Regularization of the superposition principle: Potential theory meets Fokker-Planck equations

TL;DR

This paper advances the superposition principle for Fokker-Planck equations by constructing associated Markov processes under minimal conditions, enabling probabilistic analysis of nonlinear PDEs like the porous media equation.

Contribution

It constructs a Markov process from solutions of Fokker-Planck equations under general conditions, establishing the strong Markov property and applications to PDEs and McKean-Vlasov SDEs.

Findings

Constructed right Markov processes for FPE solutions.

Proved well-posedness of the parabolic Dirichlet problem.

Introduced a Choquet capacity for FPEs.

Abstract

For a solution to a (possibly nonlinear) Fokker-Planck equation (FPE) the powerful superposition principle renders a probability measure on path space with one dimensional time marginals equal to this solution, and additionally solving the martingale problem for the Kolmogorov operator given by the FPE. The superposition principle thus reveals that such parabolic PDEs have a probabilistic counter part. The aim of this work is to go a substantial further step and, by exploiting the superposition principle, construct a full fledged Markov process, i.e. a family of path space measures for a large set of space time starting points connected by the Markov property, associated to the (linearized) FPE in the above way. Under very general (merely measurability) conditions on the coefficients of the FPE this is achieved in this paper in such a way that the resulting process is a right process, which is a particularly useful class of Markov processes, enjoying among other regularity properties the strong Markov property, which is fundamental for the analysis of the underlying FPE as a (nonlinear) parabolic PDE by probabilistic tools. As two main applications we construct fundamental flow solutions for the FPE and we prove a well-posedeness result for the parabolic Dirichlet problem through probabilistic means for more general coefficients than could be treated in the existing literature. Furthermore, we introduce a Choquet capacity for such FPEs using the corresponding right process. The validity of the strong Markov property in the context of the superposition principle was an open problem even in the linear case. In this paper we solve this also in the nonlinear case, i.e. for path laws of solutions to McKean-Vlasov SDEs with Nemytskii type coefficients. A main application here is the FPE given by the generalized porous media equation and its corresponding McKean-Vlasov SDE.