# 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!

• I remember seeing the required definition in Y.Kifer's Random Perturbation of Dynamical systems for a general elliptic operator. – Piyush Grover Mar 17 '14 at 20:32

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.

• How can we be sure that $\lambda_1$ in the above limit is finite? – rubikscube09 Oct 28 '19 at 16:21
• On a compact domain, you have a Poincaré inequality and the inverse of the Poincaré constant gives a lower bound on $\lambda_1$ by the Rayleigh principle. On the other hand, an upper bound is also obtained by the Rayleigh principle provided that there exists a non-zero function in $f\in H_0^1(\Omega)$, that is Sobolev functions. For this any bump function supported inside of $\Omega$ will do it, if $\Omega$ has non-empty interior. – André Schlichting Mar 19 at 12:28

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

• Hey Paul, thanks for the answer; the limit this paper presents on the first page doesn't seem to have any content, unless I'm misinterpreting the notation: the author claims $\lim_{t\rightarrow\infty} e^{-\lambda_1 t/2}P^x(x_t\in K)=0$, but $P^x\leq 1$ always, and for any compact domain of the type I described, $\lambda_1>0$. Am I misinterpreting something? – quick_q Mar 17 '14 at 20:11
• @quick: I can't access the paper right now, but double check the sign of the operator being used, ie whether the author is discussing an eigenvalue of the Laplacian or its negative. – Nate Eldredge Mar 17 '14 at 23:40
• @NateEldredge - thanks; they do say "the smallest eigenvalue of $-\Delta$" but you're definitely right - there's a sign error there. I found a paper, "Inequalities for Exit times and Eigenvalues of Balls", which has basically the same formula written. The author calls this "Kac's formula" with no reference, and I would really like to have a proof of this; searching "Kac's formula" always returns Feynman-Kac, which I don't think implies this limit... – quick_q Mar 18 '14 at 0:00
• Ah, it can be derived from Feynman-Kac; thanks! I am willing to post a short sketch of the proof if anyone is interested. – quick_q Mar 18 '14 at 0:59