A family of stable diffusions

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