Nonlinear Fokker-Planck-Kolmogorov equations and nonlinear Markov processes
A talk in the Cluster Group Stochastic Analysis series by
Michael Röckner
| Abstract: | Since the middle of last century a substantial part of stochastic analysis has been devoted to the
relationship between (parabolic) linear partial differential equations (PDEs), more precisely, linear
Fokker-Planck-Kolmogorov equations (FPKEs), and stochastic differential equations (SDEs), or more
generally Markov processes. Its most prominent example is the classical heat equation on one side
and the Markov process given by Brownian motion on the other. This talk is about the nonlinear
analogue, i.e., the relationship between nonlinear FPKEs on the analytic side and McKean-Vlasov
SDEs (of Nemytskii-type), or more generally, nonlinear Markov processes in the sense of McKean on
the probabilistic side. This program has been initiated by McKean already in his seminal PNAS-
paper from 1966 and this talk is about recent developments in this field. Topics will include existence
and uniqueness results for distributional solutions of the nonlinear FPKEs on the analytic side and
equivalently existence and uniqueness results for weak solutions of the McKean-Vlasov SDEs on the
probabilistic side. Furthermore, criteria for the corresponding path laws to form a nonlinear Markov
process will be presented. Among the applications are e.g. porous media equations (including such
with nonlocal operators replacing the Laplacian and possibly being perturbed by a transport term)
and their associated nonlinear Markov processes. But also the 2D Naiver-Stokes equation in vorticity
form and its associated nonlinear Markov process will be discussed.
Joint work with:
Viorel Barbu, Al.I. Cuza University and Octav Mayer Institute of Mathematics
of Romanian Academy, Ia¸si, Romania
Marco Rehmeier, Bielefeld University and SNS Pisa |