# 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!

The fundamental solution for the heat equation $$\partial_t h(t,x)=\Delta_x h(t,x),\quad h(0,x)=\delta_y(x)$$ has the interpretation of the probability density for the location of the Brownian particle at time $$t$$ that has started from $$y$$. For Neumann boundary conditions, if $$vol(\Omega)^{-\frac12}\equiv\psi_0,\psi_1,\psi_2,\dots$$ denote the normalized Laplace eigenfunctions and $$\lambda_1,\lambda_2,\dots$$ the corresponding eigenvalues, we have $$h(t,x)=\sum_{i=0}^\infty \psi_i(y)\psi_i(x)e^{-\lambda_i t}$$. As $$\lambda_0\equiv 0$$, in a connected domain the first term of asymptotics is given by a constant $$h(t,x)\equiv vol(\Omega)^{-1}$$ as $$t\to \infty$$, meaning that the particle becomes uniformly distributed over the domain. The first non-trivial eigenvalue controls the rate of convergence to the stationary distribution.
I believe a similar characterization should hold for the Neumann eigenvalues, as for the Dirichlet. If $$B(t)$$ denotes the Brownian particle which is reflected upon impact at the boundary, the formula should read $$\mu_2 = - \lim_{t\to\infty} \frac{1}{t} \ln \int_\Omega E_x\left( B(t)\right),$$ where $$E_x$$ denotes the expectation when the Brownian particle starts the motion at the point $$x$$. Here I am not sure what is the most general boundary you can address, though $$C^2$$-boundary seems way more than enough. The proof should also be the same: the main issue should be the existence of a Feynman-Kac formula, which you have in this case.