The concavity of R\`enyi entropy power

TL;DR

This paper extends the concept of entropy power to the p-th R'enyi entropy, demonstrating its concavity over time for solutions to nonlinear heat equations, generalizing Costa's inequality for Shannon entropy.

Contribution

It introduces a R'enyi entropy power that behaves concavely along solutions to nonlinear heat equations, extending classical entropy power inequalities.

Findings

R'enyi entropy power is concave over time for solutions to nonlinear heat equations.

Linear behavior of R'enyi entropy power for Barenblatt solutions.

Extension of Costa's concavity inequality to R'enyi entropies.

Abstract

We associate to the p-th R\'enyi entropy a definition of entropy power, which is the natural extension of Shannon's entropy power and exhibits a nice behaviour along solutions to the p-nonlinear heat equation in $R^n$. We show that the R\'enyi entropy power of general probability densities solving such equations is always a concave function of time, whereas it has a linear behaviour in correspondence to the Barenblatt source-type solutions. We then shown that the p-th R\'enyi entropy power of a probability density which solves the nonlinear diffusion of order p, is a concave function of time. This result extends Costa's concavity inequality for Shannon's entropy power to R\'enyi entropies.