The Kantorovich and variation distances between invariant measures of diffusions and nonlinear stationary Fokker–Planck–Kolmogorov equations

We obtain upper bounds for the total variation distance and the quadratic Kantorovich distance between stationary distributions of two diffusion processes with different drifts. More generally, our estimate holds for solutions to stationary Kolmogorov equations in the class of probability measures. This estimate is applied to nonlinear stationary Fokker–Planck–Kolmogorov equations for probability measures.