NODAL GEOMETRY, HEAT DIFFUSION AND BROWNIAN MOTION

We use tools from n-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold M. On one hand we extend a theorem of Lieb (1983) and prove that any Laplace nodal domain Omega(lambda) subset of M almost fully contains a ball of radius similar to 1/root lambda(1)(Omega(lambda)) and such a ball can be centred at any point of maximum of the Dirichlet ground state phi(lambda 1)(Omega(lambda))This also gives a slight refinement of a result by Mangoubi (2008) concerning the inradius of nodal domains. On the other hand, we also prove that no nodal domain can be contained in a reasonably narrow tubular neighbourhood of unions of finitely many submanifolds inside M.