Weighted $L^1$-semigroup approach for nonlinear Fokker--Planck equations and generalized Ornstein--Uhlenbeck processes

For the nonlinear Fokker–Planck equation $${\mathrm{\partial }}_{t}u=\mathrm{\Delta }\beta (u)-\mathrm{\nabla }\mathrm{\Phi }\cdot \mathrm{\nabla }\beta (u)-di{v}_{\varrho }(D(x)b(u)u),(t,x)\in (0,\mathrm{\infty })\times {\mathbb{R}}^d,$$ where $\varrho =\exp (-\mathrm{\Phi })$ is the density of a finite Borel measure and $\mathrm{\nabla }\mathrm{\Phi }$ is unbounded, we construct mild solutions with bounded initial data via the Crandall–Liggett semigroup approach in the weighted space $L^1({\mathbb{R}}^d,\mathbb{R};\varrho dx)$. 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.