Resolvent approaches to elliptic regularity in stationary Fokker-Planck equations

TL;DR

This work addresses local elliptic regularity for stationary Fokker-Planck equations with singular drifts by developing a resolvent-based approach. Under weak coefficient assumptions—drift $\mathbf{G}\in L^2_{loc}$ and diffusion matrix $A$ with $(a_{ij}+a_{ji})/2\in VMO_{loc}$ and ${\rm div}A\in L^2_{loc}$—the author proves that any locally bounded solution density $h$ satisfying $L^*(h\,dx)=0$ belongs to $H^{1,2}_{loc}$. The core method constructs a sub-Markovian resolvent for the principal part $L^A=\mathrm{trace}(A\nabla^2)$ and shows $h$ arises as the weak $H^{1,2}$-limit of $\alpha R_{\alpha}h$ as $\alpha\to\infty$, using energy-type estimates and mollification arguments. These regularity results enable uniqueness statements for invariant densities: any nonnegative (or bounded) solution to the stationary FP equation must be proportional to a reference density $\rho$, yielding uniqueness up to constants. The findings extend elliptic regularity theory to FP equations with rough coefficients and have implications for invariant measures of SDEs with singular coefficients.

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 diffusion matrix $A = (a_{ij})_{1 \leq i,j \leq d}$ is assumed to be locally uniformly strictly elliptic and bounded, with coefficients satisfying $\frac{a_{ij}+a_{ji}}{2} \in VMO_{loc}(U)$ 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 $L^A := \operatorname{trace}(A \nabla^2)$, 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.

Theorems & Definitions (10)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Lemma 3.1
• Theorem 3.2
• Lemma 3.3
• Theorem 3.4
• Theorem 4.1
• Example 4.2
• Example 4.3