On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$

• Date: 2026
• Main Authors: Barbu, Viorel, Röckner, Michael
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2026
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/8286
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.9 We prove that a nonlinear semigroup $S(t)$ in $L^1(\mathbb{R}^d),$ $d\ge 3,$ associated with the nonlinear Fokker–Planck equation $u_t-\Delta\beta(u)+{\rm div }(Db(u)u)=0,$ $u(0)=u_0,$ in $(0,\infty)\times\mathbb{R}^d,$ with suitable conditions imposed on the coefficients $\beta\colon \mathbb{R}\to\mathbb{R},$ $-D\colon \mathbb{R}^d\to\mathbb{R}^d,$ and $b\colon \mathbb{R}\to\mathbb{R},$ is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws for the solutions to the corresponding McKean–Vlasov stochastic differential equation. This completes the results established in [V. Barbu, M. Röckner, The invariance principle for nonlinear Fokker–Planck equations, J. Different. Equat., 315, 200–221 (2022)] on the nature of the corresponding omega-set $\omega(u_0)$ for $S(t)$ in the case where the flow $S(t)$ in $L^1(\mathbb{R}^d)$ does not have a fixed point and, hence, the corresponding stationary Fokker–Planck equation has no solutions.