An H-theorem for the Brownian motion on the hyperbolic plane

TL;DR

This paper establishes an H-theorem for Brownian motion with drift on the hyperbolic plane, using Rényi entropy instead of Boltzmann entropy, linking entropy parameter to drift value.

Contribution

It introduces an H-theorem for hyperbolic Brownian motion with drift utilizing Rényi entropy, extending previous work on entropy evolution in stochastic processes.

Findings

Proves an H-theorem for hyperbolic Brownian motion with drift

Links Rényi entropy parameter to drift value

Extends entropy-based analysis to non-Euclidean geometries

Abstract

We prove an $H-$theorem for the Brownian motion on the hyperbolic plane with a drift, as studied by Comtet and Monthus; the entropy used here is not the Boltzmann entropy but the R\'enyi entropy, the parameter of which being related in a simple way to the value of the drift.