A Variation on the Donsker-Varadhan Inequality for the Principial Eigenvalue
dc.contributor.author | Lu, Jianfeng | |
dc.contributor.author | Steinerberger, Stefan | |
dc.date.accessioned | 2017-04-23T15:40:44Z | |
dc.date.available | 2017-04-23T15:40:44Z | |
dc.date.issued | 2017-04-23 | |
dc.description.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 $\lambda_1$. | |
dc.identifier | ||
dc.identifier.uri | ||
dc.publisher | The Royal Society | |
dc.subject | math.SP | |
dc.subject | math.SP | |
dc.subject | math-ph | |
dc.subject | math.AP | |
dc.subject | math.MP | |
dc.subject | math.PR | |
dc.title | A Variation on the Donsker-Varadhan Inequality for the Principial Eigenvalue | |
dc.type | Journal article | |
duke.contributor.orcid | Lu, Jianfeng|0000-0001-6255-5165 | |
pubs.author-url | ||
pubs.organisational-group | Chemistry | |
pubs.organisational-group | Duke | |
pubs.organisational-group | Mathematics | |
pubs.organisational-group | Physics | |
pubs.organisational-group | Trinity College of Arts & Sciences |