# On the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations

## TL;DR

This paper generalizes the superposition principle for probability solutions to Fokker-Planck-Kolmogorov equations, allowing for solutions with coefficients of quadratic growth without requiring global integrability, thus broadening applicability.

## Contribution

It extends Trevisan's superposition principle by replacing integrability conditions with growth bounds, enabling analysis of solutions with quadratic growth coefficients.

## Key findings

- Established a generalized superposition principle under weaker conditions.
- Derived conditions on initial distributions for solution representation.
- Showed solutions can have quadratic growth in coefficients without global integrability.

## Abstract

We prove a generalization of the known result of Trevisan on the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation, according to which such a solution is generated by a solution to the corresponding martingale problem. The novelty is that in place of the integrability of the diffusion and drift coefficients $A$ and $b$ with respect to the solution we require the integrability of $(\|A(t,x)\|+|\langle b(t,x),x\rangle |)/(1+|x|^2)$. Therefore, in the case where there are no a priori global integrability conditions the function $\|A(t,x)\|+|\langle b(t,x),x\rangle |$ can be of quadratic growth. Moreover, as a corollary we obtain that under mild conditions on the initial distribution it is sufficient to have the one-sided bound $\langle b(t,x),x\rangle \le C+C|x|^2 \log |x|$ along with $\|A(t,x)\|\le C+C|x|^2 \log |x|$.