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

TL;DR

This paper introduces a new variation of the Donsker-Varadhan inequality that bounds the principal eigenvalue of an elliptic operator using exit time quantiles, providing a convergent approximation as the quantile approaches zero.

Contribution

It proposes a novel inequality relating the principal eigenvalue to exit time quantiles, extending the classical mean-based bound and improving understanding of eigenvalue estimates.

Findings

The new bound involves exit time quantiles rather than mean exit times.

As the quantile p approaches zero, the bound converges to the true eigenvalue.

The approach offers a potentially tighter estimate for the principal 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 $$\lambda_1 \geq \frac{1}{\sup_{x \in \Omega} \mathbb{E}_x \tau_{\Omega^c}}.$$ Instead of looking at the mean of the first exit time, we study quantiles: let $d_{p, \partial \Omega}:\Omega \rightarrow \mathbb{R}_{\geq 0}$ be the smallest time $t$ such that the likelihood of exiting within that time is $p$, then $$\lambda_1 \geq \frac{\log{(1/p)}}{\sup_{x \in \Omega} d_{p,\partial \Omega}(x)}.$$ Moreover, as $p \rightarrow 0$, this lower bound converges to $\lambda_1$.