Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	Michael Rockner，Bielefeld University
Inviter:	朱湘禅
Title:	NONLINEAR FOKKER-PLANCK-KOLMOGOROV EQUATIONS AND NONLINEAR MARKOV PROCESSES
Language:	English
Time & Venue:	2024.02.28 16:00-17:00 N202
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 PNASpaper 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 theprobabilistic 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 posssibly 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.