Non-uniqueness of nonlinear Markov processes in the sense of McKean associated with parabolic PDEs

We derive a general scheme to construct infinitely many probabilistic counterparts for solutions to nonlinear PDEs by recasting the latter as different nonlinear Fokker–Planck equations and by constructing, for each of these equations, a solution to the associated McKean–Vlasov SDE with one-dimensional time marginal densities given by the PDE solution. We utilize this scheme to prove that nonlinear Markov processes in the sense of McKean as introduced by Rehmeier–Röckner (J.,Theor.,Probab. 38, 60 (2025)) are not uniquely determined by their one-dimensional time marginals. This is in sharp contrast to the case of classical Markov processes, which are uniquely determined by their one-dimensional time marginals. We demonstrate our results by constructing a continuum of nonlinear Markov processes with one-dimensional time marginal densities given by the Barenblatt solutions to the porous medium and $p$-Laplace equations, as well as by the fundamental solution to the heat equation. This includes a novel martingale representation for the $p$-Laplace Barenblatt solutions. We also prove that a nonlinear Markov process is uniquely determined by its two-dimensional time marginals. Moreover, for the porous medium equation, we show that the different McKean–Vlasov SDEs we investigate are consistent with corresponding gradient flow interpretations of the equation in the sense of Otto calculus.