Probabilistic Methods in Geometry and Analysis

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9706910 Hsu This research proposal forms an integral part of the Principal Investigator's program of studying analytical properties of (nonflat) finite or infinite dimensional spaces (mainly Riemannian manifolds and path and loop spaces over them) by analytic and probabilistic methods. The PIs long range goal for the program is to establish a complete analytical and probabilistic theory of path and loop spaces over Riemannian manifolds, whose counterpart in (flat) Euclidean space (finite or infinite dimensional) has been known for quite some time. Successes in these efforts will contribute significantly towards our general understanding of nonlinear infinite dimensional spaces. For the analytical part, the PI will concentrate his research on further developing analytic techniques subsumed under the name "Malliavin Calculus." In particular for the period covered in this proposal, the PI will spend a substantial amount of his research resources on the equivalence of various Sobolev norms (the so-called Meyer's equivalent problem) and on the spectral properties of the generalized Ornstein-Uhlenbeck operator. For the probabilistic part, the PI intends to launch a detailed study of the Ornstein-Uhlenbeck process through sample path, especially the effect of curvature on short-time and long-time behaviors of the process. The models under study, path and loop spaces over such geometric structures such as spheres, ellipsoids, and tori, have been used extensively in recent years in the theory of relativity, quantum physics, astrophysics, and cosmology. The Principal Investigator will study these models by both traditional analytical methods and the more recent methods of probability theory. Specifically he will investigate both long-time and short-time behavior of a specific random process on path and loop spaces in the hope that it will reveal geometric properties which can be applied by theoretical scientists and engineers in their respective disciplines.