H-theorems for the Brownian motion on the hyperbolic plane

We study H-theorems associated with the Brownian motion with constant drift on the hyperbolic plane. Since this random process satisfies a linear Fokker–Planck equation, it is easy to show that, up to a proper scaling, its Shannon entropy is increasing over time. As a consequence, its distribution is converging to a maximum Shannon entropy distribution which is also shown to be related to the non-extensive statistics. In a second part, relying on a theorem by Shiino, we extend this result to the case of Tsallis entropies: we show that under a variance-like constraint, the Tsallis entropy of the Brownian motion on the hyperbolic plane is increasing provided that the non-extensivity parameter of this entropy is properly chosen in terms of the drift of the Brownian motion.