Comparison Results, Exit Time Moments, And Eigenvalues On Riemannian Manifolds With A Lower Ricci Curvature Bound

TL;DR

This paper investigates how the geometry of Riemannian manifolds influences Brownian motion exit times and eigenvalues, providing bounds and comparison results that deepen understanding of geometric analysis in curved spaces.

Contribution

It introduces new bounds for Dirichlet eigenvalues and comparison results for exit time moments on Riemannian manifolds with Ricci curvature bounds.

Findings

Established bounds for Dirichlet eigenvalues.

Derived comparison results for exit time moments.

Linked geometric properties to stochastic behavior.

Abstract

We study the relationship between the geometry of smoothly bounded domains in complete Riemannian manifolds and the associated sequence of $L^1$-norms of exit time moments for Brownian motion. We establish bounds for Dirichlet eigenvalues and, for closed manifolds, we establish a comparison result for elements of the moment sequence.