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 ( R d ) and in the space of l-differentiable compactly supported functions C c l ( 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 occurs in the optimization of Shannon’s entropy. We also study the concavity of the entropy power in 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.
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机译：我们研究了具有有限均值和方差的自相似概率密度函数，该函数可以最大化Rényi的熵。该研究仅限于Schwartz空间S（R d）和l微分紧支持函数C c l（R d）的空间。有趣的是，这种优化问题的解决方案与具有狄拉克点源的通常的多孔介质方程组的解决方案不一致，这与香农熵的优化一样。我们还使用两种不同的方法研究了R d中熵权相对于时间的凹度。第一个利用较早确定的解决方案，而第二个则基于可用于黎曼流形的设置。
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