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Generators of measure-valued jump diffusions and convergence rate of diffusive mean-field models

Xavier Erny

TL;DR

The paper analyzes diffusive mean-field particle systems with jumps in the CLT regime, proving a sharp $N^{-1/2}$ rate of convergence in distribution for empirical measures to the conditional law given common noise. It also derives explicit infinitesimal generators for two measure-valued Markov processes: the conditional laws ofMcKean–Vlasov diffusions and the empirical measures of McKean–Vlasov particle systems, using a framework of linear derivatives on probability measures and measure-valued polynomials. The results advance the understanding of fluctuations in jump-diffusion mean-field models and provide concrete tools (generators and semigroups) to study the Markov dynamics of measure-valued limits, including scenarios with jumps and common noise. The techniques enable a unified treatment of CLT-type fluctuations and the corresponding Markov structure, with potential applications to quantitative propagation of chaos and precise fluctuation analyses in high-dimensional interacting systems.

Abstract

The paper has two objectives: proving that the rate of convergence in distribution for mean-field models in CLT regime is $N^{-1/2}$, and obtaining explicit expressions for the infinitesimal generators of two types of measure-valued Markov processes (conditional law of McKean-Vlasov processes, and empirical measures of McKean-Vlasov systems). The proof of the convergence of mean-field system requires the second result about the generators, and both results need to study a notion of differentiability of measure-variable functions know as linear differentiability. Due to the particular framework that is studied, many technical difficulties arise compared to the existing literature. Two of the main problems are the following ones: the CLT regime implies that the limit measure-valued processes are not deterministic, and the empirical measure processes related to McKean-Vlasov equations with jumps are necessarily discontinuous. Both properties make the expressions of the generators more complicated than what is usually considered.

Generators of measure-valued jump diffusions and convergence rate of diffusive mean-field models

TL;DR

The paper analyzes diffusive mean-field particle systems with jumps in the CLT regime, proving a sharp rate of convergence in distribution for empirical measures to the conditional law given common noise. It also derives explicit infinitesimal generators for two measure-valued Markov processes: the conditional laws ofMcKean–Vlasov diffusions and the empirical measures of McKean–Vlasov particle systems, using a framework of linear derivatives on probability measures and measure-valued polynomials. The results advance the understanding of fluctuations in jump-diffusion mean-field models and provide concrete tools (generators and semigroups) to study the Markov dynamics of measure-valued limits, including scenarios with jumps and common noise. The techniques enable a unified treatment of CLT-type fluctuations and the corresponding Markov structure, with potential applications to quantitative propagation of chaos and precise fluctuation analyses in high-dimensional interacting systems.

Abstract

The paper has two objectives: proving that the rate of convergence in distribution for mean-field models in CLT regime is , and obtaining explicit expressions for the infinitesimal generators of two types of measure-valued Markov processes (conditional law of McKean-Vlasov processes, and empirical measures of McKean-Vlasov systems). The proof of the convergence of mean-field system requires the second result about the generators, and both results need to study a notion of differentiability of measure-variable functions know as linear differentiability. Due to the particular framework that is studied, many technical difficulties arise compared to the existing literature. Two of the main problems are the following ones: the CLT regime implies that the limit measure-valued processes are not deterministic, and the empirical measure processes related to McKean-Vlasov equations with jumps are necessarily discontinuous. Both properties make the expressions of the generators more complicated than what is usually considered.
Paper Structure (20 sections, 33 theorems, 281 equations)

This paper contains 20 sections, 33 theorems, 281 equations.

Table of Contents

  1. Main results
  2. Convergence speed of diffusive mean-field model
  3. Infinitesimal generators for measure-valued Markov processes
  4. Analysis of functions defined on spaces of probability measures
  5. Definitions and some elementary properties
  6. Chain rules
  7. Taylor's formulas and some corollaries
  8. Measure-variable polynomials
  9. Markov theory of measure-valued processes
  10. General results
  11. Proof of Theorem \ref{['barXmarkov']}
  12. Regularity of semigroups of conditional laws of diffusions
  13. Proof of Theorem \ref{['XNmarkov']}
  14. Proof of the results of Section \ref{['resultconvergence']}
  15. Proof of Theorem \ref{['mainresultdiffu']}
  16. ...and 5 more sections

Key Result

Corollary 1.1

Assume that the functions $\tilde{b},\tilde{\sigma}$ and $\tilde{f}$ are $C^5$ and that their derivatives of orders from one to five are bounded (the functions $\tilde{b}$ and $\tilde{\sigma}$ need not be bounded). In addition, we assume that $\tilde{f}$ is bounded and lower-bounded by some positive

Theorems & Definitions (78)

  • Corollary 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 68 more