Central limit theorem of Brownian motions in pinched negative curvature
In negatively curved Cartan-Hadamard manifolds, the Brownian distance has linear growth and the log of Green’s function along Brownian paths has linear decay with probability 1. In this talk, I will prove the central limit theorem for these random variables in negatively curved C-H manifolds with a non-uniform lattice. The Brownian motion gives rise to a Gibbs measure. I provide the ergodic properties of this Gibbs measure. I will conclude with a description of dynamical and probabilistic aspects of asymptotic harmonicity.