Electronic Journal of Probability
- Electron. J. Probab.
- Volume 6 (2001), paper no. 18, 19 pp.
Invariant Wedges for a Two-Point Reflecting Brownian Motion and the ``Hot Spots'' Problem
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.
Electron. J. Probab., Volume 6 (2001), paper no. 18, 19 pp.
Accepted: 14 June 2001
First available in Project Euclid: 19 April 2016
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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