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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 $(|A(t,x)|+|langle b(t,x),xrangle |)/(1+|x|^2)$. Therefore, in the case where there are no a priori global integrability conditions the function $|A(t,x)|+|langle b(t,x),xrangle |$ 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 $langle b(t,x),xrangle le C+C|x|^2 log |x|$ along with $|A(t,x)|le C+C|x|^2 log |x|$.
We prove two new results connected with elliptic Fokker-Planck-Kolmogorov equations with drifts integrable with respect to solutions. The first result answers negatively a long-standing question and shows that a density of a probability measure satis
The work concerns the superposition between the Zakai equations and the Fokker-Planck equations on measure spaces. First, we prove a superposition principle for the Fokker-Planck equations on $mR^mN$ under the integrable condition. And then by means
We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional approximations. In contr
In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued in
It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck Equations for diffusion processes. In contrast to previous such results, it is not assumed that t