Isoperimetric Conditions,Poisson Problems,and Diffusions in Riemannian Manifolds

Let (M,g) be a complete Riemannian manifold. We study exit time moments of natural diffusions from smoothly bounded domains in M with compact closure. Our results give relationships between bounds on the exit time moments and their corresponding averages (over the associated domain), and the global geometry of M. In particular, for averaged moments of Brownian motion, we prove an analog of the Faber–Krahn theorem.