[Paper Review] Conditional McKean-Vlasov Differential Equations with Common Poissonian Noise: Propagation of Chaos

Daniel Hernández–Hernández, Joshué Helí Ricalde-Guerrero|arXiv (Cornell University)|Aug 22, 2023

Mathematical Biology Tumor GrowthMathematics3 citations

TL;DR

This paper establishes the propagation of chaos for mean-field interacting particle systems driven by common Poissonian noise and Brownian motion, proving weak convergence of finite-population SDEs to a conditional McKean-Vlasov SDE with random environment. The key contribution is the rigorous derivation of the mean-field limit under general jump and diffusion dynamics, extending classical propagation of chaos to systems with common Lévy noise and conditional McKean-Vlasov dynamics.

ABSTRACT

A model for the evolution of a large population interacting system is considered in which a marked Poisson processes influences their evolution, together with a Brownian motion. Mean field McKean-Vlasov limits of such system are formulated studying first both systems individually. Letting the population size growing to infinite, the weak convergence of the solutions of such systems is proved; in other words, propagation of chaos of such systems is obtained.

Motivation & Objective

• To analyze the propagation of chaos in large interacting particle systems influenced by both common Poissonian noise and Brownian motion.
• To establish the mean-field limit of finite-population stochastic differential equations (SDEs) with empirical measure-dependent coefficients.
• To derive a limiting conditional McKean-Vlasov SDE where the law of each particle depends on the conditional distribution given a common random environment.
• To prove weak convergence of the finite-system dynamics to the mean-field limit under general integrability and regularity conditions on coefficients.
• To extend classical propagation of chaos theory to systems with common Lévy noise and non-Markovian conditional dependence structures.

Proposed method

• Formulates a finite-population SDE system with drift, diffusion, and jump coefficients depending on the empirical measure of the particle system.
• Introduces a conditional McKean-Vlasov SDE where each particle's law is conditioned on a common random environment represented by a marked Poisson process.
• Uses the Skorokhod J1 topology on path space and the 2-Wasserstein distance to define convergence of probability laws.
• Employs a canonical probability space construction with input processes: initial law, conditional law, Poisson random measure, Brownian motion, and intensity kernel.
• Applies Watanabe and Lévy characterizations to ensure independence of the Poisson and Brownian components in the canonical setting.
• Establishes existence and uniqueness of strong solutions via measurable mappings from input laws, relying on the Markov property and regularity of the intensity kernel.

Experimental results

Research questions

• RQ1 Does the empirical measure of a large system of interacting particles driven by common Poissonian noise and Brownian motion converge weakly to a deterministic limit?
• RQ2 Can the limiting dynamics be described by a conditional McKean-Vlasov SDE where each particle’s law depends on the conditional distribution given a common random environment?
• RQ3 Is the propagation of chaos preserved when the interaction is driven by a common marked Poisson process rather than independent noise?
• RQ4 How does the presence of a common random environment affect the convergence behavior of the particle system?
• RQ5 What conditions on the coefficients ensure the existence and uniqueness of the mean-field limit in this class of jump-diffusion systems?

Key findings

• The weak convergence of the finite-population SDEs to the conditional McKean-Vlasov SDE is established under standard regularity and integrability conditions on the coefficients.
• The limiting law of the particle system is characterized by a conditional McKean-Vlasov SDE where the law of each particle is adapted to a common filtration generated by the environment.
• The convergence holds in law in the space of càdlàg paths equipped with the Skorokhod topology and the 2-Wasserstein distance.
• The canonical construction ensures that the Poisson and Brownian components are independent under the limiting probability measure, preserving the structure of the noise.
• The solution to the mean-field SDE is shown to be a measurable function of the input laws, confirming the Markovian nature of the limit.
• The result generalizes classical propagation of chaos to systems with common Lévy noise and non-Markovian conditional dependence, extending applicability to models in statistical physics and mean-field games.