Non-uniqueness of nonlinear Markov processes in the sense of McKean associated with parabolic PDEs
A talk in the Bielefeld Stochastic Afternoon series by
Marco Rehmeier
| 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--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
Within the CRC this talk is associated to the project(s): A5 |