Electronic Journal of Probability

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

Mihai Pascu

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

couplings mirror coupling reflecting Brownian motion Chavel's conjecture

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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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  • R. Atar, K. Burdzy, On Neumann eigenfunction in Lip domains, J. Amer. Math. Soc. 17 (2004), No. 2, pp. 243 - 265.
  • R. Atar, K. Burdzy, Mirror couplings and Neumann eigenfunctions, Indiana Univ. Math. J. 57 (2008), pp. 1317 - 1351.
  • R. Bañuelos, K. Burdzy, On the "hot spots" conjecture of J. Rauch, J. Funct. Anal. 164 (1999), No. 1, pp. 1 - 33.
  • R. Bass, K. Burdzy, On domain monotonicity of the Neumann heat kernel, J. Funct. Anal. 116 (1993), No. 1, pp. 215 - 224.
  • R. F. Bass, P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991), No. 2, pp. 486 - 508.
  • K. Burdzy, Neumann eigenfunctions and Brownian couplings, Proc. Potential theory in Matsue, Adv. Stud. Pure Math. 44 (2006), Math. Soc. Japan, Tokyo, pp. 11 - 23.
  • K. Burdzy, Z. Q. Chen, Coalescence of synchronous couplings, Probab. Theory Related Fields 123 (2002), No. 4, pp. 553 - 578.
  • K. Burdzy, Z. Q. Chen, Weak convergence of reflecting Brownian motions, Electron. Comm. Probab. 3 (1998), pp. 29 - 33 (electronic).
  • K. Burdzy, Z. Q. Chen, P. Jones, Synchronous couplings of reflected Brownian motions in smooth domains, Illinois J. Math. 50 (2006), No. 1 - 4, pp. 189 - 268 (electronic).
  • K. Burdzy, W. S. Kendall, Efficient Markovian couplings: examples and counterexamples, Ann. Appl. Probab. 10 (2000), No. 2, pp. 362 - 409.
  • R. A. Carmona, W. Zheng, Reflecting Brownian motions and comparison theorems for Neumann heat kernels, J. Funct. Anal. 123 (1994), No. 1, pp. 109 - 128.
  • I. Chavel, Heat diffusion in insulated convex domains, J. London Math. Soc. (2) 34 (1986), No. 3, pp. 473 - 478.
  • H. J. Englebert, W. Schmidt, On solutions of one-dimensional stochastic differential equations without drift, Z. Wahrsch. Verw. Gebiete 68 (1985), pp. 287 - 314.
  • E. Hsu, A domain monotonicity property for the Neumann heat kernel, Osaka Math. J. 31 (1994), pp. 215 - 223.
  • K. Itô, H. P. McKean, Diffusion processes and their sample paths, Second edition, Springer-Verlag, Berlin-New York, 1974.
  • W. S. Kendall, Coupled Brownian motions and partial domain monotonicity for the Neumann heat kernel, J. Funct. Anal. 86 (1989), No. 2, pp. 226 - 236.
  • M. N. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc. 354 (2002), No. 11, pp. 4681 - 4702.
  • M. N. Pascu, N. R. Pascu, A note on pathwise uniqueness for a degenerate stochastic differential equation (to appear).