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