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) +| b(t, x), x |)/(1+|x| 2 ). Therefore, in the case where there are no a priori global integrability conditions the function A(t, x) + | b(t, x), x | 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 b(t, x), x ≤ C + C|x| 2 log |x| along with A(t, x) ≤ C + C|x| 2 log |x|.
AMS MSC 2010: 60J60; 35Q84where it is assumed that a ij and b i are locally (i.e., on compact sets in [0, T ] × R d ) integrable with respect to the measure µ t dt:This measure can be identified with the solution, which is also denoted by {µ t }.Recall that the weak topology on P(R d ) is generated by the seminorms