Weighted L¹-semigroup approach for nonlinear Fokker--Planck equations and generalized Ornstein--Uhlenbeck processes

For the nonlinear Fokker--Planck equation ∂ₜu = Δβ(u)-∇ Φ· ∇ β(u) - div_(D(x)b(u)u), (t,x) ∈ (0,∞)× Rᵈ, where = exp(-Φ) is the density of a finite Borel measure and ∇ Φ is unbounded, we construct mild solutions with bounded initial data via the Crandall--Liggett semigroup approach in the weighted space L¹(Rᵈ,R; dx). By the superposition principle, we lift these solutions to weak solutions to the corresponding McKean--Vlasov SDE, which can be considered a model for generalized nonlinear perturbed Ornstein--Uhlenbeck processes. Finally, for these solutions we prove the nonlinear Markov property in the sense of McKean.