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

Mihai N Pascu (Transilvania University of Brasov)

Abstract


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 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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 504-530

Publication Date: March 17, 2011

DOI: 10.1214/EJP.v16-859

References

  1. R. Atar, K. Burdzy, On Neumann eigenfunction in Lip domains, J. Amer. Math. Soc. 17 (2004), No. 2, pp. 243 &ndash 265. MR2051611
  2. R. Atar, K. Burdzy, Mirror couplings and Neumann eigenfunctions, Indiana Univ. Math. J. 57 (2008), pp. 1317 &ndash 1351. MR2429094
  3. R. Bañuelos, K. Burdzy, On the "hot spots'' conjecture of J. Rauch, J. Funct. Anal. 164 (1999), No. 1, pp. 1 &ndash 33. MR1694534
  4. R. Bass, K. Burdzy, On domain monotonicity of the Neumann heat kernel, J. Funct. Anal. 116 (1993), No. 1, pp. 215 &ndash 224. MR1237993
  5. 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 &ndash 508. MR1106272
  6. K. Burdzy, Neumann eigenfunctions and Brownian couplings, Proc. Potential theory in Matsue, Adv. Stud. Pure Math. 44 (2006), Math. Soc. Japan, Tokyo, pp. 11 &ndash 23. MR2277819
  7. K. Burdzy, Z. Q. Chen, Coalescence of synchronous couplings, Probab. Theory Related Fields 123 (2002), No. 4, pp. 553 &ndash 578. MR1921013
  8. K. Burdzy, Z. Q. Chen, Weak convergence of reflecting Brownian motions, Electron. Comm. Probab. 3 (1998), pp. 29 &ndash 33 (electronic). MR1625707
  9. K. Burdzy, Z. Q. Chen, P. Jones, Synchronous couplings of reflected Brownian motions in smooth domains, Illinois J. Math. 50 (2006), No. 1 &ndash 4, pp. 189 &ndash 268 (electronic). MR2247829
  10. K. Burdzy, W. S. Kendall, Efficient Markovian couplings: examples and counterexamples, Ann. Appl. Probab. 10 (2000), No. 2, pp. 362 &ndash 409. MR1768241
  11. R. A. Carmona, W. Zheng, Reflecting Brownian motions and comparison theorems for Neumann heat kernels, J. Funct. Anal. 123 (1994), No. 1, pp. 109 &ndash 128. MR1279298
  12. I. Chavel, Heat diffusion in insulated convex domains, J. London Math. Soc. (2) 34 (1986), No. 3, pp. 473 &ndash 478. MR0864450
  13. H. J. Englebert, W. Schmidt, On solutions of one-dimensional stochastic differential equations without drift, Z. Wahrsch. Verw. Gebiete 68 (1985), pp. 287 &ndash 314. MR0771468
  14. E. Hsu, A domain monotonicity property for the Neumann heat kernel, Osaka Math. J. 31 (1994), pp. 215 &ndash 223. MR1262798
  15. K. Itô, H. P. McKean, Diffusion processes and their sample paths, Second edition, Springer-Verlag, Berlin-New York, 1974. MR0345224
  16. W. S. Kendall, Coupled Brownian motions and partial domain monotonicity for the Neumann heat kernel, J. Funct. Anal. 86 (1989), No. 2, pp. 226 &ndash 236. MR1021137
  17. M. N. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc. 354 (2002), No. 11, pp. 4681 &ndash 4702. MR1926894
  18. M. N. Pascu, N. R. Pascu, A note on pathwise uniqueness for a degenerate stochastic differential equation (to appear).
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.