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Mean-field limits à la Tanaka and large deviations for particle systems with network interactions

Louis-Pierre Chaintron, Antoine Diez

TL;DR

This work generalizes Tanaka's fixed-point construction to non-exchangeable mean-field particle systems with network interactions by introducing a digraph-measure parameter that governs inter-particle connectivity. It proves well-posedness, mean-field convergence, and a large deviations principle in an abstract fixed-point framework, and shows how network structure yields tractable PDE characterizations of the limit in constant, pathwise, and adaptive graph settings. The results include a new LDP for the interaction measure and a PDE-based description when possible, with explicit treatment of constant and evolving networks via the digraph-limit formalism. The approach accommodates both deterministic and stochastic networks and clarifies the role of graph regularity and Lipschitz conditions in deriving mean-field limits and limit PDEs. Overall, it provides a unified pathway from graph-based particle interactions to continuum descriptions and probabilistic large deviations for networked mean-field systems.

Abstract

This article proposes a unified framework to study non-exchangeable mean-field particle systems with some general interaction mechanisms. The starting point is a fixed-point formulation of particle systems originally due to Tanaka that allows us to prove mean-field limit and large deviation results in an abstract setting. While it has been recently shown that such formulation encompasses a large class of exchangeable particle systems, we propose here a setting for the non-exchangeable case, including the case of adaptive interaction networks. We introduce sufficient conditions on the network structure that imply the mean-field limit and a new large deviations principle for the interaction measure. Finally, we formally highlight important models for which it is possible to derive a closed PDE characterization of the limit.

Mean-field limits à la Tanaka and large deviations for particle systems with network interactions

TL;DR

This work generalizes Tanaka's fixed-point construction to non-exchangeable mean-field particle systems with network interactions by introducing a digraph-measure parameter that governs inter-particle connectivity. It proves well-posedness, mean-field convergence, and a large deviations principle in an abstract fixed-point framework, and shows how network structure yields tractable PDE characterizations of the limit in constant, pathwise, and adaptive graph settings. The results include a new LDP for the interaction measure and a PDE-based description when possible, with explicit treatment of constant and evolving networks via the digraph-limit formalism. The approach accommodates both deterministic and stochastic networks and clarifies the role of graph regularity and Lipschitz conditions in deriving mean-field limits and limit PDEs. Overall, it provides a unified pathway from graph-based particle interactions to continuum descriptions and probabilistic large deviations for networked mean-field systems.

Abstract

This article proposes a unified framework to study non-exchangeable mean-field particle systems with some general interaction mechanisms. The starting point is a fixed-point formulation of particle systems originally due to Tanaka that allows us to prove mean-field limit and large deviation results in an abstract setting. While it has been recently shown that such formulation encompasses a large class of exchangeable particle systems, we propose here a setting for the non-exchangeable case, including the case of adaptive interaction networks. We introduce sufficient conditions on the network structure that imply the mean-field limit and a new large deviations principle for the interaction measure. Finally, we formally highlight important models for which it is possible to derive a closed PDE characterization of the limit.

Paper Structure

This paper contains 14 sections, 14 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Particle systems as fixed points
  3. Abstract setting
  4. Particle systems
  5. Well-posedness and abstract mean-field limit
  6. Applications to particle networks
  7. Setting
  8. Mean-field limit and Large Deviations
  9. Characterization of the limit
  10. Constant network
  11. Pathwise network
  12. Adaptive network
  13. Useful results on Large Deviations
  14. Labeled Sanov theorem

Key Result

Theorem 3.1

Assume that the function $b$ is globally bounded in all its arguments. Given $\alpha\in\mathcal{A}$, if there exists $L_b^\mathcal{B} = L^\mathcal{B}_b (\alpha) >0$ such that for all $(t,\omega) \in [0,T] \times \Omega$, then eq:Xa has a unique solution $\mathbb{X}^\alpha$ in $\sigma + \mathcal{B}$. Moreover, $\mathbb{X}^\alpha$ enjoys the following regularity properties.

Theorems & Definitions (30)

  • Remark 2.1
  • Theorem 3.1: Well-posedness and regularity
  • proof
  • Corollary 3.2: Mean-field limit
  • proof
  • Corollary 3.3: Large deviations
  • proof
  • Remark 3.4: Central limit theorem
  • Remark 3.5: Classical McKean-Vlasov system
  • Lemma 4.1
  • ...and 20 more