2024/12/19 by Haesung Lee, Lee, Haesung · 2 citations
Economics, Econometrics and Finance · Mathematics · #31C25 #35D30 #35Q84 #60J35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Primary: 35B65 #Probability (math.PR) #Secondary: 60J60 #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2412.14636
openalex publication_date 2024/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we show that, for a solution to the stationary Fokker-Planck equation with general coefficients, defined as a measure with an L2-density, this density not only exhibits H1,2-regularity but also Hölder continuity. To achieve this, we first construct a reference measure μ=ρdx 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 H2,2-regularity. Furthermore, through calculations based on the Dirichlet form and the H2,2-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 H1,2-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.