The Evolution to Equilibrium of Solutions to Nonlinear Fokker-Planck Equation

One proves the H-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation (1) u(t) - Delta beta(u) + div(E(x)b(u)u) = 0, t >= 0, x is an element of R-d, and-under appropriate hypotheses on beta, E and b-the convergence in L-loc(1) (R-d), L-1(R-d), respectively, for some t(n) -> infinity of the solution u(t(n)) to an equilibrium state of the equation for a large set of nonnegative initial data in L-1. These results are new in the literature on nonlinear Fokker-Planck equations arising in the mean field theory and are also relevant to the theory of stochastic differential equations. As a matter of fact, by the above convergence result, it follows that the solution to the McKean-Vlasov stochastic differential equation corresponding to (1), which is a nonlinear distorted Brownian motion, has this equilibrium state as its unique invariant measure.