Geodesic Flows on Compact Spaces · Question 1

Let $\tilde{M}$ be contractible negatively curved Riemannian manifold, and $\Gamma \subset \operatorname{Isom}(\tilde{M})$ be a discrete group of isometries which acts cocompactly.

• Consider a nearest-neighbor random walk on $\Gamma$, which produces an exit measure on the boundary $\partial\Gamma$. Is this measure a Gibbs state?

• If $\tilde{M}$ is nonpositively curved, what can be said about the measures induced on the boundary by Brownian motion? Is it a Gibbs state? What else can be said about it? (In the negative curvature case, this is known by //////)