The evolution to equilibrium of solutions to nonlinear Fokker-Planck equation

TL;DR

This paper proves the $H$-theorem and convergence to equilibrium for solutions of a nonlinear Fokker-Planck equation, establishing new results relevant to mean field theory and stochastic differential equations.

Contribution

It introduces new convergence results for nonlinear Fokker-Planck equations with applications to stochastic differential equations and mean field theory.

Findings

Proves the $H$-theorem for mild solutions.

Establishes convergence to equilibrium in various $L^p$ spaces.

Shows the solution's equilibrium state as the unique invariant measure.

Abstract

One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$ u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, \ t\geq0, \ x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and $b$ the convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, for some $t_n\to\infty$ 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. Keywords: Fokker-Planck equation, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, McKean-Vlasov stochastic differential equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject Classification: 35B40, 35Q84, 60H10.