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Mirror coupling of reflecting Brownian motion and an application to Chavel’s conjecture. (English) Zbl 1228.60087
Summary: 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 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 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 [J. Lond. Math. Soc., II. Ser. 34, 473–478 (1986; Zbl 0622.35025)], respectively W. S. Kendall [J. Funct. Anal. 86, No. 2, 226–236 (1989; Zbl 0684.60060)], 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.

##### MSC:
 60J65 Brownian motion 60H20 Stochastic integral equations 35K05 Heat equation
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