Stochastic Processes in non-Euclidean spaces

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Brownian motion on finite dimensional Riemannian manifolds is well studied, and the deep relationship between the Laplace-Beltrami operator and the geometry of a space and the properties of Brownian motion and its heat kernel measure on that space are well understood.This proposal is devoted to the study of certain generalizations of this paradigm, including the study of diffusions on sub-Riemannian manifolds and certain infinite dimensional Lie groups, Levy processes on Lie groups, and Brownian motion on ``tree space,'' a continuous space with geometric and combinatorial structures which has biological applications. These studies lie in the intersection of analysis, geometry, and probability, and the study of solutions to stochastic differential equations and their generators is a uniting framework of many of the problems considered.Probability provides a powerful tool in analysis and geometry, and stochastic processes give tractable models for many physical and biological phenomena. For example, Brownian motion gives a way of understanding heat flow on a space. The PI will investigate properties of several stochastic processes on spaces which occur naturally in some physical or biological applications. Sub-Riemannian manifolds arise in classical and quantum mechanics, and certain geometric quantities are best understood in this setting.Infinite dimensional spaces appear in physics in quantum field theory and string theory. Levy processes have recently been a subject of intense research, due in part to new applications in finance. ``Tree space'' models the space of all phylogenetic trees.The research during this grant period should have implications in various mathematical disciplines, such as harmonic analysis, functional analysis, and mathematical physics, and should find applications in other scientific fields, such as physics and biology.