Electronic Journal of Probability

Invariant Wedges for a Two-Point Reflecting Brownian Motion and the ``Hot Spots'' Problem

Rami Atar

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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.

Article information

Source
Electron. J. Probab., Volume 6 (2001), paper no. 18, 19 pp.

Dates
Accepted: 14 June 2001
First available in Project Euclid: 19 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1461097648

Digital Object Identifier
doi:10.1214/EJP.v6-91

Mathematical Reviews number (MathSciNet)
MR1873295

Zentralblatt MATH identifier
1125.60310

Subjects
Primary: 60J30

Keywords
Reflecting Brownian motion Neumann eigenvalue problem convex domains

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

Citation

Atar, Rami. Invariant Wedges for a Two-Point Reflecting Brownian Motion and the ``Hot Spots'' Problem. Electron. J. Probab. 6 (2001), paper no. 18, 19 pp. doi:10.1214/EJP.v6-91. https://projecteuclid.org/euclid.ejp/1461097648


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References

  • R. Banuelos and K. Burdzy. On the “hot spots” conjecture of J. Rauch, J. Funct. Anal. 164, 1–33 (1999)
  • R. F. Bass. Probabilistic Techniques in Analysis, Series of Prob. Appl. (1995)
  • R. F. Bass and K. Burdzy. Fiber Brownian motion and the “hot spots” problem, Duke Math. J. 105 (2000), no. 1, 25–58.
  • 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)
  • K. Burdzy and W. Werner. A counterexample to the “hot spots” conjecture, Ann. Math. 149 309–317 (1999)
  • 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)
  • E. B. Davies. Heat Kernels and Spectral Theory, Cambridge University Press, 1989.
  • 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)
  • 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.
  • 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
  • B. Kawohl. Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, Vol. 1150 (1985)
  • 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)
  • 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)
  • P. L. Lions and A. S. Sznitman. Stochastic differential equations with reflecting boundary conditions, Comm. Pure appl. Math. Vol. 37, 511–537 (1984)
  • 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)
  • N. S. Nadirashvili. On the multiplicity of the eigenvalues of the Neumann problem, Soviet Math. Dokl. 33 281–282 (1986)