On Sullivan's construction of eigenfunctions via exit times of Brownian motion

Abstract

The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. Let $\lambda_1 = \lambda_1(X) < 0$ be the supremum of the spectrum of the Laplacian on $L^2(X)$, and let $D \subset X$ be a bounded domain in $X$ with smooth boundary. Let $(B_t)_{t \geq 0}$ be Brownian motion on $X$ and let $\tau = \tau_D$ be the first exit time of Brownian motion from $D$. For each $\lambda \in \mathbb{C}$ with $\hbox{Re } \ \lambda > \lambda_1$ and $x \in D$, we show that for any continuous function $\phi : \partial D \to \mathbb{C}$, the function $$ h(x) = \mathbb{E}_x(e^{-\lambda \tau} \phi(B_{\tau})) \ , \ x \in D, $$ is an eigenfunction of the Laplacian on $D$ with eigenvalue $\lambda$ and boundary value $\phi$.