The ergodicity of nonlinear Fokker-Planck flows in $L^1(\mathbb R^d)$

TL;DR

This paper proves that the nonlinear Fokker-Planck flows in $L^1(R^d)$ are mean ergodic under certain conditions, extending understanding of their long-term behavior and implications for related stochastic differential equations.

Contribution

It establishes the mean ergodicity of nonlinear Fokker-Planck semigroups in $L^1(R^d)$, including cases without fixed points, advancing the theory of nonlinear PDEs and stochastic processes.

Findings

Proves mean ergodicity of nonlinear Fokker-Planck flows in $L^1(R^d)$.

Shows implications for the ergodic behavior of solutions to McKean-Vlasov SDEs.

Completes previous results on omega-limit sets for flows without fixed points.

Abstract

One proves in this work that the nonlinear semigroup $S(t)$ in $L^1(\mathbb R^d)$, $d\geq 3$, associated with the nonlinear Fokker-Planck equation $u_t-\Delta\beta(u)+\text{div}(Db(u)u){=}0$, $u(0)=u_0$ in $(0,\infty)\times\mathbb R^d$, under suitable conditions on the coefficients $\beta:\mathbb R\to\mathbb R$, $D:\mathbb R^d\to\mathbb R^d$ and $b:\mathbb R\to\mathbb R$, is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws of the solutions to the corresponding McKean-Vlasov stochastic differential equation. This completes the results established in [7] on the nature of the corresponding omega-set $\omega(u_0)$ for $S(t)$ in the case where the flow $S(t)$ in $L^1(\mathbb R^d)$ has not a fixed point and so the corresponding stationary Fokker-Planck equation has no solutions.