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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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