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

TL;DR

The paper demonstrates that nonlinear Markov processes associated with certain PDEs are not uniquely determined by their one-dimensional marginals, contrasting with classical Markov processes, and constructs multiple such processes with explicit examples.

Contribution

It introduces a scheme to construct infinitely many nonlinear Markov processes for solutions of nonlinear PDEs, showing non-uniqueness in their one-dimensional marginals and establishing new representations.

Findings

Nonlinear Markov processes are not uniquely determined by their one-dimensional marginals.

Constructed a continuum of nonlinear Markov processes with explicit PDE solutions.

Proved that two-dimensional marginals uniquely determine the process.

Abstract

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--R\"ockner (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.