Martin boundary and local limit theorem of Brownian motion on negatively-curved manifolds

Let $p(t,x,y)$ be the heat kernel on the universal cover $\widetilde{M}$ of a compact Riemannian manifold of negative curvature. We show that $$C(x,y) = \lim_{t \to \infty} e^{\lambda_0 t} t^{3/2} p(t,x,y) $$ is a positive function depending only on $x,y \in \widetilde{M}$, where $\lambda_0$ is the bottom of the spectrum. The function $C(x,y)$ can be described in terms of a Patterson-Sullivan density on $\partial \widetilde{M}$. We also show that $\lambda_0$-Martin boundary of $\widetilde{M}$ coincides with its topological boundary. We will explain how Margulis argument for counting geodesics as well as a uniform version of Dolgopyat's rapid-mixing of the geodesic flow are used to prove the results. This is a joint work with François Ledrappier.

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