Linearization and a superposition principle for deterministic and stochastic nonlinear Fokker-Planck-Kolmogorov equations

TL;DR

This paper establishes a superposition principle linking nonlinear and linearized Fokker-Planck-Kolmogorov equations, providing new insights into existence, uniqueness, and stochastic perturbations of measure-valued equations.

Contribution

It introduces a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations and their stochastic counterparts, connecting deterministic and stochastic measure equations through linearized forms.

Findings

Superposition principle for nonlinear Fokker-Planck equations

Equivalence of existence and uniqueness results

Extension to stochastically perturbed equations

Abstract

We prove a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations on Euclidean spaces and their corresponding linearized first-order continuity equation over the space of Borel (sub-)probability measures. As a consequence, we obtain equivalence of existence and uniqueness results for these equations. Moreover, we prove an analogous result for stochastically perturbed Fokker-Planck-Kolmogorov equations. To do so, we particularly show that such stochastic equations for measures are, similarly to the deterministic case, intrinsically related to linearized second-order equations on the space of Borel (sub-)probability measures.