One-Dimensional Fokker–Planck Equations and Functional Inequalities for Heavy Tailed Densities

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We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker–Planck equations modeling socio-economic problems, and one-dimensional functional inequalities of the type of Poincaré, Wirtinger and logarithmic Sobolev, with weight, for probability densities with polynomial tails. As main examples, we consider inequalities satisfied by inverse Gamma densities, taking values on mathbb R_+, and Cauchy-type densities, taking values on mathbb R.