Many basic Markov processes evolve on a state space carrying a related geometric structure. Brownian motion on a Riemannian manifold, random walks on Cayley graphs of finitely generated groups and finite Markov chains on complex combinatorial structures such as trees or matchings are all primary examples. This proposal focuses on the relationships between the behavior of such processes and the properties of the underlying geometric structure. It involves problems at the interface between analysis, geometry and probability with a major role played by groups 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 and human 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 behavior of 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 the fundamental properties of such stochastic processes and on how they relate to the global structure of the environment.
