On the ergodicity of nonlinear Fokker–Planck flows in L^1(R^d) - Barbu, Viorel; Röckner, Michael (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.9 We prove that a nonlinear semigroup S(t) in L^1(R^d), d3, associated with the nonlinear Fokker–Planck equation u_t-Δβ(u)+div (Db(u)u)=0, u(0)=u_0, in (0,∞)×R^d, with suitable conditions imposed on the coefficients βR→R, -DR^d→R^d, and bR→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(R^d) does not have a fixed point and, hence, the corresponding stationary Fokker–Planck equation has no solutions.