Consider a C 8 closed connected Riemannian manifold pM, gq with negative curvature. The unit tangent bundle SM is foliated by the (weak) stable foliation W s of the geodesic flow. Let ∆ s be the leafwise Laplacian for W s and let X be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each ρ, the operator Lρ :" ∆ s`ρ X generates a diffusion for W s. We show that, as ρ Ñ´8, the unique stationary probability measure for the leafwise diffusion of Lρ converge to the normalized Liouville measure on SM .