Exit Time Moments and Eigenvalue Estimates

TL;DR

This paper investigates bounds on the principal Dirichlet eigenvalue of a domain in a Riemannian manifold using exit time moments of Brownian motion, extending classical inequalities.

Contribution

It generalizes Polya's classical inequality by relating eigenvalues to exit time moments in a broader geometric setting.

Findings

Derived new bounds for eigenvalues using exit time moments

Extended classical inequalities to Riemannian manifolds

Provided a unified framework for eigenvalue estimates

Abstract

We study estimates involving the principal Dirichlet eigenvalue associated to a smoothly bounded domain in a complete Riemannian manifold and L1-norms of exit time moments of Brownian motion. Our results generalize a classical inequality of Polya.