Kinetic Theory with Fluctuations: Strong Well-Posedness of the Vlasov-Fokker-Planck-Dean-Kawasaki System

TL;DR

This paper proves the strong well-posedness of the Vlasov-Fokker-Planck-Dean-Kawasaki equation, modeling fluctuating particle systems with complex interactions and stochastic noise, using advanced analytical techniques.

Contribution

It establishes the well-posedness of a complex kinetic equation with correlated noise, combining kinetic semigroup estimates and renormalized solutions.

Findings

Proves strong well-posedness of the VFPDK equation.

Handles irregularities from square-root noise coefficients.

Develops a novel analytical framework for kinetic equations with fluctuations.

Abstract

The strong well-posedness of the Vlasov-Fokker-Planck-Dean-Kawasaki (VFPDK) equation with correlated noise is established. This equation can be interpreted as the fluctuating mean-field limit of second-order Newtonian particle systems, combining kinetic theory with stochastic fluctuations. It includes bounded nonlocal interactions and a diffusion coefficient exhibiting a square-root structure. 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.