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 $\hat{I}{”}^s$ 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 $\hat{I}»$, the operator ${\mathcal{L}}_{\hat{I}}»:=\hat{I}{”}^s+\hat{I}»\overline{X}$ generates a diffusion for ${\mathcal{W}}^s$. We show that, as $\hat{I}»\to -\infty$, the unique stationary probability measure for the leafwise diffusion of ${\mathcal{L}}_{\hat{I}}»$ converges to the normalized Lebesgue measure on $SM$.

The proof of Proposition 3.1 was incomplete. We correct it in this version