Mirror coupling of reflecting Brownian motion and an application to Chavel’s conjecture∗
نویسنده
چکیده
In a series of papers, Burdzy et al. introduced the mirror coupling of reflecting Brownian motions in a smooth bounded domain D ⊂ Rd , 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 D1, D2 ⊂ Rd . 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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