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Invariant wedges for a two-point reflecting Brownian motion and the “hot spots” problem. (English) Zbl 1125.60310

Summary: We consider domains \(D\) of \(\mathbb{R}^d\), \(d\geq 2\) with the property that there is a wedge \(V\subset \mathbb{R}^d\) 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 \(u(x)-u(y)\geq 0\) 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.

MSC:

60J65 Brownian motion
35B50 Maximum principles in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35K05 Heat equation
35K20 Initial-boundary value problems for second-order parabolic equations
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