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2011 Mirror Coupling of Reflecting Brownian Motion and an Application to Chavel's Conjecture
Mihai Pascu
Author Affiliations +
Electron. J. Probab. 16: 504-530 (2011). DOI: 10.1214/EJP.v16-859


In a series of papers, Burdzy et al. introduced the <em>mirror coupling</em> 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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Mihai Pascu. "Mirror Coupling of Reflecting Brownian Motion and an Application to Chavel's Conjecture." Electron. J. Probab. 16 504 - 530, 2011.


Accepted: 17 March 2011; Published: 2011
First available in Project Euclid: 1 June 2016

zbMATH: 1228.60087
MathSciNet: MR2781844
Digital Object Identifier: 10.1214/EJP.v16-859

Primary: 60J65
Secondary: 35K05, 60H20, 60H30


Vol.16 • 2011
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