Problems in Stochastic Processes: Hyperbolic structures, Bayesian nonparametric estimation, and spatial epidemic and interspecies competition models

University Of Chicago, Chicago IL

Investigators

Abstract

The primary focus of the project will be the study of random processes random walks and branching random walks, contact processes, Ising model, and competition processes on hyperbolic groups and graphs. The last four of these processes exhibit multiple phases, including an intermediate phase of weak survival that is not seen on integer lattices. A major objective of the research will be to understand how hyperbolic geometry constrains the upper phase transition between weak and strong survival, and how this in turn is related to asymptotic behavior of the Green's functions of random walk. This will involve development of techniques centered around Ancona inequalities, symbolic dynamics and infinite-dimensional Perron-Frobenius theory, and the Lyapunov-Schmidt reduction of infinite systems of algebraic functional equations. Secondary foci of the project will be (a) mathematical problems arising in Bayesian nonparametric function estimation, and (b) stochastic models of epidemics and interspecies competition. Stochastic interacting particle systems are widely used as models in various areas of science, especially in statistical physics and in population biology, but also in the social sciences. It is hoped that studying such processes in infinite hyperbolic geometries will ultimately contribute to understanding their behavior in finite "expander" and "small-worlds" networks, which may better model interactions in various social and ecological systems.