Random Walks and Diffusions and Their Geometries

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Randomness plays a role in many aspects of science and human activities. A familiar yet complex and mathematically interesting example is card shuffling, which serves as a model for many important phenomena involving the idea of mixing. More generally, randomness is used in modeling a wide range of systems, from polymers and DNA to image restoration and recognition, communication and social networks, and the behavior of financial markets. Random processes are also used as important tools for efficient computations and simulation. In all these applications, strong structural constraints associated with the complex combinatorial or geometric structure underlying the problem determine the behavior of the process. This research project is concerned with the fundamental properties of such stochastic processes and their relationship to global structures.The project focusses on Markov processes that are defined in terms of a related geometric or algebraic structure. The long-term and global properties of these processes are determined by and often reflect the global structure of the underlying space. The project involves questions at the interface between analysis, geometry, and probability with a major role played by groups and their actions. Partial differential equations and potential theory, i.e., the study of harmonic functions and, more generally, of solutions of the heat equation, are also at the center of many of these considerations. Brownian motion on a Riemannian manifold and random walks on Cayley graphs of finitely generated groups provide key examples.