MATJ5126 MA1: Nonlinear Fokker-Planck Flows and their Probabilistic Counterparts (JSS35) (2 op)

Already in 1966 in his visionary paper in PNAS, H.P. McKean, jr., formulated a programme to construct a probabilistic counterpart to nonlinear parabolic partial differential equations (PDEs) in the form of nonlinear Markov processes, in the same way as was being done at that time in the linear case. The aim was to exploit this relationship to transform problems in analysis to their probabilistic counterparts and vice versa as well as to have two associated tool boxes at hand for their better understanding and for developing respective solution strategies in both fields. While the linear theory was widely developed in the past 60 years with great success, documented in a huge literature up to today, McKean’s nonlinear case was, however, much less developed and for quite some time many standard nonlinear parabolic PDEs were not covered because of too strong assumptions on the coefficients. Starting from around 2018 the situation substantially changed and by employing a new technique, that is, the (nonlinear) superposition principle, the said restrictions on the coefficients could be considerably weakened and a number of nonlinear parabolic PDEs, such as the viscous Burgers equation, the generalized (possibly in space nonlocal) porous media equations, 2D vorticity Navier-Stokes equations and, more recently, the (doubly nonlinear) Leibenson equation, could be shown to have a nonlinear Markov process as its probabilistic counterpart. The Leibenson equation contains the parabolic p-Laplace equation as a special case, in which one thus obtains a complete analogue of classical Brownian motion, which is the linear Markov process associated to the classical heat equation (= parabolic 2-Laplace equation), namely the p-Brownian motion as the nonlinear Markov process associated to the parabolic p-Laplace equation. In this lecture course the underlying general technique will be presented, i.e.,

(i) Identify the nonlinear parabolic PDE as a nonlinear Fokker-Planck-Kolmogorov equation (FPKE) and solve it;

(ii) Solve the corresponding McKean-Vlasov stochastic differential equation (MVSDE) by linearization and applying the superposition principle;

(iii) Prove that the path laws of the solutions to the MVSDE (for a suitable class of initial conditions) form a nonlinear Markov process in the sense of McKean;

Obviously, a crucial point to implement this technique is to construct solutions to nonlinear FPKEs in (i). For illustration a corresponding general existence theorem including its proof, which applies to quite a large class of FPKEs, will also be part of the lecture course.

References:

Barbu/Rehmeier/R: arXiv: 2409.18744v2, AOP 2025+

Barbu/Grube/Rehmeier/R: arXiv: 2508.12979

Barbu/R: Springer LN 2024

Barbu/R/Deng Zhang: arXiv: 2309.13910, JEMS 2025+

Barbu/R: PTRF 2024

Barbu/R: AOP 2020 and SIAM 2018

McKean: PNAS 1966

Trevisan: EJP 2016  

Among other things the students will learn:

(a) A fundamental way how to connect analysis and probability;

(b) How to solve a fairly large class of nonlinear parabolic PDEs;

(c) How to solve McKean-Vlasov SDEs with merely measurability conditions on the coefficients (in particular, in their probability measure-valued variable);

(d) About the (in probability fundamental) notion of a (linear and) nonlinear Markov process;

(e) How to prove the crucial Markov property without having well-posedness, neither for the considered nonlinear parabolic PDE (FPKE) nore the associated McKean-Vlasov SDE.   

- basic knowledge in probability and measure theory;

- basic knowledge in stochastic analysis (Itô-formula, weak solutions to SDEs, martingale problem);

- basic knowledge in functional analysis related to linear PDEs (Hilbert spaces, weak topology, Lax-Milgram theorem, Schauder fix point theorem, Sobolev embeddings)