The PI will investigate analytical and geometrical properties of the path and loop spaces over a Riemannian manifold. The analysis is based on the Wiener measure on these spaces, which plays a similar role as the Lebesgue measure in the analysis of finite dimensional manifolds. Since the Wiener measure gives rise to Brownian motion and Brownian bridge, the probabilistic methods (stochastic differential equations, diffusion theory, etc.) will be used extensively in our analysis. The geometric and analytical properties of path and loop spaces will be studied through the probabilistic properties of the so-called Ornstein-Uhlenbeck process. The PI will discover generalizations of integration by parts formula for the gradient operator in various geometric settings (mainly for manifolds with boundary) and compute the familiar geometric objects such as torsion and curvature tensors of the path and loop spaces (as Hilbert manifolds) in terms of stochastic integrals involving the usual torsion and curvature tensors of the underlying Riemannian manifold. The long-term goal is to develop an intrinsic, geometric Malliavin calculus and to investigate hypercontractivity, logarithmic Sobolev inequality, and Meyer's equivalence in our new geometric setting and their interaction with the Riemannian structure of the base manifold. Interdisciplinary research is the current trend of scientific research. Probability theory is a branch of mathematics which studies random behavior of collective phenomena. In the last two decades probability theory has been applied with great success to problems from classical mathematical subjects such as partial differential equations and geometry. This new probabilistic point of view not only stimulated research in these classical subjects but also opened new avenues of research such as stochastic differential geometry and diffusion theory. The PI will use probabilistic methods to study properties of an important class of geometric objects called loop spaces (for example, the collection of closed paths on a sphere), which just began to gain importance in modern physics. He will show how the curvature of the base space (the sphere in the above case) affects the behavior of certain random processes associated with loop spaces and give both qualitative and quantitative descriptions of the interaction between the geometry of the space and the underlying random processes.
