Mirror couplings and Neumann eigenfunctions

Rami Atar; Krzysztof Burdzy

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Mirror couplings and Neumann eigenfunctions

Rami Atar·Krzysztof Burdzy

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Abstract

We analyze a pair of reflected Brownian motions in a planar domain $D$, for which the increments of both processes form mirror images of each other when the processes are not on the boundary. We show that for $D$ in a class of smooth convex planar domains, the two processes remain ordered forever, according to a certain partial order. This is used to prove that the second eigenvalue is simple for the Laplacian with Neumann boundary conditions for the same class of domains.

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Rami Atar, Krzysztof Burdzy. 2006-07-18. Mirror couplings and Neumann eigenfunctions. https://arxiv.org/abs/math/0607400

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