H-theorems for the Brownian motion on the hyperbolic plane

► Existence of an H-theorem for the Brownian motion with constant external drift on the hyperbolic plane. ► The associated entropy is of Rényi type. ► The parameter of this entropy depends on the value of the external drift. ► Both representations of the constant negative curvature space as the Poincaré upper half-plane and the disk are considered.