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

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 $\mathrm{\Omega }$ by the largest mean first exit time of the associated drift-diffusion process via $${\lambda }_1\ge \frac{1}{\underset{x\in \mathrm{\Omega }}{\sup }{\mathbb{E}}_x{\tau }_{{\mathrm{\Omega }}^c}}\mathrm{.}$$ Instead of looking at the mean of the first exit time, we study quantiles: let $d_{p, \partial \Omega}:\Omega \rightarrow $\mathbb{R}${\geq 0}$bethesmallesttime$t$suchthatthelikelihoodofexitingwithinthattimeis$p$,then$$\lambda_1 \geq \frac{\log{(1/p)}}{\sup{x \in \Omega} d_{p,\partial \Omega}(x)}.$$Moreover,as$p \rightarrow 0$,thislowerboundconvergesto$\lambda_1$.