A family of stable diffusions

Abstract

Consider a $C^{\infty}$ closed connected Riemannian manifold $(M, g)$ with negative curvature. The unit tangent bundle $SM$ is foliated by the (weak) stable foliation $\mathcal{W}^s$ of the geodesic flow. Let $Δ^s$ be the leafwise Laplacian for $\mathcal{W}^s$ and let $\overline{X}$ be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each $λ$, the operator $\mathcal{L}_λ:=Δ^s+λ\overline{X}$ generates a diffusion for $\mathcal{W}^s$. We show that, as $λ\to -\infty$, the unique stationary probability measure for the leafwise diffusion of $\mathcal{L}_λ$ converges to the normalized Lebesgue measure on $SM$.