The Seminar on Stochastic Processes is an ongoing series of conferences.
Many of the participants work on such topics as Markov processes,
Brownian motion, Superprocesses, and Stochastic Analysis. SSP encourages
interaction between young researchers and more senior ones, and
typically devotes half its program to informal and problem sessions.
An alphabetical list of former speakers is available.

From 1981 to 1992, the Proceedings of SSP were published as volumes
in Birkhäuser's Progress in Probability series. For more information
see: http://www.math.yorku.ca/Probability/ssparch.html

**Speaker list **

**Ted Cox **(Syracuse)

**Robert Griffiths **(Oxford)

**Chuck Newman **(Courant Institute)

Kavita Ramanan (Carnegie Mellon)**
**

Balasz Szegedy (University of Toronto)**
**

**Short presentations**

Michelle Boue, Trent University

Dimitris Cheliotis, University of Toronto

Nikolai Dokuchaev, Trent University

Anna Savu, University of Toronto

Deniz Sezer,York University

David Steinsaltz, Queen's University

Benedek Valko (University of Toronto)

Vladimir Vinogradov, (Ohio University)

**Michelle Boue**, Trent University

*Phase transitions for some interacting particle systems with
movement*

We consider a system of interacting moving particles on the d-dimensional
lattice. This sytem introduces disorder to a model proposed by Kesten
and Sidoravicius for the spread of epidemics. We will discuss and
compare the phase transitions of both systems.

**Dimitris Cheliotis** (University of
Toronto)

*The noise of perturbed random walk on some regular graphs*

We consider random walk on mildly random environment on finite transitive
d-regular graphs of increasing girth. After centering and scaling,
the analytic spectrum of the transition matrix converges in distribution
to a Gaussian noise. An interesting phenomenon occurs at d=2: as
the limiting object changes from a regular tree to the integers,
the noise becomes localized. The talk is based on joint work with
Balint Virag.

**Ted Cox** (University of Syracuse)

*Survival and coexistence for a stochastic Lotka-Volterra model*

In 1999, Neuhauser and Pacala introduced a stochastic spatial version
of the Lotka-Volterra model for interspecific competition. In this
talk I will discuss some recent work with Ed Perkins analyzing this
model. Our approach, which works when the parameters of the process
are "nearly critical," is to (1) show that suitably scaled
Lotka-Volterra models converge to super-Brownian motion, and (2)
"transfer" information from the super-Brownian motion
back to the Lotka-Volterra models. We are able to show survival
and coexistence for certain parameter values this way.

**Deniz Sezer,** (York University)

*Conditioning super-Brownian motion on its exit measure*

Let $X$ be a super-Brownian motion defined on a domain $E$ in
the euclidean space and $(X_D)$ be its exit measures indexed by
sub-domains of $E$. We pick a sub-domain $D$ and condition the super-Brownian
motion inside this domain on its exit measure $X_D$. We give an
explicit construction of the resulting conditional law in terms
of a particle system, which we call the ``backbone'', along which
a mass is created uniformly. In the backbone, each particle is assigned
a measure $\nu$ at its birth. The spatial motion of the particle
is an h-transform of Brownian motion, where $h$ is a potential that
depends on $\nu$. $\nu$ represents the particle's contribution to
the exit measure. At the particle's death two new particles are
born and $\nu$ is passed to the newborns by fragmentation into two
bits. (Joint work with Tom Salisbury.)

**Nikolai Dokuchaev**, Trent University

*Mean-reverting market model: Novikov condition, speculative opportunities,
and non-arbitrage*

We study arbitrage opportunities and possible speculative opportunities
for diffusion mean-reverting market models. We found that the Novikov
condition is satisfied for any time interval and for any set of
parameters. It is non-trivial because the appreciation rate has
Gaussian distribution converging to a stationary limit. It follows
that the mean-reverting model is arbitrage free for any finite time
interval. However, we found that this model still allows some speculative
opportunities: a gain for a wide enough set of expected utilities
can be achieved for a strategy that does not require any hypothesis
on market parameters and does not use estimation of these parameters.

**Bob Griffiths** (University of Oxford)

*Diffusion processes and coalescent trees.*

Diffusion process models for evolution of neutral genes have the
coalescent process underlying them. Models are reversible with transition
functions having a diagonal expansion in orthogonal polynomial eigenfunctions
of dimension greater than one, extending classical one-dimensional
diffusion models with Beta stationary distribution and Jacobi polynomial
expansions to models with Dirichlet or Poisson Dirichlet stationary
distributions. Another form of the transition functions is as a
mixture depending on the mutant and non-mutant families represented
in the leaves of the infinite- leaf coalescent tree.

**Charles Newman** (Courant Institute)

*Percolation methods for spin glasses*

Percolation methods, e.g., those based on the Fortuin-Kasteleyn
random cluster representation (of vacant and occupied bonds), have
been enormously important in the mathematical analysis of ferromagnetic
Ising models. There exists a Fortuin-Kasteleyn representation for
non-ferromagnetic Ising models (including spin glasses) but to date
that has not been terribly useful in the non-ferromagnetic context.
We will discuss why this is so and the prospects for this to change
in the future. Although our motivation is to study short-range models,
we may also describe the percolation situation in the mean-field
Sherrington-Kirkpatrick spin glass. Much of the talk is joint work
with Jon Machta and Dan Stein.

**Kavita Ramanan **(Carnegie Mellon)

*Measure-valued Process Limits of Some Stochastic Networks*

Markovian representations of certain classes of stochastic networks
give rise naturally to measure-valued processes. In the context
of two specific examples, we will describe some techniques that
have proved useful in obtaining limit theorems for such processes.
In particular, we will discuss the role of certain mappings, which
can be viewed as a generalization to the measure-valued setting
of the Skorokhod map that has been used to analyze stochastic networks
admitting a finite-dimensional representation. This talk is mainly
based on various joint works with Haya Kaspi, Lukasz Kruk, John
Lehoczky and Steven Shreve.

**Anna Savu** (University of Toronto)

*Convergence of a process of Wishart matrices to free Poisson
process*

Free Poisson process is the free analogue of the classical Poisson
process and can be obtained as a limit of a process of Wishart matrices
of size arbitraly large. A large deviation principle for this convergence
is studied. The analogous large deviation principle for the convergence
of the Hermitian Brownian motion towards the free Brownian motion
has been obtained by P. Biane, M. Capitaine and A. Guionnet.

**David Steinsaltz**, Queen's University

M*easure-valued dynamical systems, with applications to the evolution
of aging*

We consider an infinite population, described at any time by a probability
distribution on a space of "genotypes", each of which
is a subset of a space of possible "mutations". The probability
distribution changes in time according to a mutation rule, which
augments the genotypes with more mutations, and a selection rule,
which reduces the frequency of genotypes with more deleterious mutations.
The Feynman-Kac formula enables us to write down a closed-form solution
to this dynamical system, amenable to various kinds of approximations.

This talk is based on joint work with Steve Evans and Ken Wachter.

**Balasz Szegedy** (University of Toronto)

*Limits of Discrete Structures*

Take a family of discrete objects, define a limit notion on them
and take the topological closure of the family. We study the discrete
objects through the analytic properties of their closure. An example
for this strategy is classical ergodic theory by Furstenberg where
the discrete structures are subsets of intervals of the integers
and the limit objects are certain group invariant measures. This
theory, in particular, leads to various strengthenings of the famous
theorem by Szemeredi on arithmetic progressions. We present analogous
theories where the discrete objects are graphs or hypergraphs. Among
the applications we show various results about group invariant random
processes. The talk is based on joint works with Gábor Elek
and László Lovász.

**Benedek Valko** (University of Toronto)

*t^{1/3} Superdiffusivity of Finite-Range Asymmetric Exclusion
Processes on Z*

We give bounds on the diffusivity of finite-range asymmetric
exclusion processes on Z with non-zero drift. We use the resolvent
method to make a direct comparison with the totally asymmetric simple
exclusion process, for which the recent works of Ferrari and Spohn,
and Balazs and Seppalainen provide sharp bounds.

**Vladimir Vinogradov**, Ohio University

*On local limit theorems related to Levy-type branching mechanism*

We prove local limit theorems for total masses of two branching-diffusing
particle systems which converge to discontinuous $(2,d,\beta)$-superprocess.
We establish new properties of the total mass for these superprocesses.
Both particle systems are characterized by the same heavy-tailed
branching mechanism. One of them starts from a Poisson field, whereas
the initial number of particles for the other system is non-random.
The poissonization is related to Gnedenko's method of accompanying
infinitely divisible laws. We observe a worse discrepancy between
the extinction probabilities than in the continuous case.

### Presentation Requests:

If you wish to schedule a short presentation, please e-mail Balint
Virag at balint@math.toronto.edu

** **

### Schedule

**Thursday March 15**

9:30 coffee

10:10 - 11:00 Kavita Ramanan

11:10 - 12:00 Bob Griffiths

2:30-4:00 Problem/Informal session

4:00-4:30 Coffee

4:30-6:00 Short presentations

4:30-4:55 David Steinsaltz

5:00-5:25 Nikolai Dokuchaev

5:30-5:55 Vladimir Vinogradov

8:00-10:00 Banquet, Bright Pearl Restaurant

(tickets must be reserved in advance)

**Friday March 16**

9:30 coffee

10:10 - 11:00 Balazs Szegedy

11:10 - 12:00 Chuck Newman

2:30-4:00 Problem/Informal session

4:00-4:30 Coffee

4:30-6:00 Short presentations

4:30-4:55 Dimitris Cheliotis

5:00-5:25 Michelle Boue

5:30-5:55 Denis Sezer

**Saturday March 17**

9:30 coffee

10:10 - 11:00 Ted Cox

11:10 - 12:10 Short presentations

11:10-11:35 Ana Savu

11:40-12:05 Benedek Valko

### Banquet

There will be a banquet on Thursday March 15 at the Bright Pearl
Restaurant. Please contact gensci@fields.utoronto.ca

or 416-348-9710 ext. 3018 to reserve a ticket @$30.00 each. **All
tickets must be picked up and paid for at Fields by Thursday March
15.**

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