Mirror couplings and Neumann eigenfunctions
  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. 
Rami Atar and Krzysztof Burdzy