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Probability Theory / Théorie des probabilités
(D. Dawson and G. Slade, Organizers)

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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 hit-and-run samplers

I will review the main convergence properties of the hit-and-run sampler. An example exhibiting arbitrarily slow convergence will be presented. I will also discuss a variation of the hit-and-run 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 hit-and-run sampler converges to its target distribution.

T. COX, Department of Mathematics, Syracuse University, Syracuse, New York  13214, USA
Measure-valued limits of a Lotka-Volterra model

A stochastic spatial version of the Lotka-Volterra model was introduced and studied by Neuhauser and Pacala. We show that suitably rescaled versions of the process converge weakly to super-Brownian 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 well-known mean-field 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 super-Brownian 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 super-Brownian 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 reaction-diffusion equations

The approximation of spatial stochastic processes by well-constructed 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 reaction-diffusion 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 3-dimensional 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 non-standard scaling.

A. SAKAI, University of British Columbia, Vancouver, British Columbia  V6T 1Z2
High-dimensional graphical networks of self-avoiding walks

A single self-avoiding walk (SAW) is often used as a model of a linear polymer in a good solution. Networks of mutually-avoiding 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 spread-out 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 spread-out SAW's 2-point function in Z^d, d > 4.

GORDON SLADE, Department of Mathematics, University British Columbia, Vancouver, British Columbia  V6T 1Z2
The incipient infinite cluster for high-dimensional oriented percolation

We use the lace expansion to construct the incipient infinite cluster for sufficiently spread-out 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 Z³, Z², and pseudo-one-dimensional subgraphs (infinite tubes) of Z³. Rigorous results and open questions relating to the density of vertices of degree greater than two, the density of nearest-neighbour contacts, and the entanglement complexity (probability of knotting and linking) of these embeddings will be discussed.

GEORGE STOICA, University of New Brunswick-Saint 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 time-dependent limited contact hypothesis. The latter allows a relaxation of the mean-ellipticity condition for degenerate parabolic second-order time-dependent 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.