  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
$\lambda_1$.
