In the unit open ball of the complex plane, we can describe harmonic functions from the boundary information by the Poisson formula. For example, a bounded harmonic measure is represented as the integration of a bounded measurable function on the unit circle with the Poisson kernel. In this talk, we consider such boundary value problems on manifolds with negative curvature. We recall some properties of Brownian motions on manifolds with negative curvature, verify the boundary problems in terms of Brownian motions, and how they are proved. And then we discuss our ongoing project on Teichmüller space with the Weil-Petersson metric. This is joint work with Inhyeok Choi (KIAS).