# Probability Abstracts 68

This document contains abstracts 1907-1959. They have been mailed on April 30, 2002.

### 1907. BROWNIAN EXCURSIONS AND PARISIAN BARRIER OPTIONS: A NOTE

Michael Schr\"{o}der

This note re-addresses the Paris barrier options proposed by Yor and
collaborators and their valuation using the Laplace transform approach. The
notion of Paris barrier options, based on excursion theory and using the
Brownian meander, is extended such that their valuation is now possible at any
point during their lifespan. The pertinent Laplace transforms are modified when
necessary.

schroder@euklid.math.uni-mannheim.de

### 1908. EXISTENCE OF QUASI-STATIONARY MEASURES FOR ASYMMETRIC ATTRACTIVE PARTICLE SYSTEMS ON $\ZZ^D$

A. Asselah, F. Castell

We show the existence of non-trivial quasi-stationary measures for
conservative attractive particle systems on $\ZZ^d$ conditioned on avoiding an
increasing local set $\A$. Moreover, we exhibit a sequence of measures
$\{\nu_n\}$, whose $\omega$-limit set consists of quasi-stationary measures.
For zero range processes, with stationary measure $\nur$, we prove the
existence of an $L^2(\nur)$ nonnegative eigenvector for the generator with
Dirichlet boundary on $\A$, after establishing a priori bounds on the
$\{\nu_n\}$.

fabienne.castell@cmi.univ-mrs.fr

### 1909. LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS WITH FUNCTIONAL BOUNDARY CONDITIONS

Aureli Alabert, Marco Ferrante

We consider linear n-th order stochastic differential equations on [0,1],
with linear boundary conditions supported by a finite subset of [0,1]. We study
some features of the solution to these problems, and especially its conditional
independence properties of Markovian type.

alabert@mat.uab.es

### 1910. ASYMPTOTIC OF THE HEAT KERNEL IN GENERAL BENEDICKS DOMAINS

P.Collet, S.Martinez, J.San Martin

Using a new inequality relating the heat kernel and the probability of
survival, we prove asymptotic ratio limit theorems for the heat kernel (and
survival probability) in general Benedicks domains. In particular, the
dimension of the cone of positive harmonic measures with Dirichlet boundary
condition can be derived from the rate of convergence to zero of the heat
kernel (or the survival probability).

collet@pth.polytechnique.fr

### 1911. LARGE DEVIATIONS OF BIVARIATE EMPIRICAL MEASURES FOR TWO SYSTEMS COUPLED BY A SYMMETRIC INTERACTION

Wlodek Bryc

In this paper we study empirical measures which can be thought as a decoupled
version of the empirical measures generated by random matrices. We prove the
large deviation principle with the rate function, which is finite only on
product measures and hence is non-convex. As a corollary, we derive a large
deviations principle for (univariate) average empirical measures with the rate
function that superficially resembles the rate function of random matrices, but
may be concave.

brycw@math.uc.edu

### 1912. RANDOM MATRICES, NON-COLLIDING PROCESSES AND QUEUES

Neil O'Connell

This is survey of some recent results connecting random matrices,
non-colliding processes and queues.

neil.o.connell@ens.fr

### 1913. A PATH-TRANSFORMATION FOR RANDOM WALKS AND THE ROBINSON-SCHENSTED CORRESPONDENCE

Neil O'Connell

In [O'Connell and Yor (2002)] a path-transformation G was introduced with the
property that, for X belonging to a certain class of random walks on the
integer lattice, the transformed walk G(X) has the same law as that of the
original walk conditioned never to exit a type-A Weyl chamber. In this paper,
we show that G is closely related to the Robinson-Schensted algorithm, and use
this connection to give a new proof of the above representation theorem. The
new proof is valid for a larger class of random walks and yields additional
information about the joint law of X and G(X). The corresponding results for
the Brownian model are recovered by Donsker's theorem. These are connected with
Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix
theory. The connection we make between the path-transformation G and the RS
algorithm also provides a new formula and interpretation for the latter. This
can be used to study properties of the RS algorithm and, moreover, extends
easily to a continuous setting.

neil.o.connell@ens.fr

### 1914. THE BROWNIAN WEB

L. R. G. Fontes, M. Isopi, C. M. Newman, K. Ravishankar

Arratia, and later T\'oth and Werner, constructed random processes that
formally correspond to coalescing one-dimensional Brownian motions starting
from every space-time point. We extend their work by constructing and
characterizing what we call the {\em Brownian Web} as a random variable taking
values in an appropriate (metric) space whose points are (compact) sets of
paths. This leads to general convergence criteria and, in particular, to
convergence in distribution of coalescing random walks in the scaling limit to
the Brownian Web.

isopi@mat.uniroma1.it

### 1915. WEAK-INTERACTION LIMITS FOR ONE-DIMENSIONAL RANDOM POLYMERS

R. van der Hofstad, F. den Hollander, W. Koenig

In this paper we present a new and flexible method to show that, in one
dimension, various self-repellent random walks converge to self-repellent
Brownian motion in the limit of weak interaction after appropriate space-time
scaling. Our method is based on cutting the path into pieces of an
appropriately scaled length, controlling the interaction between the different
pieces, and applying an invariance principle to the single pieces. In this way
we show that the self-repellent random walk large deviation rate function for
the empirical drift of the path converges to the self-repellent Brownian motion
large deviation rate function after appropriate scaling with the interaction
parameters. The method is considerably simpler than the approach followed in
our earlier work, which was based on functional analytic arguments applied to
variational representations and only worked in a very limited number of
situations.
We consider two examples of a weak interaction limit: (1) vanishing
self-repellence, (2) diverging step variance. In example (1), we recover our
earlier scaling results for simple random walk with vanishing self-repellence
and show how these can be extended to random walk with steps that have zero
mean and a finite exponential moment. Moreover, we show that these scaling
results are stable against adding self-attraction, provided the self-repellence
dominates. In example (2), we prove a conjecture by Aldous for the scaling of
self-avoiding walk with diverging step variance. Moreover, we consider
self-avoiding walk on a two-dimensional horizontal strip such that the steps in
the vertical direction are uniform over the width of the strip and find the
scaling as the width tends to infinity.

koenig@math.tu-berlin.de

### 1916. LARGE DEVIATIONS FOR THE ONE-DIMENSIONAL EDWARDS MODEL

R. van der Hofstad, F. den Hollander, W. Koenig

In this paper we prove a large deviation principle for the empirical drift of
a one-dimensional Brownian motion with self-repellence called the Edwards
model. Our results extend earlier work in which a law of large numbers,
respectively, a central limit theorem were derived. In the Edwards model a path
of length $T$ receives a penalty $e^{-\beta H_T}$, where $H_T$ is the
self-intersection local time of the path and $\beta\in(0,\infty)$ is a
parameter called the strength of self-repellence. We identify the rate function
in the large deviation principle for the endpoint of the path as $\beta^{\frac 23} I(\beta^{-\frac 13}\cdot)$, with $I(\cdot)$ given in terms of the principal
eigenvalues of a one-parameter family of Sturm-Liouville operators. We show
that there exist numbers $0<b^{**}<b^*<\infty$ such that: (1) $I$ is linearly
decreasing on $[0,b^{**}]$; (2) $I$ is real-analytic and strictly convex on
$(b^{**},\infty)$; (3) $I$ is continuously differentiable at $b^{**}$; (4) $I$
has a unique zero at $b^*$. (The latter fact identifies $b^*$ as the asymptotic
drift of the endpoint.) The critical drift $b^{**}$ is associated with a
crossover in the optimal strategy of the path: for $b\geq b^{**}$ the path
assumes local drift $b$ during the full time $T$, while for $0\leq b<b^{**}$ it
assumes local drift $b^{**}$ during time $\frac{b^{**}+b}{2b^{**}}T$ and local
drift $-b^{**}$ during the remaining time $\frac{b^{**}-b}{2b^{**}}T$. Thus, in
the second regime the path makes an overshoot of size $\frac{b^{**}-b}{2}T$ in
order to reduce its intersection local time.

koenig@math.tu-berlin.de

### 1917. FLOWS, COALESCENCE AND NOISE

Yves Le Jan and Olivier Raimond

We are interested in stationary fluid'' random evolutions with independent
increments. Under some mild assumptions, we show they are solutions of a
stochastic differential equation (SDE). There are situations where these
evolutions are not described by flows of diffeomorphisms, but by coalescing
flows or by flows of probability kernels.
In an intermediate phase, for which there exists a coalescing flow and a flow
of kernels solution of the SDE, a classification is given: All solutions of the
SDE can be obtained by filtering the coalescing motion with respect to a
sub-noise containing the Gaussian part of its noise. Thus, the coalescing
motion cannot be described by a white noise.

olivier.raimond@math.u-psud.fr

### 1918. CONVERGENCE IN ENERGY-LOWERING (DISORDERED) STOCHASTIC SPIN SYSTEMS

Emilio De Santis, Charles M. Newman

We consider stochastic processes, S^t \equiv (S_x^t : x \in Z^d), with each
S_x^t taking values in some fixed finite set, in which spin flips (i.e.,
changes of S_x^t) do not raise the energy. We extend earlier results of
Nanda-Newman-Stein that each site x has almost surely only finitely many flips
that strictly lower the energy and thus that in models without zero-energy
flips there is convergence to an absorbing state. In particular, the assumption
of finite mean energy density can be eliminated by constructing a
percolation-theoretic Lyapunov function density as a substitute for the mean
energy density. Our results apply to random energy functions with a
translation-invariant distribution and to quite general (not necessarily
Markovian) dynamics.

desantis@mat.uniroma1.it

### 1919. FREE L\'EVY PROCESSES ON DUAL GROUPS

Uwe Franz

We give a short introduction to the theory of L\'evy processes on dual
groups. As examples we consider L\'evy processes with additive increments and
L\'evy processes on the dual affine group.

franz@mail.uni-greifswald.de

### 1920. SCALED BOOLEAN ALGEBRAS

Michael Hardy

Scaled Boolean algebras are a category of mathematical objects that arose
from attempts to understand why the conventional rules of probability should
hold when probabilities are construed, not as frequencies or proportions or the
like, but rather as degrees of belief in uncertain propositions. This paper
separates the study of these objects from that not-entirely-mathematical
problem that motivated them. That motivating problem is explicated in the first
section, and the application of scaled Boolean algebras to it is explained in
the last section. The intermediate sections deal only with the mathematics. It
is hoped that this isolation of the mathematics from the motivating problem
makes the mathematics clearer.

hardy@math.mit.edu

### 1921. FIRST PASSAGE PERCOLATION HAS SUBLINEAR DISTANCE VARIANCE

Itai Benjamini, Gil Kalai, Oded Schramm

Let $0<a<b<\infty$, and for each edge $e$ of $Z^d$ let $\omega_e=a$ or
$\omega_e=b$, each with probability 1/2, independently. This induces a random
metric $\dist_\omega$ on the vertices of $Z^d$, called first passage
percolation. We prove that for $d>1$ the distance $dist_\omega(0,v)$ from the
origin to a vertex $v$, $|v|>2$, has variance bounded by $C |v|/\log|v|$, where
$C=C(a,b,d)$ is a constant which may only depend on $a$, $b$ and $d$. Some
related variants are also discussed

schramm@microsoft.com

### 1922. FUNCTIONAL CENTRAL LIMIT THEOREMS FOR VICIOUS WALKERS

Makoto Katori and Hideki Tanemura

We consider the diffusion scaling limit of the vicious walkers, which is a
model of nonintersecting random walks. We show a functional central limit
theorem for the model and derive two types of nonintersecting Brownian motions,
in which we impose nonitersecting condition in the finite time interval $(0,T]$
(resp. in the infinite time interval $(0,\infty)$) for the first-type (resp.
second-type). The first-type is a temporally inhomogeneous diffusion, and the
second-type is a temporally homogeneous diffusion called Dyson's model of
Brownian motions. We also study the vicious walkers with wall restriction and
prove the functional central limit theorem in the diffusion scaling limit.

katori@phys.chuo-u.ac.jp

### 1923. GEOMETRIC SINGULAR PERTURBATION THEORY FOR STOCHASTIC DIFFERENTIAL EQUATIONS

Nils Berglund and Barbara Gentz

We consider slow-fast systems of differential equations, in which both the
slow and fast variables are perturbed by additive noise. When the deterministic
system admits a uniformly asymptotically stable slow manifold, we show that the
sample paths of the stochastic system are concentrated in a neighbourhood of
the slow manifold, which we construct explicitly. Depending on the dynamics of
the reduced system, the results cover time spans which can be exponentially
long in the noise intensity squared (that is, up to Kramers' time). We give
exponentially small upper and lower bounds on the probability of exceptional
paths. If the slow manifold contains bifurcation points, we show similar
concentration properties for the fast variables corresponding to
non-bifurcating modes. We also give conditions under which the system can be
approximated by a lower-dimensional one, in which the fast variables contain
only bifurcating modes.

berglund@univ-tln.fr

### 1924. A GENERAL HSU-ROBBINS-ERDOS TYPE ESTIMATE OF TAIL PROBABILITIES OF SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES

Alexander R. Pruss

Let $X_1,X_2,...$ be a sequence of independent and identically distributed
random variables, and put $S_n=X_1+...+X_n$. Under some conditions on the
positive sequence $\tau_n$ and the positive increasing sequence $a_n$, we give
necessary and sufficient conditions for the convergence of $\sum_{n=1}^\infty \tau_n P(|S_n|\ge \epsilon a_n)$ for all $\epsilon>0$, generalizing Baum and
Katz's (1965) generalization of the Hsu-Robbins-Erdos (1947, 1949) law of large
numbers, also allowing us to characterize the convergence of the above series
in the case where $\tau_n=n^{-1}$ and $a_n=(n\log n)^{1/2}$ for $n\ge 2$,
thereby answering a question of Spataru. Moreover, some results for
non-identically distributed independent random variables are obtained by a
recent comparison inequality. Our basic method is to use a central limit
theorem estimate of Nagaev (1965) combined with the Hoffman-Jorgensen
inequality (1974).

pruss@imap.pitt.edu

### 1925. A CHAOTIC DECOMPOSITION FOR L\'EVY PROCESSES ON MANIFOLDS WITH AN APPLICATION TO PROCESSES OF MEIXNER'S TYPE

E. Lytvynov, D. Mierzejewski

It is well known that between all processes with independent increments,
essentially only the Brownian motion and the Poisson process possess the
chaotic decomposition property (CRP). Thus, a natural question appears: What is
an appropriate analog of the CRP in the case of a general L\'evy process. At
least three approaches are possible here. The first one is due to It\^o. It
uses the CRP of the Brownian motion and the Poisson process, as well as the
representation of a L\'evy process through those processes. The second approach
is due to Nualart and Schoutens and consists in representing any
square-integrable random variable as a sum of multiple stochastic integrals
constructed with respect to a family of orthogonalized centered power jumps
processes. Finally, the third approach uses the idea of orthogonalization of
polynomials with respect to a probability measure defined on the dual of a
nuclear space. The main aims of the present paper are to develop the three
approaches in the case of a general (${\mathbb R}$-valued) L\'evy process on a
Riemannian manifold and (what is more important) to understand a relationship
between these approaches. We apply the obtained results to the processes of
Meixner's type--the gamma, Pascal, and Meixner processes. In this case, the
analysis related to the orthogonalized polynomials becomes essentially simpler
and reacher than in the general case.

lytvynov@wiener.iam.uni-bonn.de

### 1926. A SINGULAR PARABOLIC ANDERSON MODEL

Carl Mueller and Roger Tribe

We give a new example of a measure-valued process without a density, which
arises from a stochastic partial differential equation with a multiplicative
noise term. This process has some unusual properties. We work with the heat
equation with a random potential: u_t=Delta u+kuF. Here k>0 is a small number,
and x lies in d-dimensional Euclidean space with d>2. F is a Gaussian noise
which is uncorrelated in time, and whose spatial covariance equals |x-y|^(-2).
The exponent 2 is critical in the following sense. For exponents less than 2,
the equation has function-valued solutions, and for exponents higher than 2, we
do not expect solutions to exist. This model is closely related to the
parabolic Anderson model; we expect solutions to be small, except for a
collection of high peaks. This phenomenon is called intermittency, and is
reflected in the singular nature of our process. Solutions exist as singular
measures, under suitable assumptions on the initial conditions and for
sufficiently small k. We investigate various properties of the solutions, such
as dimension of the support and long-time behavior. As opposed to the
super-Brownian motion, which satisfies a similar equation, our process does not
have compact support, nor does it die out in finite time. We use such tools as
scaling, self-duality and moment formulae.

cmlr@math.rochester.edu

### 1927. AN ASYMPTOTIC LINK BETWEEN LUE AND GUE AND ITS SPECTRAL INTERPRETATION

Yan Doumerc

We use a matrix central-limit theorem which makes the Gaussian Unitary
Ensemble appear as a limit of the Laguerre Unitary Ensemble together with an
observation due to Johansson in order to derive new representations for the
eigenvalues of GUE. For instance, it is possible to recover the celebrated
equality in distribution between the maximal eigenvalue of GUE and a
last-passage time in some directed brownian percolation. Similar identities for
the other eigenvalues of GUE also appear.

doumerc@clipper.ens.fr

### 1928. GIBBS MEASURES AND SEMICLASSICAL APPROXIMATIONS TO ACTION MINIMIZING MEASURES

N. Anantharaman

We study the approximation of action minimizing measures, for a mechanical
Lagrangian on the torus, by the probability densities given by quasi-periodic
eigenfunctions of the Schrodinger operator.

Nalini.ANANTHARAMAN@umpa.ens-lyon.fr

### 1929. HAUSDORFF DIMENSIONS FOR $SLE_6$

Vincent Beffara

We prove that the Hausdorff dimension of the trace of $SLE_6$ is almost
surely 7/4 and give a more direct derivation of the result (due to
Lawler-Schramm-Werner) that the dimension of its boundary is 4/3. We also prove
that for all $\kappa<8$ the $SLE_\kappa$ trace has cut-points.

vincent.beffara@math.u-psud.fr

### 1930. EXPLICIT CONSTRUCTION OF THE BROWNIAN SELF-TRANSPORT OPERATOR

D.S.Grebenkov

Applying the technique of characteristic functions developped for
one-dimensional regular surfaces (curves) with compact support, we obtain the
distribution of hitting probabilities for a wide class of finite membranes on
square lattice. Then we generalize it to multi-dimensional finite membranes on
hypercubic lattice. Basing on these distributions, we explicitly construct the
Brownian self-transport operator which governs the Laplacian transfer. In order
to verify the accuracy of the distribution of hitting probabilities, numerical
analysis is carried out for some particular membranes.

dg@pmc.polytechnique.fr

### 1931. ON EXPONENTIAL STABILITY OF WONHAM FILTER

P. Chigansky, R. Liptser

We give elementary proof of a stability result concerning an exponential
asymptotic ($t\to\infty$) for filtering estimates generated by wrongly
initialized Wonham filter. This proof is based on new exponential bound having
independent interest.

pavelm@eng.tau.ac.il

### 1932. COMPUTING STATIONARY PROBABILITY DISTRIBUTIONS AND LARGE DEVIATION RATES FOR CONSTRAINED RANDOM WALKS. THE UNDECIDABILITY RESULTS

David Gamarnik

Our model is a constrained homogeneous random walk in a nonnegative orthant
Z_+^d. The convergence to stationarity for such a random walk can often be
checked by constructing a Lyapunov function. The same Lyapunov function can
also be used for computing approximately the stationary distribution of this
random walk, using methods developed by Meyn and Tweedie. In this paper we show
that, for this type of random walks, computing the stationary probability
exactly is an undecidable problem: no algorithm can exist to achieve this task.
We then prove that computing large deviation rates for this model is also an
undecidable problem. We extend these results to a certain type of queueing
systems. The implication of these results is that no useful formulas for
computing stationary probabilities and large deviations rates can exist in
these systems.

gamarnik@watson.ibm.com

### 1933. ON THE SCALING LIMIT OF PLANAR SELF-AVOIDING WALK

Gregory F. Lawler, Oded Schramm, Wendelin Werner

A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in
the square lattice with no self-intersection. A planar self-avoiding polygon
(SAP) is a loop with no self-intersection. In this paper we present conjectures
for the scaling limit of the uniform measures on these objects. The conjectures
are based on recent results on the stochastic Loewner evolution and
non-disconnecting Brownian motions. New heuristic derivations are given for the
critical exponents for SAWs and SAPs.

lawler@polygon.math.cornell.edu

### 1934. ON DIFFUSION APPROXIMATION WITH DISCONTINUOUS COEFFICIENTS

N. V. Krylov, R. Liptser

Convergence of stochastic processes with jumps to diffusion processes is
investigated in the case when the limit process has discontinuous coefficients.
An example is given in which the diffusion approximation of a queueing model
yields a diffusion process with discontinuous diffusion and drift coefficients.

liptser@eng.tau.ac.il

### 1935. ON THE UNIVERSALITY OF THE PROBABILITY DISTRIBUTION OF THE PRODUCT $B^{-1}X$ OF RANDOM MATRICES

Joshua Feinberg

Consider random matrices $A$, of dimension $m\times (m+n)$, drawn from an
ensemble with probability density $f(\rmtr AA^\dagger)$, with $f(x)$ a given
appropriate function. Break $A = (B,X)$ into an $m\times m$ block $B$ and the
complementary $m\times n$ block $X$, and define the random matrix $Z=B^{-1}X$.
We calculate the probability density function $P(Z)$ of the random matrix $Z$
and find that it is a universal function, independent of $f(x)$. Universality
of $P(Z)$ is, essentially, a consequence of rotational invariance of the
probability ensembles we study. More generally, $P(Z)$ must be independent, of
course, of any common scale of the distribution functions of $B$ and $X$. As an
application, we study the distribution of solutions of systems of linear
equations with random coefficients, and extend a classic result due to Girko.

joshua@physics.technion.ac.il

### 1936. OCCUPATION DENSITIES FOR SPDE'S WITH REFLECTION

Lorenzo Zambotti

We consider the solution (u,\eta) of the white-noise driven stochastic
partial differential equation with reflection on the space interval [0,1]
introduced by Nualart and Pardoux. First, we prove that at any fixed time t>0,
the measure \eta([0,t]\times d\theta) is absolutely continuous w.r.t. the
Lebesgue measure d\theta on (0,1). We characterize the density as a family of
additive functionals of u, and we interpret it as a renormalized local time at
0 of (u(t,\theta))_{t\geq 0}. Finally we study the behaviour of \eta at the
boundary of [0,1]. The main technical novelty is a projection principle from
the Dirichlet space of a Gaussian process, vector-valued solution of a linear
SPDE, to the Dirichlet space of the process u.

zambotti@mail.sns.it

### 1937. A NOTE ON EDGE ORIENTED REINFORCED RANDOM WALKS AND RWRE

N. Enriquez and C. Sabot

This work introduces the notion of edge oriented reinforced random walk which
proposes in a general framework an alternative understanding of the annealed
law of random walks in random environment.

sabot@ccr.jussieu.fr

### 1938. STATIONARY DETERMINANTAL PROCESSES: PHASE TRANSITIONS, BERNOULLICITY, ENTROPY, AND DOMINATION

Russell Lyons and Jeffrey E. Steif

We study a class of stationary processes indexed by $\Z^d$ that are defined
via minors of $d$-dimensional Toeplitz matrices. We obtain necessary and
sufficient conditions for the existence of a phase transition (phase
multiplicity) analogous to that which occurs in statistical mechanics. The
absence of a phase transition is equivalent to the presence of a strong $K$
property, a particular strengthening of the usual $K$ (Kolmogorov) property. We
show that all of these processes are Bernoulli shifts (isomorphic to i.i.d.\
processes in the sense of ergodic theory). We obtain estimates of their
entropies and we relate these processes via stochastic domination to product
measures.

rdlyons@indiana.edu

### 1939. DETERMINANTAL PROBABILITY MEASURES

Russell Lyons

Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We initiate a detailed study of the
discrete analogue, the most prominent example of which has been the uniform
spanning tree measure. Our main results concern relationships with matroids,
stochastic domination, negative association, completeness for infinite
matroids, tail triviality, and a method for extension of results from
orthogonal projections to positive contractions. We also present several new
avenues for further investigation, involving Hilbert spaces, combinatorics,
homology, and group representations, among other areas.

rdlyons@indiana.edu

### 1940. ON MARKOVIAN COCYCLE PERTURBATIONS IN CLASSICAL AND QUANTUM PROBABILITY

G.G. Amosov

We introduce Markovian cocycle perturbations of the groups of transformations
associated with the classical and quantum stochastic processes with stationary
increments, which are characterized by a localization of the perturbation to
the algebra of events of the past. It is namely the definition one needs
because the Markovian perturbations of the Kolmogorov flows associated with the
classical and quantum noises result in the perturbed group of transformations
which can be decomposed in the sum of a part associated with deterministic
stochastic processes lying in the past and a part associated with the noise
isomorphic to the initial one. This decomposition allows to obtain some analog
of the Wold decomposition for classical stationary processes excluding a
nondeterministic part of the process in the case of the stationary quantum
stochastic processes on the von Neumann factors which are the Markovian
perturbations of the quantum noises. For the classical stochastic process with
noncorrelated increaments it is constructed the model of Markovian
perturbations describing all Markovian cocycles up to a unitary equivalence of
the perturbations. Using this model we construct Markovian cocyclies
transformating the Gaussian state $\rho$ to the Gaussian states equivalent to
$\rho$.

gramos@volterra.mat.uniroma2.it

### 1941. ON NODAL LINES OF NEUMANN EIGENFUNCTIONS

Krzysztof Burdzy

We present a new method for locating the nodal line
of the second eigenfunction for the Neumann problem
in a planar domain. The technique is based on the
mirror coupling' of reflected Brownian motions.

burdzy@math.washington.edu

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

### 1942. DUALITIES FOR THE DOMANY-KINZEL MODEL

Makoto Katori, Norio Konno, Aidan Sudbury and Hideki Tanemura

We study the Domany-Kinzel model, which is
a class of discrete-time Markov processes in one-dimension with two parameters
$(p_{1}, p_{2}) \in [0,1]^{2}.$ When $p_{1}=\alpha \beta$ and $p_{2}=\alpha ( 2 \beta-\beta^{2})$ with $(\alpha, \beta) \in [0,1]^{2}$, the process
can be identified with the mixed site-bond oriented percolation model
on a square lattice with probabilities $\alpha$ of a site being open and
$\beta$ of a bond being open.
This paper treats dualities for the Domany-Kinzel model $\xi^A _t$
and the DKdual $\eta^A _t$ starting from $A.$ We prove that
(i) $E(x^{|\xi^A _t \cap B|}) = E(x^{|\xi^B _t \cap A|})$ if $x=1 - (2p_1-p_2)/p_1 ^2$,
(ii) $E(x^{|\xi^A _t \cap B|}) = E(x^{|\eta^B _t \cap A|})$ if $x=1 - (2p_1-p_2)/p_1$,
and
(iii) $E(x^{|\eta^A _t \cap B|}) = E(x^{|\eta^B _t \cap A|})$ if $x=1 - (2p_1-p_2),$
as long as one of $A, B$ is finite and $p_2\leq p_{1}$.

katori@phys.chuo-u.ac.jp  norio@mathlab.sci.ynu.ac.jp
Aidan.Sudbury@sci.monash.edu.au  tanemura@math.s.chiba-u.ac.jp

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

### 1943. TWO WORKLOAD PROPERTIES FOR BROWNIAN NETWORKS

M. Bramson and R. J. Williams

As one approach to dynamic scheduling problems for open
stochastic processing networks, J. M. Harrison has proposed
the use of formal heavy traffic approximations known as
Brownian networks. A key step in this approach is a
reduction in dimension of a Brownian network, due to
Harrison and Van Mieghem, in which the queue length' process
is replaced by a workload' process.
In this paper, we establish two properties of these workload
processes. Firstly, we derive a formula for
the dimension of such processes. For a given Brownian
network, this dimension is unique. However, there are
infinitely many possible choices for the workload process.
Harrison has proposed a canonical' choice, which
reduces the  possibilities to a finite number.
Our second result provides sufficient conditions for this
canonical choice to be valid and for it to yield a
non-negative workload process. The assumptions and proofs
for our results involve only first-order model parameters.

williams@math.ucsd.edu   bramson@math.umn.edu



### 1944. DIFFUSIONS ON THE SIMPLEX FROM BROWNIAN MOTIONS ON HYPERSURFACES

Steven N. Evans

The $(n-1)$-dimensional simplex is the collection of
probability measures on a set with $n$ points. Many applied
situations result in simplex-valued data or in stochastic
processes that have the simplex as their state space.
In this paper we study a large class of simplex-valued
diffusion processes that are constructed by first
coordinatising'' the simplex with the points of a smooth
hypersurface in such a way that several points on the
hypersurface may correspond to a given point on the simplex,
and then mapping forward the canonical Brownian
motion on the hypersurface.  For example, a particular
instance of the Fleming-Viot process on $n$ points
arises from Brownian motion on the $(n-1)$-dimensional
sphere.  The Brownian motion on the hypersurface has the
normalised Riemannian volume as its equilibrium
distribution.  It is straightforward to compute the
corresponding distribution on the simplex, and this provides
a large class of interesting probability measures on the
simplex.

evans@stat.berkeley.edu

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

### 1945. MAXIMAL BRANCHING PROCESSES AND THEIR GENERALIZATIONS

Alexey V. Lebedev

We consider maximal branching processes (MBP) introduced by
J.Lamperti (1970) which resemble the Galton-Watson branching processes,
but with one difference: every time we get maximum of offspring numbers
instead of their sum. We generalize MBP from $Z_+$ to $R_+$
(by analogy with Jirina processes). Ergodic theorem is proved, some
properties are studied, examples are given. Limit theorems for stationary
distributions are obtained. Applications in the queueing theory are shown
(namely, for gated infinite-server queues researched by S.Browne et all.(1992)).

alebedev@mech.math.msu.su



### 1946. POISSON PROCESS PARTITION CALCULUS WITH APPLICATIONS TO EXCHANGEABLE MODELS AND BAYESIAN NONPARAMETRICS

Lancelot F. James

Fubini Calculus for Poisson processes. The approach is an
amplification of Bayesian techniques developed in Lo and
Weng for gamma/Dirichlet processes. Applications to models
are considered which fall within an inhomogeneous spatial
extension of the size biased framework used in Perman,
Pitman and Yor. Among some of the results; an explicit
partition based calculus is then developed for such models,
which also includes a series of important exponential
change of measure formulae. These results
are then applied
to solve the mostly unknown calculus for spatial Levy-Cox
moving average models.
The analysis then proceeds to exploit a structural feature
of a scaling operation which arises in
Brownian excursion theory.
From this a series of new mixture representations
and posterior characterizations for large classes of
random measures, including probability measures, are
given.These results are applied to yield new
results/identities related to the large class of
two-parameter Poisson Dirichlet models.
The results also yields easily perhaps
the most general and certainly quite
informative characterizations of extensions of the Markov-Krein
correspondence exhibited by linear functionals of
extension of Doksum's Neutral to the Right priors  (NTR)
to a spatial setting. NTR models are practically
synonymous with exponential functions of subordinators and
arise in Bayesian nonparametric survival models. It is shown
that manipulation of the exponential formulae makes
what has been formidable analysis transparent. Additional
interesting identities related to the the Dirichlet process
and other measures are developed. Based on practical
considerations, computational procedures which are
extensions of the Chinese restaurant process are also
developed.

lancelot@ust.hk



### 1947. INTERSECTIONS OF BROWNIAN MOTIONS

Davar Khoshnevisan

intersections of Brownian motion paths. Among
other things, we present a truly elementary
proof of a classical theorem of A. Dvoretzky,
P. Erdös and S. Kakutani. This proof is motivated
by old ideas of P. Lévy that were originally used
to investigate the curve of
planar Brownian motion.

davar@math.utah.edu

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

### 1948. STABILITY OF PARABOLIC HARNACK INEQUALITIES

Richard Bass and Martin Barlow

Let $(G,E)$ be a graph with weights $\{a_{xy}\}$ for which a
parabolic Harnack inequality holds with  space-time scaling
exponent $\beta\ge 2$.
Suppose $\{a'_{xy}\}$ is another
set of weights that are comparable to $\{a_{xy}\}$.
We prove that this parabolic Harnack inequality
also holds for $(G,E)$ with the weights $\{a'_{xy}\}$.
We also give necessary and sufficient conditions
for this parabolic Harnack inequality to hold.

bass@math.uconn.edu  barlow@math.ubc.ca

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### 1949. EINSTEIN RELATION FOR A CLASS OF INTERFACE MODELS

Roberto H. Schonmann

A class of SOS interface models which can be seen as simplified
stochastic Ising model interfaces is studied.
In the absence of an external field the long-time fluctuations
of the interface are shown to behave as Brownian motion with
diffusion coefficient $(\sigma^{\text{GK}})^2$ given by a Green-Kubo
formula.
When a small external field $h$ is applied, it is shown that
the shape of the interface converges exponentially fast to a stationary
distribution and the interface moves with an asymptotic velocity $v(h)$.
The mobility is shown to exist and to satisfy the
Einstein relation: $(dv/dh)(0) = \beta (\sigma^{\text{GK}})^2$,
where $\beta$ is the inverse temperature.

rhs@math.ucla.edu

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### 1950. ON EDGEWORTH EXPANSIONS FOR DEPENDENCY-NEIGHBORHOODS-CHAIN STRUCTURES AND STEIN'S METHOD

Yosef Rinott  and Vladimir Rotar

Let $W$ be the sum of dependent random variables, and $h(x)$ be a function. This paper provides an Edgeworth expansion
of an arbitrary length'' for $E\{h(W)\}$ in terms of
certain characteristics of dependency, and of the
smoothness of $h$ and/or the distribution of $W$. The core
of the class of dependency structures for which these
characteristics are meaningful is the local dependency, but
in fact, the class is essentially wider. The remainder is
estimated in terms of Lyapunov's ratios. The proof is based
on a Stein's method. The most typical example for the
results of this paper is mixing on graphs, that is when the
parameter indexing the summands, which is usually thought
of as a time'' or space'' parameter, has values which
may be identified with vertices of a graph. If the graph is
a usual integer valued lattice in $\QTR{Bbb}{Z}^{k}$, with
edges connecting only nearest vertices, we deal with the
usual mixing scheme for random fields, and for $k=1$ - with
a process on a line. If the graph is arbitrary, the scheme
is more complicated. This is especially true when the graph
is random, and its structure may depend on the values of
summands. In this case the above dependency neighborhoods
may be random too.

rotar@sciences.sdsu.edu

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### 1951. ON OPTIMALITY IN PROBABILITY AND ALMOST SURELY FOR CONTROLLED STOCHASTIC PROCESSES WITH A COMMUNICATION PROPERTY

Tatiana Belkina and Vladimir Rotar

The paper concerns both controlled diffusion processes, and
processes in discrete time. We establish conditions under
which the strategy minimizing the expected value of a cost
functional has a much stronger property; namely, it minimizes
the random cost functional itself for all
realizations of the controlled process from a set, the
probability of which is close to one for large time
horizons. The main difference of the conditions mentioned
from those obtained earlier is that the former do not deal
with strategies optimal in the mean themselves but concern
a possibility of transition of the controlled process from
one state to another in a time with a finite expectation.
It makes the verification of these conditions in a number
of situations much easier.

rotar@sciences.sdsu.edu

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

### 1952. DIFFUSION PROCESSES ON FRACTAL FIELDS: HEAT KERNEL ESTIMATES AND LARGE DEVIATIONS

Ben M. Hambly and Takashi Kumagai

A fractal field is a collection of fractals with, in general, different
Hausdorff dimensions, embedded in ${\bf R}^2$. We will construct diffusion
processes on such fields which behave as Brownian motion in ${\bf R}^2$
outside the fractals and as the appropriate fractal diffusion within
each fractal component of the field. We will discuss the properties of
the diffusion process in the case where the fractal components tile
${\bf R}^2$. By working in a suitable shortest path metric we will
establish heat kernel bounds and large deviation estimates which
determine the trajectories followed by the diffusion over short times.

hambly@maths.ox.ac.uk    kumagai@kurims.kyoto-u.ac.jp

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• Or here.

### 1953. LARGE DEVIATIONS FOR STOCHASTIC REACTION-DIFFUSION SYSTEMS WITH MULTIPLICATIVE NOISE AND NON-LIPSCHITZ REACTION TERM

S. Cerrai, M. Roeckner

Following classical work by M.I. Freidlin and subsequent works
by R. Sowers and S. Peszat, we prove large deviation estimates
for the small noise limit of systems of stochastic
reaction-diffusion equations with globally Lipschitz but
unbounded diffusion coefficients, however,
assuming the reaction terms to be only locally Lipschitz
with polynomial growth. This generalizes results of the above
mentioned authors.
Our results apply, in particular, to systems of stochastic
Ginzburg-Landau equations with multiplicative noise.

cerrai@cce.unifi.it
roeckner@mathematik.uni-bielefeld.de



### 1954. ON L^P-UNIQUENESS AND ESSENTIAL SELF-ADJOINTNESS OF SYMMETRIC DIFFUSION OPERATORS ON RIEMANNIAN MANIFOLDS

V.I. Bogachev, M. Roeckner

It is proved that the symmetric diffusion operator
$\mathcal{L}f = \Delta f + <b, \nabla>$ on a complete
Riemannian Manifold of dimension $d$ is $L^p$-unique
provided that the vector field $b$ is locally in $L^p$
with respect to the Riemannian volume and $p>d$.
An analogous statement is proved for for elliptic
operators with non constant second order part.

bibos@physik.uni-bielefeld.de

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

### 1955. PERTURBATIONS OF GENERALIZED MEHLER SEMIGROUPS AND APPLICATIONS TO STOCHASTIC HEAT EQUATIONS WITH LEVY NOISE AND SINGULAR DRIFT

P. Lescot, M. Roeckner

In this paper we solve the Kolmogorov equation and, as a
consequence, the martingale problem corresponding to a stochastic
differential equation of type
$dX_t=AX_t\,dt+b(X_t)\,dt+dY_t,$
on a Hilbert space $E$, where $(Y_t)_{t\ge0}$ is a Levy process on
$E$, $A$ generates a $C_0$-semigroup on $E$ and $b:E\to E$. Our main
point is to allow unbounded $A$ and also singular (in particular,
non-continuous) $b$.  Our approach is based on perturbation theory
of $C_0$-semigroups, which we apply to generalized Mehler semigroups
considered on $L^2(\mu)$, where $\mu$ is their respective invariant
measure. We apply our results, in particular, to stochastic heat
equations with Levy noise and singular drift.

paul.lescot@insset.u-picardie.fr
roeckner@mathematik.uni-bielefeld.de



### 1956. SCALING LIMIT OF STOCHASTIC DYNAMICS IN CLASSICAL CONTINUOUS SYSTEMS

M. Grothaus, Y.G. Kondratiev, E. Lytvynov, M. Roeckner

We investigate a scaling limit of gradient
stochastic dynamics associated to Gibbs states
in classical continuous systems on $\RR^d$,
$d \geq 1$. For these dynamics several scalings
have already been studied, see e.g. [Bro80].
The aim is to derive macroscopic quantities
from a given micro- or mesoscopic system. The
scaling we consider has been investigated in
[Bro80] and [Ros81]. Assuming that the underlying
potential is smooth, compactly supported and
positive, convergence of the generators of the
scaled stochastic dynamics, averaged with respect
to time, has been analyzed in [Spo86].
Another approach has been proposed in [GP85],
where the idea has been to prove convergence of
the corresponding resolvents.
We prove that the Dirichlet forms of the scaled
stochastic dynamics converge on a core of
functions to the Dirichlet form of a generalized
Ornstein-Uhlenbeck process. The proof is based
on the analysis and geometry on the configuration
space which was developed in [AKR98a],[AKR98b],
and works for general Gibbs measures of Ruelle
type. Hence, the underlying potential may have
a singularity at the origin, only has to be
bounded from below, and may not be compactly
supported. Therefore, singular interactions of
physical interest are covered, as e.g. the one
given by the Lennard-Jones potential, which is
studied in the theory of fluids. Furthermore,
using the Lyons-Zheng decomposition we give a
simple proof for the tightness of the scaled
processes. We also prove that the corresponding
generators, however, do not converge in the $L^2$-
sense. This settles a conjecture formulated in
[Bro80], [Ros81], [Spo86].

bibos@physik.uni-bielefeld.de

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### 1957. SINGULAR DISSIPATIVE STOCHASTIC EQUATIONS IN HILBERT SPACES

G.d. Prato, M. Roeckner

Existence of solutions to martingale problems
corresponding to singular dissipative stochastic
equations in Hilbert spaces are proved for any
initial condition. The solutions for the single
starting points form a conservative diffusion
process whose transition semigroup is shown to be
strong Feller. Uniqueness in a generalized sense
is proved also, and a number of applications is
presented.

roeckner@mathematik.uni-bielefeld.de



### 1958. ON THE SPECTRUM OF A CLASS OF (NONSYMMETRIC) DIFFUSION OPERATORS

M. Roeckner, F.-Y. Wang

In terms of the upper bounds of a second order
elliptic operator acting on specific Lyapunov-type
functions with compact level sets, sufficient
conditions are presented for the corresponding
Dirichlet form to satisfy the Poincar\'e and
the super-Poincar\'e inequalities. Here the
elliptic operator is assumed to be symmetric on
$L^2(\mu)$ for some probability measure $\mu$.
As applications, we prove for a class of (non-
symmetric) diffusion operators generating
$C_0$-semigroups on $L^1(\mu)$ that their
$L^p(\mu)$-essential spectrum is empty. This
follows since we prove that their $C_0$-semigroups
are compact.

roeckner@mathematik.uni-bielefeld.de



### 1959. SURFACE MEASURES AND TIGHTNESS OF CAPACITIES ON POISSON SPACE

V.I. Bogachev, O.V. Pugachev, M. Roeckner

We prove tightness of $(r,p)$--Sobolev capacities
on configuration spaces equipped with Poisson measure.
By using this result we construct surface measures
on configuration spaces in the spirit of
the Malliavin calculus.
A related Gauss--Ostrogradskii formula is obtained.

bibos@physik.uni.bielefeld.de

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stefano . iacus at unimi . it