Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains

TL;DR

This paper develops a comprehensive theory for weak and Perron-Wiener-Brelot solutions to the stationary Kramers-Fokker-Planck equation in bounded domains, extending existence results and unifying solution concepts.

Contribution

It introduces the first unified framework for weak and Perron solutions in bounded domains for the stationary Kramers-Fokker-Planck equation, using recent advances in rough coefficient theory.

Findings

Existence of weak solutions in product domains.

Equivalence of weak and Perron solutions in well-behaved domains.

Foundation for Perron-Wiener-Brelot solutions in arbitrary bounded domains.

Abstract

In this paper, we investigate weak solutions and Perron-Wiener-Brelot solutions to the linear stationary Kramers-Fokker-Planck equation in bounded domains. We establish the existence of weak solutions in product domains by applying the Lions-Lax-Milgram theorem and the vanishing viscosity method. Furthermore, we show that these solutions coincide in well-behaved domains. Building on the existence of weak solutions in product domains, we develop the foundational theory of Perron-Wiener-Brelot solutions in arbitrary bounded domains. Our results rely on recent advancements in the theory of kinetic Fokker-Planck equations with rough coefficients.