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.