This project is devoted to the study of stochastic analysis in infinite dimensions. The main topic is stochastic differential equations (SDEs) in infinite-dimensional spaces, such as infinite-dimensional groups, loop groups and path spaces, non-commutative $L^p$-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 such as an infinite-dimensional Heisenberg group and the Virasoro group. 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. 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.
The intellectual merit of this proposal is in providing a better understanding of Gaussian-type measures on infinite-dimensional curved spaces. In particular, the proposed research will connect diverse fields: stochastic analysis, geometric analysis and mathematical physics. This research project has broader impacts on diverse areas of mathematics, and it involves activities which help to disseminate the knowledge of new findings in the field. The proposed research is motivated by several subjects. Infinite-dimensional spaces such as loop groups and path spaces appear in physics, for example, in quantum field theory and string 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 two graduate students of the PI.
