  We study nonlinear Markov processes in the sense of McKean's seminal work
[30] and present a large class of new examples. Our notion of nonlinear Markov
property is in McKean's spirit, but, in fact, more general in order to include
examples of such processes whose one-dimensional time marginal densities solve
a nonlinear parabolic PDE, more precisely, a nonlinear
Fokker--Planck--Kolmogorov equation, such as Burgers' equation, the porous
media equation on $\mathbb{R}^d$ for all $d \geq 1$, or variants of the latter
with transport-type drift, and the $2D$ vorticity Navier--Stokes equation. We
show that the associated nonlinear Markov process is given by path laws of weak
solutions to a corresponding distribution-dependent stochastic differential
equation where both the diffusion and drift coefficient depends singularly
(i.e. Nemytskii-type) on its one-dimensional time marginals. We stress that for
this the nonlinear Fokker--Planck--Kolmogorov equations do not necessarily have
to be well-posed. Thus, we establish a one-to-one correspondence between
solution flows of a large class of nonlinear parabolic PDEs and nonlinear
Markov processes. As a particular example we obtain, for all $d \geq 1$, that
the solution flow of the porous media equation on $\mathbb{R}^d$ is associated
to a nonlinear Markov process, in the same way as the heat flow is associated
to Brownian motion.
