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

Abstract

For the nonlinear Fokker--Planck equation $${\partial }_{t}u=\Delta \beta (u)-\nabla \Phi \cdot \nabla \beta (u)-di{v}_{\varrho }(D(x)b(u)u),(t,x)\in (0,\infty )\times {\mathbb{R}}^d,$$ where $\varrho =\exp (-\Phi )$ is the density of a finite Borel measure and $\nabla \Phi$ 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.

18 pages