On nonlinear Markov processes in the sense of McKean

Abstract

We study nonlinear time-inhomogeneous Markov processes in the sense of McKean's seminal work [32]. These are given as families of laws $\mathbb{P}_{s,\zeta}$, $s\geq 0$, on path space, where $\zeta$ runs through a set of admissible initial probability measures on $\mathbb{R}^d$. In this paper, we concentrate on the case where every $\mathbb{P}_{s,\zeta}$ is given as the path law of a solution to a McKean-Vlasov SDE, where the latter is allowed to have merely measurable coefficients, which in particular are not necessarily weakly continuous in the measure variable. Our main result is the identification of general and checkable conditions on such general McKean-Vlasov SDEs, which imply that the path laws of their solutions form a nonlinear Markov process. Our notion of nonlinear Markov property is in McKean's spirit, but more general in order to include 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 and variants thereof with transport-type drift, and also the very recently studied $2D$ vorticity Navier-Stokes equation and the $p$-Laplace equation. In all these cases, the associated McKean-Vlasov SDEs are such that both their diffusion and drift coefficients singularly depend (i.e. Nemytskii-type) on the one-dimensional time marginals of their solutions. We stress that for our main result the nonlinear Fokker-Planck-Kolmogorov equations do not 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.