  We associate a coupled nonlinear Fokker-Planck equation on $\R^d$, i.e. with
solution paths in $\scr P$, to a linear Fokker-Planck equation for probability
measures on the product space $\R^d\times \scr P$, i.e. with solution paths in
$\scr P(\R^d\times\scr P)$. We explicitly determine the corresponding linear
Kolmogorov operator $\tilde{\mathbf{L}}_t$ using the natural tangent bundle
over $\scr P$ with corresponding gradient operator $\nabla^\scr P$. Then it is
proved that the diffusion process generated by $\tilde{\mathbf{L}}_t$ on
$\R^d\times\scr P$ is intrinsically related to the solution of a McKean-Vlasov
stochastic differential equation (SDE). We also characterize the ergodicity of
the diffusion process generated by $\tilde{\mathbf{L}}_t$ in terms of
asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another
main result of the paper is that the restricted well-posedness of the
non-linear Fokker-Planck equation and its linearized version imply the
(restricted) well-posedness of the McKean-Vlasov equation and that in this case
the laws of the solutions have the Markov property. All this is done under
merely measurebility conditions on the coefficients in their measure
dependence, hence in particular applies if the latter is of "Nemytskii-type".
As a consequence, we obtain the restricted weak well-posedness and the Markov
property of the so-called nonlinear distorted Brownian motion, whose associated
nonlinear Fokker-Planck equation is a porous media equation perturbed by a
nonlinear transport term. This realizes a programme put forward by McKean in
his seminal paper of 1966 for a large class of nonlinear PDEs. As a further
application we obtain a probabilistic representation of solutions to
Schr\"odinger type PDEs on $\R^d\times\scr P_2$, through the Feynman-Kac
formula for the corresponding diffusion processes.
