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

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 lowe...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2023-04
Main Authors: Banuelos, Rodrigo, Phanuel Mariano, Wang, Jing
Format: Article
Language:English
Subjects:
Brownian motion
Dirichlet problem
Domains
Eigenvalues
Lower bounds
Upper bounds
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422