Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

TL;DR

This paper establishes universal bounds relating the first Dirichlet eigenvalue of the Laplacian to the moments of Brownian exit times in domains, identifying extremal domains and asymptotic sharpness of bounds.

Contribution

It provides new universal bounds for the product of the Laplacian's spectral bottom and Brownian exit time moments, including extremal domain existence and sharpness results.

Findings

Lower bounds are sharp for integer p.

Upper bounds are asymptotically sharp as dimension increases.

Existence of extremal convex, symmetric domains is proven.

Abstract

For domains in $\mathbb{R}^d$, $d\geq 2$, we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power $p>0$ and the supremum over all starting points of the $p$-moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of $p$ and that for $p \geq 1$, the upper bound is asymptotically sharp as $d\to\infty$. For all $p>0$, we prove the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For this class of domains we conjecture that the cube is extremal.