  In this paper, we investigate weak solutions and Perron-Wiener-Brelot
solutions to the linear kinetic Fokker-Planck equation in bounded domains. We
establish the existence of weak solutions by applying the Lions-Lax-Milgram
theorem and the vanishing viscosity method in product domains. Additionally, we
demonstrate the regularity of weak solutions and establish a strong maximum
principle. Furthermore, we construct a Perron solution and provide examples of
barriers in arbitrary bounded domains. Our findings are based on recent
advancements in the theory of kinetic Fokker-Planck equations with rough
coefficients, particularly focusing on the characterization of a weaker notion
of trace and the convolution-translation.
