Probability Abstracts 66

This document contains abstracts 1833-1870. They have been mailed on December 31, 2001.

1833. SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS

Dmitry Dolgopyat, Vadim Kaloshin, Leonid Koralov

We consider a stochastic flow driven by a finite dimensional Brownian motion.
We show that almost every realization of such a flow exhibits strong
statistical properties such as the exponential convergence of an initial
measure to the equilibrium state and the central limit theorem. The proof uses
new estimates of the mixing rates of the multipoint motion.

kaloshin@math.mit.edu

This article is available from the xxx mathematics archive as math.PR/0111011.

1834. FISHER INFORMATION INEQUALITIES AND THE CENTRAL LIMIT THEOREM

Oliver Johnson and Andrew Barron

We give conditions for an O(1/n) rate of convergence of Fisher information
and relative entropy in the Central Limit Theorem. We use the theory of
projections in L2 spaces and Poincare inequalities, to provide a better
understanding of the decrease in Fisher information implied by results of
Barron and Brown. We show that if the standardized Fisher information ever
becomes finite then it converges to zero.

o.johnson@statslab.cam.ac.uk

This article is available from the xxx mathematics archive as math.PR/0111020.

1835. A CONDITIONAL ENTROPY POWER INEQUALITY FOR DEPENDENT VARIABLES

Oliver Johnson

We provide a condition under which a version of Shannon's Entropy Power
Inequality will hold for dependent variables. We provide information
inequalities extending those found in the independent case.

o.johnson@statslab.cam.ac.uk

This article is available from the xxx mathematics archive as math.PR/0111021.

1836. APPROXIMATE DISTRIBUTION OF HITTING PROBABILITIES FOR A FINITE REGULAR MEMBRANE IN 2D

D.S.Grebenkov

Generalizing the well-known relations on characteristic functions on a plane
to the case of a finite regular membrane, we have established the implicit
equations for these functions. After solving the combinatorial problems, we
introduced an approximation allowing us to reduce them to a set of linear
equations for a finite number of unknown functions. Imposing the natural
conditions, we have obtained a system of linear equations which can be solved
for a given membrane. Its solutions can be used to approximate the distribution
of hitting probabilities for a finite regular membrane.
  In order to verify the accuracy of the approximation numerical analysis was
carried out for some particular shapes.

dg@pmc.polytechnique.fr

This article is available from the xxx mathematics archive as math.PR/0111024.

1837. FLUCTUATIONS IN THE COMPOSITE REGIME OF A DISORDERED GROWTH MODEL

Janko Gravner, Craig A. Tracy, Harold Widom

We continue to study a model of disordered interface growth in two
dimensions. The interface is given by a height function on the sites of the
one--dimensional integer lattice and grows in discrete time: (1) the height
above the site $x$ adopts the height above the site to its left if the latter
height is larger, (2) otherwise, the height above $x$ increases by 1 with
probability $p_x$. We assume that $p_x$ are chosen independently at random with
a common distribution $F$, and that the initial state is such that the origin
is far above the other sites. Provided that the tails of the distribution $F$
at its right edge are sufficiently thin, there exists a nontrivial composite
regime in which the fluctuations of this interface are governed by extremal
statistics of $p_x$. In the quenched case, the said fluctuations are
asymptotically normal, while the annealed case they satisfy the appropriate
extremal limit law.

gravner@math.ucdavis.edu

This article is available from the xxx mathematics archive as math.PR/0111036.

1838. SHOT NOISE DISTRIBUTIONS AND SELFDECOMPOSABILITY

Aleksander M.Iksanov, Zbigniew J.Jurek

Stationary (limiting) distributions of shot noise processes, with exponential
response functions, form a large subclass of positive selfdecomposable
distributions that we illustrate by many examples. These shot noise
distributions are described among selfdecomposable ones via the regular
variation at zero of their distribution functions. However, slow variation at
the origin of (an absolutely continuous) distribution function is incompatible
with selfdecomposability and this is shown in three examples.

iksan@unicyb.kiev.ua

This article is available from the xxx mathematics archive as math.PR/0111069.

1839. UNIFORM EXPONENTIAL ERGODICITY OF STOCHASTIC DISSIPATIVE SYSTEMS

Beniamin Goldys and Bohdan Maslowski

We study ergodic properties of stochastic dissipative systems with additive
noise. We show that the system is uniformly exponentially ergodic provided the
growth of nonlinearity at infinity is faster than linear. The abstract result
is applied to the stochastic reaction diffusion equation in $\mathbb R^d$ with
$d\le 3$.

beng@maths.unsw.edu.au

This article is available from the xxx mathematics archive as math.PR/0111143.

1840. APPROXIMATING DISTRIBUTION FUNCTIONS BY ITERATED FUNCTION SYSTEMS

Stefano M. Iacus and Davide La Torre

In this paper an iterated function system on the space of distribution
functions is built. The inverse problem is introduced and studied by convex
optimization problems. Some applications of this method to approximation of
distribution functions and to estimation theory are given.

jago@mclink.it

This article is available from the xxx mathematics archive as math.PR/0111152.

1841. STATISTICAL ANALYSIS OF STOCHASTIC RESONANCE WITH ERGODIC DIFFUSION NOISE

Stefano M. Iacus

A subthreshold signal is transmitted through a channel and may be detected
when some noise -- with known structure and proportional to some level -- is
added to the data. There is an optimal noise level, called stochastic
resonance, that corresponds to the highest Fisher information in the problem of
estimation of the signal. As noise we consider an ergodic diffusion process and
the asymptotic is considered as time goes to infinity. We propose consistent
estimators of the subthreshold signal and we solve further a problem of
hypotheses testing. We also discuss evidence of stochastic resonance for both
estimation and hypotheses testing problems via examples.

jago@mclink.it

This article is available from the xxx mathematics archive as math.PR/0111153.

1842. APPROACH TO FIXATION FOR ZERO-TEMPERATURE STOCHASTIC ISING MODELS ON THE HEXAGONAL LATTICE

Federico Camia, Charles M. Newman, Vladas Sidoravicius

We investigate zero-temperature dynamics on the hexagonal lattice H for the
homogeneous ferromagnetic Ising model with zero external magnetic field and a
disordered ferromagnetic Ising model with a positive external magnetic field h.
We consider both continuous time (asynchronous) processes and, in the
homogeneous case, also discrete time synchronous dynamics (i.e., a
deterministic cellular automaton), alternating between two sublattices of H.
The state space consists of assignments of -1 or +1 to each site of H, and the
processes are zero-temperature limits of stochastic Ising ferromagnets with
Glauber dynamics and a random (i.i.d. Bernoulli) spin configuration at time 0.
We study the speed of convergence of the configuration $\sigma^t$ at time t to
its limit $\sigma^{\infty}$ and related issues.

fc276@scires.nyu.edu

This article is available from the xxx mathematics archive as math.PR/0111170.

1843. CRITICAL RESONANCE IN THE NON-INTERSECTING LATTICE PATH MODEL

Richard W. Kenyon and David B. Wilson

We study the phase transition in the honeycomb dimer model (equivalently,
monotone non-intersecting lattice path model). At the critical point the system
has a strong long-range dependence; in particular, periodic boundary conditions
give rise to a ``resonance'' phenomenon, where the partition function and other
properties of the system depend sensitively on the shape of the domain.

dbwilson@microsoft.com

This article is available from the xxx mathematics archive as math.PR/0111199.

1844. THE GAMBLER'S RUIN PROBLEM IN PATH REPRESENTATION FORM

Oscar Bolina

We consider the classical one-dimensional random walk of a particle on the
right-half real line. We assume that the particle is initially at position x=k,
k > 0, and moves to the right with probability p or to the left with
probability 1-p. We consider that the particle is absorbed at the origin
without fixing the number of steps needed to get there. We calculate the
probability P(x=k) that the particles end up at the origin, given that it
starts at x=k, by means of a geometric representation of this random walk in
terms of paths on a two-dimensional lattice.

bolina@fma.if.usp.br

This article is available from the xxx mathematics archive as math.PR/0111242.

1845. LOGARITHMIC FLUCTUATIONS FOR THE INTERNAL DIFFUSION LIMITED AGGREGATION

Sebastien Blachere

The Internal Diffusion Limited Aggregation (Internal DLA) is a growth model
on an infinite set, associated to a random walk on this set. It was introduced
by Diaconis and Fulton in 1991. Lawler, Bramson, and Griffeath (1992) studied
this model for the simple random walk on $Z^d$. They proved that the limiting
shape of the cluster generated by this model is the trace of the Euclidean
balls. Later (1995), Lawler gave bounds of order $n^{1/3}$ (with logarithmic
corrections) on the fluctuations around this limiting shape. Here, we prove
that the fluctuations are at most logarithmic, using an induction based on
Lawler's proof.

sebastien.blachere@epfl.ch

This article is available from the xxx mathematics archive as math.PR/0111253.

1846. ORNSTEIN-ZERNIKE THEORY FOR THE FINITE RANGE ISING MODELS ABOVE T_C

M.Campanino, D.Ioffe, Y.Velenik

We derive precise Ornstein-Zernike asymptotic formula for the decay of the
two-point function in the general context of finite range Ising type models on
Z^d. The proof relies in an essential way on the a-priori knowledge of the
strict exponential decay of the two-point function and, by the sharp
characterization of phase transition due to Aizenman, Barsky and Fernandez,
goes through in the whole of the high temperature region T > T_c. As a
byproduct we obtain that for every T > T_c, the inverse correlation length is
an analytic and strictly convex function of direction.

velenik@cmi.univ-mrs.fr

This article is available from the xxx mathematics archive as math.PR/0111274.

1847. RANDOM WALKS ON RANDOMLY ORIENTED LATTICES

Massimo Campanino and Dimitri Petritis

Simple random walks on various types of partially horizontally oriented
regular lattices are considered. The horizontal orientations of the lattices
can be of various types (deterministic or random) and depending on the nature
of the orientation the asymptotic behaviour of the random walk is shown to be
recurrent or transient. In particular, for randomly horizontally oriented
lattices the random walk is almost surely transient.

dimitri.petritis@univ-rennes1.fr

This article is available from the xxx mathematics archive as math.PR/0111305.

1848. NONCOMMUTATIVE EXTENSIONS OF THE FOURIER TRANSFORM AND ITS LOGARITHM

Romuald Lenczewski

We introduce and study noncommutative extensions of the Fourier transform and
its logarithm to the algebra of functions on the free semigroup FS(2) on two
generators with the convolution multiplication. These extensions are new types
of moment and cumulant generating functions, respectively, the latter
corresponding to the cumulants which are additive under the so-called filtered
convolution on the free *-algebra on two generators. This algebra plays the
role of a ``noncommutative plane'' built on the ``classical real line'' and the
``boolean real line''. The restrictions of the cumulant generating function to
the commutative subsemigroups generated by single generators give the logarithm
of the Fourier transform and the K-transform in the boolean case, respectively.
In turn, the moment generating function is a ``semigroup interpolation''
between the Fourier transform and the Cauchy transform. Using suitable weight
function $W$ on the semigroup, both generating functions become elements of the
Banach algebra $l^{1}(FS(2),W)$. The main results of the paper are based on the
new combinatorics developed for the cumulants, the Moebius function and the
moment-cumulant formulas.

lenczew@mazur.im.pwr.wroc.pl

This article is available from the xxx mathematics archive as math.PR/0112019.

1849. THE DIAMETER OF A LONG RANGE PERCOLATION GRAPH

Don Coppersmith, David Gamarnik, Maxim Sviridenko

We consider the following long range percolation model: an undirected graph
with the node set $\{0,1,...,N\}^d$, has edges $(\x,\y)$ selected with
probability $\approx \beta/||\x-\y||^s$ if $||\x-\y||>1$, and with probability
1 if $||\x-\y||=1$, for some parameters $\beta,s>0$. This model was introduced
by Benjamini and Berger, who obtained bounds on the diameter of this graph for
the one-dimensional case $d=1$ and for various values of $s$, but left cases
$s=1,2$ open. We show that, with high probability, the diameter of this graph
is $\Theta(\log N/\log\log N)$ when $s=d$, and, for some constants
$0<\eta_1<\eta_2<1$, it is at most $N^{\eta_2}$, when $s=2d$ and is at least
$N^{\eta_1}$ when $d=1,s=2,\beta<1$ or $s>2d$. We also provide a simple proof
that the diameter is at most $\log^{O(1)}N$ with high probability, when
$d<s<2d$, established previously by Berger and Benjamini.

gamarnik@watson.ibm.com

This article is available from the xxx mathematics archive as math.PR/0112029.

1850. ASYMPTOTIC NORMALITY OF KERNEL TYPE DECONVOLUTION ESTIMATORS

A.J. van Es, H.-W. Uh

We derive asymptotic normality of kernel type deconvolution estimators of the
density, the distribution function at a fixed point, and of the probability of
an interval. We consider the so called super smooth case where the
characteristic function of the known distribution decreases exponentially.
  It turns out that the limit behavior of the pointwise estimators of the
density and distribution function is relatively straightforward while the
asymptotics of the estimator of the probability of an interval depends in a
complicated way on the sequence of bandwidths.

vanes@science.uva.nl

This article is available from the xxx mathematics archive as math.PR/0112032.

1851. LAPLACE OPERATORS IN DERHAM COMPLEXES ASSOCIATED WITH MEASURES

S. Albeverio, A. Daletskii, Y. Kondratiev, and Eugene Lytvynov

Let $\Gamma_X$ denote the space of all locally finite configurations in a
complete, stochastically complete, connected, oriented Riemannian manifold $X$,
whose volume measure $m$ is infinite. In this paper, we construct and study
spaces $L^2_\mu\Omega^n$ of differential $n$-forms over $\Gamma_X$ that are
square integrable with respect to a probability measure $\mu$ on $\Gamma_X$.
The measure $\mu$ is supposed to satisfy the condition $\Sigma_m'$ (generalized
Mecke identity) well known in the theory of point processes. On
$L^2_\mu\Omega^n$, we introduce bilinear forms of Bochner and deRham type. We
prove their closabilty and call the generators of the corresponding closures
the Bochner and deRham Laplacian, respectively. We prove that both operators
contain in their domain the set of all smooth local forms. We show that, under
a rather general assumption on the measure $\mu$, the space of all
Bochner-harmonic $\mu$-square integrable forms on $\Gamma_X$ consists only of
the zero form. Finally, a Weitzenb\"ock type formula connecting the Bochner and
deRham Laplacians is obtained. As examples, we consider (mixed) Poisson
measures, Ruelle type measures on $\Gamma_{{\Bbb R}^d}$, and Gibbs measures in
the low activity--high temperature regime, as well as Gibbs measures with a
positive interaction potential on $\Gamma_X$.

lytvynov@wiener.iam.uni-bonn.de

This article is available from the xxx mathematics archive as math.PR/0112055.

1852. DISCRETE SPACINGS

Chris A.J. Klaassen and J. Theo Runnenburg

Consider a string of $n$ positions, i.e. a discrete string of length $n$.
Units of length $k$ are placed at random on this string in such a way that they
do not overlap, and as often as possible, i.e. until all spacings between
neighboring units have length less than $k$. When centered and scaled by
$n^{-1/2}$ the resulting numbers of spacings of length $1, 2,..., k-1$ have
simultaneously a limiting normal distribution as $n\to\infty$. This is proved
by the classical method of moments.

chrisk@science.uva.nl

This article is available from the xxx mathematics archive as math.PR/0112056.

1853. RANDOM WALKS IN RANDOM ENVIRONMENT ON TREES AND MULTIPLICATIVE CHAOS

Mikhail Menshikov and Dimitri Petritis

We study random walks in a random environment on a regular, rooted, coloured
tree. The asymptotic behaviour of the walks is classified for
ergodicity/transience in terms of the geometric properties of the matrix
describing the random environment. A related problem, with only one type of
vertices and quite stringent conditions on the transition probabilities but on
general trees has been considered previously in the literature LyoPem. In the
presentation we give here, we restrict the study of the process on a regular
graph instead of the irregular graph used in LyoPem. The close connection
between various problems on random walks in random environment and the so
called multiplicative chaos martingale is underlined by showing that the
classification of the random walk problem can be drawn by the corresponding
classification for the multiplicative chaos, at least for those situations
where both problems have been solved by independent methods. The chaos
counterpart of the problem we considered here has not yet been solved. The
results we obtain for the random walk problem localise the position of the
critical point. We conjecture that the additional conditions needed for the
chaos problem to have non trivial solutions will be the same as the ones needed
for the random walk to be null recurrent.

dimitri.petritis@univ-rennes1.fr

This article is available from the xxx mathematics archive as math.PR/0112103.

1854. GROWTH FLUCTUATIONS IN A CLASS OF DEPOSITION MODELS

Marton Balazs

We compute the growth fluctuations in equilibrium of a wide class of
deposition models. These models also serve as general frame to several
nearest-neighbor particle jump processes, e.g. the simple exclusion or the zero
range process, where our result turns to current fluctuations of the particles.
We use martingale technique and coupling methods to show that, rescaled by
time, the variance of the growth as seen by a deterministic moving observer has
the form |V-C|*D, where V and C is the speed of the observer and the second
class particle, respectively, and D is a constant connected to the equilibrium
distribution of the model. Our main result is a generalization of Ferrari and
Fontes' result for simple exclusion process. Law of large numbers and central
limit theorem are also proven. We need some properties of the motion of the
second class particle, which are known for simple exclusion and are partly
known for zero range processes, and which are proven here for a type of
deposition models and also for a type of zero range processes.

balazs@math.bme.hu

This article is available from the xxx mathematics archive as math.PR/0112143.

1855. OPTIMAL TAIL ESTIMATES FOR DIRECTED LAST PASSAGE SITE PERCOLATION WITH GEOMETRIC RANDOM VARIABLES

Jinho Baik, Percy Deift, Ken McLaughlin, Peter Miller and Xin Zhou

In this paper, we obtain optimal uniform lower tail estimates for the
probability distribution of the properly scaled length of the longest up/right
path of the last passage site percolation model considered by Johansson in
[12]. The estimates are used to prove a lower tail moderate deviation result
for the model. The estimates also imply the convergence of moments, and also
provide a verification of the universal scaling law relating the longitudinal
and the transversal fluctuations of the model.

jbaik@math.princeton.edu

This article is available from the xxx mathematics archive as math.PR/0112162.

1856. ENHANCED INTERFACE REPULSION FROM QUENCHED HARD-WALL RANDOMNESS

Daniela Bertacchi, Giambattista Giacomin

We consider the harmonic crystal on the d-dimensional lattice, d larger or
equal to 3, that is the centered Gaussian field $\phi$ with covariance given by
the Green function of the simple random walk on $Z^d$. Our main aim is to
obtain quantitative information on the repulsion phenomenon that arises when we
condition the field to be larger than an IID field $\sigma$ (which is also
independent of $\phi$), for every x in a large region $D_N=ND\cap \Z^d$, with N
a positive integer and $D \subset\R^d$. We are mostly motivated by results for
given typical realizations of the $\sigma$ (quenched set-up), since the
conditioned harmonic crystal may be seen as a model for an equilibrium
interface, constrained not to go below a inhomogeneous substrate that acts as a
hard wall. We consider various types of substrate and we observe that the
interface is pushed away from the wall much more than in the case of a flat
wall as soon as the upward tail of $\sigma$ is heavier than Gaussian, while
essentially no effect is observed if the tail is sub--Gaussian. In the critical
case, that is the one of approximately Gaussian tail, the interplay of the two
sources of randomness, $\phi$ and $\sigma$, leads to an enhanced repulsion
effect of additive type.

bertacchi@matapp.unimib.it

This article is available from the xxx mathematics archive as math.PR/0112225.

1857. A GENERAL PROOF OF THE DYBVIG-INGERSOLL-ROSS-THEOREM: LONG FORWARD RATES CAN NEVER FALL

Friedrich Hubalek, Irene Klein, Josef Teichmann

A general proof of the Dybvig-Ingersoll-Ross Theorem on the monotonicity of
long forward rates is presented. Some inconsistencies in the original proof of
this theorem are discussed.

josef.teichmann@fam.tuwien.ac.at

This article is available from the xxx mathematics archive as math.PR/0112230.

1858. CONFORMAL INVARIANCE OF PLANAR LOOP-ERASED RANDOM WALKS AND UNIFORM SPANNING TREES

Gregory F. Lawler, Oded Schramm, Wendelin Werner

We prove that the scaling limit of loop-erased random walk in a simply
connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular,
the limit exists and is conformally invariant. It follows that the scaling
limit of the uniform spanning tree in a Jordan domain exists and is conformally
invariant. Assuming that the boundary of the domain is a $C^1$ simple closed
curve, the same method is applied to show that the scaling limit of the uniform
spanning tree Peano curve, where the tree is wired along a proper arc $A$ on
the boundary, is the chordal SLE(8) path in the closure of $D$ joining the
endpoints of $A$. A by-product of this result is that
 SLE(8) is almost surely generated by a continuous path. The results and proofs
are not restricted to a particular choice of lattice.

schramm@microsoft.com

This article is available from the xxx mathematics archive as math.PR/0112234.

1859. MONTE CARLO TESTS OF SLE PREDICTIONS FOR THE 2D SELF-AVOIDING WALK

Tom Kennedy

The conjecture that the scaling limit of the two-dimensional self-avoiding
walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE)
with $\kappa=8/3$ leads to explicit predictions about the SAW. A remarkable
feature of these predictions is that they yield not just critical exponents,
but probability distributions for certain random variables associated with the
self-avoiding walk. We test two of these predictions with Monte Carlo
simulations and find excellent agreement, thus providing numerical support to
the conjecture that the scaling limit of the SAW is SLE$_{8/3}$.

tgk@math.arizona.edu

This article is available from the xxx mathematics archive as math.PR/0112246.

1860. BROWNIAN BRIDGE ASYMPTOTICS FOR THE SUBCRITICAL BERNOULLI BOND PERCOLATION

Yevgeniy Kovchegov 

For the d-dimensional model of a subcritical bond percolation (p<p_c) and a
point \vec{a} in Z^d, we prove that a cluster conditioned on connecting points
(0,...,0) and n\vec{a} if scaled by 1/(n|vec{a}|) along \vec{a} and by
1/sqrt{n} in the orthogonal direction converges asymptotically to Time x
(d-1)-dimensional Brownian Bridge.

yevgeniy@math.stanford.edu

This article is available from the xxx mathematics archive as math.PR/0112272.

1861. ANNUITIES UNDER RANDOM RATES OF INTEREST - REVISITED

K. Burnecki, A. Marciniuk and A. Weron

In the article we consider accumulated values of annuities-certain with
yearly payments with independent random interest rates. We focus on annuities
with payments varying in arithmetic and geometric progression which are
important basic varying annuities (see Kellison, 1991). They appear to be a
generalization of the types studied recently by Zaks (2001). We derive, via
recursive relationships, mean and variance formulae of the final values of the
annuities. As a consequence, we obtain moments related to the already discussed
cases, which leads to a correction of main results from Zaks (2001).

burnecki@mazur.im.pwr.wroc.pl

This article is available from the xxx mathematics archive as math.PR/0112298.

1862. SMALL AND LARGE TIME SCALE ANALYSIS OF A NETWORK TRAFFIC MODEL

Krishanu Maulik and Sidney Resnick

Empirical studies of the internet and WAN traffic have
observed multifractal behavior at time scales below a few hundred 
milliseconds. There have been some attempts to model this
phenomenon, but there is no model to connect the small time scale 
behavior with behavior observed at large time scales of bigger 
than a few hundred milliseconds. There have been separate analyses 
of models for  high speed data transmissions, which show that
appropriate approximations to large time scale behavior of
cumulative traffic are either fractional Brownian motion or stable 
L\'evy motion, depending on the input rates assumed. We bridge this 
gap and develop and analyze a model
offering an explanation of both the small and large time scale
behavior of a network traffic model based on the infinite source 
Poisson model. Previous studies of this model have usually assumed that 
transmission rates are constant and deterministic. We consider 
a non-constant, multifractal, random transmission rate at the user 
level which results in cumulative traffic exhibiting
multifractal behavior on small time scales and self-similar
behavior on large time scales. 

km75@cornell.edu  sid@orie.cornell.edu

To see a preprint or other information provided by the author click here.

1863. REPRESENTATIONS OF THE BROWNIAN SNAKE WITH DRIFT

Romain Abraham and Laurent Serlet

We consider a path-valued process which is a generalization of
the classical Brownian snake introduced by Le Gall. More
precisely we add a
drift term $b$ to the lifetime process, which may depends on the
spatial process. This consequently introduce a coupling between
the lifetime process and the spatial motion. This
process can be obtained from the standard Brownian snake by
Girsanov's theorem or by killing of the spatial motion. It can also
be viewed as the limit of discrete snakes or, in some special
cases, as conditioned Brownian snakes.
We also use this process
to describe the solutions of the non-linear partial differential
equation $\Delta u =4u^{2}+4b\,u$.

Romain.Abraham@math-info.univ-paris5.fr   Laurent.Serlet@math-info.univ-paris5.fr

To see a preprint or other information provided by the author click here.

1864. CENSORED STABLE PROCESSES

K. Bogdan, K. Burdzy and Z.-Q. Chen

We present several constructions of a `censored
stable process' in an open set $D$, i.e., a symmetric
stable process which is not allowed to jump outside $D$.
We address the question of whether the process will
approach the boundary of $D$ in a finite time - we
give sharp conditions for such approach in terms
of the stability index and the `thickness'
of the boundary. As a corollary, new results are
obtained concerning Besov spaces on non-smooth domains,
including the critical exponent case.
We also study the decay rate of the corresponding harmonic
functions which vanish on a part of the boundary.
We derive a boundary Harnack principle in $C^{1,1}$ open 
sets.

bogdan@im.pwr.wroc.pl burdzy@math.washington.edu
zchen@math.washington.edu

To see a preprint or other information provided by the author click here.
Or here.

1865. RADONIFICATION OF CYLINDRICAL SEMIMARTINGALES BY A SINGLE HILBERT-SCHMIDT OPERATOR

Adam Jakubowski, Stanislaw Kwapien, Paul Raynaud de Fitte and Jan Rosinski

It is proved that in Hilbert
spaces a single Hilbert-Schmidt operator radonifies cylindrical
semimartingales to strong semimartingales. This improves a result
due to Badrikian and Ustunel (also L. Schwartz), who
needed composition of three Hilbert-Schmidt operators.

adjakubo@mat.uni.torun.pl kwapstan@mimuw.edu.pl prf@univ-rouen.fr rosinski@math.utk.edu

1866. MARKOV PROCESSES ON VERMICULATED SPACES

Martin T. Barlow and Steven N. Evans

A general technique is given for constructing
new Markov processes from existing ones.
The new process and its state space are both projective 
limits of sequences built by an iterative scheme.
The space at each stage in the scheme is obtained by taking
disjoint copies of the space at the previous stage
and quotienting to identify certain distinguished points.
Away from the distinguished points, the process at each 
stage evolves like the one constructed at the previous 
stage on some copy of the previous state space, but when 
the process hits a distinguished point it enters at random
another of the copies ``pinned'' at that point.
Special cases of this construction produce diffusions on
fractal-like objects that have been studied recently.

barlow@math.ubc.ca evans@stat.berkeley.edu

To see a preprint or other information provided by the author click here.
Or here.

1867. ORNSTEIN-ZERNIKE THEORY FOR THE FINITE RANGE ISING MODELS ABOVE T_c

Massimo Campanino, Dmitry Ioffe and Yvan Velenik

We derive precise Ornstein-Zernike asymptotic formula
for the decay of the two-point function 
$\langle \sigma_0\sigma_x\rangle_\beta$ in the general context of
finite range Ising type models on  $\bbZ^d$. The proof
relies in an essential way on the a-priori knowledge 
of the  strict exponential decay of the two-point function and,   
by the sharp characterization of phase transition due
to Aizenman, Barsky and   Fern\'{a}ndez, goes through in the
whole of the high temperature region $\beta<\beta_c$.  
As a byproduct we obtain that for every $\beta <\beta_c$, the
inverse correlation length $\xi_\beta$ is an analytic and
strictly convex function of direction.

campanin@dm.unibo.it ieioffe@ie.technion.ac.il velenik@cmi.univ-mrs.fr

1868. LOCALIZATION TRANSITION OF (d+1)-FRIENDLY WALKERS

Hideki Tanemura and Nobuo Yoshida

Friendly walkers is a stochastic model obtained from 
independent one-dimensional simple random walks 
$\{ S^k_n \}_{n\ge 0}$, $k=1,2,\dots, d+1$ by 
introducing ``non-crossing condition'': 
$S^1_j \ge S^2_j \ge \ldots \ge S^{d+1}_j,j=1,2,\dots, n$ 
and ``reward for collisions'' characterized by parameters 
$\beta_1, \ldots, \beta_d \ge 0$. Here, the reward for 
collisions is described as follows. If there are exactly 
$m$ collisions at time $j$, i.e., 
$m=\sharp \{1\le k \le d:S^k_j = S^{k+1}_j \} \ge 1$, 
then the probabilistic weight for the walkers increases by 
multiplicative factor $\exp (\beta_m )\ge 1$.
We study the localization transition of this model 
in terms of the positivity of the free energy. 
In particular, we prove the existence of the critical 
surface in the $d$-dimensional space for the parameters 
$(\beta_1, \ldots, \beta_d)$.

tanemura@math.s.chiba-u.ac.jp  nobuo@kusm.kyoto-u.ac.jp

To see a preprint or other information provided by the author click here.

1869. ON NEUMANN EIGENFUNCTIONS IN LIP DOMAINS

Rami Atar and Krzysztof Burdzy

A `lip domain' is a planar set lying between
graphs of two Lipschitz functions with constant 1.
We show that the second Neumann eigenvalue
is simple in every lip domain except the square.
The corresponding eigenfunction attains its
maximum and minimum at the boundary points at the
extreme left and right. Two conjectures of Jerison 
and Nadirashvili are special cases of our main result. 
Our techniques are probabilistic
in nature and may have independent interest.
We prove existence and uniqueness of mirror couplings
in piecewise smooth domains. We also prove
a `parabolic boundary Harnack principle'
for such couplings.

atar@ee.technion.ac.il   burdzy@math.washington.edu

To see a preprint or other information provided by the author click here.
Or here.

1870. UNE INEGALITE DE DECOUPLAGE ERGODIQUE

Michel Weber

We use a decoupling inequality in ergodic theory for maximal operators. We
apply this inequality to the study of the property for a set of functions
to be a Donsker class. The sets we examin are built from a sequence of
$L^2$-operators, and appear naturally in the study of the almost sure
regularity properties of these-ones. We obtain new individual necessary
conditions (for a given $\LLf$ ) and new global necessary conditions. The
latter conditions are of uniform type and have a natural translation on the
regularity properties of the canonical Gaussian process $Z$ defined on
$\LL$. We also apply the decoupling inequality to provide new necessary
conditions in the study of the regularity properties of a sequence of
operators.

weber@math.u-strasbg.fr


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