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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