Brownian Motion on Riemannian Manifolds
Department of Mathematics
We begin with the Laplace operator on a complete Riemannian manifold. The isotropic transport process is a Markov process on the tangent bundle; in the limit of small mean free path, the process converges weakly to a Markov process on the basic manifold. A number of analytic problems can be discussed using the small ball power series of the Laplace operator. These include the mean exit time, the distribution of hitting place, and the principal eigenvalue of a small ball.