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

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 $\|A(t,x)\|\le C+C|x|^2 \log |x|$.