




Probability Theory / Théorie des probabilités (D. Dawson and G. Slade, Organizers)
 MARTIN BARLOW, University of British Columbia
Random walk on supercritical percolation clusters

Consider bond percolation on Z^{d}. It is well known that for
p > p_{c} there exists (except for a set of w with probability
zero) a unique infinite cluster C(w), which has positive
density.
I will discuss the behaviour of the simple random walk on C(w).
The problem divides into two parts. The first is to use fairly well
known properties of supercritical percolation to control the volume and
spectral gap of balls in C(w). The second is to use `heat
kernel' methods, which have mainly been developed for very regular
graphs, in this situation, where there are small local irregularities.
 CLAUDE BELISLE, Laval University, Quebec, Quebec G1Y 2W6
Convergence properties of hitandrun samplers

I will review the main convergence properties of the hitandrun
sampler. An example exhibiting arbitrarily slow convergence will be
presented. I will also discuss a variation of the hitandrun sampler
where the sequence of update directions is now a deterministic sequence
on the unit sphere. When the set of cluster points of that sequence is
full dimensional, the resulting hitandrun sampler converges to its
target distribution.
 T. COX, Department of Mathematics, Syracuse University, Syracuse,
New York 13214, USA
Measurevalued limits of a LotkaVolterra model

A stochastic spatial version of the LotkaVolterra model was introduced
and studied by Neuhauser and Pacala. We show that suitably rescaled
versions of the process converge weakly to superBrownian motion with
drift.
 D. DAWSON, Carleton University, Ottawa, Ontario K1S 5B6
Hierarchical random walks and HMF limits

This talk will be an introduction to the idea of hierarchical mean
field limit as an extension of the wellknown meanfield approximation.
In addition applications to critical phenomena for some classes of
population models will be described. A brief discussion of related
questions of transience and recurrence properties of hierarchical
random walks will be included.
 W. HONG, School of Mathematics and Statistics, Carleton University, Ottawa,
Ontario K1S 5B6
Large deviations for the superBrownian motion

Large deviation principles are established in dimensions d ³ 3 for
the super Brownian motion with random immigration X_{t}^{r},
where the immigration rate is governed by the trajectory of another
superBrownian motion r. The speed function is t for d ³ 4 and t^{1/2} for d=3, compared with the existing results, the
interesting phenomenon happened in d=4 with speed t ( although only
the upper large deviation bound is derived here ) is just because the
structure of this new model: the random immigration ``smooth'' the
critical dimension in some sense. The rate function are characterized
by an evolution equation.
 MICHAEL KOURITZIN, Department of Mathematical and Statistical Sciences, University of
Alberta,
Edmonton, Alberta T6G 2G1
Markov chain approximations of stochastic
reactiondiffusion equations

The approximation of spatial stochastic processes by wellconstructed
Markov chains is important for simulating and filtering. Based upon joint
works with Hongwei Long and Wei Sun, I shall discuss the merits and
convergence of a Markov chain construction for approximating stochastic
reactiondiffusion equations with Poisson measure noise. Most basically, our
construction relies upon the random time change method as the previous
constructions of Arnold and Theodosopulu or Kotelenez did. However, I shall
also discuss our innovations to reduce the Markov chain rates, yielding a
hundred fold speed increase in our simulations, and to include a broader
class of operators and reaction functions. The proof techniques for
quenched convergence, where a Poisson noise source path is fixed, and
annealed convergence, where the Poisson source is considered as a random
medium, will both be discussed.
 NEAL MADRAS, Department of Mathematics and Statistics,
York University, Toronto, Ontario M3J 1P3
Decomposition of Markov chains

The speed of convergence of a Markov chain to its equilibrium
distribution has been a subject of intense study in recent years. Much
progress has been made for chains with nice structure, but one often
has to deal with chains that are not quite so nice. This talk will
describe a method that analyzes convergence rates by decomposing a
Markov chain into smaller pieces. The idea is that if the chain
equilibrates rapidly on each piece, and if the chain moves from piece
to piece efficiently, then the entire chain equilibrates rapidly. We
will give some examples of this method in action. This talk describes
joint work with Dana Randall and with Zhongrong Zheng.
 C. MUELLER, Department of Mathematics, University of Rochester, Rochester,
New York 14627, USA
The 3dimensional wave equation with a singular random potential

We consider the wave equation is 3 dimensions, with a potential term
which is a Gaussian noise that is white in time, but correlated in
space. The solution is just on the edge of existing; it exists as a
generalized function. If the noise were a little smoother, the
solution would be a function. This is joint work with R. Tribe,
building on earlier work on the heat equation with potential.
 J. QUASTEL, University of Toronto, Toronto, Ontario M5S 3G3
Time decay and fluctuations in conservative particle systems

We will discuss central limit theorems for additive functionals of
Markov processes consisting of infinitely many interacting particles,
where slow decay of correlations leads to nonstandard scaling.
 A. SAKAI, University of British Columbia, Vancouver,
British Columbia V6T 1Z2
Highdimensional graphical networks of selfavoiding walks

A single selfavoiding walk (SAW) is often used as a model of a linear
polymer in a good solution. Networks of mutuallyavoiding SAWs can be
used to model networks of polymers containing monomers capable of
making more than two chemical bonds, leading to branching. In this
talk, we consider networks of critical spreadout SAWs with the
topology of a general graph that may contain cycles or multiple lines,
and prove its Gaussian behaviour in Z^{d}, d > 4, as the
network's branch points are widely separated from each other. The
proof is based on the lace expansion on a tree and the Gaussian
behavior for the critical spreadout SAW's 2point function in
Z^{d}, d > 4.
 GORDON SLADE, Department of Mathematics, University British Columbia,
Vancouver, British Columbia V6T 1Z2
The incipient infinite cluster for highdimensional
oriented percolation

We use the lace expansion to construct the incipient infinite cluster
for sufficiently spreadout oriented percolation in d+1 dimensions,
with d > 4. This is joint work with Remco van der Hofstad and Frank
den Hollander.
 CHRIS SOTEROS, Mathematical Sciences Group, University of Saskatchewan,
Saskatoon, Saskatchewan S7N 5E6
The statistics of embeddings of eulerian graphs in Z^{d}

Recent rigorous results regarding embeddings of eulerian graphs
(connected graphs with only even degree vertices) in Z^{d} will be
presented. The study of these embeddings is motivated by viewing each
embedding as a possible conformation of a polymer network; for a fixed
size (number of edges), distinct (up to translation) embeddings are
considered equally likely. The focus will be on embeddings in Z^{3},
Z^{2}, and pseudoonedimensional subgraphs (infinite tubes) of Z^{3}.
Rigorous results and open questions relating to the density of vertices
of degree greater than two, the density of nearestneighbour contacts,
and the entanglement complexity (probability of knotting and linking)
of these embeddings will be discussed.
 GEORGE STOICA, University of New BrunswickSaint John Campus, Department of
Mathematical
Sciences, Saint John, New Brunswick E2L 4L5
Case study of a nonlinear stochastic delay equation

We compute the rate of growth at infinity for the solution
of a stochastic delay equation with nonlinear diffusion coefficient
and a drift term linear in the space variable. We also prove that the
density of the above process is regular under a timedependent
limited contact hypothesis. The latter allows a relaxation of the
meanellipticity condition for degenerate parabolic secondorder
timedependent diffusion operators.
 J. WALSH, UBC
To be announced

 X. ZHOU, Department of Mathematics, Concordia University, Montreal,
Quebec H4B 1R6
Dualities involving coalescing Brownian motion

Two kinds of dualities are discussed in this talk. We first point out a
dual relationship between coalescing Brownian motions and annihilating
Brownian motions. This duality is similar to the one in voter model.
The other duality is between two coalescing Brownian motions. More
precisely, the distribution of a coalescing Brownian motion at time t
can be determined by another coalescing Brownian motion running
backward. Such a duality can be used to formulate martingale problems
for coalescing Brownian motions.

