NONLINEAR FOKKER-PLANCK-KOLMOGOROV EQUATIONS AND NONLINEAR MARKOV PROCESSES

Michael Ro ̈ckner (Bielefeld University)

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

References on which the talk is based:

[1] Barbu, V., R ̈ockner, M., Nonlocal, nonlinear Fokker–Planck equations and nonlin- ear martingale problems, arXiv:2308.06388 (2023)

[2] Barbu, V., R ̈ockner, M., Nonlinear Fokker–Planck equations with fractional Lapla- cian and McKean–Vlasov SDEs with L ́evy noise, arXiv:2210.05612v2

[3] Rehmeier, M., R ̈ockner, M., On nonlinear Markov processes in the sense of McKean, arXiv:2212.12424v2

Further underlying references from our group:
[4] Barbu, V., R ̈ockner, M., From Fokker–Planck equations to solutions of distribution

dependent SDE, Ann. of Probab., 48 (2020), 1902-1920.

[5] Barbu, V., R ̈ockner, M., Solutions for nonlinear Fokker–Planck equations with measures as initial data and McKean–Vlasov equations, J. Funct. Anal., 280 (7) (2021), 1-35.

[6] Barbu, V., R ̈ockner, M., Uniqueness for nonlinear Fokker–Planck equations and for McKean–Vlasov SDEs: the degenerate case, J. Funct. Anal., 285 (4), 2023.

[7] Barbu, V., R ̈ockner, M., The evolution to equilibrium of solutions to nonlinear Fokker–Planck equations, Indiana Univ. Math. J., 72 (1) (2023).

[8] R ̈ockner, M., Xie, L., Zhang, X., Superposition principle for nonlocal Fokker– Planck–Kolmogorov operators, Probab. Theory Rel. Fields, 178 (3-4) (2020), 699-733.

[9] Ren, P., R ̈ockner, M., Wang, F.Y., Linearization of nonlinear Fokker–Planck equa- tions and applications, J. Differential Equations, 322 (2022), 1-37.