASS MODEL
Philippe Carmona, Yueyun Hu
We show that the limiting free energy in Sherrington-Kirkpatrick's Spin Glass
Model does not depend on the environment.
2533. EXACT CONVERGENCE RATES IN THE CENTRAL LIMIT THEOREM FOR A CLASS OF
MARTINGALES
Mohamed El Machkouri, Lahcen Ouchti
We give optimal convergence rates in the central limit theorem for a large
class of martingale difference sequences with bounded third moments. The rates
depend on the behaviour of the conditional variances and for stationary
sequences the rate $n^{-1/2}\log n$ is reached. We give interesting examples of
martingales with unbounded increments which belong to the considered class.
2534. ON THE RATE OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM FOR MARTINGALE
DIFFERENCE SEQUENCES
Lahcen Ouchti
We established the rate of convergence in the central limit theorem for
stopped sums of a class of martingale difference sequences.
2535. LIMIT OF NORMALIZED QUADRANGULATIONS: THE BROWNIAN MAP
J. F. Marckert and A. Mokkadem
Consider $q_n$ a random pointed quadrangulation chosen equally likely among
the pointed quadrangulations with $n$ faces. In this paper, we show that, when
$n$ goes to $+\infty$, $q_n$ suitably normalized converges in a certain sense
to a random limit object, which is continuous and compact, and which we name
the Brownian map. The same result is shown for a model of rooted
quadrangulations and for some models of rooted quadrangulations with random
edge lengths. A metric space of abstract maps (that contains the model of
discrete rooted quadrangulations and the model of Brownian map) is defined. The
weak convergence holds in this metric space.
2536. NON-BACKTRACKING RANDOM WALKS AND COGROWTH OF GRAPHS
Ronald Ortner and Wolfgang Woess
Let X be a locally finite, connected graph without vertices of degree 1.
Non-backtracking random walk moves at each step with equal probability to one
of the "forward" neighbours of the actual state, i.e., it does not go back
along the preceding edge to the preceding state. This is not a Markov chain,
but can be turned into a Markov chain whose state space is the set of oriented
edges of X. Thus we obtain for infinite X that the n-step non-backtracking
transition probabilities tend to zero, and we can also compute their limit when
X is finite. This provides a short proof of old results concerning cogrowth of
groups, and makes the extension of that result to arbitrary regular graphs
rigorous. Even when X is non-regular, but "small cycles are dense" in X, we
show that the graph X is non-amenable if and only if the non-backtracking
n-step transition probabilities decay exponentially fast. This is a partial
generalization of the cogrowth criterion for regular graphs which comprises the
original cogrowth criterion for finitely generated groups of Grigorchuk and
Cohen.
2537. ON THE LARGEST SINGULAR VALUES OF RANDOM MATRICES WITH INDEPENDENT
CAUCHY ENTRIES
Alexander Soshnikov and Yan V. Fyodorov
We apply the method of determinants to study the distribution of the largest
singular values of large real rectangular random matrices with independent
Cauchy entries. We show that statistical properties of the largest singular
values are different from the Tracy-Widom law. Among other corollaries of our
method we show an interesting connection between the mathematical expectations
of the determinants of complex rectangular $ m \times n $ standard Wishart
ensemble and real rectangular $ 2m \times 2n $ standard Wishart ensemble.
2538. ASYMPTOTIC LAWS FOR COMPOSITIONS DERIVED FROM TRANSFORMED SUBORDINATORS
A. Gnedin, J. Pitman and M. Yor
A random composition of $n$ appears when the points of a random closed set
$\widetilde{\cal R}\subset [0,1]$ are used to separate into blocks $n$ points
sampled from the uniform distribution. We study the number of parts $K_n$ of
this composition and other related functionals under the assumption that
$\widetilde{\cal R}=\phi(S_t)$ where $(S_t,t\geq 0)$ is a subordinator and
$\phi:[0,\infty]\to [0,1]$ is a diffeomorphism. We derive the asymptotics of
$K_n$ for the case when the L{\'e}vy measure of subordinator is regularly
varying at 0 with positive index. Specialising to the case of exponential
function $\phi(x)=1-e^{-x}$ we establish a connection of the asymptotics of
$K_n$ to the exponential functional of subordinator.
2539. STRONG DISORDER FOR A CERTAIN CLASS OF DIRECTED POLYMERS IN A RANDOM
ENVIRONMENT
Philippe Carmona, Francesco Guerra, Yueyun Hu and Olivier Mejane
We study a model of directed polymers with an exponentially recurrent Markov
chain and an indefinitely divisible random environment. We prove that the
normalized partition function converges exponentially fast towards zero at all
temperatures.
2540. NATURAL DECOMPOSITION OF PROCESSES AND WEAK DIRICHLET PROCESSES
Jean M\'{e}min, Fran\c{c}ois Coquet, Adam Jakubowski, Leszek Slominski
A class of stochastic processes, called "weak Dirichlet processes", is
introduced and its properties are investigated in detail. This class is much
larger than the class of Dirichlet processes. It is closed under
$C^1$-transformations and under absolutely continuous change of measure. If a
weak Dirichlet process has finite energy, as defined by Graversen and Rao, its
Doob-Meyer type decomposition is unique. The developed methods have been
applied to a study of generalized martingale convolutions.
2541. SOLUTION OF THE MONGE-AMPERE EQUATION ON WIENER SPACE FOR LOG-CONCAVE
MEASURES
D. Feyel and A.S. Ustunel
In this work we prove that the unique 1-convex solution of the Monge problem
contructed from the solution of the Monge-Kantorovitch problem between the
Wiener measure and a target measure which has a log-concave density w.r.to the
Wiener measure is also the strong solution of the Monge-Ampere equation in the
frame of infinite dimensional Frechet spaces. We enhance also the polar
factorization results of the mappings which transform a spread measure to
another one of finite Wasserstein distance. Finally we calculate the
semimartingale decomposition of the transport process with respect to its
natural filtration and make the connection between the curved Brownian motion
and the polar decomposition of the corresponding shifts.
2542. RELATIVISTIC DIFFUSIONS
Jacques Franchi and Yves Le Jan
The purpose of this note is to introduce and study a relativistic motion
whose acceleration, in proper time, is given by a white noise. We begin with
the flat case of special relativity, continue with the case of general
relativity, and finally consider more closely the example of the Schwarzschild
space.
2543. PDE'S FOR THE JOINT DISTRIBUTIONS OF THE DYSON, AIRY AND SINE PROCESSES
Mark Adler & Pierre van Moerbeke
In a celebrated paper, Dyson shows that the spectrum of a n x n random
Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves
as n non-colliding Brownian motions held together by a drift term. The
universal edge and bulk scalings for Hermitian random matrices, applied to the
Dyson process, lead to the Airy and Sine processes. In particular, the Airy
process is a continuous stationary process, describing the motion of the
outermost particle of the Dyson Brownian motion, when the number of particles
gets large, with space and time appropriately rescaled.
In this paper, we answer a question posed by Kurt Johansson, to find a PDE for
the joint distribution of the Airy Process at two different times. Similarly we
find a PDE satisfied by the joint distribution of the Sine process. This hinges
on finding a PDE for the joint distribution of the Dyson process, which itself
is based on the joint probability of the eigenvalues for coupled Gaussian
Hermitian matrices. The PDE for the Dyson process is then subjected to an
asymptotic analysis, consistent with the edge and bulk rescalings. The PDE's
enable one to compute the asymptotic behavior of the joint distribution and
their correlation at different times t_1 and t_2, when t_2-t_1 tends to
infinity, as illustrated in this paper for the Airy process.
2544. FUNCTIONAL CENTRAL LIMIT THEOREMS FOR A LARGE NETWORK IN WHICH CUSTOMERS
JOIN THE SHORTEST OF SEVERAL QUEUES
Carl Graham
We consider N single server infinite buffer queues with service rate \beta.
Customers arrive at rate N\alpha, choose L queues uniformly, and join the
shortest. We study the processes R^N for large N, where R^N_t(k) is the
fraction of queues of length at least k at time t. Laws of large numbers (LLNs)
are known. We consider certain Hilbert spaces with the weak topology. First, we
prove a functional central limit theorem (CLT) under the a priori assumption
that the initial data R^N_0 satisfy the corresponding CLT. We use a
compactness-uniqueness method, and the limit is characterized as an
Ornstein-Uhlenbeck (OU) process. Then, we study the R^N in equilibrium under
the stability condition \alpha < \beta, and prove a functional CLT with limit
the OU process in equilibrium. We use ergodicity and justify the inversion of
limits for large times and for large sizes N by a compactness-uniqueness
method. We deduce a posteriori the CLT for R^N_0 under the invariant laws, an
interesting result in its own right. The main tool for proving tightness of the
implicitly defined invariant laws in the CLT scaling and ergodicity of the
limit OU process is a global exponential stability result for the nonlinear
dynamical system obtained in the functional LLN limit.
2545. PERCOLATION, FIRST-PASSAGE PERCOLATION, AND COVERING TIMES FOR
RICHARDSON'S MODEL ON THE N-CUBE
James Allen Fill, Robin Pemantle
Percolation with edge-passage probability p and first-passage percolation are
studied for the n-cube B_n ={0,1}^n with nearest neighbor edges. For oriented
and unoriented percolation, p=e/n and p=1/n are the respective critical
probabilities. For oriented first-passage percolation with i.i.d. edge-passage
times having a density of 1 near the origin, the percolation time (time to
reach the opposite corner of the cube) converges in probability to 1 as
n->infty. This resolves a conjecture of David Aldous. When the edge-passage
distribution is standard exponential, the (smaller) percolation time for
unoriented edges is at least 0.88. These results are applied to Richardson's
model on the (unoriented) n-cube. Richardson's model, otherwise known as the
contact process with no recoveries, models the spread of infection as a Poisson
process on each edge connecting an infected node to an uninfected one. It is
shown that the time to cover the entire n-cube is bounded between 1.41 and
14.05 in probability as n->infty.
2546. AN ASYMPTOTIC OPTIMALITY OF THE TRANSPOSITION RULE FOR LINEAR LISTS
David Gamarnik and Petar Momcilovic
The transposition rule is an algorithm for self-organizing linear lists. Upon
a request for a given item, the item is transposed with the preceding one. The
cost of a request is the distance of the requested item from the beginning of
the list. An asymptotic optimality of the rule with the respect to the optimal
static arrangement is demonstrated for two families of request distributions.
The result is established by considering an associated constrained asymmetric
exclusion process.
2547. GENERAL STATE SPACE MARKOV CHAINS AND MCMC ALGORITHMS
Gareth O. Roberts and Jeffrey S. Rosenthal
This paper surveys various results about Markov chains on general
(non-countable) state spaces. It begins with an introduction to Markov chain
Monte Carlo (MCMC) algorithms, which provide the motivation and context for the
theory which follows. Then, sufficient conditions for geometric and uniform
ergodicity, along with quantitative bounds on the rate of convergence to
stationarity in terms of minorisation and drift conditions, are presented. Many
of these results are proved using direct coupling constructions. Necessary and
sufficient conditions for Central Limit Theorems (CLTs) are also presented, in
some cases proved via the Poisson Equation or direct regeneration
constructions. Finally, optimal scaling and weak convergence results for
Metropolis-Hastings algorithms are discussed. None of the results presented is
new, though many of the proofs are. We also describe some Open Problems.
2548. VERTEX-REINFORCED RANDOM WALK
Robin Pemantle
This paper considers a class of non-Markovian discrete-time random processes
on a finite state space {1,...,d}. The transition probabilities at each time
are influenced by the number of times each state has been visited and by a
fixed a priori likelihood matrix, R, which is real, symmetric and nonnegative.
Let S_i(n) keep track of the number of visits to state i up to time n, and form
the fractional occupation vector, V(n), where v_i(n)=S_i(n)/(sum_{j=1}^d
S_j(n)). It is shown that V(n) converges to a set of critical points for the
quadratic form H with matrix R, and that under nondegeneracy conditions on R,
there is a finite set of points such that with probability one, V(n)->p for
some p in the set. There may be more than one p in this set for which
P(V(n)->p)>0. On the other hand P(V(n)->p)=0 whenever p fails in a strong
enough sense to be maximum for H.
2549. CRITICAL RWRE ON TREES AND TREE-INDEXED RANDOM WALKS
Robin Pemantle, Yuval Peres
We study the behavior of Random Walk in Random Environment (RWRE) on trees in
the critical case left open in previous work. Representing the random walk by
an electrical network, we assume that the ratios of resistances of neighboring
edges of a tree Gamma are i.i.d.random variables whose logarithms have mean
zero and finite variance. Then the resulting RWRE is transient if simple random
walk on Gamma is transient, but not vice versa. We obtain general transience
criteria for such walks, which are sharp for symmetric trees of polynomial
growth. In order to prove these criteria, we establish results on boundary
crossing by tree-indexed random walks. These results rely on comparison
inequalities for percolation processes on trees and on some new estimates of
boundary crossing probabilities for ordinary mean-zero finite variance random
walks in one dimension, which are of independent interest.
2550. CHOOSING A SPANNING TREE FOR THE INTEGER LATTICE UNIFORMLY
Robin Pemantle
Consider the nearest neighbor graph for the integer lattice Z^d in d
dimensions. For a large finite piece of it, consider choosing a spanning tree
for that piece uniformly among all possible subgraphs that are spanning trees.
As the piece gets larger, this approaches a limiting measure on the set of
spanning graphs for Z^d. This is shown to be a tree if and only if d=<4. In
this case, the tree has only one topological end, i.e. there are no doubly
infinite paths. When d>=5 the spanning forest has infinitely many components
almost surely, with each component having one or two topological ends.
2551. DOMINATION BETWEEN TREES AND APPLICATION TO AN EXPLOSION PROBLEM
Robin Pemantle, Yuval Peres
We define a notion of stochastic domination between trees, where one tree
dominates another if when the vertices of each are labeled with independent,
identically distributed random variables, one tree is always more likely to
contain a path with a specified property. Sufficient conditions for this kind
of domination are (1) more symmetry and (2) earlier branching. We apply these
conditions to the problem of determining how fast a tree must grow before
first-passage percolation on the tree exhibits an explosion, that is to say,
infinitely many vertices are reached in finite time. For a tree in which each
vertex at distance n-1 from the root has f(n) offspring, f nondecreasing, an
explosion occurs with exponentially distributed passage times if and only if
sum f(n)^{-1}<infty.
2552. RANDOM WALK IN A RANDOM ENVIRONMENT AND FIRST-PASSAGE PERCOLATION ON
TREES
Robin Pemantle, Russell Lyons
We show that the transience or recurrence of a random walk in certain random
environments on an arbitrary infinite locally finite tree is determined by the
branching number of the tree, which is a measure of the average number of
branches per vertex. This generalizes and unifies previous work of the authors.
It also shows that the point of phase transition for edge-reinforced random
walk is likewise determined by the branching number of the tree. Finally, we
show that the branching number determines the rate of first-passage percolation
on trees, also known as the first-birth problem. Our techniques depend on
quasi-Bernoulli percolation and large deviation results.
2553. THE CONTACT PROCESS ON TREES
Robin Pemantle
The contact process on an infinite homogeneous tree is shown to exhibit at
least two phase transitions as the infection parameter lambda is varied. For
small values of lambda a single infection eventually dies out. For larger
lambda the infection lives forever with positive probability but eventually
leaves any finite set. (The survival probability is a continuous function of
lambda, and the proof of this is much easier than it is for the contact process
on d-dimensional integer lattices.) For still larger lambda the infection
converges in distribution to a nontrivial invariant measure. For an n-ary tree,
with n large, the first of these transitions occurs when lambda~1/n and the
second occurs when 1/2 sqrt{n}<lambda<e/sqrt{n}. Nonhomogeneous trees whose
vertices have degrees varying between 1 and n behave essentially as homogeneous
n-ary trees, provided that vertices of degree n are not too rare. In
particular, letting n go to infty, Galton-Watson trees whose vertices have
degree n with probability that does not decrease exponentially with n may have
both phase transitions occur together at lambda=0. The nature of the second
phase transition is not yet clear and several problems are mentioned in this
regard.
2554. ON PATH INTEGRALS FOR THE HIGH-DIMENSIONAL BROWNIAN BRIDGE
Robin Pemantle, Mathew Penrose
Let v be a bounded function with bounded support in R^d, d>=3. Let x,y in
R^d. Let Z(t) denote the path integral of v along the path of a Brownian bridge
in R^d which runs for time t, starting at x and ending at y. As t->infty, it is
perhaps evident that the distribution of Z(t) converges weakly to that of the
sum of the integrals of v along the paths of two independent Brownian motions,
starting at x and y and running forever.Here we prove a stronger result, namely
convergence of the corresponding moment generating functions and of moments.
This result is needed for applications in physics.
2555. LOCAL CHARACTERISTICS, ENTROPY AND LIMIT THEOREMS FOR SPANNING TREES AND
DOMINO TILINGS VIA TRANSFER-IMPEDANCES
Robert Burton, Robin Pemantle
Let G be a finite graph or an infinite graph on which Z^d acts with finite
fundamental domain. If G is finite, let T be a random spanning tree chosen
uniformly from all spanning trees of G; if G is infinite, known methods show
that this still makes sense, producing a random essential spanning forest of G.
A method for calculating local characteristics (i.e. finite-dimensional
marginals) of T from the transfer-impedance matrix is presented. This differs
from the classical matrix-tree theorem in that only small pieces of the matrix
(n-dimensional minors) are needed to compute small (n-dimensional) marginals.
Calculation of the matrix entries relies on the calculation of the Green's
function for G, which is not a local calculation. However, it is shown how the
calculation of the Green's function may be reduced to a finite computation in
the case when G is an infinite graph admitting a Z^d-action with finite
quotient. The same computation also gives the entropy of the law of T. These
results are applied to the problem of tiling certain lattices by dominos - the
so-called dimer problem. Another application of these results is to prove
modified versions of conjectures of Aldous on the limiting distribution of
degrees of a vertex and on the local structure near a vertex of a uniform
random spanning tree in a lattice whose dimension is going to infinity.
Included is a generalization of moments to tree-valued random variables and
criteria for these generalized moments to determine a distribution.
2556. CRITICAL RANDOM WALK IN RANDOM ENVIRONMENT ON TREES OF EXPONENTIAL
GROWTH
Robin Pemantle
This paper studies the behavior of RWRE on trees in the critical case left
open in previous work. For trees of exponential growth, a random perturbation
of the transition probabilities can change a transient random walk into a
recurrent one. This is the opposite of what occurs on trees of sub-exponential
growth.
2557. PLANAR FIRST-PASSAGE PERCOLATION TIMES ARE NOT TIGHT
Robin Pemantle, Yuval Peres
We consider first-passage percolation on the two-dimensional integer lattice
Z^2 with passage times that are IID exponentials of mean one. It has been
conjectured, based on numerical evidence, that the variance of the time T(0,n)
to reach the vertex (0,n) is of order n^{2/3}. Kesten showed that the variance
of T(0,n) is at O(n). He also noted that the variance is bounded away from
zero. This note improves the lower bound on the variance of T(0,n) to C log n.
Simultaneously and independently, Newman and Piza have achieved the same result
for {0,1}-valued passage times. Their methods extend to more general passage
times, while ours work only for exponential times. On the other hand, our
theorem shows that the variance comes from fluctuations of nonvanishing
probability in the sense that, as n->infty, the law of T(0,n) is not tight
about its median. Very recently, Newman and Piza showed that the log n may be
improved to a power of n for directions in which the shape is not flat (it is
not known whether the shape can be flat in any direction). As pointed out to us
by Harry Kesten, in the exponential case this may also be obtained via the
method given here.
2558. A SHUFFLE THAT MIXES SETS OF ANY FIXED SIZE MUCH FASTER THAN IT MIXES
THE WHOLE DECK
Robin Pemantle
Consider an n by n array of cards shuffled in the following manner. An
element x of the array is chosen uniformly at random; Then with probability 1/2
the rectangle of cards above and to the left of x is rotated 180 degrees, and
with probability 1/2 the rectangle of cards below and to the right of x is
rotated 180 degrees. It is shown by an eigenvalue method that the time required
to approach the uniform distribution is between n^2/2 and cn^2 ln n for some
constant c. On the other hand, for any k it is shown that the time needed to
uniformly distribute a set of cards of size k is at most c(k)n, where c(k) is a
constant times k^3 ln(k)^2. This is established via coupling; no attempt is
made to get a good constant.
2559. GALTON-WATSON TREES WITH THE SAME MEAN HAVE THE SAME POLAR SETS
Robin Pemantle, Yuval Peres
Evans defines a notion of what it means for a set B to be polar for a process
indexed by a tree. The main result herein is that a tree picked from a
Galton-Watson measure whose offspring distribution has mean m and finite
variance will almost surely have precisely the same polar sets as a
deterministic tree of the same growth rate. This implies that deterministic and
nondeterministic trees behave identically in a variety of probability models.
Mapping subsets of Euclidean space to trees and polar sets to capacity
criteria, it follows that certain random Cantor sets are capacity-equivalent to
each other and to deterministic Cantor sets. An extension to branching
processes in varying environment is also obtained.
2560. MARTIN CAPACITY FOR MARKOV CHAINS
Itai Benjamini, Robin Pemantle, Yuval Peres
The probability that a transient Markov chain, or a Brownian path, will ever
visit a given set Lambda, is classically estimated using the capacity of Lambda
with respect to the Green kernel G(x,y). We show that replacing the Green
kernel by the Martin kernel G(x,y)/G(0,y) yields improved estimates, which are
exact up to a factor of 2. These estimates are applied to random walks on
lattices, and also to explain a connection found by R. Lyons between capacity
and percolation on trees.
2561. AN ALMOST SURE INVARIANCE PRINCIPLE FOR THE RANGE OF PLANAR RANDOM WALKS
Richard F. Bass and Jay Rosen
For a symmetric random walk in $Z^2$ with $2+\delta$ moments, we represent
$|\mathcal{R}(n)|$, the cardinality of the range, in terms of an expansion
involving the renormalized intersection local times of a Brownian motion. We
show that for each $k\geq 1$ \[ (\log n)^k [ \frac{1}{n} |\mathcal{R}(n)|
+\sum_{j=1}^k (-1)^j (\textstyle{\frac1{2\pi}}\log n +c_X)^{-j}
\gamma_{j,n}]\to 0, \qquad a.s. \] where $W_t$ is a Brownian motion,
$W^{(n)}_t=W_{nt}/\sqrt n$, $\gamma_{j,n}$ is the renormalized intersection
local time at time 1 for $W^{(n)}$, and $c_X$ is a constant depending on the
distribution of the random walk.
2562. COVARIANCE MATRICES OF SELF-AFFINE MEASURES
K. Zajkowski
In this paper we derive a formula for a covariance matrix of any self-affine
measure.
2563. CONCEPTUAL PROOFS OF L LOG L CRITERIA
Russell Lyons, Robin Pemantle, Yuval Peres
The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth
of a supercritical branching process, showing that an L log L condition is
decisive. In critical and subcritical cases, results of Kolmogorov and later
authors give the rate of decay of the probability that the process survives at
least n generations. We give conceptual proofs of these theorems based on
comparisons of Galton-Watson measure to another measure on the space of trees.
This approach also explains Yaglom's exponential limit law for conditioned
critical branching processes via a simple characterization of the exponential
distribution.
2564. MAXIMUM VARIATION OF TOTAL RISK
Robin Pemantle
Let Z>0 be a random time. The total risk of discovering Z in the next time
interval (t,t+dt) is never more variable than an exponential of mean one, which
is achieved when the information up to time t is sigma(Z wedge t).
2565. RANDOM WALKS IN VARYING DIMENSIONS
Itai Benjamini, Robin Pemantle, Yuval Peres
We establish recurrence criteria for sums of independent random variables
which take values in Euclidean lattices of varying dimension. In particular, we
describe transient inhomogenous random walks in the plane which interlace two
symmetric step distributions of bounded support.
2566. ON WHICH GRAPHS ARE ALL RANDOM WALKS IN RANDOM ENVIRONMENTS TRANSIENT?
Robin Pemantle, Yuval Peres
An infinite graph G has the property that a random walk in random environment
on G defined by i.i.d. resistances with any common distribution is almost
surely transient, if and only if for some p<1, simple random walk is transient
on a percolation cluster of G under bond percolation with parameter p.
2567. THE TRACE OF SPATIAL BROWNIAN MOTION IS CAPACITY-EQUIVALENT TO THE UNIT
SQUARE
Robin Pemantle, Yuval Peres, Jonathan W. Shapiro
We show that with probability 1, the trace B[0,1] of Brownian motion in
space, has positive capacity with respect to exactly the same kernels as the
unit square. More precisely, the energy of occupation measure on B[0,1] in the
kernel f(|x-y|), is bounded above and below by constant multiples of the energy
of Lebesgue measure on the unit square. (The constants are random, but do not
depend on the kernel.) As an application, we give almost-sure asymptotics for
the probability that an alpha-stable process approaches within epsilon of
B[0,1], conditional on B[0,1]. The upper bound on energy is based on a strong
law for the approximate self-intersections of the Brownian path. We also prove
analogous capacity estimates for planar Brownian motion and for the zero-set of
one-dimensional Brownian motion.
2568. DIFFUSION LIMITED AGGREGATION ON A TREE
Martin T. Barlow, Robin Pemantle, Edwin A. Perkins
We study the following growth model on a regular d-ary tree. Points at
distance n adjacent to the existing subtree are added with probabilities
proportional to alpha^{-n}, where alpha<1 is a positive real parameter. The
heights of these clusters are shown to increase linearly with their total size;
this complements known results that show the height increases only
logarithmically when alpha>=1. Results are obtained using stochastic
monotonicity and regeneration results which may be of independent interest. Our
motivation comes from two other ways in which the model may be viewed: as a
problem in first-passage percolation, and as a version of diffusion-limited
aggregation (DLA), adjusted so that `fingering' occurs.
2569. SHARPNESS OF SECOND MOMENT CRITERIA FOR BRANCHING AND TREE-INDEXED
PROCESSES
Robin Pemantle
A class of branching processes in varying environments is exhibited which
become extinct almost surely even though the means M_n grow fast enough so that
sum M_n^{-1} is finite. In fact, such a process is constructed for every
offspring distribution of infinite variance, and this establishes the converse
of a previously known fact: that if a distribution has finite variance then sum
M_n^{-1}=infty is equivalent to almost sure extinction. This has as an
immediate consequence the converse to a theorem on equipolarity of
Galton-Watson trees.
2570. ON NEAR-CRITICAL AND DYNAMICAL PERCOLATION IN THE TREE CASE
Olle Haggstrom, Robin Pemantle
Consider independent bond percolation with retention probability p on a
spherically symmetric tree Gamma. Write theta_Gamma(p) for the probability that
the root is in an infinite open cluster, and define the critical value
p_c=inf{p:theta_Gamma(p)>0}. If theta_Gamma(p_c)=0, then the root may still
percolate in the corresponding dynamical percolation process at the critical
value p_c, as demonstrated recently by Haggstrom, Peres and Steif. Here we
relate this phenomenon to the near-critical behaviour of theta_Gamma(p) by
showing that the root percolates in the dynamical percolation process if and
only if int_{p_c}^1 (theta_Gamma(p))^{-1}dp<infty. The ``only if'' direction
extends to general trees, whereas the ``if'' direction fails in this
generality.
2571. ROBUST PHASE TRANSITIONS FOR HEISENBERG AND OTHER MODELS ON GENERAL
TREES
Robin Pemantle, Jeffrey E. Steif
We study several statistical mechanical models on a general tree. Particular
attention is devoted to the classical Heisenberg models, where the state space
is the d-dimensional unit sphere and the interactions are proportional to the
cosines of the angles between neighboring spins. The phenomenon of interest
here is the classification of phase transition (non-uniqueness of the Gibbs
state) according to whether it is robust. In many cases, including all of the
Heisenberg and Potts models, occurrence of robust phase transition is
determined by the geometry (branching number) of the tree in a way that
parallels the situation with independent percolation and usual phase transition
for the Ising model. The critical values for robust phase transition for the
Heisenberg and Potts models are also calculated exactly. In some cases, such as
the q>=3 Potts model, robust phase transition and usual phase transition do not
coincide, while in other cases, such as the Heisenberg models, we conjecture
that robust phase transition and usual phase transition are equivalent. In
addition, we show that symmetry breaking is equivalent to the existence of a
phase transition, a fact believed but not known for the rotor model on Z^2.
2572. MOMENT CONDITIONS FOR A SEQUENCE WITH NEGATIVE DRIFT TO BE UNIFORMLY
BOUNDED IN L^R
Robin Pemantle, Jeffrey S. Rosenthal
Suppose a sequence of random variables {X_n} has negative drift when above a
certain threshold and has increments bounded in L^p. When p>2 this implies that
EX_n is bounded above by a constant independent of n and the particular
sequence {X_n}. When p=<2 there are counterexamples showing this does not hold.
In general, increments bounded in L^p lead to a uniform L^r bound on X_n^+ for
any r<p-1, but not for r>=p-1. These results are motivated by questions about
stability of queueing networks.
2573. SOJOURN TIMES FOR BROWNIAN SHEET
Davar Khoshnevisan, Robin Pemantle
This paper is dedicated to Professor Endre Csakion the occasion of his 65th
birthday.
2574. TOWARDS A THEORY OF NEGATIVE DEPENDENCE
Robin Pemantle
The FKG theorem says that the POSITIVE LATTICE CONDITION, an easily checkable
hypothesis which holds for many natural families of events, implies POSITIVE
ASSOCIATION, a very useful property. Thus there is a natural and useful theory
of positively dependent events. There is, as yet, no corresponding theory of
negatively dependent events. There is, however, a need for such a theory. This
paper, unfortunately, contains no substantial theorems. Its purpose is to
present examples that motivate a need for such a theory, give plausibility
arguments for the existence of such a theory, outline a few possible directions
such a theory might take, and state a number of specific conjectures which
pertain to the examples and to a wish list of theorems.
2575. NON-AMENABLE PRODUCTS ARE NOT TREEABLE
Robin Pemantle, Yuval Peres
Let X and Y be infinite graphs, such that the automorphism group of X is
nonamenable, and the automorphism group of Y has an infinite orbit. We prove
that there is no automorphism-invariant measure on the set of spanning trees in
the direct product X times Y. This implies that the minimal spanning forest
corresponding to i.i.d. edge-weights in such a product, has infinitely many
connected components almost surely.
2576. WHERE DID THE BROWNIAN PARTICLE GO?
Robin Pemantle, Yuval Peres, Jim Pitman, Marc Yor
Consider the radial projection onto the unit sphere of the path a
d-dimensional Brownian motion W, started at the center of the sphere and run
for unit time. Given the occupation measure mu of this projected path, what can
be said about the terminal point W(1), or about the range of the original path?
In any dimension, for each Borel set A subseteq S^{d-1}, the conditional
probability that the projection of W(1) is in A given mu(A) is just mu(A).
Nevertheless, in dimension d>=3, both the range and the terminal point of W can
be recovered with probability 1 from mu. In particular, for d>=3 the
conditional law of the projection of W(1) given mu is not mu. In dimension~2 we
conjecture that the projection of W(1) cannot be recovered almost surely from
mu, and show that the conditional law of the projection of W(1) given mu is not
mu.
2577. A PHASE TRANSITION IN RANDOM COIN TOSSING
David A. Levin, Robin Pemantle, Yuval Peres
Suppose that a coin with bias theta is tossed at renewal times of a renewal
process, and a fair coin is tossed at all other times. Let mu_\theta be the
distribution of the observed sequence of coin tosses, and let u_n denote the
chance of a renewal at time n. Harris and Keane showed that if sum_{n=1}^infty
u_n^2=\infty, then mu_theta and \mu_0 are singular, while if sum_{n=1}^{infty}
u_n^2<infty and theta is small enough, then mu_theta is absolutely continuous
with respect to mu_0. They conjectured that absolute continuity should not
depend on theta, but only on the square-summability of {u_n}. We show that in
fact the power law governing the decay of {u_n} is crucial, and for some
renewal sequences {u_n}, there is a {phase transition at a critical parameter
theta_c in (0,1): for |theta|<theta_c the measures mu_theta$ and mu_0 are
mutually absolutely continuous, but for |theta|>theta_c, they are singular. We
also prove that when u_n=O(n^{-1}), the measures mu_theta for theta in [-1,1]
are all mutually absolutely continuous.
2578. UNIFORM RANDOM SPANNING TREES
Robin Pemantle
There are several good reasons you might want to read about uniform spanning
trees, one being that spanning trees are useful combinatorial objects. Not only
are they fundamental in algebraic graph theory and combinatorial geometry, but
they predate both of these subjects, having been used by Kirchoff in the study
of resistor networks. This article addresses the question about spanning trees
most natural to anyone in probability theory, namely what does a typical
spanning tree look like?
2579. TREE-INDEXED PROCESSES
Robin Pemantle
This article examines a recent body of work on stochastic processes indexed
by a tree. Emphasis is on the application of this new framework to existing
probability models. Proofs are largely omitted, with references provided.
2580. A DYNAMIC MODEL OF SOCIAL NETWORK FORMATION
Brian Skyrms, Robin Pemantle
We consider a dynamic social network model in which agents play repeated
games in pairings determined by a stochastically evolving social network.
Individual agents begin to interact at random, with the interactions modeled as
games. The game payoffs determine which interactions are reinforced, and the
network structure emerges as a consequence of the dynamics of the agents'
learning behavior. We study this in a variety of game-theoretic conditions and
show that the behavior is complex and sometimes dissimilar to behavior in the
absence of structural dynamics. We argue that modeling network structure as
dynamic increases realism without rendering the problem of analysis
intractable.
2581. CYCLES IN RANDOM K-ARY MAPS AND THE POOR PERFORMANCE OF RANDOM RANDOM
NUMBER GENERATION
Robin Pemantle
Knuth shows that iterations of a random function perform poorly on average as
a random number generator. He proposes a generalization in which the next value
depends on two or more previous values. This note demonstrates, via an analysis
of the cycle length of a random k-ary map, the equally poor performance of a
random instance in Knuth's more general model.
2582. A PROBABILISTIC MODEL FOR THE DEGREE OF THE CANCELLATION POLYNOMIAL IN
GOSPER'S ALGORITHM
Robin Pemantle
Milenkovic and Compton in 2002 gave an analysis of the run time of Gosper's
algorithm applied to a random input. The main part of this was an asymptotic
analysis of the random degree of the cancellation polynomial c(k) under various
stipulated laws for the input. Their methods use probabilistic transform
techniques. Here, a more general classof input distributions is considered, and
limit laws of the type proved by Milenkovic and Compton are shown to follow
from a general functional central limit theorem. The methods herein are
probabilistic and elementary and may be used to compute the means of the
limiting distributions.
2583. WHAT IS THE PROBABILITY OF INTERSECTING THE SET OF BROWNIAN DOUBLE
POINTS?
Robin Pemantle, Yuval Peres
We give potential theoretic estimates for the probability that a set A
contains a double point of planar Brownian motion run for unit time. Unlike the
probability for A to intersect the range of a Markov Process, this cannot be
estimated by a capacity of the set A. Instead, we introduce the notion of a
capacity with respect to two gauge functions simultaneously. We also give a
polar decomposition of A into a set that never intersects the set of Brownian
double points and a set for which intersection with the set of Brownian double
points is the same as intersection with the Brownian path.
2584. NETWORK FORMATION BY REINFORCEMENT LEARNING: THE LONG AND MEDIUM RUN
Robin Pemantle, Brian Skyrms
We investigate a simple stochastic model of social network formation by the
process of reinforcement learning with discounting of the past. In the limit,
for any value of the discounting parameter, small, stable cliques are formed.
However, the time it takes to reach the limiting state in which cliques have
formed is very sensitive to the discounting parameter. Depending on this value,
the limiting result may or may not be a good predictor for realistic
observation times.
2585. TIME TO ABSORPTION IN DISCOUNTED REINFORCEMENT MODELS
Robin Pemantle, Brian Skyrms
Reinforcement schemes are a class of non-Markovian stochastic processes.
Their non-Markovian nature allows them to model some kind of memory of the
past. One subclass of such models are those in which the past is exponentially
discounted or forgotten. Often, models in this subclass have the property of
becoming trapped with probability~1 in some degenerate state. While previous
work has concentrated on such limit results, we concentrate here on a contrary
effect, namely that the time to become trapped may increase exponentially in
1/x as the discount rate, 1-x, approaches~1. As a result, the time to become
trapped may easily exceed the lifetime of the simulation or of the physical
data being modeled. In such a case, the quasi-stationary behavior is more
germane. We apply our results to a model of social network formation based on
ternary (three-person) interactions with uniform positive reinforcement.
2586. DETERMINANTAL PROCESSES WITH NUMBER VARIANCE SATURATION
Kurt Johansson
Consider Dyson's Hermitian Brownian motion model after a finite time S, where
the process is started at N equidistant points on the real line. These N points
after time S form a determinantal process and has a limit as N tends to
infinity. This limting determinantal proceess has the interesting feature that
it shows number variance saturation. The variance of the number of particles in
an interval converges to a limiting value as the length of the interval goes to
infinity. Number variance saturation is also seen for example in the zeros of
the Riemann zeta function (Odlyzko, Berry). The process can also be constructed
using non-intersecting paths and we consider several variants of this
construction. One construction leads to a model which shows a transition from a
non-universal behaviour with number variance saturation to a universal
sine-kernel behaviour as we go up the line.
2587. FINITE AND COUNTABLE INFINITE PRODUCTS OF PROBABILISTIC NORMED SPACES
Bernardo Lafuerza-Guillen
In this work we first give for PN spaces results parallel to those obtained
by Egbert for the product of PM spaces, and generalize results by Alsina and
Schweizer in order to study non-trivial products and the product of $m$--
transforms of several PN spaces.
In addition we present a detailed study of $\alpha$--simple product PN spaces
and, finally, the product topologies in PN spaces which are products of
countable families of PN spaces.
2588. MIXING TIMES FOR RANDOM WALKS ON FINITE LAMPLIGHTER GROUPS
Yuval Peres, David Revelle
Given a finite graph G, a vertex of the lamplighter graph consists of a
zero-one labeling of the vertices of G, and a marked vertex of G. For
transitive graphs G, we show that, up to constants, the relaxation time for
simple random walk in corresponding lamplighter graph is the maximal hitting
time for simple random walk in G, while the mixing time in total variation on
the lamplighter graph is the expected cover time on G. The mixing time in the
uniform metric on the lamplighter graph admits a sharp threshold, and equals
|G| multiplied by the relaxation time on G, up to a factor of log |G|.
For the lamplighter group over the discrete two dimensional torus of
sidelength n, the relaxation time is of order n^2 log n, the total variation
mixing time is of order n^2 log^2 n, and the uniform mixing time is of order
n^4. In dimension d>2, the relaxation time is of order n^d, the total variation
mixing time is of order n^d log n, and the uniform mixing time is of order
n^{d+2}. These are the first examples we know of of finite transitive graphs
with uniformly bounded degrees where these three mixing time parameters are of
different orders of magnitude.
2589. GROWTH OF THE BROWNIAN FOREST
Jim Pitman and Matthias Winkel
Trees in Brownian excursions have been studied since the late 1980s. Forests
in excursions of Brownian motion above its past minimum are a natural extension
of this notion. In this paper we study a forest-valued Markov process which
describes the growth of the Brownian forest. The key result is a composition
rule for binary Galton-Watson forests with i.i.d. exponential branch lengths.
We give elementary proofs of this composition rule and explain how it is
intimately linked with Williams' decomposition for Brownian motion with drift.
2590. LARGE DEVIATIONS FOR A CLASS OF NONHOMOGENEOUS MARKOV CHAINS
Zach Dietz and Sunder Sethuraman
Large deviation results are given for a class of ``perturbed''
time-nonhomogeneous Markov chains on finite state space which includes some
stochastic optimization algorithms.
2591. A MARTINGALE PROOF OF DOBRUSHIN'S THEOREM FOR NON-HOMOGENEOUS MARKOV
CHAINS
Sunder Sethuraman and S.R.S. Varadhan
In 1956, Dobrushin proved a definitive central limit theorem for
non-homogeneous Markov chains. In this note, a shorter and different proof
elucidating more the assumptions is given through martingale approximation.
2592. BI-POISSON PROCESS
Wlodzimierz Bryc and Jacek Wesolowski
We study a two parameter family of processes with linear regressions and
linear conditional variances. We give conditions for the unique solution of
this problem, and point out the connection between the resulting Markov
processes and the generalized convolutions introduced by Bo\.zejko and
Speicher.
2593. TRACKING OF HISTORICAL VOLATILITY
L.Goldentayer, F.Klebaner, R.Liptser
We propose an adaptive algorithm for tracking of historical volatility. The
algorithm is built under the assumption that the historical volatility function
belongs to the Stone-Ibragimov-Khasminskii class of $k$ times differentiable
functions with bounded highest derivative and its subclass of functions
satisfying a differential inequalities. We construct an estimator of the Kalman
filter type and show optimality of the estimator's convergence rate to zero as
sample size $n\to\infty$. This estimator is in the framework of GARCH design,
but a tuning procedure of its parameters is faster than with traditional GARCH
techniques.
2594. HOW NON-GIBBSIANNESS HELPS A METASTABLE MORITA MINIMIZER TO PROVIDE A
STABLE FREE ENERGY
Christof Kuelske
We analyze a simple approximation scheme based on the Morita-approach for the
example of the mean field random field Ising model where it is claimed to be
exact in some of the physics literature. We show that the approximation scheme
is flawed, but it provides a set of equations whose metastable solutions
surprisingly yield the correct solution of the model. We explain how the same
equations appear in a different way as rigorous consistency equations. We
clarify the relation between the validity of their solutions and the almost
surely discontinuous behavior of the single-site conditional probabilities.
2595. COMPOSITIONS OF RANDOM TRANSPOSITIONS
Oded Schramm
Let $Y=(y_1,y_2,...)$, $y_1\ge y_2\ge...$, be the list of sizes of the cycles
in the composition of $c n$ transpositions on the set $\{1,2,...,n\}$. We prove
that if $c>1/2$ is constant and $n\to\infty$, the distribution of $f(c)Y/n$
converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where
the function $f$ is known explicitly. A new proof is presented of the theorem
by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that the PD(1) measure is
the unique invariant measure for the uniform coagulation-fragmentation process.
2596. SHUFFLING BY SEMI-RANDOM TRANSPOSITIONS
Elchanan Mossel and Yuval Peres and Alistair Sinclair
In the cyclic-to-random shuffle, we are given n cards arranged in a circle.
At step k, we exchange the k'th card along the circle with a uniformly chosen
random card. The problem of determining the mixing time of the cyclic-to-random
shuffle was raised by Aldous and Diaconis in 1986. Recently, Mironov used this
shuffle as a model for the cryptographic system known as ``RC4'' and proved an
upper bound of O(n log n) for the mixing time. We prove a matching lower bound,
thus establishing that the mixing time is indeed of order $\Theta(n \log n)$.
We also prove an upper bound of O(n log n) for the mixing time of any
``semi-random transposition shuffle'', i.e., any shuffle in which a random card
is exchanged with another card chosen according to an arbitrary (deterministic
or random) rule. To prove our lower bound, we exhibit an explicit
complex-valued test function which typically takes very different values for
permutations arising from the cyclic-to-random-shuffle and for uniform random
permutations; we expect that this test function may be useful in future
analysis of RC4. Perhaps surprisingly, the proof hinges on the fact that the
function exp(z)-1 has nonzero fixed points in the complex plane. A key insight
from our work is the importance of complex analysis tools for uncovering
structure in nonreversible Markov chains.
2597. INDIFFERENCE PRICING AND HEDGING IN STOCHASTIC VOLATILITY MODELS
M. R. Grasselli and T. R. Hurd
We apply the concepts of utility based pricing and hedging of derivatives in
stochastic volatility markets and introduce a new class of "reciprocal affine"
models for which the indifference price and optimal hedge portfolio for pure
volatility claims are efficiently computable. We obtain a general formula for
the market price of volatility risk in these models and calculate it explicitly
for the case of an exponential utility.
2598. LOGARITHMIC SOBOLEV INEQUALITY FOR ZERO-RANGE DYNAMICS: INDEPENDENCE OF
THE NUMBER OF PARTICLES
Paolo Dai Pra and Gustavo Posta
We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a
box of diameter L may depend on L but not on the number of particles. This is a
first, but relevant and quite technical step, in the proof that this
logarithmic-Sobolev constant grows as L^2, that will be presented in a
forthcoming paper.
2599. INTEGRATION BY PARTS ON THE LAW OF THE REFLECTING BROWNIAN MOTION
Lorenzo Zambotti
We prove an integration by parts formula on the law of the reflecting
Brownian motion $X:=|B|$ in the positive half line, where $B$ is a standard
Brownian motion. In other terms, we consider a perturbation of $X$ of the form
$X^\epsilon = X+\epsilon h$ with $h$ smooth deterministic function and
$\epsilon>0$ and we differentiate the law of $X^\epsilon$ at $\epsilon=0$. This
infinitesimal perturbation changes drastically the set of zeros of $X$ for any
$\epsilon>0$. As a consequence, the formula we obtain contains an infinite
dimensional generalized functional in the sense of Schwartz, defined in terms
of Hida's renormalization of the squared derivative of $B$ and in terms of the
local time of $X$ at 0. We also compute the divergence on the Wiener space of a
class of vector fields not taking values in the Cameron-Martin space.
2600. ITERATED BROWNIAN MOTION IN PARABOLA-SHAPED DOMAINS
Erkan Nane
Iterated Brownian motion $Z_{t}$ serves as a physical model for diffusions in
a crack. If $\tau_{D}(Z) $ is the first exit time of this processes from a
domain $D \subset \RR{R}^{n}$, started at $z\in D$, then $P_{z}[\tau_{D}(Z)>t]$
is the distribution of the lifetime of the process in $D$. In this paper we
determine the large time asymptotics of $P_{z}[\tau_{P_{\alpha}}(Z) > t]$ which
gives exponential integrability of $\tau_{P_{\alpha}}(Z) $ for parabola-shaped
domains of the form $ P_{\alpha}=\{(x,Y)\in \RR{R} \times \RR{R}^{n-1}: x>0,
|Y|<Ax^{\alpha} \}$, for $ 0<\alpha <1$, $A>0.$ We also obtain similar results
for twisted domains in $\RR{R}^{2}$ as defined in \cite{DSmits}. In particular,
for a planar iterated Brownian motion in a parabola $\mathcal{P}=\{(x,y): x>0,
|y|< \sqrt{x} \}$ we find that for $z\in \mathcal{P}$
$$\lim_{t\to\infty} t^{-{1/7}} \log P_{z}[\tau_{\mathcal{P}}(Z) >t]= -
\frac{7 \pi ^{2}}{2^{25/ 7}}. $$
2601. COARSENING, NUCLEATION, AND THE MARKED BROWNIAN WEB
L.R.G. Fontes, M. Isopi, C.M. Newman, K. Ravishankar
Coarsening on a one-dimensional lattice is described by the voter model or
equivalently by coalescing (or annihilating) random walks representing the
evolving boundaries between regions of constant color and by backward (in time)
coalescing random walks corresponding to color genealogies. Asympotics for
large time and space on the lattice are described via a continuum space-time
voter model whose boundary motion is expressed by the {\it Brownian web} (BW)
of coalescing forward Brownian motions. In this paper, we study how small noise
in the voter model, corresponding to the nucleation of randomly colored
regions, can be treated in the continuum limit. We present a full construction
of the continuum noisy voter model (CNVM) as a random {\it quasicoloring} of
two-dimensional space time and derive some of its properties. Our construction
is based on a Poisson marking of the {\it backward} BW within the {\it double}
(i.e., forward and backward) BW.
2602. A NOTE ON THE TOP LYAPUNOV EXPONENT FOR THE ZAKAI EQUATION
P. Chigansky
An exact expression is derived for the top Laypunov exponent of the Zakai
equation, arising in the nonlinear filtering of finite state Markov chains.
2603. A GENERALIZED MODEL OF MUTATION-SELECTION BALANCE WITH APPLICATIONS TO AGING
David Steinsaltz, Steven N. Evans, Kenneth W. Wachter
A probability model is presented for the dynamics of
mutation-selection balance in a haploid infinite-
population infinite-sites setting sufficiently general to
cover mutation-driven changes in full age-specific
demographic schedules. The model accommodates epistatic as
well as additive selective costs. Closed form
characterizations are obtained for solutions in finite
time, along with proofs of convergence to stationary
distributions and a proof of the uniqueness of solutions
in a restricted case. Examples are given of applications
to the biodemography of aging, including instabilities in
current formulations of mutation accumulation.
evans@stat.berkeley.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
2604. BUFFON'S NEEDLE PROBLEM
Harald Schröer
We accidentally let a needle fall on a table. On the table are
drawn parallel lines with a distance. We introduce the probability
that the needle intersects one parallel.
Then we generalize the Buffon's needle problem. We view a general
convex set. We calculate the probability that the needle intersects
or touchs the boundary. We have to solve a two-dimensional integral.
There is an english and a german edition.
uzsjzj@uni-bonn.de
- To see a preprint or other
information provided by the author
click here.
- Or
here.
2605. ADAPTATION WITH SERIES OF MEASUREMENTS
Harald Schröer
Here two new methods, that refine an approximative law of
a series of measurements, are described. These methods can be
used, if the form of the law is changed insignificantly. Later
vector-valued series of measurements are treated, too. In the
nature of science this is important for the following reason:
In known laws, there are often variables that have a certain meaning.
There is an english and a german edition.
uzsjzj@uni-bonn.de
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
2606. LARGE DEVIATIONS FOR PROCESSES WITH DISCONTINUOUS STATISTIC
Irina Ignatiou-Robert
This paper is devoted to the problem of sample path large deviations
for the Markov processes on an orthant R^N_+ having a constant
but different transition mechanism on each boundary set of the
orthant. The global sample path large deviation principle and
an integral representation of the rate function are derived
from local large deviation estimates. Our results complete
the proof of Dupuis and
Ellis of the sample path large deviation principle for
Markov processes describing a general class of queueing networks.
Irina.Ignatiouk@math.u-cergy.fr
2607. HOLDER CONTINUITY OF HARMONIC FUNCTIONS WITH RESPECT TO OPERATORS
OF VARIABLE ORDER
Richard F. Bass and Moritz Kassmann
We consider integro-differential operators associated with jump
processes and their corresponding harmonic functions. Under mild
assumptions on the jump measure we prove a priori estimates
and establish Holder continuity of bounded functions that are
harmonic in a domain.
bass@math.uconn.edu kassmann@math.uconn.edu
- To see a preprint or other
information provided by the author
click here.
2608. MINIMAL SPANNING TREES AND DICKMAN-TYPE DISTRIBUTIONS
Mathew D. Penrose and Andrew Wade
In Bhatt and Roy's minimal directed spanning tree construction
for $n$ random points in the unit square, all edges must be
in a southwesterly direction and there must be a directed path
from each vertex to the root placed at the origin. We identify
the limiting distributions (for large $n$) for the total
length of rooted edges, and also for the maximal length of all
edges in the tree. These limit distributions have been seen
previously in analysis of the Poisson-Dirichlet distribution
and elsewhere; they are expressed in terms of Dickman's
function, and their properties are discussed in some detail.
m.d.penrose@bath.ac.uk
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
2609. EXACT AND APPROXIMATE RESULTS FOR DEPOSITION AND ANNIHILATION
PROCESSES ON GRAPHS
Mathew D. Penrose and Aidan Sudbury
We consider random sequential adsorption processes where
the initially empty sites of a graph are irreversibly
occupied, in random order,either by monomers which block
neighbouring sites, or by dimers. We also consider a
process where initially occupied
sites annihilate their neighbours at random times.
We verify that these processes are well-defined on
infinite graphs, and derive forward equations governing
joint vacancy/occupation probabilities. Using these,
we derive exact formulae for occupation probabilities
and pair correlations in Bethe lattices. For the blocking
and annihilation processes we also prove positive
correlations between sites an even distance apart, and
for blocking we derive rigorous lower bounds for the site
occupation probability in lattices, including a lower bound
of $1/3$ for $Z^2$.We also give normal approximation results
for the number of occupied sites in a large finite graph.
m.d.penrose@bath.ac.uk
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
2610. NORMAL APPROXIMATION IN GEOMETRIC PROBABILITY
Mathew D. Penrose and J. E. Yukich
Statistics arising in geometric probability can often be
expressed as sums of stabilizing functionals, that is
functionals which satisfy a local dependence structure.
In this note we show that stabilization leads to nearly
optimal rates of convergence in the CLT for statistics
such as total edge length and total number of edges of
graphs in computational geometry and the total number of
particles accepted in random sequential packing models.
These rates also apply to the 1-dimensional marginals
of the random measures associated with these statistics.
m.d.penrose@bath.ac.uk
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
2611. ON AN INVARIANVCE PRINCIPLE FOR PHASE SEPARATION LINES
Lev Greenberg Dmitry Ioffe
We prove invariance principles for phase separation lines in the
two dimensional nearest neighbour Ising model up to the
critical temperature and for connectivity
lines in the general context of high temperature finite range
ferromagnetic Ising models.
D.Ioffe@statslab.cam.ac.uk
2612. ONE-DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS
WITH SINGULAR AND DEGENERATE COEFFICIENTS
Richard F. Bass and Zhen-Qing Chen
We show the existence of strong solutions and pathwise uniqueness
for two types of one-dimensional stochastic differential equations.
The first type allows singular drifts:
$$X_t=X_0+ \int_0^t a(X_t) dW_t +\int_{\R} L^w_t(X) \mu(dw),$$
where $L^w$ is the local time at $w$ for the semimartingale $X$.
The second type is the equation
$$dX_t= (X_t)^\al dW_t+dL_t,$$
where $L$ is a continuous non-decreasing process that increases
only when $X$ is at 0, $\al\in (0,\frac12)$, and $X_t\geq 0$ for all $t$.
Although this second equation does not have a unique solution, it does
have a pathwise unique solution if one restricts attention to those solutions
that spend zero time at 0.
bass@math.conn.edu zchen@math.washington.edu
- To see a preprint or other
information provided by the author
click here.
2613. SEMIMARTINGALE ATTRACTORS FOR ALLEN-CAHN SPDEs DRIVEN
BY SPACE-TIME WHITE NOISE I: EXISTENCE AND FINITE DIMENSIONAL ASYMPTOTIC BEHAVIOR
Hassan Allouba and Jose A. Langa
We delve deeper into the study of semimartingale attractors that
we recently introduced. In this article we focus on second order
SPDEs of the Allen-Cahn type. After proving existence, uniqueness,
and detailed regularity results for our SPDEs and for corresponding
random PDEs of Allen-Cahn type; we prove the existence of
semimartingale global attractors for these equations. We also give
some results on the finite dimensional asymptotic behavior of the solutions.
In particular, we show the finite fractal dimension of these random attractors;
and we give a result on determining modes, both in the forward and the
pullback senses.
allouba@math.kent.edu langa@us.es
- To see a preprint or other
information provided by the author
click here.
2614. THE RANDOM INTEGRAL REPRESENTATION HYPOTHESIS REVISTED: NEW CLASSES OF
S-SELFDECOMPOSABLE LAWS
Zbigniew J. Jurek
The class of s-selfdecomposable laws is defined using some
non-linear shrinking operations (in short: s-operation). It coincides with
a class of limit distributions. Here we introduce a continuos family of
subclasses of the class of s-selfdecomposable laws. Each class is represented
as a class of some random integrals and is described via characteristic
functionals. Furthermore, their spectral Levy functions satisfy some
differential equations.
zjjurek@math.uni.wroc.pl
2615. SELFDECOMPOSABLE LAWS ASSOCIATED WITH HYPERBOLIC FUNCTIONS
Zbigniew J. Jurek and Marc Yor
It is shown that the hyperbolic sine and cosine functions can be
assocoated with selfdecomposable characteristic functions. Consequently, they
admit associated with them background driving prosess $Y$ ( BDLP $Y$). We
interpret the distribution of $Y(1)$ via bessel square processes, Besel
bridges and local times. Our arguments rely on the Shiga-Watanabe Theorem.
zjjurek@math.uni.wroc.pl
