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Entropic Chaos of Mixed Mean-Field Jump Processes

Tau Shean Lim, Shuoning Zhang

TL;DR

The paper develops a comprehensive entropy-based framework for mixed mean-field jump processes, including PDMPs, to prove entropic propagation of chaos as the number of particles grows. It introduces an abstract generator $\mathcal{L}(\mu)=\mathcal{K}+\mathcal{A}(\mu)$ with nonlinear mean-field dependence, and establishes well-posedness of mean-field evolution alongside explicit $O(1/N)$ relative-entropy bounds between the $N$-particle law and the mean-field product law. The core is a generalized concentration inequality (via a second-order bound) that closes the entropy method under first- and second-order bounded-difference conditions on the mean-field kernel, enabling quantitative propagation of chaos. The results apply to both PDMP-type and jump-driven mean-field dynamics, include averaged-in-N analyses, and provide explicit conditions under which entropic chaos and L1-chaos hold, with illustrative multi-body and $\mathbb{R}^k$-parametrized kernels.

Abstract

This paper studies a class of mixed mean-field jump processes on an abstract state space $Π$, together with their associated $N$-particle systems. The dynamics consist of the superposition of an independent Markovian component and a bounded mean-field jump interaction; in particular, piecewise deterministic Markov processes (PDMPs) with mean-field interactions are covered by this framework. Under a second-order bounded difference condition on the mean-field jump kernel, we establish entropic propagation of chaos as $N \to \infty$. In particular, we obtain an explicit qualitative bound on the relative entropy between the law of the $N$-particle system and the product measure induced by the mean-field limit. The proof relies on the second-order concentration inequality introduced in Götze and Sambale, 2020.

Entropic Chaos of Mixed Mean-Field Jump Processes

TL;DR

The paper develops a comprehensive entropy-based framework for mixed mean-field jump processes, including PDMPs, to prove entropic propagation of chaos as the number of particles grows. It introduces an abstract generator with nonlinear mean-field dependence, and establishes well-posedness of mean-field evolution alongside explicit relative-entropy bounds between the -particle law and the mean-field product law. The core is a generalized concentration inequality (via a second-order bound) that closes the entropy method under first- and second-order bounded-difference conditions on the mean-field kernel, enabling quantitative propagation of chaos. The results apply to both PDMP-type and jump-driven mean-field dynamics, include averaged-in-N analyses, and provide explicit conditions under which entropic chaos and L1-chaos hold, with illustrative multi-body and -parametrized kernels.

Abstract

This paper studies a class of mixed mean-field jump processes on an abstract state space , together with their associated -particle systems. The dynamics consist of the superposition of an independent Markovian component and a bounded mean-field jump interaction; in particular, piecewise deterministic Markov processes (PDMPs) with mean-field interactions are covered by this framework. Under a second-order bounded difference condition on the mean-field jump kernel, we establish entropic propagation of chaos as . In particular, we obtain an explicit qualitative bound on the relative entropy between the law of the -particle system and the product measure induced by the mean-field limit. The proof relies on the second-order concentration inequality introduced in Götze and Sambale, 2020.

Paper Structure

This paper contains 34 sections, 32 theorems, 319 equations.

Table of Contents

  1. Introduction
  2. Settings and Main Results
  3. Settings and Notations
  4. Notations for pairings
  5. Notations in High Dimensions
  6. Adjoint Markov semigroups
  7. Markov jump kernels, jump generators, and adjoint
  8. The space of bounded signed kernels
  9. Markov jump kernels
  10. Adjoint kernels and adjoint generators
  11. Mixed mean-field jump processes: evolution equations and its well-posedness
  12. Mean-field jump kernels, generators, and evolution equations
  13. Empirical jump kernels/generators
  14. Mean-field $N$-particle systems
  15. Entropic estimate of the law of mean-field $N$-particle systems
  16. ...and 19 more sections

Key Result

Proposition 2.2

Let $\{S_t\}_{t\ge 0}$ be a $C_0$-semigroup on $L^1(\nu)$, and let $(\mathcal{K}^*, D(\mathcal{K}^*))$ denote its infinitesimal generator. The following statements are equivalent:

Theorems & Definitions (68)

  • Definition 2.1: Adjoint Markov semigroup and generator
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Definition 2.5: The space of bounded kernels
  • Proposition 2.6
  • Definition 2.7: Adjoint jump kernel
  • Example 2.8: Reversible case
  • Example 2.9: Absolutely continuous kernels
  • Proposition 2.10
  • ...and 58 more