Mirror coupling of reflecting Brownian motion and an application to Chavel's conjecture

TL;DR

This paper extends the mirror coupling of reflecting Brownian motions to different domains and uses it to prove key results related to Chavel's conjecture on the Neumann heat kernel's domain monotonicity.

Contribution

It introduces a novel extension of mirror coupling to Brownian motions in different domains and applies it to prove Chavel's conjecture.

Findings

Unified proof of Chavel's conjecture

Extension of mirror coupling to different domains

Insights into Neumann heat kernel properties

Abstract

In a series of papers, Burdzy et. al. introduced the \emph{mirror coupling} of reflecting Brownian motions in a smooth bounded domain $D\subset \mathbb{R}^{d}$, and used it to prove certain properties of eigenvalues and eigenfunctions of the Neumann Laplaceian on $D$. In the present paper we show that the construction of the mirror coupling can be extended to the case when the two Brownian motions live in different domains $D_{1},D_{2}\subset \mathbb{R}^{d}$. As an application of the construction, we derive a unifying proof of the two main results concerning the validity of Chavel's conjecture on the domain monotonicity of the Neumann heat kernel, due to I. Chavel (\cite{Chavel}), respectively W. S. Kendall (\cite{Kendall}).