Self-similar solutions of Rényi's entropy and the concavity of its entropy power

Abstract

We study the class of self-similar probability density functions with finite mean and variance which maximize Rényi'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.

18 pages, 2 figures

References in corpus (1)

• The concavity of Rènyi entropy power