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 $SM$.
Categories: math.DS math.PR