On Nonlinear Markov Processes in the Sense of McKean

Abstract We study nonlinear time-inhomogeneous Markov processes in the sense of McKean’s (Proc Natl Acad Sci USA 56(6):1907–1911, 1966) seminal work. These are given as families of laws $$\mathbb {P}_{s,\zeta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ζ</mml:mi> </mml:mrow> </mml:msub> </mml:math> , $$s\ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , on path space, where $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> runs through a set of admissible initial probability measures on $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . In this paper, we concentrate on the case where every $$\mathbb {P}_{s,\zeta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ζ</mml:mi> </mml:mrow> </mml:msub> </mml:math> is given as the path law of a solution to a McKean–Vlasov stochastic differential equation (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 partial differential equation, 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 two-dimensional 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.