 
Summary: Mirror Couplings and Neumann Eigenfunctions
RAMI ATAR & KRZYSZTOF BURDZY
ABSTRACT. We analyze a pair of reflected Brownian motions in a pla
nar 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 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.
1. INTRODUCTION
We will prove that the second eigenvalue for the Laplacian with Neumann bound
ary conditions is simple for a class of planar convex domains. We will also present
some geometric properties of the corresponding eigenfunctions. The main tool
that we use is a coupling of a pair of reflected Brownian motions in the domain,
for which the increments of both processes form mirror images of each other when
both processes are not on the boundary. This coupling, referred to as a mirror cou
pling, has been used before to study properties of Neumann Laplacian eigenfunc
tions (see [4], [7] and references therein) and, in particular, has been used in [3]
to determine whether the second eigenvalue is simple. That paper was concerned
with "lip domains" defined as follows. A lip domain is a bounded planar domain
