Resolvent approaches to elliptic regularity in stationary Fokker-Planck equations

TL;DR

This paper establishes local regularity of solutions to stationary Fokker-Planck equations under relaxed coefficient assumptions, showing that bounded solutions are actually in the Sobolev space $H^{1,2}_{loc}$ using resolvent methods.

Contribution

It introduces a resolvent-based approach to prove elliptic regularity for Fokker-Planck equations with minimal coefficient regularity assumptions.

Findings

Bounded solutions belong to $H^{1,2}_{loc}$.

Construction of a sub-Markovian resolvent for the elliptic operator.

Solutions can be approximated as limits of resolvent images.

Abstract

This paper investigates the local regularity of solutions to stationary Fokker-Planck equations on an open set $U \subset \mathbb{R}^d$ with $d \geq 2$. 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. $\mathbf{G} \in L^2_{loc}(U, \mathbb{R}^d)$, the symmetric diffusion matrix $A = (a_{ij})_{1 \leq i,j \leq d}$ is assumed to be locally uniformly strictly elliptic and bounded, with coefficients satisfying $a_{ij} \in VMO_{loc}(U)$ for all $1 \leq i,j \leq d$ and ${\rm div}A \in L^2_{loc}(U, \mathbb{R}^d)$. Our main result shows that any locally bounded function $h \in L^\infty_{loc}(U)$ satisfying the stationary Fokker-Planck equation $L^*(h\, dx) = 0$ must in fact belong to the local Sobolev space $H^{1,2}_{loc}(U)$. The proof is based on the construction of a sub-Markovian resolvent associated with the principal elliptic operator $L^A := {\rm trace}(A \nabla^2)$, 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.