Probability Abstracts 36
This document contains abstracts 771--803.
They have been mailed on December 2, 1996.
771. THE PROBABILITY THAT BROWNIAN MOTION ALMOST CONTAINS A LINE
Robin Pemantle
Let $R_\epsilon$ be the Wiener sausage of radius $\epsilon$
about a planar Brownian motion run to time 1. Let $P_\epsilon$ be
the probability that $R_\epsilon$ contains the line segment $[0,1]$
on the $x$-axis.
Upper and lower estimates are given for $P_\epsilon$. The upper
estimate implies that the range of a planar Brownian motion
contains no line segment. The lower estimate is order
$$ \exp (- C \log^2 \epsilon) $$
and the upper estimate is similar but with a poly-loglog correction
in the denominator of the exponent.
pemantle@math.wisc.edu
772. LARGE DEVIATIONS AND CONTINUUM LIMIT IN THE 2D ISING MODEL
C.E. Pfister and Y. Velenik
We study the 2D Ising model in a rectangular box of linear size O(L). We
determine the exact asymptotic behaviour of the large deviations of the
magnetization for values of the parameters of the model corresponding to the
phase coexistence region, where the order parameter is strictly positive. We
study in particular boundary effects due to an arbitrary real-valued boundary
magnetic field. Using the self-duality of the model a large part of the
analysis consists in deriving properties of the covariance function at dual
values of the parameters of the model. To do this analysis we establish new
results about the high--temperature representation of the model. These results
are valid for dimensions greater or equal 2 and up to the critical temperature.
We construct the continuum limit of the model and describe the limit by the
solutions of a variational problem of isoperimetric type, similar but not
identical to the Wulff construction.
cpfister@eldp.epfl.ch velenik@eldp.epfl.ch
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information provided by the author
click here.
773. SETS AVOIDED BY BROWNIAN MOTION
Omer Adelman, Krzysztof Burdzy and Robin Pemantle
A fixed 2-dimensional projection of a 3-dimensional Brownian
motion is almost surely neighborhood recurrent; is this
simultaneously true of all the 2-dimensional projections
with probability one? Equivalently: 3-dimensional Brownian
motion hits any infinite cylinder with probability one;
does it hit all cylinders? This papers shows that the answer
is no. Brownian motion in three dimensions avoids random
cylinders and in fact avoids bodies of revolution that
grow almost as fast as cones.
burdzy@math.washington.edu, pemantle@math.wisc.edu
774. DIFFUSIONS ON PATH AND LOOP SPACES: EXISTENCE, FINITE
DIMENSIONAL APPROXIMATION AND HOELDER CONTINUITY
Andreas Eberle
We construct Ornstein-Uhlenbeck processes and more general diffusion processes
on path and loop spaces of Riemannian manifolds by finite dimensional approxi-
mation. We also show Hoelder continuity of the sample paths w.r.t. the supre-
mum norm. The proofs are based on the Lyons-Zheng decomposition.
eberle@mathematik.uni-bielefeld.de
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information provided by the author
click here.
775. ONE-DIMENSIONAL LOSS NETWORKS AND CONDITIONED $M/G/\infty$ QUEUES
Pablo A. Ferrari and Nancy Lopes Garcia
We study one dimensional continuous loss networks with length
distribution $G$ and cable capacity $C$. We prove that the unique stationary
distribution $\eta_L$ of the network for which the restriction on the number
of calls to be less than $C$ is imposed only in the segment $[-L,L]$ is the
same as the distribution of a stationary $M/ G / \infty$ queue conditioned
to be less than $C$ in the time interval $[-L,L]$. For distributions $G$
which are of phase-type ($=$ absorbing times of finite state Markov
processes) we show that the limit as $L\to\infty$ of $\eta_L$ exists and is
unique. The limiting distribution turns out to be invariant for the infinite
loss network. This was conjectured by Kelly (1991).
pablo@ime.usp.br
776. A SUPER-BROWNIAN MOTION WITH A LOCALLY INFINITE CATALYTIC MASS
Klaus Fleischmann and Carl Mueller
A super-Brownian motion $X$ in $\bf{R}$ with "hyperbolic" branching
rate $\rho(x)=x^{-2}$ is constructed. Heuristically, the process
could be described by the equation
$dX=(1/2)\Delta Xdt+\sqrt{2\rho X}dW$, where $W=W(t,x)$ is
2-parameter white noise. We show that if the process starts with
mass 1 concentrated at the point $x=a$, then no mass can ever reach
the catalytic center $x=0$. This phenomenon could never occur with
the usual superprocesses considered so far by Dynkin and others.
Our proof relies on the historical process, which keeps track of
the historical paths of the catalytic process.
fleischmann@wias-berlin.de (postscript file available)
777. THE WILLS FUNCTIONAL AND GAUSSIAN PROCESSES
R.A. Vitale
We show that the Wills functional from the theory of
lattice point enumeration has a surprising, and apparently
deep, connection with Gaussian processes. Together with the
Alexandrov--Fenchel inequality for convex bodies, it yields
the following exponential bound for mean-zero Gaussian processes
$$Ee^{\sup_t(X_t - (1/2)\sigma_t^2)}\leq e^{E\sup_tX_t},$$
where $\sigma_t^2=EX_t^2$. An application is a new proof of the
deviation inequality for the supremum of a Gaussian process
above its mean
$$P(\sup_t X_t-E\sup_tX_t \geq a)\leq e^{-(1/2)a^2/\sigma^2},$$
where $a>0$ and $\sigma^2=\sup_t \sigma_t^2$.
rvitale@uconnvm.uconn.edu (hardcopy or LaTeX file available)
778. AVOIDING-PROBABILITIES FOR BROWNIAN SNAKES
AND SUPER-BROWNIAN MOTION
Romain Abraham and Wendelin Werner
We investigate the asymptotic behaviour of the probability that
a normalized $d$-dimensional Brownian snake (for instance when
the life-time process is an excursion of height 1)
avoids 0 when starting at distance $\eps$ from the origin.
In particular we show that when $\eps$ tends to 0,
this probability respectively behaves (up to multiplicative constants) like
$\eps^4$, $\eps^{2\sqrt{2}}$ and $\eps^{(\sqrt {17}-1)/2}$,
when $d=1$, $d=2$ and $d=3$.
Analogous results are derived for super-Brownian motion
started from $\delta_x$ (conditioned to survive until some time) when
the modulus of $x$ tends to 0.
abraham@math-info.univ-paris5.fr, wwerner@dmi.ens.fr
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information provided by the author
click here.
779. INDEPENDENCE AND DETERMINATION OF PROBABILITIES
Zhiqiang Chen, Herman Rubin, and Richard A. Vitale
Suppose that $P$ and $Q$ are two probability measures on the same
measurable space. We say that they {\it have identical independent
events} if, for any pair of events $A$ and $B$,
$P(AB) = P(A)P(B)$ if and only if $Q(AB) = Q(A)Q(B)$. Two such
probability measures may be quite unrelated, as, for example,
in simple point mass cases. A natural question then arises: ``What
properties, if any, are shared by two probability measures with
identical independent events?'' We show somewhat unexpectedly that
the probability measures must actually {\it coincide} when point
masses are disallowed, or, more precisely, in the nonatomic case.
rvitale@uconnvm.uconn.edu (Hard copy or LaTeX file available)
780. LE THEOREME DE RAY-KNIGHT A TEMPS FIXE
Christophe Leuridan
Let $B$ be a brownian motion in ${\bf R}$ staring at 0, and
$(L_t^x)$ its local times. We give the joint law of $B_{\tau}$
and $(L_{\tau}^x)_{x in {\bf R}}$ when $\tau$ is a fixed time
$t$, and when $\tau$ is an inverse time of sojourn time of $B$
in ${\bf R}_+$.
leuridan@fourier.ujf-grenoble.fr
781. CROSSINGS AND OCCUPATION MEASURES FOR A CLASS
OF SEMIMARTINGALES
Gonzalo Perera and Mario Wschebor
Let $W$ be a Brownian Motion,
$X_t =\int_0^t a_s dW_s +\int_0^t b_s ds $ with
$a$ and $b$ regular and adapted processes and $X_\varepsilon$
the approximation of $X$ by convolution with a smooth kernel.
If $N_t^g(u)$ stands for the number of crossings
of the function $g$ with the level $u$ in $(0,t]$
and $f$ is smooth, we show that the asymptotic distribution
(for $\varepsilon$ tending to $0$) of
$\frac{1}{\sqrt{\varepsilon}} \{ \int\limits_{-\infty}^\infty
f(u) k_{\varepsilon}N_\tau ^{X_\varepsilon}(u)du -\int_0^\tau
f(X_t) a_t dt \}$ is$$c \int_0^{\tau} f(X_t) a_t \; dB_t $$
with $B$ a Brownian Motion independent of $W$
and $c$ a constant depending on the kernel.
perera@cristal.math.u-psud.fr
782. ZENO'S WALK: A RANDOM WALK WITH REFINEMENTS
David Steinsaltz
A one-dimensional self-modifying random walk is defined as follows: We
begin with a random walk on the integers, starting at 0 and going right or
left with probability p and 1-p. The unusual feature is that at each step
a new vertex is interpolated at the midpoint of the edge just crossed. For
p strictly between 1/3 and 2/3 the process converges almost surely to a
real-valued random variable whose fractal-like distribution may be
precisely described. In particular, we may compute the Hausdorff dimension
of its support, which is less than 1. By generating function techniques we
may also calculate the exponential rate of convergence of the process to
its limit point, which serves as a bound for the convergence of the measure
in the Wasserstein metric.
The process is equivalent to the following random dynamical system: we pick
sequentially at random from the following two piecewise-linear functions:
f^+(x)={x+2 if x<-2; x/2+1 if -2<=x<=0; x+1 if x>0}, and f^-(x)={x-1 if
x<0; x/2-1 if 0<=x<=2; x-2 if x>2}. The choices are i.i.d., with
probability p of choosing f^+ and 1-p of choosing f^-. The function f_n is
formed by composing the sequence from left to right in the order in which
they were chosen. Then the sequence f_n(0) has the distribution of the
random walk defined above.
dstein@math.tu-berlin.de
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information provided by the author
click here.
783. MIXING PROPERTIES OF THE GENERALIZED $T,T^{-1}$--PROCESS
Frank den Hollander and Jeff Steif
Consider a general random walk on $Z^d$ together with an i.i.d. random
coloring of $Z^d$. The $T,T^{-1}$--process is the one where time is indexed
by $Z$, and at each unit of time we see the step taken by the walk together
with the color of the newly arrived at location. S. Kalikow proved that
if $d=1$ and the random walk is simple, then this process is not Bernoulli.
We generalize his result by proving that it is not Bernoulli in $d=2$,
Bernoulli but not Weak Bernoulli in $d=3$ and $4$, and Weak Bernoulli
in $d\ge 5$. These properties are related to the intersection behavior
of the past and the future of simple random walk. We obtain similar results
for general random walks on $Z^d$, leading to an almost complete
classification. For example, in $d=1$, if a step of size $x$ has probability
proportional to $1/|x|^\alpha$ ($x \neq 0$), then the $T,T^{-1}$--process is
not Bernoulli when $\alpha\ge 2$, Bernoulli but not Weak Bernoulli when
$3/2 \leq \alpha < 2$ and Weak Bernoulli when $1 < \alpha < 3/2$.
denholla@sci.kun.nl steif@math.chalmers.se
784. AN OBSTRUCTION FOR FINITARY CODING OF MARKOV RANDOM FIELDS
Jeff Steif
In this paper, we show that there is an obstruction for the existence
of a finitary isomorphism from an i.i.d. process to a large class of Markov
random fields, many of which are Bernoulli Shifts. This obstruction is that
there is a phase transition in the sense that the given Markov random field
is not the unique Markov random field with its conditional probabilities.
This obstruction also prevents the existence of a finitary homomorphism from
an i.i.d. process onto the given Markov random field. This obstruction will
immediately be obtained by combining known results in ergodic theory and
statistical mechanics. This will also be generalized to general Gibbs states
(with possibly infinite range interactions).
steif@math.chalmers.se
785. CONSISTENT ESTIMATION OF PERCOLATION QUANTITIES
Ronald Meester and Jeff Steif
In this paper, we consider ordinary independent bond percolation and construct
consistent estimators for most of the important quantities associated to this
model.
meester@math.ruu.nl steif@math.chalmers.se
786. PROBABILITY ON TREES
Russell Lyons
This book is still being written. Most parts that are available (in
PostScript) are in close-to-finished form, but some are definitely in
progress. The book will be published by Cambridge University Press,
perhaps around January 1998. I expect Yuval Peres to join as a coauthor.
Chapter Titles:
Some Highlights
Random Walks and Electric Networks
Percolation
Hausdorff Dimension
Capacity
Random Spanning Trees
Limit Theorems for Galton-Watson Processes
Speed of Random Walks
Harmonic Measure on Galton-Watson Trees
Tree-Indexed Processes (to be written)
Ising Models and Recursions (to be written)
Fractal Percolation (to be written)
Target Percolation (to be written)
Isoperimetric Inequalities (to be written)
Comments on Exercises
Bibliography
Glossary of Notation
Index (to be written)
rdlyons@indiana.edu (Please pick up the book from the url.)
- To see a preprint or other
information provided by the author
click here.
787. SPECTRAL REGULARIZATION INEQUALITIES
Mikhail Lifshits and Michel Weber
We develop a recent idea of spectral regularization introduced
by M.Talagrand in the study of covering numbers of averages of
contractions in a Hilbert space. We show that this idea can be
concentrated in one inequality, which turns out to be a suitable
tool for the study of other characteristics of the set of averages
and yields many useful corollaries. In particular, we recover
recent results for the Littlewood-Paley square functions and
oscillation in ergodic theory due to R.Jones et al. We also easily
deduce original Talagrand's estimate of covering numbers and
provide better estimates for geometric subsequences of the
averages. Similar results for the bilateral Hilbert transform
are also obtained. In the last section, we prove a new criterion
of the a.s. convergence of random sequences under suitable
incremental conditions. Combining this criterion with our
inequalities, we obtain as a corollary the classical theorem of
Rademacher-Menshov on orthogonal series and Gaposhkin spectral
criterion for the strong law of large numbers for weakly
stationary processes.
mikhail@lifshits.spb.su, weber@math.u-strasbg.fr
788. ERGODICITY FOR THE STOCHASTIC DYNAMICS OF QUASI--INVARIANT MEASURES
WITH APPLICATIONS TO GIBBS STATES
Sergio Albeverio, Yuri G. Kondratiev, Michael R\"ockner
The convex set $\calM^a$ of quasi--invariant measures on a locally convex space
$E$ with given ''shift''--Radon-Nikodym derivatives (i.e., cocycles)
$a=(a_{tk})_{k\in K_0,\, t\in\bbbr}$ is analyzed. The extreme points of
$\calM^a$ are characterized and proved to be non-empty. A specification (of
lattice type) is constructed so that $\calM^a$ coincides with the set of the
corresponding Gibbs states. As a consequence, via a well-known method due to
Dynkin--F\"ollmer a unique representation of an arbitrary element in $\calM^a$
in terms of extreme ones is derived. Furthermore, the corresponding classical
Dirichlet forms $(\calE_\nu,D(\calE_\nu))$ and their associated semigroups
$(T_t^\nu)_{t>0}$ on $L^2(E;\nu)$ are discussed. Under a mild positivity
condition it is shown that $\nu\in\calM^a$ is extreme if and only if
$(\calE_\nu,D(\calE_\nu))$ is irreducible or equivalently,
$(T_t^\nu)_{t>0}$ is
ergodic. This implies time--ergodicity of associated
diffusions. Applications to Gibbs states of classical and quantum lattice
models as well as those occuring in Euclidean quantum field theory are
presented. In particular, it is proved that the stochastic quantization of a
Guerra--Rosen--Simon Gibbs state on $\calD '(\bbbr^2)$ in
{\em infinite volume} with polynomial interaction is ergodic, if the Gibbs
state is extreme (i.e., is a
pure phase) which solves a long-standing open problem.
roeckner@Mathematik.Uni-Bielefeld.de
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information provided by the author
click here.
789. DIFFERENTIAL GEOMETRY OF POISSON SPACES
S. Albeverio, Yu.G. Kondratiev, M.R\"ockner
We identify the differential geometry of the Poisson measure over
$\R^{d}$. We prove an integration by parts formula and determine explicitly
the corresponding gradient, divergence, Dirichlet form, generalized
Laplacian and heat semigroup. We also briefly
describe immediate generalizations, for example to Poisson measures over
(infinite dimensional) manifolds or other random fields (resp. Gibbs
measures).
roeckner@Mathematik.Uni-Bielefeld.de
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information provided by the author
click here.
790. CANONICAL DIRICHLET OPERATOR AND DISTORTED BROWNIAN MOTION
ON POISSON SPACES
S. Albeverio, Yu.G. Kondratiev, M.R\"ockner
We derive the precise relation of the {\it internal} differential geometry
of Poisson spaces developed in \cite{AKR} to the corresponding {\it
external} geometry given by the associated Fock structure. In particular,
we identify the (internal) Laplace-Beltrami operator $H_{\pi_{\s}}^{\G}$
on Poisson space as the second quantization of the Laplacian on $\R^{d}$.
Furthermore, the diffusion process generated by
$H_{\pi_{\s}}^{\G}$, which is a kind of distorted Brownian motion on the
configuration space, is described explicitly.
All this generalizes to the case where the underlying
Euclidean space $\R^{d}$ is replaced by an (infinite-dimensional) manifold.
roeckner@Mathematik.Uni-Bielefeld.de
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information provided by the author
click here.
791. UNIQUENESS OF GIBBS STATES ON LOOP LATTICES
S. Albeverio, Yu.G. Kondratiev, M. R\"ockner, T.V. Tsikalenko
We prove uniqueness of Gibbs states for certain quantum lattice systems with
unbounded spins, based on Dobrushin's fundamental uniqueness
criterion. The necessary
estimates for the Vasershtein distance between one-point conditional
distributions, with boundary conditions differing only at one site, are obtained
from the Log-Sobolev inequality which holds on the infinite dimensional single
spin ($=$ loop) spaces. Detailed proofs are contained in \cite{AKRT1} resp.
\cite{AKPT2}. The latter deals with the more general
case of nonharmonic pair potentials with possibly infinite radius of
interaction.
roeckner@Mathematik.Uni-Bielefeld.de
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information provided by the author
click here.
792. DOBRUSHIN'S UNIQUENESS FOR QUANTUM LATTICE SYSTEMS WITH
NONLOCAL INTERACTION
S. Albeverio, Yu.G. Kondratiev, M. R\"ockner, T.V. Tsikalenko
Based on Dobrushin's fundamental criterion, we prove uniqueness of Euclidean
Gibbs states for a certain class of quantum lattice systems with unbounded
spins, nonharmonic pair potentials and infinite radius of interaction. The
necessary estimates on Dobrushin's coefficients are obtained from the
Log-Sobolev inequality which holds for the one-point conditional
distributions on the infinite dimensional single spin ( $=$ loop) spaces.
roeckner@Mathematik.Uni-Bielefeld.de
793. CENTRAL LIMIT THEOREM FOR A WEAKLY INTERACTING RANDOM POLYMER
R. van der Hofstad, F. den Hollander and W. K\"onig
The Domb-Joyce model in one dimension is a transformed path measure
for simple random walk on $\Z$ in which an $n$-step path gets a penalty
$e^{-2\beta_n}$ for every self-intersection. Here $\beta_n$ is the strength
of repellence, which may depend on $n$. We prove a central limit theorem
for the end-to-end distance of the path in the case where
$\beta_n \rightarrow 0$
and $n^{\frac32} \beta_n \rightarrow \infty$ as $n \rightarrow \infty$. It
turns out that the mean grows like $b^* \beta_n^{\frac13} n$ and the standard
deviation like $c^* \sqrt{n}$, where $b^*$ and $c^*$ are constants that can be
identified in terms of a Sturm-Liouville problem. The asymptotic mean shows an
interpolation between ballistic behavior ($\beta_n \equiv \beta$) and
diffusive behavior ($\beta_n = \beta n^{-\frac 32}$).
Strikingly, the asymptotic standard
deviation is {\it independent} of $\beta_n$. Our result is closely related to
the central limit theorem for the Edwards model (the continuous space-time
analogue of the Domb-Joyce model based on Brownian motion on $\R$),
which is proved in a separate paper.
koenig@sci.kun.nl (ps-file available)
794. A SCALING LIMIT PROCESS FOR THE AGE-REPRODUCTION
STRUCTURE IN A MARKOV POPULATION
A. Bose and I. Kaj
The purpose of this paper is to introduce a model for a reproducing
Markov population, set within terms of a fairly general Markovian
framework and with arbitrary offspring mechanism, and study the time
evolution of the joint empirical distribution of age and reproduction
numbers. A possible interpretation is that the model describes the
evolving status of messages in a typical e-mail reader.
The particular objective is to apply a diffusion approximation scaling
as the population size grows and establish the existence of a non-
degenerate measure-valued Markov process in the scaling limit. In
general the limit process is <em>not</em> a superprocess. Still it
may be characterized by its log-Laplace function being the unique
solution of a non-linear integral equation.
ikaj@math.uu.se
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information provided by the author
click here.
795. SUDAKOV MINORATION PRINCIPLE AND SUPREMUM OF SOME PROCESSES
Rafa{\l} Lata{\l}a
Let $X_{i}$ be a sequence of independent symmetric real random variables with
logarithmically concave tails (i.e. such that the functions
$N_{i}(x)=-\ln P(|X_{i}|\geq x$) are convex. We find a new version of Sudakov
minoration principle for estimating from below $E\sup_{t\in T}\sum t_{i}X_{i}$.
If we moreover assume that the functions $N_{i}$ are of moderate growth (i.e.
N_{i}(2x)\leq CN_{i}(x) for all $i$ and $x\geq 0$), then this enables us
to give geometric conditions for set $T$, which are equivalent to boundedness
of the process $(\sum t_{i}X_{i})_{t\in T}$. This generalize previous results
of Talagrand, proved in the case of $N_{i}(x)=x^{p}$ for all $i$ and some
$1\leq p<\infty$.
rlatala@mimuw.edu.pl (LaTeX file available)
796. LARGE DEVIATIONS FOR A RANDOM WALK IN DYNAMICAL RANDOM ENVIRONMENT
Irina Ignatiouk-Robert
We consider a random walk $X_t\in \Z^d, t\in \Z_+$, in a dynamical
random environment $(\xi_t(x), x\in \Z^d), t\in \Z_+$, with a mutual
interaction with each other. The Markov process $(X_t,\xi_t(x),x\in
\Z^d)$ is a perturbation of a process for which the random walk $X_t$
and the
environment $\xi_t(x),x\in \Z^d$ are independent, $X_t, t\in \Z_+$ is
a homogeneous random walk in $\Z^d$ and the environment $
\xi_t(x),x\in \Z^d$ evolves independently in each site as an ergodic
Markov chain. For the perturbated process we assume that
\item The interaction between the position of the particle and the
environment is local;
\item The influence of the environment on the particle $X_{t}$ is
small;
\item The particle modifies the environment of its location (it
cancels the memory of the environment).
We consider a large deviation problem for the random walk $X_t, t\in
\Z_+$. We prove that a large deviation principle holds for this random
walk with a good rate function which is analytic with respect to the
parameter of interaction in a neighborhood of 0.
Irina.Ignatiouk@u-cergy.fr
797. ALMOST SURE CENTRAL LIMIT THEOREM FOR DIRECTED
POLYMERS AND RANDOM CORRECTIONS
C.Boldrighini, R.A. Minlos, A.Pellegrinotti
We consider a model of directed plymers on the lattice
$\zn$, depending on a small parameter $\ep$ . We prove that
if $\nu\ge 3$, for $\ep$ small enough, the central limit
theorem holds for the random walk associated to the
plymer model, as the natural parameter T goes to
infinity, almost surely with respect to the possible
histories of the field (environment), and that the
asympotics corresponding to the local central limit
theorem is given by an explicit functional of the
"backward field". We also prove that the random
correction of order $\frac {1}{\sqt }$ to the average
value of the displacement for $\nu\ge 5$ and of order
$\frac {1}{ T}$ to the covariance matrix for $\nu\ge 7$
are given by some functionals of the field. This allows
to determine the random corrections to the central limit
theorem of order $\frac {1}{\sqt}$ for $\nu\ge 5$ and
of order $\frac {1}{ T}$ for $\nu\ge 7$.
pellegri@axcasp.caspur.it pellegri@matrm3.mat.uniroma3.it
798. A SCALING LIMIT FOR QUEUES IN SERIES
Timo Seppalainen
We derive a law of large numbers for a tagged particle
in the totally asymmetric simple exclusion process under
a scaling different from the usual Euler scaling. By
interpreting the particles as the servers of a series
of queues we use this result to verify an open conjecture
about the scaling behavior of the departure times from a
long series of queues.
seppalai@iastate.edu (AMS-TeX file and hardcopy available)
799. EXACT LIMITING SHAPE FOR A SIMPLIFIED MODEL OF
FIRST-PASSAGE PERCOLATION ON THE PLANE
Timo Seppalainen
We derive the limiting shape for the following model of
first-passage bond percolation on the two-dimensional
integer lattice: The percolation is directed in the
sense that admissible paths are nondecreasing in both
coordinate directions. The passage times of horizontal
bonds are Bernoulli distributed, while the passage times
of vertical bonds are all equal to a deterministic constant.
To analyze the percolation model we couple it with a
one-dimensional interacting particle system. This particle
process has nonlocal dynamics in the sense that the movement
of any given particle can be influenced by far-away particles.
We prove a law of large numbers for a tagged particle in this
process, and the shape result for the percolation is obtained
as a corollary.
seppalai@iastate.edu (Postscript, Ams-Tex file and hardcopy available)
800. FAST EQUILIBRIUM SELECTION BY RATIONAL PLAYERS
LIVING IN A CHANGING WORLD
Krzysztof Burdzy, David M. Frankel and Ady Pauzner
This is an economics paper with a substantial
mathematical component. The mathematical part is
concerned with the equation
- k X(t) if B(t) > f(X(t))
dX/dt =
k (1 - X(t)) if B(t) < f(X(t)).
Here B(t) is a Brownian motion, k is a fixed
positive constant, and f is a given increasing
Lipshitz function. Existence and uniqueness of
the solutions is proved. The main results needed
to solve the economics model are asymptotic estimates
of the bifurcation time and bifurcation direction.
burdzy@math.washington.edu (Latex file and hardcopy available)
- To see a preprint or other
information provided by the author
click here.
801. ASYMPTOTIC BEHAVIOR OF HITTING RATES FOR ABSORBING STABLE
MOTIONS IN A HALF SPACE
Seiji Hiraba
An "Absorbing Stable Motion" in a half space is the time changed process
of an absorbing Brownian motion in a half space
by an increasing stable process in a half line, which are independent.
This process is different from absorbing stable processes considered
in usual sense, i.e,, the killed processes when the original processes
in the hole space jump into the other half space.
Because the absorbing stable motion will be dead
even if the path doesn't jump into the other half space,
when the absorbing Brownian motion is killed.
We consider the stationary Kuznetsov measure associated with
the absorbing stable motion and the canonical process under the measure.
We shall give the asymptotic behavior of a hitting rate
by a time point for a compact subset in the half space
under the measure as the time goes to infinity.
Moreover by applying the obtained result to the equilibrium process
with immigration associated with the stationary Kuznetsov measure
we shall also give some limit theorems related with the number of
particles hitting the compact subset by a time point
as the time goes to infinity.
hiraba@sci.osaka-cu.ac.jp
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802. RATE ESTIMATION FOR THE WEAK CONVERGENCE OF DIFFUSION PROCESSES
Weian Zheng
Let $L=\nabla (A\nabla )$ and $\tilde L=\nabla (\tilde A\nabla )$ be two
uniformly elliptic operators with measurable coefficients $A=(a^{ij})$ and
$\tilde A=(\tilde a^{ij})$ on $R^2,$ and let $(X_t)$ and $(\tilde X_t)$ be
two diffusion processes with $L$ and $\tilde L$ as their generators. We
show that the Kantorovich-Wasserstein's distance of these two processes
are dominated by
$$ C\{\log[2+\log^-(\| A-\tilde A\|_{L^2_{loc}})]\}^{-\gamma }$$
where $\gamma $ and $ C$ are positive constants depending only on the
uniform ellipticity. Our argument works also for higher dimensional case.
In particular, this quantitative result implies the previous qualitative
tightness results for symmetric diffusion processes.
wzheng@uci.edu
803. RANDOM-CLUSTER REPRESENTATIONS IN THE STUDY OF PHASE TRANSITIONS
Olle H\"aggstr\"om
The Fortuin--Kasteleyn random-cluster model has, during the last 10 years,
proved to be a highly useful probabilistic device for studying the phase
transition behaviour of Ising and Potts models. In this survey paper, a
detailed description is given of how this is accomplished. It is then shown
how the same set of ideas can be used to study phase transitions in various
other systems: the Ising model with random interactions, the Ashkin--Teller
model, subshifts of finite type, and certain point processes. The tools
used include coupling, stochastic domination, and percolation.
olleh@math.chalmers.se
