In this paper, we show that, for a solution to the stationary Fokker–Planck equation with general coefficients, defined as a measure with an $L^{2}$ -density, this density not only exhibits $H^{1,2}$ -regularity but also Hölder continuity. To achieve this, we first construct a reference measure $\mu =\rho dx$ by utilizing existence and elliptic regularity results, ensuring that the given divergence-type operator corresponds to a sectorial Dirichlet form. By employing elliptic regularity results for homogeneous boundary value problems in both divergence and non-divergence type equations, we demonstrate that the image of the resolvent operator associated with the sectorial Dirichlet form has $H^{2,2}$ -regularity. Furthermore, through calculations based on the Dirichlet form and the $H^{2,2}$ -regularity of the resolvent operator, we prove that the density of the solution measure for the stationary Fokker–Planck equation is, indeed, the weak limit of $H^{1,2}$ -functions defined via the resolvent operator. Our results highlight the central role of Dirichlet form theory and resolvent approximations in establishing the regularity of solutions to stationary Fokker–Planck equations with general coefficients.
