Let \(\widetilde{M}\) denote a complete, simply connected manifold with sectional curvature \(K \leq 0\), and let \(\widetilde{M}(\infty)\) denote the boundary sphere of \(\widetilde{M}\) that consists of equivalence classes of asymptotic unit speed geodesics of \(\widetilde{M}\). The manifold \(\widetilde{M}\) is said to have rank 1 if there exists a geodesic \(\gamma: \mathbb{R} \to \widetilde{M}\) such that for every perpendicular, parallel vector field \(E(t)\) along \(\gamma\) there exists \(t_ 0 \in \mathbb{R}\) with \(K(E(t_ 0),\gamma'(t_ 0)) < 0\). In a previous paper W. Ballmann showed that the Dirichlet problem in \(\widetilde{M}(\infty)\) can be solved if \(\widetilde{M}\) has rank 1 and admits a compact quotient manifold \(M\). More precisely, if \(f: \widetilde{M}(\infty) \to \mathbb{R}\) is any continuous function, then there exists a bounded harmonic function \(h : \widetilde{M} \to \mathbb{R}\) such that \(h\) extends continuously to \(f\) on \(\widetilde{M}(\infty)\) relative to the cone topology on \(\widetilde{M} \cup \widetilde{M}(\infty)\). Earlier, several authors had solved the Dirichlet problem on \(\widetilde{M}(\infty)\) in the case that the sectional curvatures of \(\widetilde{M}\) lie between two negative constants. The Brownian motion starting at a point \(p\) in \(\widetilde{M}\) converges almost surely at \(\widetilde{M}(\infty)\) and defines a hitting measure \(\nu_ p\) on \(\widetilde M(\infty)\). The family of measures \(\{\nu_ p: p\in \widetilde{M}\}\) defines the harmonic measure class \(\nu^*\) on \(\widetilde{M}(\infty)\), and the bounded harmonic extension \(h: \widetilde{M}\to \mathbb{R}\) of a continuous function \(f: \widetilde{M}(\infty) \to \mathbb{R}\) is given by the formula (1) \(h(p) = \int_{\widetilde{M}(\infty)} f(x)d\nu_ p(x)\) for all \(p \in \widetilde{M}\). If \(f: \widetilde{M}(\infty) \to \mathbb{R}\) is a bounded \(\nu^*\)-measurable function, then (1) also defines a bounded harmonic function \(h: \widetilde{M} \to \mathbb{R}\). The first main result of this paper says that all bounded harmonic functions \(h\) on \(\widetilde{M}\) arise in this fashion. The first main result is a corollary of a more general result involving random walks defined by a probability measure \(\mu\) on a group \(\Gamma\), a construction introduced by T. Lyons and D. Sullivan. A function \(h: \Gamma \to \mathbb{R}\) is said to be \(\mu\)-harmonic if \(h(\gamma) = \sum_{\gamma \in \Gamma} h(\gamma)\mu(\gamma')\). The random walk on \(\Gamma\) defined by \(\mu\) also defines a hitting probability measure \(\nu\) on \(\widetilde{M}(\infty)\) such that \(\nu = \sum_{\gamma \in \Gamma} \mu(\gamma) \gamma\nu\). If \(f: \widetilde{M}(\infty) \to \mathbb{R}\) is continuous, then the \(\mu\)-harmonic extension \(h: \Gamma \to \mathbb{R}\) is given by (2) \(h(\gamma) = \sum_{\widetilde{M}(\infty)} f(\gamma x) d\nu(x)\). The second main result of the article, which implies the first, says that (2) defines an isomorphism between bounded \(\nu\)-measurable functions \(f: \widetilde M(\infty) \to \mathbb{R}\) and \(\mu\)-harmonic functions \(h:\Gamma \to \mathbb{R}\).