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 $U\subset {\mathbb{R}}^d$ with $d\ge 2$. A central objective is to relax the classical assumptions on the coefficients by focusing on the case where the drift vector field $\mathbf{G}$ is only assumed to be locally square-integrable, i.e. $\mathbf{G}\in L_{loc}^2(U,{\mathbb{R}}^d)$, the symmetric diffusion matrix $A=(a_{ij}{)}_{1\le i,j\le d}$ is assumed to be locally uniformly strictly elliptic and bounded, with coefficients satisfying $a_{ij}\in VM{O}_{loc}(U)$ for all $1\le i,j\le d$ and $\mathrm{div}A\in L_{loc}^2(U,{\mathbb{R}}^d)$. Our main result shows that any locally bounded function $h\in L_{loc}^{\infty }(U)$ satisfying the stationary Fokker-Planck equation $L^{\ast }(hdx)=0$ must in fact belong to the local Sobolev space $H_{loc}^{1,2}(U)$. The proof is based on the construction of a sub-Markovian resolvent associated with the principal elliptic operator $L^A:=\mathrm{trace}(A{\nabla }^2)$, combined with delicate energy-type inequalities. In particular, we show that the density $h$ can be realized as the weak $H^{1,2}$-limit of images of resolvent operators.

19 pages