Asymptotic estimates on the time derivative of $\Phi$-entropy on Riemannian manifolds

TL;DR

This paper derives asymptotic estimates for the time derivative of $\

Contribution

It provides new asymptotic bounds for the $\\Phi$-entropy's time derivative on Riemannian manifolds, including hyperbolic space and Heisenberg group.

Findings

Time derivative of $\\Phi$-entropy is non-increasing in hyperbolic space and Heisenberg group.

Sharp asymptotic bounds for the $\\Phi$-entropy derivative in the Heisenberg group.

Estimates are expressed in terms of Bakry-Emery curvature lower bounds.

Abstract

In this note, we obtain the asymptotic estimate for the time derivative of the $\Phi$-entropy in terms of the lower bound on the Bakry-Emery $\Gamma_2$ curvature. In the cases of Hyperbolic space and Heisenberg group, we show that the time derivative of the $\Phi$-entropy is non-increasing, and we also get sharp asymptotic bound for the time derivative of the entropy in the Heisenberg group.