Regularity of solutions to Fokker–Planck–Kolmogorov equations

V. I. Bogachev^ab, S. V. Shaposhnikov<sup>ab

a</sup> Lomonosov Moscow State University
^b National Research University Higher School of Economics

Abstract: A survey of the regularity properties of solutions to Fokker-Planck-Kolmogorov equations of elliptic and parabolic type is presented. Conditions for the existence of the densities of solutions and some local properties of such densities, such as boundedness, continuity, Hölder continuity, and Sobolev continuity are discussed, as well as some global properties such as estimates on the whole space, high integrability, and membership in Sobolev classes in the whole space. New results on properties of solutions in the case of low regularity of coefficients are also presented.

Keywords: Fokker–Planck–Kolmogorov equation, Cauchy problem, elliptic equation, parabolic equation.

UDC: 517.5+519.2

Received: 05.11.2025

DOI: 10.4213/rm10293