Fokker--Planck equations and n--dimensional Poincar\'e inequalities for isotropic densities

TL;DR

This paper explores the relationship between the convergence to equilibrium in n-dimensional Fokker--Planck equations and weighted Poincaré inequalities, introducing new theoretical connections inspired by wealth distribution models.

Contribution

It establishes novel links between Fokker--Planck equations and Poincaré inequalities in multiple dimensions, extending one-dimensional insights to more complex systems.

Findings

Derived new n-dimensional Poincaré inequalities for isotropic densities

Connected Fokker--Planck dynamics with weighted inequalities

Provided theoretical framework for wealth distribution evolution

Abstract

We consider new connections between the problem of trend to equilibrium for the n-dimensional Fokker--Planck equation of statistical physics, and weighted Poincar\'e 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.