On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ - Barbu, Viorel; Röckner, Michael (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.9We prove that a nonlinear semigroup $S(t)$ in $L^1(\mathbb{R}^d),$ $d\ge 3,$ associated with the nonlinear Fokker–Planck equation $u_t-Δβ(u)+{\rm div }(Db(u)u)=0,$ $u(0)=u_0,$ in $(0,∞)×\mathbb{R}^d,$ with suitable conditions imposed on the coefficients $β\colon \mathbb{R}→\mathbb{R},$ $-D\colon \mathbb{R}^d→\mathbb{R}^d,$ and $b\colon \mathbb{R}→\mathbb{R},$ 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 $ω(u_0)$ 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.