Isoperimetric Conditions and Diffusions in Riemannian Manifolds

We study diiusions in Riemannian manifolds and properties of their exit time moments from smoothly bounded domains with compact closure. For any smoothly bounded domain with compact closure, ; and for each positive integer k; we characterize the kth exit time moment of Brownian motion, averaged over the domain with respect to the metric density, using a variational quotient. We prove that for Riemannian manifolds satisfying an isoperimetric condition, the averaged k-th exit time moment of Brownian motion from domains of a xed volume is bounded above. For the case of Euclidean space, we establish similar boundedness properties for a larger class of diiusions.