Stochastic Analysis and Related Topics

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This project is devoted to the study of stochastic analysis in infinite dimensions. The main topic is stochastic differential equations (SDEs) 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. These solutions will be used to construct and study heat kernel measures (a non-commutative analogue of Gaussian or Wiener measure) on infinite-dimensional manifolds. In general these infinite-dimensional spaces do not have an analogue of the Lebesgue measure or a Haar measure in the group case. The PI intends to study Cameron-Martin type quasi-invariance of these measures. 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 an energy representation of path groups. It is proposed to study properties of square-integrable holomorphic functions, including non-linear analogues of the Segal-Bargmann transform and bosonic Fock space representations. Levy processes in Lie groups will be studied.The proposed research will connect diverse fields: stochastic analysis, geometric analysis, representation theory and mathematical physics. This research project has broader impacts on diverse areas of mathematics such as stochastic analysis, representation theory, geometric analysis etc, and it involves activities which help to disseminate the knowledge of new findings in the field. The motivation comes from several subjects.Infinite-dimensional spaces such as loop groups and path spaces appear in physics, for example, in quantum field theory. The PI proposes to formalize and study some of the notions used in physics, such as measures on certain infinite-dimensional spaces. In addition, it has a significant educational component, namely, it involves graduate students of the PI.