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

Invited Speaker

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.

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.

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.