Kinetic Theory with Fluctuations: Well-Posedness of The Vlasov--Fokker--Planck--Dean--Kawasaki Equations

Abstract

We study Vlasov--Fokker--Planck--Dean--Kawasaki equations driven by correlated conservative noise. For regular noise coefficients and bounded nonlocal interactions, we establish probabilistically strong existence and uniqueness in a renormalized kinetic framework. For the square-root coefficient, we treat the non-interacting case and construct a probabilistically weak solution. Key challenges stem from the complexity of the kinetic operator and the irregularity introduced by the conservative noise with square-root-type coefficients. The proof relies on a novel combination of kinetic semigroup estimates and the framework of renormalized kinetic solutions.

The previous version contained an error in Section 6, which revealed an additional challenge in the whole-space setting. This issue has been addressed in the revised version, and the resulting conclusion differs from that of the previous version