# Probabilistic characterization of first Neumann eigenvalue

In this MO post, a question has been asked (and answered) about the probabilistic interpretation of the first Dirichlet eigenvalue of the Laplacian in terms of boundary hitting times. I wish to ask the exact same question for the first non-zero eigenvalue of the Laplacian on a bounded domain $$\Omega \subset \mathbb{R}^n$$ with Neumann boundary conditions.

A few comments: I have heard that the Brownian motion corresponding to the Neumann boundary condition has to be reflected at the boundary, as opposed to being killed on impact in the Dirichlet case. Observe that we can talk about Brownian motion, because if $$\mu$$ and $$\psi (x)$$ are the first non-zero Neumann eigenvalue and eigenfunction respectively, then $$e^{-\mu t}\psi (x)$$ solves the heat equation $$\partial_t u = \Delta u$$ with initial condition $$\psi (x)$$. So we can assume that the boundary $$\partial \Omega$$ is smooth. I would really appreciate a reference. Thanks in advance!