On an open Riemannian manifold of negative curvature, the L^2-spectrum and the positive spectrum of the Laplace-Beltrami operator are closely related by a theorem of Sullivan. Positive spectrum are used to investigate the behavior of Green function at the bottom of the L^2-spectrum. We show that Martin boundary at the bottom of the spectrum coincides with the geometric boundary, and we will explain how ergodic theory of the geodesic flow on a closed Riemannian manifold M of negative curvature can be used to give an asymptotics of the heat kernel on the universal cover of M. This is a joint work with François Ledrappier.