A strengthened entropy power inequality for log-concave densities

We show that Shannon's entropy--power inequality admits a strengthened version in the case in which the densities are log-concave. In such a case, in fact, one can extend the Blachman--Stam argument to obtain a sharp inequality for the second derivative of Shannon's entropy functional with respect to the heat semigroup.Comment: 21 page

The concavity of R\`enyi entropy power

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

Boltzmann legacy and wealth distribution

We briefly review results on nonlinear kinetic equation of Boltzmann type which describe the evolution of wealth in a simple agents market. The mathematical structure of the underlying kinetic equations allows to use well-known techniques of wide use in kinetic theory of rarefied gases to obtain information on the process of relaxation to a stationary profile, as well as to identify simple interaction rules which are responsible of the formation of Pareto tails

Heat equation and convolution inequalities

It is known that many classical inequalities linked to convolutions can be obtained by looking at the monotonicity in time of convolutions of powers of solutions to the heat equation, provided that both the exponents and the coefficients of diffusions are suitably chosen and related. This idea can be applied to give an alternative proof of the sharp form of the classical Young's inequality and its converse, to Brascamp--Lieb type inequalities, Babenko's inequality and Pr\'ekopa--Leindler inequality as well as the Shannon's entropy power inequality. This note aims in presenting new proofs of these results, in the spirit of the original arguments introduced by Stam to prove the entropy power inequality.Comment: 29 page

The fractional Fisher information and the central limit theorem for stable laws

A new information-theoretic approach to the central limit theorem for stable laws is presented. The main novelty is the concept of relative fractional Fisher information, which shares most of the properties of the classical one, included Blachman-Stam type inequalities. These inequalities relate the fractional Fisher information of the sum of $n$ independent random variables to the information contained in sums over subsets containing $n-1$ of the random variables. As a consequence, a simple proof of the monotonicity of the relative fractional Fisher information in central limit theorems for stable law is obtained, together with an explicit decay rate

An information-theoretic proof of Nash's inequality

We show that an information-theoretic property of Shannon's entropy power, known as concavity of entropy power, can be fruitfully employed to prove inequalities in sharp form. In particular, the concavity of entropy power implies the logarithmic Sobolev inequality, and Nash's inequality with the sharp constant

The information-theoretic meaning of Gagliardo--Nirenberg type inequalities

Gagliardo--Nirenberg inequalities are interpolation inequalities which were proved independently by Gagliardo and Nirenberg in the late fifties. In recent years, their connections with theoretic aspects of information theory and nonlinear diffusion equations allowed to obtain some of them in optimal form, by recovering both the sharp constants and the explicit form of the optimizers. In this note, at the light of these recent researches, we review the main connections between Shannon-type entropies, diffusion equations and a class of these inequalities

Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays

In this paper, we consider functionals based on moments and non-linear entropies which have a linear growth in time in case of source-type so-lutions to the fast diffusion or porous medium equations, that are also known as Barenblatt solutions. As functions of time, these functionals have convexity properties for generic solutions, so that their asymptotic slopes are extremal for Barenblatt profiles. The method relies on scaling properties of the evo-lution equations and provides a simple and direct proof of sharp Gagliardo-Nirenberg-Sobolev inequalities in scale invariant form. The method also gives refined estimates of the growth of the second moment and, as a consequence, establishes the monotonicity of the delay corresponding to the best matching Barenblatt solution compared to the Barenblatt solution with same initial sec-ond moment. Here the notion of best matching is defined in terms of a relative entropy

Explicit equilibria in a kinetic model of gambling

We introduce and discuss a nonlinear kinetic equation of Boltzmann type which describes the evolution of wealth in a pure gambling process, where the entire sum of wealths of two agents is up for gambling, and randomly shared between the agents. For this equation the analytical form of the steady states is found for various realizations of the random fraction of the sum which is shared to the agents. Among others, Gibbs distribution appears as steady state in case of a uniformly distributed random fraction, while Gamma distribution appears for a random fraction which is Beta distributed. The case in which the gambling game is only conservative-in-the-mean is shown to lead to an explicit heavy tailed distribution