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A Cordes framework for stationary Fokker--Planck--Kolmogorov equations

Timo Sprekeler

TL;DR

This work develops a Cordes-type framework for stationary FPK equations under periodic and Dirichlet boundary conditions by renormalizing the differential operator to be near the Laplacian in the Campanato sense. It provides existence and uniqueness of $L^2$ very weak solutions via a Lax–Milgram reduction and introduces a constructive FE approach that avoids $H^2$-conforming spaces by using a vector potential $\rho$, yielding explicit formulas $u=C\gamma\tilde{u}$ (periodic) or $u=\gamma\tilde{u}$ (Dirichlet) with $\tilde{u}=-\Delta\psi= -\nabla\cdot\rho$. The framework extends to both periodic and Dirichlet settings, delivering stable numerical schemes with provable error control and practical steps for computing $\rho_h$ and the solution $u_h$. The results support robust FE approximations for FPK-type equations with discontinuous coefficients, relevant in homogenization and stochastic settings, where standard elliptic theory may fail. The approach provides a unified, constructive route from Cordes-type conditions to implementable numerical methods in low-regularity contexts.

Abstract

We first review the Cordes condition for nondivergence-form differential operators through the lens of Campanato's theory of near operators. We then survey a recently proposed Cordes framework that guarantees the existence and uniqueness of $L^2$ solutions to stationary Fokker--Planck--Kolmogorov equations subject to periodic boundary conditions, and that allows for the construction of a simple finite element method for its numerical approximation. Finally, we propose a Cordes framework for stationary Fokker--Planck--Kolmogorov-type equations subject to a homogeneous Dirichlet boundary condition.

A Cordes framework for stationary Fokker--Planck--Kolmogorov equations

TL;DR

This work develops a Cordes-type framework for stationary FPK equations under periodic and Dirichlet boundary conditions by renormalizing the differential operator to be near the Laplacian in the Campanato sense. It provides existence and uniqueness of very weak solutions via a Lax–Milgram reduction and introduces a constructive FE approach that avoids -conforming spaces by using a vector potential , yielding explicit formulas (periodic) or (Dirichlet) with . The framework extends to both periodic and Dirichlet settings, delivering stable numerical schemes with provable error control and practical steps for computing and the solution . The results support robust FE approximations for FPK-type equations with discontinuous coefficients, relevant in homogenization and stochastic settings, where standard elliptic theory may fail. The approach provides a unified, constructive route from Cordes-type conditions to implementable numerical methods in low-regularity contexts.

Abstract

We first review the Cordes condition for nondivergence-form differential operators through the lens of Campanato's theory of near operators. We then survey a recently proposed Cordes framework that guarantees the existence and uniqueness of solutions to stationary Fokker--Planck--Kolmogorov equations subject to periodic boundary conditions, and that allows for the construction of a simple finite element method for its numerical approximation. Finally, we propose a Cordes framework for stationary Fokker--Planck--Kolmogorov-type equations subject to a homogeneous Dirichlet boundary condition.
Paper Structure (28 sections, 12 theorems, 96 equations)

This paper contains 28 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Cordes condition and Campanato nearness
  3. The Cordes condition and its geometric interpretation
  4. Bijectivity of \ref{['-A:D2']} under Cordes condition via Campanato nearness
  5. Cordes-type condition for operators with lower-order terms
  6. Beyond
  7. A Cordes framework for FPK equations: analysis
  8. Cordes-type condition
  9. Nearness of $L$ to $-\Delta$ after renormalization
  10. Construction of very weak solutions to the FPK problem
  11. The renormalized FPK problem
  12. The original FPK problem
  13. A Cordes framework for FPK equations: FE approximation
  14. Relating the operator $\tilde{L}$ to $-\mathrm{div}$ after renormalization
  15. The function $\rho$
  16. ...and 13 more sections

Key Result

Theorem 2.1

Let $L_1,L_2:X \rightarrow Y$ be two maps from a nonempty set $X$ to a real Banach space $(Y,\|\cdot\|)$. Suppose that $L_2$ is bijective and that $L_1$ is near $L_2$, that is, there exist constants $\alpha > 0$ and $K\in [0,1)$ such that Then, $L_1$ is bijective, i.e., for any $f\in Y$ there exists a unique solution $u\in X$ to Further, for this solution $u$ we have the bound

Theorems & Definitions (12)

  • Theorem 2.1: nearness of operators Cam94
  • Theorem 2.2: bijectivity of $(-A:D^2)$ under the Cordes condition Tal65SS13
  • Theorem 2.3: bijectivity of $(-A:D^2- b\cdot \nabla + c)$ in a Cordes setting SS14
  • Lemma 3.1: well-posedness and solution structure of the renormalized FPK problem SSZ25
  • Theorem 3.1: well-posedness and solution structure of the FPK problem SSZ25
  • Theorem 4.1: another solution formula for the FPK problem SSZ25
  • Theorem 4.2: realization of Step 1 SSZ25
  • Theorem 5.1: bijectivity of $L$
  • Lemma 5.1: well-posedness and solution structure of the renormalized problem
  • Theorem 5.2: well-posedness and solution structure of the original problem
  • ...and 2 more