Self-similar solutions of R\'enyi's entropy and the concavity of its entropy power

TL;DR

This paper explores self-similar solutions that maximize Reny's entropy, revealing differences from Shannon entropy solutions and analyzing the concavity of entropy power in Euclidean and Riemannian settings.

Contribution

It characterizes self-similar solutions for Reny's entropy maximization and examines the entropy power's concavity using novel methods.

Findings

Self-similar solutions differ from porous medium equation solutions with Dirac sources.

The entropy power exhibits concavity properties in Euclidean space.

Two methods are developed to analyze entropy power on Riemannian manifolds.

Abstract

We study the class of self-similar probability density functions with finite mean and variance which maximize R\'{e}nyi's entropy. The investigation is restricted in the Schwartz space $S(\mathbb{R}^d)$ and in the space of $l$-differentiable compactly supported functions $C_c^l(\mathbb{R}^d)$. Interestingly the solutions of this optimization problem do not coincide with the solutions of the usual porous medium equation with a Dirac point source, as it occurs in the optimization of Shannon's entropy. We also study the concavity of the entropy power in $\mathbb{R}^d$ with respect to time using two different methods. The first one takes advantage of the solutions determined earlier while the second one is based on a setting that could be used for Riemannian manifolds.