Diffusions and jump processes on groups and manifolds

Modeling scientific experiments or human activities often involves randomness. Card shuffling procedures provide a familiar, yet complex and mathematically interesting example that serves as a model for many mixing phenomena. Randomness is used to understand image restoration and recognition, communication and social networks, the behavior of financial markets, as well as in the analysis of large data sets in general. It is an important tool in the study of efficient computations and scientific simulations. In all these applications, strong structural constraints associated with the complex combinatorial or geometric structure underlying the problem determine the behavior. This project is concerned with the fundamental properties of basic stochastic processes and how the behavior of these processes relates to the global geometric structure of their different environments. Postdoctoral associates, graduate students, and undergraduate students will be mentored and trained as part of this project. The funded research focusses on random processes that are defined by a related geometric or algebraic structure (e.g., Riemannian manifolds and groups). The global behaviors of these processes are determined by this underlying structure. In some cases, these behaviors can provide information on the underlying space and its structure. These explorations are at the interface between analysis, geometry, and probability, with the notion of group structure playing a key part. Partial differential equations and potential theory, i.e. the study of harmonic functions and solutions of the heat equation, are also central. Brownian motion on Riemannian Manifolds and random walks on Cayley graphs of finitely generated groups provide key examples. The notion of stable-like processes on nilpotent groups is also studied. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.