9626590 Lalley The main focus of the investigation will be on how the coarse geometry of a state space constrains the behavior of stochastic processes supported by the state space, in particular, on categorizing the behaviors possible for random walks and random growth processes (such as branching random walks and contact processes) in hyperbolic geometries. Particular attention will be paid to behaviors that cannot occur in Euclidean state spaces, such as exponential population growth with eventual extinction in every compact subset. Related problems regarding the mixing rate for various finite state Markov chains will also be investigated, with the immediate aim of contributing to the understanding of the so-called "cutoff phenomenon". Random growth processes serve as crude models of population growth in structured environments, and thus are of natural interest in population biology. They are also important as models in various areas of theoretical physics, in particular, statistical mechanics. Problems concerning mixing rates of finite-state Markov chains are of practical importance because of the increasing use of "Markov chain Monte Carlo" methods in simulation studies. ***
