PUB - Publikationen an der Universität Bielefeld

We prove a generalization of the known result of Trevisan on the Ambrosio-FigalliTrevisan 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. is represented by a probability measure P. on the path space such that P. solves the corresponding martingale problem and mu t is the one-dimensional distribution of P. 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 casewhere 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 vertical bar x vertical bar.