Probabilistic Interpretation of First Dirichlet Eigenvalue? 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!
 A: The first eigenvalue is the exponential rate for hitting the boundary, i.e.
$$
  \lambda_1 = - \lim_{t\to\infty} \frac{1}{t} \log P_{u_0}(\tau > t) ,
$$
where $\tau := \inf\{X_t \in \partial \Omega\}$ and $\mathrm{law}\, X_0 = u_0$ for a Brownian motion $X_t$ killed at $\partial \Omega$. This probability is also given in terms of the density of $X_t$ which solves the pde
$$\begin{align}
  \partial_t u &= \Delta u & \text{in } \Omega \\
    u &= 0  &\text{on } \partial \Omega \\
    u(t=0,\cdot) &= u_0(\cdot) .
\end{align}
$$
Therewith, the probability of not yet hit the boundary is the probability to be still inside of $\Omega$, that is
$$ 
  P_{u_0}(\tau > t ) = \int_\Omega u(t,x) \;\mathrm{d} x .
$$
Assuming now that $0<\lambda_1 <\lambda_2 \leq \dots$, it follows by the semigroup representation of the solution to the pde
$$
 \int u(t,\cdot) = e^{-\lambda_1 t} \left( \langle u_0, \psi_1 \rangle \int \psi_1 + O(e^{-(\lambda_2-\lambda_1)t})\right),
$$
where $\psi_1$ is the first eigenfunction. From here the claim follows.
A: The First eigenvalue determine the behavior of the heat equation and then of the brownian motion, see reference therein
http://www.actamath.com/Jwk_sxxb_en/EN/article/downloadArticleFile.do?attachType=PDF&id=6458
Edit:
The new link is Free? Else I can send you the pdf.
Of course there is the marvelous article of Kac, where the role of the eigenvalue in the diffusion process is make so clearly
https://www.math.ucdavis.edu/~saito/courses/ACHA.READ.F03/kac-drum.pdf 
