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We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker–Planck equations, and one-dimensional functional inequalities of the type of Poincaré and Wirtinger, with weight, for probability densities in the form of one-dimensional Barenblatt solutions of the porous medium equation. It it also shown that Hardy's classical one-dimensional inequality can be obtained resorting to the differential expression of the Cauchy-type density, as given by a related Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

We investigate the relaxation to equilibrium of the solution of a class of one-dimensional linear Fokker–Planck type equations that have been recently considered in connection with the study of addiction phenomena in a system of individuals. The steady states of these equations belong to the class of generalized Gamma densities. As a by-product of the relaxation analysis, we prove new weighted Poincaré and logarithmic Sobolev type inequalities for this class of densities. [ABSTRACT FROM AUTHOR]

In a recent paper we introduced the concept of fractional score, a generalization of the linear score function, well-known in theoretical statistics. As the Gaussian density is closely related to the linear score, the fractional score function allows to identify Lévy stable laws as the (unique) probability densities for which the score of a random variable X is proportional to - X . We use this analogy to extend to stable laws the classical Poincaré inequality for Gaussian densities. Application of this inequality allows to obtain bounds on moments of stable laws, and a sharp one-dimensional version of Hardy–Poincaré inequality. [ABSTRACT FROM AUTHOR]

Generalizations of the linear score function, a well-known concept in theoretical statistics, are introduced. As the Gaussian density and the classical Fisher information are closely related to the linear score, nonlinear (respectively fractional) score functions allow to identify generalized Gaussian densities (respectively Lévy stable laws) as the (unique) probability densities for which the score of a random variable X is proportional to $$-X$$ . In all cases, it is shown that the variance of the relative to the generalized Gaussian (respectively Lévy) score provides an upper bound for $$L^1$$ -distance from the generalized Gaussian density (respectively Lévy stable laws). Connections with nonlinear and fractional Fokker-Planck type equations are introduced and discussed. [ABSTRACT FROM AUTHOR]