Electronic Journal of Probability

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

Mihai Pascu

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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

Permanent link to this document
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

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60H20: Stochastic integral equations 35K05: Heat equation 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
couplings mirror coupling reflecting Brownian motion Chavel's conjecture

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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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References

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