The Principal Investigator (PI) of this proposal plans to study various analytic and stochastic properties of path and loop spaces over a Riemannian manifold. The goal is to break new ground in this area of mathematics by proposing and solving new problems and, at the same time, devising a substantial research program for further investigation. Path and loop spaces over a Riemannian manifold are naturally equipped with the basic ingredients for an analytic theory, namely, the Wiener measure and a differentiable structure. These are important infinite dimensional mathematical and physical models. The PI will study the following problems concerning these models: (1) Integration by parts formula for manifolds with the Neumann boundary conditions (adiabatic case); (2) Logarithmic Sobolev and related inequalities for simply-connected manifolds; (3) Couplings of Brownian motions on manifolds, especially the problem of maximal couplings; (4) Infinite dimensional mass transportation problems related to the Wiener measure. The importance of mathematical objects, such as path and loop spaces over Riemannian manifolds, has been recognized by mathematicians and physicists for quite some time. The need for their study is motivated by a variety of problems in applicable mathematics and theoretical and applied physics. A better understanding of these models from the mathematical as well as from the physical points of view will contribute to our current knowledge in these two fields. The practical aspects of these investigations will become significant in the future development of mathematical and physical sciences.
