Fokker--Planck equations and n--dimensional Poincaré inequalities for isotropic densities

We consider new connections between the problem of trend to equilibrium for the n-dimensional Fokker–Planck equation of statistical physics, and weighted Poincaré inequality. To this aim we consider a class of n-dimensional Fokker–Planck equations with variable isotropic coefficient of diffusion and drift, inspired by the analogous one-dimensional Fokker–Planck equation appearing when studying the evolution of wealth distribution.