The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to $$ -\Delta\psi = \lambda\psi\ \ \ \text{in}\ \Omega\\ \psi=0\ \ \ \text{on}\ \partial\Omega. $$

I remember hearing that there is a very concrete interpretation of $\lambda_1(\Omega)$ in terms of (expected) boundary hitting times of random walks, but I can't seem to find this written down explicitly anywhere, and I can't seem to come up with it on my own.

I would greatly appreciate a reference, or if this is not true that would be helpful to know as well!