Abstract: 
We develop a theory of nonlinear Markov processes in the sense of McKean`s seminal work from 1966 and provide a general construction of such processes with onedimensional time marginals given by solutions to nonlinear FokkerPlanckKolmogorov equations. These processes consist of path laws of weak solutions to the corresponding McKeanVlasov stochastic differential equation. We stress that neither of these equations needs to be wellposed. Our theory applies to a large new class of examples, e.g. the porous media equation (PME) on $\R^d$, $d \geq 1$, with general diffusivity and transporttype drift, which includes the classical PME with its Barenblatt solutions as the onedimensional time marginal densities of the corresponding nonlinear Markov process. 
