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 Electronic Journal of Probability > Vol. 6 (2001) > Paper 18 open journal systems 


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

Rami Atar, Technion - Israel Institute of Technology


Abstract
We consider domains D of Rd, dge 2 with the property that there is a wedge Vsubset Rd 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)ge 0 whenever x-yin 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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Pages: 1-19

Published on: June 14, 2001
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Electronic Journal of Probability. ISSN: 1083-6489