Resolvent approaches to elliptic regularity in stationary Fokker–Planck equations

Abstract This paper investigates the local regularity of solutions to stationary Fokker–Planck equations on an open set U ⊂ R d $U\subset {\mathbb{R}}^{d}$ with d ≥ 2. A central objective is to relax the classical assumptions on the coefficients by focusing on the case where the drift vector field G is only assumed to be locally square-integrable, i.e. G ∈ L loc 2 ( U , R d ) $\mathbf{G}\in {L}_{\text{loc}}^{2}\left(U,{\mathbb{R}}^{d}\right)$ , the symmetric diffusion matrix A = ( a i j ) 1 ≤ i , j ≤ d $A={\left({a}_{ij}\right)}_{1\le i,j\le d}$ is assumed to be locally uniformly strictly elliptic and bounded, with coefficients satisfying a ij ∈ VMO loc ( U ) for all 1 ≤ i , j ≤ d and div A ∈ L loc 2 ( U , R d ) $\text{div}A\in {L}_{\text{loc}}^{2}\left(U,{\mathbb{R}}^{d}\right)$ . Our main result shows that any locally bounded function h ∈ L loc ∞ ( U ) $h\in {L}_{\text{loc}}^{\infty }\left(U\right)$ satisfying the stationary Fokker–Planck equation L *( h d x ) = 0 must in fact belong to the local Sobolev space