On the Ambrosio–Figalli–Trevisan Superposition Principle for Probability Solutions to Fokker–Planck–Kolmogorov Equations

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 $\{\mu _t\}$ with initial distribution $\nu$ is represented by a probability measure $P_\nu$ on the path space such that $P_\nu$ solves the corresponding martingale problem and $\mu _t$ is the one-dimensional distribution of $P_\nu$ 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 $(\Vert A(t,x)\Vert +|\langle b(t,x),x\rangle |)/(1+|x|^2)$. Therefore, in the case where there are no a priori global integrability conditions the function $\Vert A(t,x)\Vert +|\langle b(t,x),x\rangle |$ 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 $\langle b(t,x),x\rangle \le C+C|x|^2 \log |x|$ along with $\Vert A(t,x)\Vert \le C+C|x|^2 \log |x|$.