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

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Date
2017-04-23
Authors
Lu, Jianfeng
Steinerberger, Stefan
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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
Journal article
Subject
math.SP
math.SP
math-ph
math.AP
math.MP
math.PR
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https://hdl.handle.net/10161/14045
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Scholars@Duke

Lu

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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