UW Math Probability Seminar, Spring 2002

UW Math Probability Seminar, 2001-2002

Organized by Timo Seppalainen, University of Wisconsin

SPRING SEMESTER 2002

Thursdays in 901 Van Vleck Hall at 2:25 pm unless otherwise noted

SCHEDULE AND ABSTRACTS

Thursday, January 31, ROOM CHANGE: 23 Ingraham

Moo Chung, University of Wisconsin - Madison
Random Fields in Brain Imaging

Since the introduction of the maximum t statistic to neuroscientists in 1992 (Worsley et al, 1992), random fields have become an integral part of brain imaging. Random fields in higher dimension and on manifolds will be introduced and a simple method for determining an approximate p-value for the maximum random fields will be explained (Adler, 1981; Worsley et al, 1996). Time permitting, the concept of isotropic diffusion filtering in relation to random fields will be also presented (Chung et al, 2002).

References:

Adler, R.J. (1981) The geometry of random fields, Wiley, New York.

Chung, M.K., Worsley, K.J., Paus, T., Robbins, S., Taylor, J., Giedd, J.N., Rapoport, J.L., Evans, A.C. (2002) Tensor-based surface morphometry, Department TR 1049 (http://www.stat.wisc.edu/~mchung).

Worsley, K.J., Evans, A.C., Marrett, S., Neelin, P. (1992) A Three-Dimensional Statistial Analysis for CBF Activation Studies in Human Brain. Human Brain Mapping, 4:58-73.

Worsley, K.J., Marrett, S., Neelin, P., Vandal, A.C., Friston, K.J., Evans, A.C. (1996) A unified statistial approach for determining significant signals in iames of cerebral activation. Human Brain Maping, 4:58-73.

There will be a brief orgnizational meeting after the seminar.

Thursday, February 7

David Aldous, University of California - Berkeley
The Asymmetric One-Dimensional Constrained Ising Model: Rigorous Results

We study a one-dimensional spin (interacting particle) system, with product Bernoulli(p) stationary distribution, in which a site can flip only when its left neighbor is in state +1. Such models have been studied in physics as simple exemplars of systems exhibiting slow relaxation. In our "`East" model the natural conjecture is that the relaxation time \tau(p), that is 1/(spectral gap), satisfies \log \tau(p) ~ {\log^2 (1/p)}/{\log 2} as p decreases to 0. We prove this up to a factor of 2.

Our proof combines several ingredients:

(i) The Poincare comparison method, whose use involves analyzing the combinatorial structure of transition paths.

(ii) Invention of a "long-range wave" process for comparison purposes, and its analysis via coupling and exponential martingales.

(iii) For the lower bound, invention of an approximate "dual" process of coalescing random jumps and its analysis.

Thursday, February 14

Federico Camia, New York University
Zero-Temperature Dynamics and Critical Percolation Scaling Limits

Zero-temperature stochastic Ising models are interacting particle systems on a regular lattice that can be obtained as a suitable zero-temperature limit of a stochastic Ising model. Their behaviour varies drastically according to the lattice, and for certain lattices they can be used to define dependent percolation models with interesting properties.

I will give a brief survey of some results, conjectures and open problems concerning zero-temperature stochastic Ising models, and then focus on the analysis of the continuum scaling limit of some two-dimensional dependent percolation models generated by such interacting particle systems.

The scaling limit of those percolation models is shown to be the same as that of critical (independent) site percolation on the triangular lattice, whose existence and conformal invariance properties have been recently proved by S. Smirnov.

Thursday, February 21

Timo Seppalainen, University of Wisconsin - Madison
Second-class Particles in Asymmetric Exclusion Processes

We describe what second-class particles are in exclusion processes, and discuss two interesting things about them:

(1) Paths of second-class particles converge to characteristics and shocks of the macroscopic p.d.e. of the system, and

(2) Second-class particles can be used to prove central limit theorems for the process.

Thursday, February 28 (rescheduled from last fall)

Thomas Kurtz, University of Wisconsin - Madison
Stochastic Equations for Spatial Birth and Death Processes, part I

Spatial birth and death processes of the type to be considered were first studied systematically by Preston in the 1970s. They began to play a central role in spatial statistics a few years later when Ripley recognized that important classes of spatial point processes (Gibbs distributions) could be obtained as stationary distributions of birth and death processes and that this characterization provided a computationally feasible method of simulation of the spatial point process (Markov chain Monte Carlo). In the mathematical arena, the infinite population versions provide a class of nonlattice interacting particle systems. Stochastic equations driven by Poisson random measures will be formulated for a large class of birth and death processes. The relationship to the corresponding martingale problem will be discussed. Conditions for existence and uniqueness and ergodicity in the infinite population setting will be given.

Thursday, March 7

Steven Lalley, University of Chicago
The Weak/Strong Survival Transition in the Contact Process on a Tree

The contact process on a homogeneous tree of degree three or greater is known to have three distinct phases: an extinction phase, a weak survival phase, and a strong survival phase. The existence of two qualitatively different survival phases is the most striking feature of the process, as the contact process on the integer lattice, in any dimension, exhibits only one survival phase (strong survival). Thus, the contact process on a homogeneous tree exhibits a phase transition, from weak to strong survival, of a different character than the phase transition for the contact process on the integer lattices. I shall discuss certain features of this phase transition, and compare it with similar phase transitions in other stochastic processes.

Thursday, March 14

Ziyu Zheng, University of Wisconsin - Milwaukee
Reflected Backward Stochastic Differential Equations,
Variational Inequalities and Dirichlet Problems

We study the connections between stochastic exit time problems and possibly discontinuous viscosity solutions to a second order semi-linear Dirichlet problems. We also study the connections between stochastic exit time optimal control problems and possibly discontinuous viscosity solutions to a second order fully nonlinear Hamilton-Jacobi-Bellman equation up to the boundary. These equations admit maximal and minimal solutions, which have two sorts of probabilistic interpretations: as value functions associated to stopping time problems on the boundary, or as value functions associated to related Dynkin games and mixed control problems. The only hypothesis on the domain O is that it is the interior of its closure. To this end, we developed the theory of reflected backward stochastic differential equations (RBSDE's) with random terminal time, the theory of degenerate variational inequalities with regular obstacles, and a new convergence methodology to obtain our results.

Thursday, March 21

Esa Nummelin, University of Helsinki
Large Deviations of Price Equilibria of Random Economic Systems

We consider an economic system comprising a large number of economic agents. We will be concerned with laws of large numbers (unconditional and conditional) for the associated random equilibrium prices. The tool will be the theory of large deviations.

Thursday, April 4

Alex de Acosta, Case Western Reserve University
Refinement of a General Non-convex Large Deviations Result and Applications

We relax certain assumptions in our recent large deviation result for a sequence of random elements in a vector space satisfying an "abstract exponential martingale condition." We give applications to large deviations of sequences of D[0,1] - valued random elements defined by certain recursive schemes.

Thursday, April 11

Thomas Kurtz, University of Wisconsin - Madison
Stochastic Equations for Spatial Birth and Death Processes, part II

(continuation; see February 18 abstract)

Thursday, April 18

Alexander Soshnikov, University of California - Davis
Random Matrices and Determinantal Random Point Fields

The purpose of the talk is to give an introduction to determinantal random point fields. Determinantal random point fields appear naturally in random matrix theory, probability theory, quantum mechanics, combinatorics, representation theory, and some other areas of mathematics and physics. The first part of the talk will be devoted to examples. In the second part we will concentrate on CLT type results for linear statistics. In particular, we will talk about the Costin-Lebowitz theorem for the counting function of particles.

Thursday, April 25

Elena Kosygina, Northwestern University
Homogenization of stochastic Hamilton-Jacobi equations

I shall briefly recall the classical results of P. L. Lions, G. Papanicolaou, S. R. S. Varadhan on homogenization of Hamilton-Jacobi equations with periodic Hamiltonian and also the results of F. Rezakhanlou and J. E. Tarver on the homogenization of stochastic Hamilton-Jacobi equations. Next I plan to consider the viscous version of the stochastic Hamilton-Jacobi equation and explain the connection between a special case of such a homogenization problem and large deviations for the quenched Brownian motion among Poissonian obstacles (A. Sznitman).

Thursday, May 2 in B219 Van Vleck, ROOM CHANGE: B219 Van Vleck

Peter Glynn, Stanford University
Brownian Traffic Modeling

The use of Brownian approximations in the analysis of queueing systems has become a standard tool for queueing theorists. Such Brownian models offer good numerical approximations to conventional queueing systems ( fed by renewal-type traffic ) when the system is heavily utilized. In addition, even when the Brownian limit is analytically intractable, the temporal and spatial scalings associated with the corresponding limit theory offer tremendous insight into the behavior of the underlying queueing network. The existing literature has primarily focused on modeling contexts in which the inputs are renewal-like, by which we mean that the input processes have the time-homogeneity, short-range dependence, and light tails necessary to place the input stream in the " domain of attraction " of Brownian motion. In this talk, we will discuss several modeling environments in which more complex approximating processes are appropriate, and we shall describe some of the associated performance implications. In particular, we shall discuss Brownian traffic models for non-stationary queues ( incorporating " time-of-day " effects ), scheduled arrival processes, and perturbed renewal processes.