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Local elliptic regularity for solutions to stationary Fokker-Planck equations via Dirichlet forms and resolvents

Haesung Lee

TL;DR

The paper addresses local elliptic regularity for stationary Fokker-Planck equations with general coefficients, proving that the density of an infinitesimally invariant measure gains local Sobolev regularity and Hölder continuity. It employs a Dirichlet-form framework, constructing a reference measure $\mu=\rho dx$ so that the divergence-type operator becomes part of a sectorial Dirichlet form, and analyzes the associated resolvent to obtain $H^{2,2}$-regularity. The main result shows that the density $h$ lies in $H^{1,p\wedge q}_{loc}(U)$ (and Hölder when $q>d$), and that $h$ is the weak $H^{1,2}_{loc}$-limit of resolvent approximations, enabling robust regularity conclusions for general coefficients. These findings bolster the deterministic understanding of invariant measures for SDEs, inform numerical schemes, and showcase the central role of Dirichlet-form theory in analyzing stationary FP equations with rough data.

Abstract

In this paper, we show that, for a solution to the stationary Fokker-Planck equation with general coefficients, defined as a measure with an $L^2$-density, this density not only exhibits $H^{1,2}$-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 $H^{2,2}$-regularity. Furthermore, through calculations based on the Dirichlet form and the $H^{2,2}$-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 $H^{1,2}$-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.

Local elliptic regularity for solutions to stationary Fokker-Planck equations via Dirichlet forms and resolvents

TL;DR

The paper addresses local elliptic regularity for stationary Fokker-Planck equations with general coefficients, proving that the density of an infinitesimally invariant measure gains local Sobolev regularity and Hölder continuity. It employs a Dirichlet-form framework, constructing a reference measure so that the divergence-type operator becomes part of a sectorial Dirichlet form, and analyzes the associated resolvent to obtain -regularity. The main result shows that the density lies in (and Hölder when ), and that is the weak -limit of resolvent approximations, enabling robust regularity conclusions for general coefficients. These findings bolster the deterministic understanding of invariant measures for SDEs, inform numerical schemes, and showcase the central role of Dirichlet-form theory in analyzing stationary FP equations with rough data.

Abstract

In this paper, we show that, for a solution to the stationary Fokker-Planck equation with general coefficients, defined as a measure with an -density, this density not only exhibits -regularity but also Hölder continuity. To achieve this, we first construct a reference measure 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 -regularity. Furthermore, through calculations based on the Dirichlet form and the -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 -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.

Paper Structure

This paper contains 6 sections, 22 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Dirichlet forms approach
  4. Main results
  5. Auxiliary results
  6. Conclusions and discussion

Key Result

Theorem 1.1

Let $U$ be a (possibly unbounded) open subset of $\mathbb{R}^d$ with $d \geq 3$. Let $\mathbf{H} \in L^p_{loc}(U, \mathbb{R}^d)$ with $p \in (d, \infty)$, and let $A = (a_{ij})_{1 \leq i,j \leq d}$ be a (possibly non-symmetric) matrix of functions, where $a_{ij} \in VMO_{loc}(U)$ for all $1 \leq i,j As in Proposition convdivnon, consider a partial differential operator $(\mathcal{L}, C_0^{\infty}(

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Proposition 3.5
  • ...and 17 more