Stochastic Analysis in Differential Geometry

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The investigator will study various analytic and probabilistic problems related to finite and infinite differential geometry, especially Riemannian manifolds with boundary and path and loop spaces over such manifolds. In the case of Riemannian manifold with boundary, he will try to obtain a useful formulation of the Feynman-Kac formula for vector bundles (mainly differential forms) which can be used effectively in a number of problems. A formula of this type was obtained before but is not easy to be applied to the problems. He will prove that the derivatives of the heat semigroup can be bounded explicitly in terms of the Ricci curvature and the second fundamental form of the boundary. The approach he will adopt for the finite dimensional problems is intimately related to stochastic analysis on path spaces over Riemannian manifolds. In the infinite dimensional setting, he hopes to break new ground in path and loop space analysis by studying the case when the manifold has a boundary, thus breaking new ground. Specifically he will try to prove an integration by parts formula in the path space for this case. This is a reasonable starting point for investigating manifolds with boundary, for as it is well known that integration by parts formula lies at the center of many interesting problems in path and loop spaces. Stochastic analysis in geometry is an active research area in probability theory. Its goal is to use stochastic methods (as opposed to analytic methods) to investigate models with well defined geometric structures. Many models studied in this area (e.g., path and loop spaces) are mathematical abstractions of concrete models in physics and other related areas of science and engineering (especially high energy physics and space technology). An understanding of the mathematical structure of these models is usually the first step towards their practical applications.