Atar, Rami Invariant wedges for a two-point reflecting Brownian motion and the “hot spots” problem. (English) Zbl 1125.60310 Electron. J. Probab. 6, Paper No. 18, 19 p. (2001). 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. Cited in 5 Documents 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 Keywords:reflecting Brownian motion; Neumann eigenvalue problem; convex domains PDFBibTeX XMLCite \textit{R. Atar}, Electron. J. Probab. 6, Paper No. 18, 19 p. (2001; Zbl 1125.60310) Full Text: DOI EuDML EMIS