2019/03/26 by Bogachev, Vladimir I., Röckner, Michael, Shaposhnikov, Stanislav V. · 4 citations
#35Q84 #60J60 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1903.10834
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|.