On the Spectral Gap of Brownian Motion with Jump Boundary

TL;DR

This paper provides a new probabilistic proof for the convergence rate of one-dimensional Brownian motion with jump boundary and explores how the spectral gap depends on jump distribution in higher dimensions.

Contribution

It introduces a novel probabilistic coupling approach for analyzing spectral gaps and addresses the open question of jump distribution effects in multi-dimensional cases.

Findings

Established the exact convergence rate for 1D Brownian motion with jump boundary.

Demonstrated the dependence of spectral gap on jump distribution in multi-dimensional settings.

Provided a new proof technique differing from previous methods.

Abstract

In this paper we consider the Brownian motion with jump boundary and present a new proof of a recent result of Li, Leung and Rakesh concerning the exact convergence rate in the one-dimensional case. Our methods are different and mainly probabilistic relying on coupling methods adapted to the special situation under investigation. Moreover, we answer a question raised by Ben-Ari and Pinsky concerning the dependence of the spectral gap on the jump distribution in a multi-dimensional setting.