Scaling coupling of reflecting Brownian motions and the hot spots problem

Abstract

We introduce a new type of coupling of reflecting Brownian motions in smooth planar domains, called <italic>scaling coupling</italic>. We apply this to obtain monotonicity properties of antisymmetric second Neumann eigenfunctions of convex planar domains with one line of symmetry. In particular, this gives the proof of the hot spots conjecture for some known types of domains and some new ones.