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First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation

Alessandro Pinzi

TL;DR

The paper develops a framework to evolve curves of random probability measures $M_t$ via a first-order, non-local operator $\mathcal{K}_{b_t,a_t}$, and proves a nested superposition principle that represents $M_t$ as both a superposition of solutions to the non-linear Kolmogorov-Fokker-Planck equation and a superposition of weak solutions to the McKean-Vlasov SDE. It defines a measurable lifting structure with $\Lambda$ on path space and $\mathfrak{L}$ on curves of SDE paths, establishing precise compatibility and measurability properties and a mechanism to transfer existence and uniqueness from the random-measure equation to the KFP and MV equations under a linearized uniqueness assumption. The work introduces integral-metric tools to handle convergence in the space of random measures and provides a rigorous treatment of filtrations and measurability needed for the stochastic-analytic arguments. Together, these results extend stochastic and measure-valued reformulations of interacting particle systems, offering a pathway to well-posedness analyses under low-regularity coefficients and informing selection principles in infinite dimensions.

Abstract

The goal of this paper is to define an evolution equation for a curve of random probability measures $(M_t)_{t\in[0,T]}\subset \mathcal{P}(\mathcal{P}(\mathbb{R}^d))$ associated to a non-local drift $b:[0,T]\times\mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$ and a non-local diffusion term $a:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \operatorname{Sym}_+(\mathbb{R}^{d\times d})$. Then, we show that any solution to that equation can be lifted to a superposition of solutions to a non-linear Kolmogorov-Fokker-Planck equation and also to a superposition of weak solutions to the McKean-Vlasov equations. Finally, we use this superposition result to show how existence and uniqueness can be transferred from the equation on random measures to the associated non-linear Kolmogorov-Fokker-Planck equation and to the McKean-Vlasov equation, assuming uniqueness of the linearized KFP.

First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation

TL;DR

The paper develops a framework to evolve curves of random probability measures via a first-order, non-local operator , and proves a nested superposition principle that represents as both a superposition of solutions to the non-linear Kolmogorov-Fokker-Planck equation and a superposition of weak solutions to the McKean-Vlasov SDE. It defines a measurable lifting structure with on path space and on curves of SDE paths, establishing precise compatibility and measurability properties and a mechanism to transfer existence and uniqueness from the random-measure equation to the KFP and MV equations under a linearized uniqueness assumption. The work introduces integral-metric tools to handle convergence in the space of random measures and provides a rigorous treatment of filtrations and measurability needed for the stochastic-analytic arguments. Together, these results extend stochastic and measure-valued reformulations of interacting particle systems, offering a pathway to well-posedness analyses under low-regularity coefficients and informing selection principles in infinite dimensions.

Abstract

The goal of this paper is to define an evolution equation for a curve of random probability measures associated to a non-local drift and a non-local diffusion term . Then, we show that any solution to that equation can be lifted to a superposition of solutions to a non-linear Kolmogorov-Fokker-Planck equation and also to a superposition of weak solutions to the McKean-Vlasov equations. Finally, we use this superposition result to show how existence and uniqueness can be transferred from the equation on random measures to the associated non-linear Kolmogorov-Fokker-Planck equation and to the McKean-Vlasov equation, assuming uniqueness of the linearized KFP.

Paper Structure

This paper contains 16 sections, 24 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical topology over spaces of measures and the space of continuous curves
  4. Space of continuous curves
  5. Spaces of measures
  6. -Wasserstein distance
  7. Space of random measures
  8. A note on filtrations and measurability
  9. Martingale problem, KFP equation and superposition principle in
  10. Integral metrics
  11. Integral metrics over in duality with smooth functions
  12. Integral metrics over in duality with smooth cylinder functions
  13. Equation on random measures and nested superposition principle
  14. Superposition principle: from to
  15. Nested superposition principle: from to
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $b:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$ and $a:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \operatorname{Sym}_+(\mathbb{R}^{d\times d})$ be Borel measurable maps. Let $\boldsymbol{M} = (M_t)_{t\in[0,T]} \in C([0,T],\mathcal{P}(\mathcal{P}( Then, there exist $\Lambda \in \mathcal{P}(C([0,T],\mathcal{P}(\mathbb{R}^d)))$ and $\mathfrak{L}\i

Theorems & Definitions (54)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 44 more