A restricted superposition principle for (non-)linear Fokker-Planck-Kolmogorov equations on Hilbert spaces

TL;DR

This paper extends the superposition principle to certain solutions of Fokker-Planck-Kolmogorov equations on infinite-dimensional Hilbert spaces and adapts it to nonlinear cases, advancing the mathematical understanding of these equations.

Contribution

It introduces a restricted superposition principle applicable to a subclass of solutions on Hilbert spaces and extends it to nonlinear equations, which was not previously established.

Findings

Superposition principle proven for a subclass of solutions

Extension of the principle to nonlinear Fokker-Planck-Kolmogorov equations

Applicable on separable infinite-dimensional Hilbert spaces

Abstract

We prove a version of the Ambrosio-Figalli-Trevisan superposition principle for a restricted subclass of solutions to the Fokker-Planck-Kolmogorov equation, that is valid on separable infinite-dimensional Hilbert spaces. Furthermore, we transfer this restricted superposition principle into a nonlinear setting.