Stochastic Analysis on Riemannian Manifolds
Dr. Elton P. Hsu
3:00 Friday, 1 March 2013
Shelby Center 218
The fundamental solution of the heat equation on a Riemannian manifold associated with the Laplace-Beltrami operator can be served as the transition density function of a stochastic process called Brownian motion on the manifold. Many geometric properties of the manifold are reflected in the random behavior of Brownian motion. In this talk I will discuss several such results, in which stochastic techniques are used to prove results of purely geometric nature. On the other hand, Brownian motion on a Riemannian manifold can be regarded as a measure on the path space over the Riemannian manifold, a good example of infinite dimensional Hilbert manifold. This point of view gives rise to functional analysis on the path space. We will use this point of view to discuss logarithmic Sobolev, transportation cost, and other functional inequalities on the path space. The talk is oriented towards general audience with a liberal education in classical and geometric analysis.