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We provide a method to select flows of solutions to the Cauchy problem forlinear and nonlinear Fokker--Planck--Kolmogorov equations (FPK equations) formeasures on Euclidean space. In the linear case, our method improves similarresults of a previous work of the author. Our consideration of flow selectionsfor nonlinear equations, including the particularly interesting case ofNemytskii-type coefficients, seems to be new. We also characterize the(restricted) well-posedness of FPK equations by the uniqueness of such(restricted) flows. Moreover, we show that under suitable assumptions in thelinear case such flows are Markovian, i.e. they fulfill the Chapman-Kolmogorovequations.

Flow selections for (nonlinear) Fokker-Planck-Kolmogorov equations