Mirror couplings and Neumann eigenfunctions Rami AtarKrzysztof Burdzy 35J0560H30Neumann eigenfunctionsreflected Brownian motioncouplings 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. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3222 10.1512/iumj.2008.57.3222 en Indiana Univ. Math. J. 57 (2008) 1317 - 1352 state-of-the-art mathematics http://iumj.org/access/