Distances between Stationary Distributions of Diffusions and Solvability of Nonlinear Fokker--Planck--Kolmogorov Equations

Abstract

This paper is concerned with investigation of stationary distributions of diffusion processes. We obtain estimates for the Kantorovich, Prohorov, and total variation distances between stationary distributions of diffusions with different diffusion matrices and different drift coefficients. Applications are given to nonlinear stationary Fokker--Planck--Kolmogorov equations, for which new conditions for the existence and uniqueness of probability solutions are found; moreover, these conditions are optimal in a sense.