Mirror Coupling of Reflecting Brownian Motion and an Application to Chavel's Conjecture

Mihai N Pascu (Transilvania University of Brasov)


In a series of papers, Burdzy et al. introduced the 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 Laplacian 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 applications 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 ([12]), respectively W. S. Kendall ([16]), and a new proof of Chavel's conjecture for domains satisfying the ball condition, such that the inner domain is star-shaped with respect to the center of the ball.

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Pages: 504-530

Publication Date: March 17, 2011

DOI: 10.1214/EJP.v16-859


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