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

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2017-04-23

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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 $\lambda_1$.

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### Scholars@Duke

#### Jianfeng Lu

Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science, machine learning, and other related fields.

More specifically, his current research focuses include:

High dimensional PDEs; generative models and sampling methods; control and reinforcement learning; electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis.

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