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 { μ t } with initial distribution ν is represented by a probability measure P ν on the path space such that P ν solves the corresponding martingale problem and μ t is the one-dimensional distribution of P ν at time t . 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. This is the first result in this direction that applies to unbounded coefficients without any a priori global integrability conditions. Moreover, we show 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 | .