Special Session 19: Stochastic Partial Differential Equations

On nonlinear Markov Processes in the sense of McKean

Marco Rehmeier
Bielefeld University
Co-Author(s):    Michael R\ockner
We develop a theory of nonlinear Markov processes in the sense of McKean`s seminal work from 1966 and provide a general construction of such processes with one-dimensional time marginals given by solutions to nonlinear Fokker--Planck--Kolmogorov equations. These processes consist of path laws of weak solutions to the corresponding McKean--Vlasov stochastic differential equation. We stress that neither of these equations needs to be well-posed. Our theory applies to a large new class of examples, e.g. the porous media equation (PME) on $\R^d$, $d \geq 1$, with general diffusivity and transport-type drift, which includes the classical PME with its Barenblatt solutions as the one-dimensional time marginal densities of the corresponding nonlinear Markov process.