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 $\Omega$ 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 exist 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 $\lambda_1$.
Type
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https://hdl.handle.net/10161/14045Collections
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Show full item recordScholars@Duke
Jianfeng Lu
Professor of Mathematics
Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm
development for problems from computational physics, theoretical chemistry, materials
science and other related fields.More specifically, his current research focuses include:Electronic
structure and many body problems; quantum molecular dynamics; multiscale modeling
and analysis; rare events and sampling techniques.

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