Stochastic analysis and related topics

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This project is devoted to the study of stochastic analysis in infinite dimensions. In particular, this research will provide a better understanding of Gaussian-type measures on infinite-dimensional curved spaces. The proposed research is motivated by physics. For example, infinite-dimensional spaces such as loop groups and path spaces appear in quantum field theory (QFT). The PI proposes to formalize and study some of the notions used in physics, such as measures on certain infinite-dimensional spaces. For example, it is common to see computations in QFT literature involving integrals over infinite-dimensional spaces with respect to a fictitious infinite-dimensional Lebesgue measure with a Gaussian density normalized by a constant which is infinite. Mathematically this measure can be interpreted as a Wiener measure on a flat space, or as a heat kernel measure on an infinite-dimensional curved space. In addition, this research will connect diverse fields: stochastic analysis, geometric analysis, representation theory and mathematical physics. This project is focused on elliptic and subelliptic diffusions in infinite-dimensional curved spaces, such as infinite-dimensional groups, loop groups and path spaces. The questions of existence and uniqueness of solutions of the SDEs and smoothness of solutions will be studied. In general these infinite-dimensional spaces do not have an analogue of the Lebesgue measure or a Haar measure in the group case. In addition, geometry of these spaces will be studied in connection with smoothness properties of heat kernel measures in both elliptic (Riemannian) and subelliptic (sub-Riemannian) settings. The smoothness is interpreted as a Cameron-Martin type quasi-invariance. It is an interesting question in itself, and in addition it can give rise to unitary representations of the infinite-dimensional groups. One part of the proposal is devoted to studying of Brownian and energy representations of path groups. In addition, smoothness of finite-dimensional hypo-elliptic heat kernels will be studied. This project will use new techniques coming from harmonic analysis on such spaces. The educational component of the proposal is manifold: a number of projects involve graduate students of the PI; in addition the PI is involved in mentoring and educational activities from the middle school to postdoc levels.