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


 
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1. Title Title of document Invariant Wedges for a Two-Point Reflecting Brownian Motion and the ``Hot Spots'' Problem
 
2. Creator Author's name, affiliation, country Rami Atar; Technion - Israel Institute of Technology
 
3. Subject Discipline(s) Mathematics
 
3. Subject Keyword(s) Reflecting Brownian motion, Neumann eigenvalue problem, convex domains.
 
3. Subject Subject classification 60J30
 
4. Description 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.
 
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7. Date (YYYY-MM-DD) 2001-06-14
 
8. Type Status & genre Peer-reviewed Article
 
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9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/91
 
10. Identifier Digital Object Identifier 10.1214/EJP.v6-91
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 6
 
12. Language English=en en
 
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