# On a class of Fokker–Planck equations with subcritical confinement

### Giuseppe Toscani

Università degli Studi di Pavia, Italy### Mattia Zanella

Università degli Studi di Pavia, Italy

## Abstract

We study the relaxation to equilibrium for a class of linear one-dimensional Fokker–Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker–Planck equations is that, for any given probability density $e(x)$, the diffusion coefficient can be built to have $e(x)$ as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density $e(x)$, a polynomial rate of convergence to equilibrium. Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.

## Cite this article

Giuseppe Toscani, Mattia Zanella, On a class of Fokker–Planck equations with subcritical confinement. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 3, pp. 471–497

DOI 10.4171/RLM/944