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

TL;DR

This paper reviews and extends the Cordes framework to ensure existence, uniqueness, and numerical approximation of solutions for stationary Fokker--Planck--Kolmogorov equations under various boundary conditions.

Contribution

It introduces a new Cordes framework for stationary Fokker--Planck--Kolmogorov equations with Dirichlet boundary conditions, expanding previous periodic boundary condition results.

Findings

Guarantees existence and uniqueness of solutions under the Cordes condition.

Provides a simple finite element method for numerical approximation.

Extends the framework to Dirichlet boundary conditions.

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.