Flow selections for (nonlinear) Fokker-Planck-Kolmogorov equations

TL;DR

This paper introduces a novel method for selecting solution flows for both linear and nonlinear Fokker-Planck-Kolmogorov equations, improving existing results and characterizing well-posedness through flow uniqueness.

Contribution

The paper presents a new approach to flow selection in FPK equations, including the nonlinear case with Nemytskii-type coefficients, and links flow uniqueness to well-posedness.

Findings

Improved solution flow selection method for linear FPK equations.

First characterization of flow selection for nonlinear FPK equations with Nemytskii coefficients.

Under certain conditions, these flows are Markovian, satisfying Chapman-Kolmogorov equations.

Abstract

We provide a method to select flows of solutions to the Cauchy problem for linear and nonlinear Fokker--Planck--Kolmogorov equations (FPK equations) for measures on Euclidean space. In the linear case, our method improves similar results of a previous work of the author. Our consideration of flow selections for nonlinear equations, including the particularly interesting case of Nemytskii-type coefficients, seems to be new. We also characterize the (restricted) well-posedness of FPK equations by the uniqueness of such (restricted) flows. Moreover, we show that under suitable assumptions in the linear case such flows are Markovian, i.e. they fulfill the Chapman-Kolmogorov equations.