Exit time moments and eigenvalue estimates

Abstract

We give upper bounds on the principal Dirichlet eigenvalue associated to a smoothly bounded domain in a complete Riemannian manifold; the bounds involve L1-norms of exit time moments of Brownian motion. Our results generalize a classical inequality of Pólya. We also prove lower bounds for Dirichlet eigenvalues using invariants that arise during the examination of the relationship between the heat content and exit time moments.