Mirror couplings and Neumann eigenfunctions
Rami Atar, Krzysztof Burdzy · Indiana University Mathematics Journal · 2008
We analyze a pair of reflected Brownian motions in a pla- nar domainD, for which the increments of both processes form mirror images of each other when the processes are not on the boundary. We show that forD in a class of smooth convex planar domains, the two pro- cesses 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.