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.
Categories: math-ph math.MP quant-ph