# SCIENTIFIC PROGRAMS AND ACTIVITIES

September 19, 2017

## Seminar on Stochastic Processes March 15-17, 2007 at the Fields Institute

Supported by:
Organizing Committee: Jeremy Quastel, Tom Salisbury, Balint Virag
 Invited Speakers, titles and abstracts Parking Presentation Requests Schedule Visitor Information Banquet, Thursday March 15 Housing and Hotel Scientific Advisory Committee Directions to Fields

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)

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
Measure-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.

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

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(PUT_AT_SIGN_HERE)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.**