Nonlinear Fokker–Planck–Kolmogorov equations and their probabilistic counterparts

Special Colloquium in honor of Jürgen Jost's 70th birthday.

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.