## Program in Probability and Its Applications

Seminar Series

**August 19, 1998**

*Mean-field lattice trees*

Remco van der Hofstad, Delft University of Technology

Abstract: TBA

** August 25, 1998**

*Strict concavity of the Brownian motion intersection exponent*

Greg Lawler, Duke University

Abstract: TBA

**September 9th, 1998**

*A Pattern Theorem for Lattice Clusters*

Neal Madras, York University

Abstract

In 1963, Kesten proved a Pattern Theorem for self-avoiding walks, which
says that any finite sequence of steps that can occur in the middle
of a long self-avoiding walk must in fact occur pretty often on almost
all self-avoiding walks. This result has had many applications in the
theory of self-avoiding walks, including several ratio limit theorems.
This talk will describe an analogous theorem for lattice trees and lattice
animals. A weighted version of this theorem also applies to polymer
collapse and percolation models. Applications and open questions will
be discussed.

**September 16th, 1998**

*Moment asymptotics for the Anderson model*

Wolfgang Koenig, Technische Universität Berlin / York University

Abstract: TBA

**September 23th, 1998**

*Self-diffusion for Brownian Motions with Local Interaction*

Ilie Grigorescu, McMaster University / The Fields Institute

Abstract: TBA

**September 23rd, 1998**

*Infinite Systems of Diffusions in Population Biology *

Jan Swart, University of Nijmegen

Abstract: TBA

**September 23rd, 1998**

*Ruelle's probability cascades and an abstract cavity method*

Erwin Bolthausen, University of Zurich

Abstract

The so called generalized random energy model (GREM for short) has been
introduced by Derrida as a very simple model in spin glass theory. Some
of its remarkable properties in the limit had been discussed by Ruelle.
In a joint paper with Alain-Sol Sznitman, it was found, that there is
a very simple and appealing clustering process which governs some of
the crucial properties of this model. This clustering process also allows
to formulate a continuous version of the GREM without the necessity
of cumbersome limiting procedures. Furthermore, one can give an abstract
version of the so called cavity method which plays a crucial role in
the nonrigorous discussion of the Sherrington-Kirkpatrick model.

**October 13th, 1998**

*On a Conjecture of B. Jorgensen and A.D. Wentzell: *

from Extreme Stable Laws to Tweedie Exponential Dispersion Models

Vladimir Vinogradov, University of Northern British Columbia

Abstract

The class of Tweedie exponential dispersion models includes such well-known
continuous distributions as the normal and gamma, the purely discrete
sclaled Poisson distribution, as well as the class of mixed compound
Poisson-Gamma distributions which have positive mass at zero, but are
otherwise continuous. The remaining Tweedie models are derived by exponential
tilting of extreme stable distributions. This quite heterogeneous class
of distributions was introduced in statistics by Tweedie (1984),mainly
because of the simple form of their unit variance function: V(m) = m^p.

It is known (see, e.g., Jorgensen (1997)) that Tweedie models possess
scaling properties similar to those of the stable laws and also that
they emerge as weak limits of appropriately scaled natural exponential
families. Domains of attraction to Tweedie models are described in Jorgensen,
Martinez and Tsao (1994) and Jorgensen, Martinez, Vinogradov (1998).
At the same time, Jorgensen (1997, pp. 150-151) and Wentzell (1998,personal
communication) independently conjectured that the classical theorems
on weak convergence to stable laws and those on weak convergence to
Tweedie models should be related.

In this lecture, it will be shown how the theorems on weak convergence
to Tweedie models with index p>2 can be derived from those on weak
convergence to the positive stable laws.

**October 14th, 1998**

*Large Deviation methods in Risk Theory*

Anders Martin-Lof, Stockholm University

Abstract

It will be shown that many well known estimates of e.g. ruin probabilities
for compound Poisson processes can in a natural way be derived using
L.D. techniques including second order refinements. The L.D framework
provides an interesting description of how to allocate buffer capital
to two systems in "thermodynamic equilibrium".

**November 4th, 1998**

*Random walks in a field of traps: from simple random walk on
Z^d to any Markov chain*

Stanislav Evguenievich Volkov, York University/Fields Institute

Abstract

Start with a S.R.W. Designate some points of Z^d (d>2) to be "traps"
so that if a particle hits one of them it stays in it forever. The traps
are placed at random independtently from each other. We are interested
in identifying the maximal critical density of traps for which the random
walk is still transient (that is, it eventually goes to infinity). Two
cases are considered:

A) annealed, when the set of traps is updated at each unit of time;

Q) quenched, when the set of traps is fixed once and for all.

We will show that the solutions to these two cases coincide.

Next we consider an arbitrary Markov chain on any space S and show
the equivalence of the annealed and quenched problems under uniform
boundedness of Greens function. Moreover, assuming some geometry on
the space S, a relative spherical symmetry of the density q(x) of the
field of traps implies a necessary and sufficient condition for the
transience. This condition consists in the finiteness of the sum, sum_{x
\in S} g(x_0,x)q(x).

The talk is based on two papers: one joint with F. den Hollander and
M.Menshikov, the other with R. Pemantle.

**November 4th, 1998**

*Large deviations for observation and related processes*

George L. O'Brien, York University

Abstract

Let $(X_n, n \geq 1)$ be an i.i.d.\ sequence of positive random variables
with distribution function $H$. Let $\Phi_H := \{ (n,X_n),\;n \geq 1\}$
be the associated observation process. We view $\Phi_H$ as a measure
on $E := [0,\infty) \times (0,\infty]$ where $\Phi_H(A)$ is the number
of points of $\Phi_H$ which lie in $A$. A family $(V_s, s > 0)$ of
transformations is defined on $E$ in such a way that for suitable $H$
the distributions of $(V_s\Phi_H, s > 0)$ satisfy a large deviation
principle and that a related Strassen-type law of the iterated logarithm
also holds. Some related large deviation principles and loglog laws
are then derived for extreme values and partial sums processes. Similar
results are proved for $\Phi_H$ replaced by certain planar Poisson processes,
with parallel applications to extremal processes and spectrally positive
Levy processes.

**November 18th, 1998**

*Duality for a stochastic p.d.e. with density dependent branching
noise.*

Siva Athreya, Fields Institute (Joint work with Roger Tribe)

Abstract

We establish a duality relation for the moments of the one dimensional
stochastic partial differential equation

\[ u_t = \Delta u + \sqrt{\sigma(u)} \dot W_{t,x}, \]

where $\dot W_{t,x}$ is space-time white noise and $\sigma$ is a real
analytic function satisfying certain growth and regularity conditions.
The dual process is a system of Brownian particles with an interactive
branching mechanism. In certain cases the duality relation implies weak
uniqueness for the s.p.d.e. We will also present ongoing work in applying
the same methodology towards uniqueness issues in finite dimensional
stochastic differential equations.

(Joint work with Roger Tribe)

**November 18th, 1998**

*The lace expansion and oriented percolation.*

Eric Derbez, McMaster University / Fields Institute

Abstract

The lace expansion (originated by Brydges and Spencer) has been used
to tackle many problems in statistical mechanics. This talk will consist
of an exposition of some of the results in oriented percolation and
the way in which the lace expansion can be set up to handle this model
(as per work in progress with R v d Hofsdat and G Slade). If overhead
projectors permit, Java demonstrations may be used to illustrate the
salient points.

** November 25th, 1998**

*Labyrinths*

Stanislav Volkov, York University / the Fields Institute

Abstract

First, consider the two-dimensional lattice Z^2. Let each point of it
be a "tunnel", a "\" mirror, a "/" mirror or a "normal point" independently
from the others. Once a realization of this random field is fixed, a
particle performing a random walk on Z^2 given this realization obeys
the following laws: If a point the walker hits is a tunnel, it goes
straight through it; a mirror, it reflects from it like a beam of light
would; a normal point, it behaves like a simple RW, i.e. choses one
of the four possible neighbors with equal probabilities.

This model is called a "random walk in a random labyrinth" and was
introduced by Grimmett, Menshikov and V. in 1996.

The labyrinth is called {\em recurrent} if the RW on it is recurrent,
and transient otherwise. The labyrinth is called {\em localised} if
the number of points which a walker can visit is finite.

Assuming that the density of normal points is non-zero, we show 1)
in the case of Z^2, a labyrinth is recurrent a.s. and 2) under which
conditions it is non-localized with positive probability. For a general
model on Z^d, d>2, we present sufficient conditions ensuring that
a labyrinth is transient with positive probability.

**December 9th, 1998**

*Metastability and Droplet Growth*

Frank den Hollander, University of Nijmegen

Abstract

In this talk we describe {\it metastability and droplet growth} for
the two-dimensional lattice gas with Kawasaki dynamics at low temperature
and low density. Particles perform simple exclusion on $\mathbb Z^2$,
and inside a {\it finite} but arbitrary box $\Lambda_0$ there is a binding
energy $U$ between neighboring occupied sites. The initial configuration
is chosen such that $\Lambda_0$ is empty while outside $\Lambda_0$ particles
are placed randomly with density $\rho = e^{-\beta\Delta}$ for some
$\Delta \in (U,2U)$, where $\beta$ is the inverse temperature. Since
in equilibrium $\Lambda_0$ wants to be fully occupied when $\beta$ is
large, the dynamics will tend to fill $\Lambda_0$ with particles. However,
since the particle density $\rho$ is small this will take a long time
to happen. We investigate how the transition from empty to full takes
place. In particular, we identify the size and shape of the {\it critical
droplet}, the time of its creation, and the typical trajectory prior
to its creation, in the limit as $\beta \to \infty$. The choice $\Delta
\in (U,2U)$ corresponds to the situation where the gas is supersaturated
and the critical droplet has side length $>1$.

Because particles are {\it conserved} under Kawasaki dynamics, the
analysis of metastability and droplet growth is more difficult than
for Ising spins under Glauber dynamics. The key point is to show that
at low density the gas outside $\Lambda_0$ can be treated as a reservoir
that creates and annihilates particles randomly at the boundary of $\Lambda_0$.
Once this has been achieved, the problem reduces to local metastable
behavior on $\Lambda_0$, and standard techniques from non-conservative
dynamics can be applied. Inside $\Lambda_0$ the dynamics is still conservative,
but this can be handled via local geometric arguments.

**December 9th, 1998**

*Large Time Asymptotics of the Nonlinear Filter*

Rami Atar, York University / the Fields Institute

Abstract

The nonlinear filtering equation describes the conditional law of a
Markov process $\{x_t\}$ at time $t$ conditioned on the path $y_s \int_0^s
g(x_u)du + w_s$ for $s\in[0,t]$. Here $g$ is a fixed function and $\{w_t\}$
is a Brownian motion independent of $\{x_t\}$. We study the exponential
decay rate of the variation distance between solutions that correspond
to different initial conditions. The question is related to Lyapunov
exponents, Birkhoff's contraction and large deviations.

**January 6th, 1999**

*Conditioning Super Brownian Motion*

Tom Salisbury,York University

Abstract

I'll describe some joint work with John Verzani, on conditioning super
Brownian motion (SBM) via its exit measure. In high dimensions, we condition
on the support of the exit measure hitting a given finite collection
of points, and represent this process in terms of a tree-like backbone
throwing off mass. The goal is to give an explicit dynamic description
of this tree. Along the way, I'll review LeGall's construction of the
super Brownian excursion.

**January 19th, 1999 **

*Asymptotic Behavior of Solutions of Parabolic SPDE*

Min Kang, York University / The Fields Institute

Abstract

We investigate the weak convergence as time goes to infinity of the
solutions of parabolic stochastic partial differential equations with
Dirichlet boundary conditions. We first prove a result for the linear
case, which is easy since the solution is Gaussian. Then we get a result
for non-linear drift case by proving a new comparison theorem in SDDE,
stochastic difference differential equations.

**January 20th, 1999**

*Liouville property and a conjecture of de Giorgi*

Martin Barlow, University of British Columbia

Abstract

In 1978 De Giogi conjectured that if $u(x)$, $x=(x_1, \dots, x_n) \in
R^n$ is a global bounded solution of the non-linear PDE $$ \Delta u
+ u - u^3 =0, \eqno(1)$$ which is monotone in the $x_1$ direction, then
$u$ is essentially one-dimensional. Recently Ghoussoub and Gui have
proved this conjecture for $n=2$.

It turns out that this problem can be reformulated in terms of a Liouville
property for a divergence form operator $$ {\cal L} = \nabla \sigma^2
\nabla. \eqno(2)$$ Here $\sigma(x)$, $x \in R^n$ is a positive (but
not uniformly positive) function, defined in terms of the solution $u$.
This problem can be studied using probabilistic techniques.

In this talk I will describe recent work (with R.F. Bass and C. Gui)
in which we prove a Liouville theorem for some operators of the form
(2). As a consequence, we can prove the conjecture under some additional
hypotheses.

**January 20th, 1999 **

*Generalizing the Martingale Central Limit Theorem.*

Dean Slonowsky, Fields Institute

Abstract

For each $n \geq 1$, let $X_n = \{ X_n(t) : t \in [0,1] \}$ be an $L_2$
martingale in $D([0,1])$. Roughly speaking, the martingale central limit
theorem states that if

\begin{itemize}

\item the jumps in the sample paths of $X_n$ become negligible (in a
certain sense) as $n$ gets large and

\item the predictable (or optional) quadratic variations of the $X_n$
converge to a continuous, increasing and deterministic limit $h : [0,1]
\rightarrow [0,{\infty})$ as $n$ goes to infinity

\end{itemize}

then the $X_n$ converge weakly in $D([0,1])$ to a Brownian motion $B_h$
which is ``stretched-out'' by $h$ (cf.\ Helland 82).

In this talk, we generalize the above result to the situtation in which
the latter condition holds for a sequence of {\em{general quadratic
variations}}---processes which resemble predictable/optional quadratic
variation but are not necessarily adapted. An application to the convergence
of set-indexed strong martingales will be briefly discussed.

** February 2nd, 1999**

*Graph Colouring with the Probabilistic Method*

Michael Molloy, University of Toronto** **

Abstract

The probabilistic method is a powerful technique, pioneered by Erdos,by
which one proves the existence of a combinatorial object by showing
that an attempt to produce such an object randomly will succeed with
positive probability. Often this argument will lead to an efficient
algorithm to construct the desired object. In this talk, we will survey
several applications of this technique to graph colouring. We will see
how to prove that under certain conditions, a graph has a colouring
meeting some desireable properties, by showing that a simple randomised
procedure to construct such a colouring will succeed with positive probability.
Much of the work described here is joint with Bruce Reed.

**February 10th, 1999**

*A Microscopic Model For Porous Medium Equation*

Shui Feng, McMaster University

Abstract

We introduce a stochastic lattice model that leads to porous medium
equation under diffusive scaling limit. This is a joint work with Ian
Iscoe and Timo Seppalainen.

**February 17th, 1999**

*Diffusions and heat kernel analysis on an infinite dimensional
group.*

Masha Gordina, McMaster University

Abstract

First I'll construct the heat kernel measure $\mu _t$ on an infinite
dimensional complex group $G$ using a diffusion in a Hilbert space.
Using properties of this diffusion I can prove that holomorphic polynomials
on the group are square integrable with respect to the heat kernel measure.
The closure of these polynomials, $\mathcal{H}L^2(G, \mu _t)$, is one
of two spaces of holomorphic functions I consider. Also I construct
a subgroup $G_{CM}$ of $G$ which is an analog of the Cameron-Martin
subspace. I'll describe an isometry from the first space to another
space of holomorphic functions $\mathcal{H}L^2(G_{CM})$. The main theorem
is that an infinite dimensional nonlinear analog of the Taylor expansion
defines an isometry from $\mathcal{H}L^2(G_{CM})$ into the Hilbert space
associated with a certain Lie algebra of the infinite dimensional group.
This is an extension to infinite dimensions of an isometry of B. Driver
and L. Gross for complex Lie groups. I'll explain how my results on
the Cameron-Martin subgroup are related to the work of I. Shigekawa
and H. Sugita on the complex Wiener space.

**February 17th, 1999**

*Steiner distances in random trees*

Amram Meir, York University

Abstract

Let G be a connected graph, S a subset of its vertices . The Steiner
distance d(S,G) of S in G is the minimum number of edges in any connected
subgraph of G that contains S. The total Steiner k-distance D(k;G) (k=2,3,...)
is the sum of d(S,G) over all subsets S of k vertices in G. (When k=2,
the quantity is known as Wiener index of G). The purpose of this talk
is to present asymptotic results (for large n): (i) For the expected
value M(k;n) of d(S,T) over all randomly chosen subsets of k nodes in
trees T with n nodes, which belong to certain (infinite) families F
of random rooted trees. (ii) For p(k;n,m), the probability that d(S,T)=m-1,
when |S|=k fixed, |T|=n and T belongs to a family F. (joint work with
L. Clark and J.W. Moon)

**February 24th, 1999**

*Trapping and saturation in a random media*

Alejandro F. Ramírez, École Polytechnique Fédérale de Lausanne

Abstract

We consider a model of diffusion in a random media originated from the
nuclear waste management industry. In each site of the cubic lattice
$Z^{d}$ there is an obstacle with a positive integer valued random capacity.
On the other hand, on the origin and independently of the distribution
of the obstacles, independent random walks are introduced so that at
time $t$, $N(t)$ random walks have been born, and so that they survive
until touching an obstacle. When this happens the capacity of the touched
obstacle decreases by one. An obstacle of 0 capacity disappears. For
low values of $N(t)$ we prove that with probability 1, the saturated
obstacles correspond to a sphere of volume $N(t)$. With the help of
this fact, and using the second version of the method of enlargement
of obstacles of Sznitman, we prove that there are three injection regimes,
depending on $N(t)$ and the dimension. In particular estimates on the
asymptotic behaviour of the principal Dirichlet eigenvalue of the discrete
Laplacian operator on a box of side $t$ with sites removed with positive
probability, each one independent of the others, is required. This is
a joint work with Gerard Ben Arous.

**February 24th, 1999**

*Vertex-Reinforced Random Walks: What's New?*

Stanislav Volkov, York University/Fields Institute

Abstract

Vertex-Reinforced Random Walk (VRRW) is a non-Markovian random process
which "prefers" to visit the states it has visited before. The questions
about VRRW are classical examples of problems which are very easy to
formulate but extremely hard to solve. We consider VRRW on a very general
class of infinite graphs and show that, with a positive probability,
VRRW on them visits only finitely many points. We conjecture that on
all graphs of bounded degree this takes place a.s. (this has only been
shown for the case of trees). The proof is based on identifying a "trapping"
subgraph on which VRRW can get stuck and showing how this actually happens
(and what the limiting empirical occupational measure is). An interesting
finding is that this limiting measure is dependent on whether the graph
contains triangles or not. These results represent my on-going research
and generalize those obtained by Pemantle and Volkov (1998) for $\Z^1$,
though using a different technique.

**March 3rd, 1999**

*Some results for long range percolation and maniscale models*

Mikhail Menchikov, Moscow State University

Abstract

We consider two-dimensional Bernoulli long range percolation model.
We prove that if we have a percolation for this model, we can find a
positive N such that if we cut all edges of size more then N, anyway
percolation will be. We demonstrate a conterexample for the case of
dependent percolation model. This example closely connected with discrete
and continuous multiscale models.

**March 3rd, 1999**

*Skew Convolution Semigroups and Immigration Processes*

Zenghu Li, Beijing Normal University

Abstract

The immigration phenomena associated with branching models have been
investigated quite extensively in the past decades. In the measure-valued
setting, immigration processes have been studied with different motivations
by Dawson, Dynkin, Evans, Gorostiza, Ivanoff, Shiga and others. We focus
on a special type of immigration associated with DW superprocesses formulated
by skew convolution semigroups. Transition semigroups of the immigration
processes can be characterized in terms of infinitely divisible probability
entrance laws and their trajectory structures can be investigated by
using Kuznetsov processes. The immigration processes provide a number
of new limit theorems. In particular, the fluctuation limits of them
lead to some infinite dimensional Ornstein-Uhlenbeck processes described
by the branching mechanisms of the superprocesses. Other types of limit
theorems can be obtained by randomizing the skew convolution semigroup.

**March 17th, 1999**

*Vertex Reinforced Random Walks: the techniques.*

Stas Volkov, Fields Institute/York University

Abstract

This talk is intended for people who attended my first talk on VRRW
in February AS WELL AS for those who did not have a chance to come;
I will try my best to make it interesting for both. Vetrex-Reinforced
Random Walk (VRRW) on a graph (e.g. $Z^2$) is a nearest-neighbor walk
that goes to a vertex with a probability proportional to the number
of times this vertex has been visited before. Similar models for {\em
edges} has been posed/studied by Diaconis, Davis, Durrett, Selke, Toth,
Pemantle and others. It has been shown by Pemantle and Volkov that on
$Z^1$ VRRW visits only finitely many vertices a.s. (and just 5 with
a positive probability). Recently I proved that the similar statements
hold for many other graphs (lattices, trees, etc.) In this talk I will
outline the results with the stress on HOW they can be obtained and
what are the possible approaches for this kind of problems.

**March 24th, 1999**

*How does the Vervaat process really behave?*

Ricardas Zitikis, Laboratory for Research in Statistics and Probability,
Carleton University

Abstract

It is well-known that the appropriately normalized Vervaat process asymptotically
behaves like one half times the squared empirical process. During the
talk we shall give a complete description of the strong and weak asymptotic
behaviour of the Lp-distance between these two processes. In particular,
the law of the iterated logarithm, the ``other'' law of the iterated
logarithm, weak convergence, etc. for the aforementioned distance will
be discussed in detail.

**March 24th, 1999**

*An Introduction to Set-Indexed Strong Submartingales.*

Dean Slonowsky,Fields Institute

Abstract

In their 1994 paper, G. Ivanoff and E. Merzbach introduced the notion
of a set-indexed strong submartingale, a generalization of the planar
strong submartingales defined in the 1975 paper of R. Cairoli and J.
Walsh.

After motivating their definition through simple examples, we will
present several results in the theory of set-indexed strong submartingales
including:

- A Doob-Meyer-type decomposition.

- The definition and existence of quadratic variation for square integrable
set-indexed strong martingales.

- The role of set-indexed strong martingales in a Levy-type characterization
of set-indexed Brownian motion.

Time permitting, the framework for strong martingale central limit
theorems will be discussed.

**March 31st, 1999**

*On the long-term behaviour of superprocesses with a general branching
mechanism *

Yongjin Wang, Nankai University

Abstract

The persistence criteria for branching systems and superprocesses have
been established through a variety of ways since Dawson [1977]. We here
give an analytic approach to this argument, in which a simple proof
of the criteria is available. By extending our result to a larger class
of superprocesses with a general branching mechanism, we then get a
complete characterization of the long-term behaviour of the superprocesses
on the tempered measure space as well as on the finite measure space.
Finally, using Dynkin's stopped historical superprocesses, we also show
that the extinction and non-extinction of superprocesses at large time
are heavily dependent on the branching mechanisms.

**March 31st, 1999**

*Elements of free probability*

Alexandru Nica, University of Waterloo

Abstract

The name ``free probability'' does not refer to an attitude in the practice
of probability, but to a theory initiated by D. Voiculescu about 15
years ago, with motivation from problems on free products of von Neumann
algebras. The subject has evolved since then into a kind of parallel
to basic probability theory, where tensor products and independent random
variables are replaced by free products and free (non-commuting) random
variables. The talk will avoid the von Neumann algebra side of the theory,
and will focus on the latter parallelism (free-ness for random variables,
as a non-commutative analogue for the concept of independence). For
instance -- there exist free analogues of the Gaussian and of the Poisson
distributions; the free analogue of the Gaussian is the semicircle law
of Wigner, and appears as limit in the free central limit theorem of
Voiculescu. A nice way to prove the free CLT goes by using the free
counterpart of the notion of characteristic function, which is called
``the R-transform'' (of a probability measure on the real line). The
list of free analogues could certainly be continued (e.g. free analogues
of infinitely divisible distributions and of the Levy-Khinchin theorem,
free stable laws, processes with free increments) and is a subject of
current research. A notable feature of free probability is its connection
to the theory of random matrices, which is already signaled by the occurrence
of the Wigner's law in the free CLT, but also shows up in various other
places.

**April 7th, 1999**

*p-variation, integration and stock price modelling*

Rimas Norvaisa, Institute of Mathematics and Informatics, Lithuania

Abstract

The p-variation is a generalization of the total variation of a function
by replacing the absolute value of the increment in its definition by
the p-th power of this increment. Then a.a. sample paths of a BM have
bounded p-variation for each p >2. The usefulness of p-variation
property for the Riemann-Steiljes integral was shown by L.C. Young (1936).
We plan to discuss how the p-variation could be used in stochastic calculus
and stock price modelling.

**April 7th, 1999**

*Random spanning trees of Cayley graphs and compactifications
of groups*

Steven Evans, Department of Statistics, U.C. Berkeley

Abstract

We consider a natural algorithm for producing a random spanning tree
for the Cayley graph of a finitely generated, countably infinite group.
Associated with such a tree there is a (deterministic) compactification
of the group: a sequence of group elements that is not eventually constant
is convergent if the random geodesic through the spanning tree that
joins the identity to the nth element of the sequence converges in distribution
as n goes to infinity. We identify this compactification in a number
of simple examples and present some open problems.

**April 14th, 1999**

*Assymptotic properties of integral functionals of geometric stochastic
processes.*

Mikolos Csorgo, Carleton University

Abstract

This talk is based on a forthcoming paper by E. Csaki, M. Csorgo, A.
Foldes and P. Revesz, where we study strong asymptotic properties of
two types of integral fuctionals of geometric stochastic processes.
These integral functionals are of interest in financial modeling, yielding
various Asian type option pricings via appropriate selection of the
processes in their respective integrands. We show that, under fairly
general conditions on the latter processes, the log of the integral
functionals themselves asymptotically behave like appropriate sup functionals
of the processes in the exponents of their respective integrands. In
addition to presenting these theorems in their general form, we will
also illustrate them via geometric Brownian and geometric fractional
Brownian motions.

**April 14th, 1999**

*Forward and backward integrals revisited. *

Terry Lyons, Imperial College

Abstract

In 1998 Lyons and Zheng (Asterisque,157-158) observed that the Stratonovich
integral of a one form against a reversible and stationary diffusion
process makes sense even if the one form is square integrable. Interestingly
the resulting process is not a semi-martingale. However, it is a difference
of a forward and backward martingale. The technique has lead to a number
of useful estimates and is considerably more powerful than an ItF4 approach
which would require significant smoothness.

A major failing of the techniques mentioned above, involved the need
to start the process with the stationary measure. Integrating the form
along the sample path of the process conditioned to start at a pointmight
well not make sense. (In 1990 Proc. Royal Soc. Edinburgh Sect. A 115)
it was shown by Zheng and Lyons that the integral existed if the one
form was bounded. Stoica and Lyons (Annals of Prob 1999) proved that
if the one form was in the natural conjectural Lebesgue space (coming
from the Sobolev embedding theorem). Results about the behaviour at
infinity were also given.

In this talk I will survey the basic results and, if time permits,
I will give applications.

**April 28th, 1999**

*Optimal approximation of stochastic differential equations by
adaptive step-size control*

Norbert Hofmann, University of Erlangen

Abstract

The talk will discuss the strong approximation of stochastic differential
equations with respect to the global error in the L2-norm as well as
in the uniform norm. For equations with additive noise we present sharp
lower and upper bounds for the minimal error in the class of arbitrary
methods which use a fixed number of observations of the driving Brownian
motion. We introduce Euler schemes with adaptive step-size control which
perform asymptotically optimal.

** May 5th, 1999**

*Layout problems on random geometric graphs*

Mathew Penrose, University of Durham

Abstract

Given a graph, layout problems are concerned with choosing an ordering
of its vertices so that adjacent vertices lie close together in the
ordering. For example, the aim might be to minimize the sum over all
edges of the separation of the edge's endpoints in the ordering. We
investigate problems of this type for graphs generated randomly, by
percolation or by continuum analogs. Such graphs have been used by computer
scientists for comparing heuristics for these problems. If time permits
we may also discuss some connectivity properties of these graphs.

**May 13th, 1999**

*Level sets of additive Levy processes*

Davar Khoshnevisan, University of Utah

Abstract

I will talk about recurrence issues for level sets of some multiparameter
Levy processes. This approach unifies and explains some earlier existing
results in Markov processes, and points to a notion of regeneration
which makes sense in any dimension. This is on-going work with Yimin
Xiao.

**May 19th, 1999 **

*Quasi-potential of the Fleming-Viot process with neutral mutation
and selection*

Kenji Handa, Saga Universty

Abstract

In 1998, Dawson and Feng established large deviation principles for
the Fleming-Viot processes with neutral mutation and selection, and
the corresponding reversible measures as the sampling rate tends to
0. In this talk an identity between quasi-potential defined through
the dynamical rate function and the static rate function is discussed.

**May 26th, 1999**

*Slice sampler Markov chains*

Jeffrey S. Rosenthal, University of Toronto

Abstract

The slice sampler is a recent example of a Markov chain Monte Carlo
(MCMC) algorithm, designed to approximately sample from a given probability
distribution. In this talk, we will review recent work (joint with G.O.
Roberts) on the convergence properties of this algorithm. A particular
variant, the polar slice sampler, will be shown to have strikingly good
convergence rate for a large class of distributions.

**June 2nd, 1999 **

*Relaxation to equilibrium for interacting random walks *

Jeremy Quastel, University of Toronto

Abstract

We will survey results and methods for hydrodynamic limits and decay
estimates for different models of interacting conservative systems and
discuss open problems in the field.

**June 9th, 1999**

*State Dependent Multitype Spatial Branching*

Don Dawson, The Fields Institute

Abstract:

This talk will describe joint work with Andreas Greven on spatial infinite
type branching systems indexed by a countable group, for example $Z^d$
or the hierarchical group. The space of types is [0,1] and the state
of the system at a given site is a measure on [0,1]. The spatial components
of the system interact via migration. Instead of the classical independence
assumption on the evolution of different families of the branching population,
we introduce interaction between the families through a state dependent
branching rate of individuals and state dependent mean offspring of
individuals but for most results we restrict attention to the critical
case. One objective is to establish that the large scale structure of
surviving types is related to the immortal clan of super-Brownian motion.