Invariant Wedges for a Two-Point Reflecting Brownian Motion and the ``Hot Spots'' Problem
Abstract
We consider domains $D$ of $R^d$, $d\ge 2$ with the property that there is a wedge $V\subset R^d$ which is left invariant under all tangential projections at smooth portions of $\partial D$. It is shown that the difference between two solutions of the Skorokhod equation in $D$ with normal reflection, driven by the same Brownian motion, remains in $V$ if it is initially in $V$. The heat equation on $D$ with Neumann boundary conditions is considered next. It is shown that the cone of elements $u$ of $L^2(D)$ satisfying $u(x)-u(y)\ge0$ whenever $x-y\in V$ is left invariant by the corresponding heat semigroup. Positivity considerations identify an eigenfunction corresponding to the second Neumann eigenvalue as an element of this cone. For $d=2$ and under further assumptions, especially convexity of the domain, this eigenvalue is simple.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-19
Publication Date: June 14, 2001
DOI: 10.1214/EJP.v6-91
References
- R. Banuelos and K. Burdzy. On the ``hot spots'' conjecture of J. Rauch J. Funct. Anal. 164, 1--33 (1999) Math. Review 2000m:35085
- R. F. Bass. Probabilistic Techniques in Analysis Series of Prob. Appl. (1995) Math. Review 96e:60001
- R. F. Bass and K. Burdzy. Fiber Brownian motion and the ``hot spots'' problem Duke Math. J. 105 (2000), no. 1, 25--58. Math. Review 2001g:60190
- R. F. Bass and P. Hsu. Some potential theory for reflecting Brownian motion in Holder and Lipschitz domains Ann. Prob. Vol. 19 No. 2 486--508 (1991) Math. Review 92i:60142
- K. Burdzy and W. Werner. A counterexample to the ``hot spots'' conjecture Ann. Math. 149 309--317 (1999) Math. Review 2000b:35044
- M. Cranston and Y. Le Jan. Noncoalescence for the Skorohod equation in a convex domain of R^2 Probab. Theory Related Fields 87, 241--252 (1990) Math. Review 92e:60116
- E. B. Davies. Heat Kernels and Spectral Theory Cambridge University Press, 1989. Math. Review 92a:35035
- P. Dupuis and H. Ishii. On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications Stochastics and Stochastics Reports, Vol. 35, pp. 31--62 (1991) Math. Review 93e:60110
- P. Dupuis and K. Ramanan. Convex duality and the Skorokhod Problem. I, II Probab. Theory Related Fields 115 (1999), no. 2, 153--195, 197--236. Math. Review 2001f:49041
- D. Jerison and N. Nadirashvili. The ``hot spots'' conjecture for domains with two axes of symmetry J. Amer. Math. Soc. 13 (2000), no. 4, 741--772 Math. Review 2001f:35110
- B. Kawohl. Rearrangements and Convexity of Level Sets in PDE Lecture Notes in Mathematics, Vol. 1150 (1985) Math. Review 87a:35001
- M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev. Positive Linear Systes: The Method of Positive Operators Sigma Series in Appl. Math. Vol. 5 (1989) Math. Review 91f:47051
- O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva. Linear and Quasilinear Equations of Parabolic Type Transl. Math. Monogr., Vol. 23, AMS, Providence (1968) Math. Review 39:3159b
- P. L. Lions and A. S. Sznitman. Stochastic differential equations with reflecting boundary conditions Comm. Pure appl. Math. Vol. 37, 511--537 (1984) Math. Review 85m:60105
- D. Lupo and A. M. Micheletti. On the persistence of the multiplicity of eigenvalues for some variational elliptic operator depending on the domain J. Math. Anal. Appl. 193 no. 3 990--1002 (1995) Math. Review 96h:35032
- N. S. Nadirashvili. On the multiplicity of the eigenvalues of the Neumann problem Soviet Math. Dokl. 33 281--282 (1986) Math. Review 88a:35068
This work is licensed under a Creative Commons Attribution 3.0 License.