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

TL;DR

This paper develops a semigroup approach to solve nonlinear Fokker--Planck equations with unbounded drift, constructs weak solutions for associated stochastic processes, and proves their nonlinear Markov property.

Contribution

It introduces a weighted $L^1$-semigroup method for nonlinear Fokker--Planck equations with unbounded drifts, enabling the construction of weak solutions and analysis of related stochastic processes.

Findings

Constructed mild solutions in weighted $L^1$ space.

Lifted solutions to weak solutions of McKean--Vlasov SDEs.

Proved nonlinear Markov property for the solutions.

Abstract

For the nonlinear Fokker--Planck equation $$\partial_tu = \Delta\beta(u)-\nabla \Phi \cdot \nabla \beta(u) - div_{\varrho}\big(D(x)b(u)u\big),\quad (t,x) \in (0,\infty)\times \mathbb{R}^d,$$ 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.