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

Lucian Beznea, Iulian Cîmpean, Michael Röckner

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.

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

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.
Paper Structure (36 sections, 57 theorems, 514 equations)

This paper contains 36 sections, 57 theorems, 514 equations.

Table of Contents

  1. Introduction
  2. Regularization of the superposition principle for linear time-dependent Fokker-Planck equations
  3. The superposition principle and the "simple" Markov property
  4. The "simple" Markov property
  5. Towards more regularity through generalized Dirichlet forms
  6. The main results: The corresponding right (in particular strong) Markov process
  7. Proofs of the main results
  8. Some technical lemmas needed in the proofs
  9. Proof of
  10. Proof of
  11. Consequences of the main results
  12. Existence of fundamental flow solutions to linear Fokker-Planck equations
  13. Backward Kolmogorov equations and the parabolic Dirichlet problem
  14. Lyapunov functions
  15. Adding jumps to time-dependent linear Fokker-Planck equations
  16. ...and 21 more sections

Key Result

Theorem 2.5

Let $\left(\mu_t\right)_{t\in [s,T]}$ be a weakly continuous solution to eq:LFPshort, and suppose further that Then, according to BoRoSh21, there exists $\eta\in \mathcal{P}\left(C\left([s, T];\mathbb{R}^d\right)\right)$ a solution to the canonical martingale problem associated to $({\sf L}_t)_{t\in [s,T]}$ on $[s,T]$ such that the one-dimensional time marginals $(\eta_t)_{t\in[s,T]}$ satisfy $\e

Theorems & Definitions (166)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: The superposition principle; BoRoSh21, Tr16
  • Remark 2.6
  • Proposition 2.7: Well-posedeness of the canonical MP
  • proof
  • Proposition 2.8: Tr16
  • Remark 2.9
  • ...and 156 more