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

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 $(B_t{)}_{t\ge 0}$ be Brownian motion on $X$ and let $\tau = \tau_D$bethefirstexittimeofBrownianmotionfrom$D$\mathrm{.}Foreach$\lambda \in $\mathbb{C}$$with$\hbox{Re } \ \lambda > \lambda_1$and$x \in D$, we show that for any continuous function $\varphi :\mathrm{\partial }D\to \mathbb{C}$, the function $$h(x)={\mathbb{E}}_x(e^{-\lambda \tau }\varphi (B_{\tau })),x\in D,$$ is an eigenfunction of the Laplacian on $D$ with eigenvalue $\lambda$ and boundary value $\varphi$.