Lyapunov Exponents for a Stochastic Analogue of the Geodesic Flow

Abstract

New invariants for a Riemannian manifold are defined as Lyapunov exponents of a stochastic analogue of the geodesic flow. A lower bound is given reminiscent of corresponding results for the geodesic flow, and an upper bound is given for surfaces of positive curvature. For surfaces of constant negative curvature a direct method via the Doob $h$-transform is used to determine the full Lyapunov structure relating the stable manifolds to the horocycles.