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In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain (0,T)×D×Rd, where D is either Td or Rd. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\href{arXiv:1902.04037v2}{Alb+21}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.

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.