Patrick McDonald
We study diffusions 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, $\Omega,$ and for each positive integer $k,$ we characterize the $k$th exit time moment of Brownian motion, averaged over the domain $\Omega$ 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 fixed volume is bounded above. For the case of Euclidean space, we establish similar boundedness properties for a larger class of diffusions.