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 $λ_1 = λ_1(X) < 0$ 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 $τ= τ_D$ be the first exit time of Brownian motion from $D$. For each $λ\in \mathbb{C}$ with $\hbox{Re } \ λ> λ_1$ and $x \in D$, we show that for any continuous function $φ: \partial D \to \mathbb{C}$, the function $$ h(x) = \mathbb{E}_x(e^{-λτ} φ(B_τ)) \ , \ x \in D, $$ is an eigenfunction of the Laplacian on $D$ with eigenvalue $λ$ and boundary value $φ$.