## Electronic Journal of Probability

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

Mihai Pascu

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 18, 504-530.

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

https://projecteuclid.org/euclid.ejp/1464820187

Digital Object Identifier
doi:10.1214/EJP.v16-859

Mathematical Reviews number (MathSciNet)
MR2781844

Zentralblatt MATH identifier
1228.60087

Rights

#### Citation

Pascu, Mihai. Mirror Coupling of Reflecting Brownian Motion and an Application to Chavel's Conjecture. Electron. J. Probab. 16 (2011), paper no. 18, 504--530. doi:10.1214/EJP.v16-859. https://projecteuclid.org/euclid.ejp/1464820187

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