Lu, J; Steinerberger, S, *A variation on the Donsker-Varadhan inequality for the principal eigenvalue.*,
Proceedings. Mathematical, Physical, and Engineering Sciences, vol. 473 no. 2204
(August, 2017),
pp. 20160877 [doi] .
**Abstract:**

*The purpose of this short paper 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 Ω by the largest mean first exit time of the associated drift-diffusion process via [Formula: see text]Instead of looking at the mean of the first exit time, we study quantiles: let [Formula: see text] be the smallest time t such that the likelihood of exiting within that time is p, then [Formula: see text]Moreover, as [Formula: see text], this lower bound converges to λ1.*