Markov Processes in Geometric Environments

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Many basic Markov processes evolve on a state space carryinga related geometric structure. Brownian motion on a Riemannianmanifold, random walks on Cayley graphs of finitely generatedgroups and finite Markov chains on complex combinatorial structures such as trees or matchingsare all primary examples.This proposal focuses on the relationships between the behavior of such processes and the properties ofthe underlying geometric structure. It involves problems at the interface betweenanalysis, geometry and probability with a major role played bygroups and their actions. Potential theory, i.e., the study of harmonic functions and, more generally, of solutions of the heat equation,is at the center of many of these considerations.Random processes play an important role in many aspects of science andhuman activity. The study of card shuffling procedures is an entertaining yet complex and mathematically interesting example.Various random processes are used to model complex phenomena,from polymer molecules, to DNA analysis, to image restoration, to financial markets. They are also used as crucial tools for efficient computations. In such cases, there are strong structural constraints underlying the behaviorof these stochastic processes. These constraints are expressed in terms of the environment of the process which often has a complex combinatorial or geometric nature.This proposal focuses on the study of thefundamental properties of such stochastic processesand on how they relate to the global structure of the environment.