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 $Ω$ by the largest mean first exit time of the associated drift-diffusion process via $$λ_1 \geq \frac{1}{\sup_{x \in Ω} \mathbb{E}_x τ_{Ω^c}}.$$ Instead of looking at the mean of the first exit time, we study quantiles: let $d_{p, \partial Ω}:Ω\rightarrow \mathbb{R}_{\geq 0}$ be the smallest time $t$ such that the likelihood of exiting within that time is $p$, then $$λ_1 \geq \frac{\log{(1/p)}}{\sup_{x \in Ω} d_{p,\partial Ω}(x)}.$$ Moreover, as $p \rightarrow 0$, this lower bound converges to $λ_1$.