On nonlinear Markov processes in the sense of McKean

We introduce and study nonlinear Markov processes in the sense of McKean, initiate a theory of these processes, and present a large class of new examples. More precisely, we construct nonlinear Markov processes with one-dimensional time marginals given as solution flows of nonlinear Fokker-Planck equations. These processes are given by path laws of weak solutions to the corresponding distribution-dependent stochastic differential equation. Our results apply to nonlinear Fokker-Planck equations with locally density-dependent coefficients, which includes many important nonlinear parabolic PDEs. Thus, we establish a one-to-one correspondence between solution flows of such PDEs and nonlinear Markov processes. Important examples are porous media and Burgers equation, as well as the 2D vorticity Navier-Stokes equation.