A Variation on the Donsker-Varadhan Inequality for the Principial Eigenvalue

Abstract

The purpose of this short note is to give a variation on the classical Donsker-Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain $\Omega$ by the largest mean first exit time of the associated drift-diffusion process via $${\lambda }_1\ge \frac{1}{{\sup }_{x\in \Omega }{\mathbb{E}}_x{\tau }_{{\Omega }^c}}.$$ Instead of looking at the mean of the first exit time, we study quantiles: let $d_{p,\partial \Omega }:\Omega \to {\mathbb{R}}_{\ge 0}$ be the smallest time $t$ such that the likelihood of exiting within that time is $p$, then $${\lambda }_1\ge \frac{\log (1/p)}{{\sup }_{x\in \Omega }{d}_{p,\partial \Omega }(x)}.$$ Moreover, as $p\to 0$, this lower bound converges to ${\lambda }_1$.

correction of minor typos

References in corpus (2)

• On the Diffusion Geometry of Graph Laplacians and Applications
• On the location of maximal of solutions of Schrödinger's equation

Cited by in corpus (1)

• A Local Faber-Krahn inequality and Applications to Schrödinger's Equation