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 $\lambda_1$.
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