Stochastic analysis in infinite dimensions

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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 spaces,such as noncommutative L^p-spaces, related infinite-dimensional groups, loop groups, and path spaces. The questions of existence and uniqueness of solutions of the SDEs and smoothnessof solutions will be studied. Then the solutions will be used to construct and study heat kernel measures (a noncommutative analogue of Gaussian or Wiener measure) on the infinite-dimensional manifolds. In general these infinite-dimensional groups are not locally compact and therefore do not have a Haar measure. The PI intends to study Cameron-Martin type quasi-invariance of these measures. It is an interesting questions in itself, but it also can give rise to unitary representations of the infinite-dimensional groups. It is proposed to study properties of square-integrable holomorphic functions. For example, quasi-invariance can be used to prove weak Cauchy-Riemann equations for holomorphic functions. Besides the classical infinite-dimensional stochastic analysis the PI intends to study noncommutative SDEs. 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 some of the notions used in physics, such as measures on certain infinite-dimensional spaces. The proposed problems in the field of noncommutative probability have their origins in quantum physics.