Convergence in variation of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary measures

Abstract

We study convergence in variation of probability solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary solutions. We obtain sufficient conditions for the exponential convergence of solutions to the stationary solution in case of coefficients that can have an arbitrary growth at infinity and depend on the solutions through convolutions with unbounded discontinuous kernels. In addition, we study a more difficult case where the nonlinear equation has several stationary solutions and convergence to a stationary solution depends on initial data. Finally, we obtain sufficient conditions for solvability of nonlinear Fokker–Planck–Kolmogorov equations.