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Propagation of chaos for point processes induced by particle systems with mean-field drift interaction

Nikolaos Kolliopoulos, Martin Larsson, Zeyu Zhang

Abstract

We study the asymptotics of the point process induced by an interacting particle system with mean-field drift interaction. Under suitable assumptions, we establish propagation of chaos for this point process: it has the same weak limit as the point process induced by i.i.d. copies of the solution of a limiting Mckean--Vlasov equation. This weak limit is a Poisson point process whose intensity measure is related to classical extreme value distributions. In particular, this yields the limiting distribution of the normalized upper order statistics.

Propagation of chaos for point processes induced by particle systems with mean-field drift interaction

Abstract

We study the asymptotics of the point process induced by an interacting particle system with mean-field drift interaction. Under suitable assumptions, we establish propagation of chaos for this point process: it has the same weak limit as the point process induced by i.i.d. copies of the solution of a limiting Mckean--Vlasov equation. This weak limit is a Poisson point process whose intensity measure is related to classical extreme value distributions. In particular, this yields the limiting distribution of the normalized upper order statistics.
Paper Structure (11 sections, 2 theorems, 85 equations)

This paper contains 11 sections, 2 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main result
  3. Consequences of the main result and examples
  4. Marginal point process
  5. Propagation of chaos for order statistics
  6. Propagation of chaos for the joint distribution of the top $k$ order statistics
  7. Propagation of chaos when $k\to\infty$
  8. Two examples
  9. Proof of Theorem \ref{['preprocess']}
  10. Proof of Theorem \ref{['T_main_precise']}
  11. Proof of condition \ref{["T2_claim_1'"]}:

Key Result

Theorem 2.3

Suppose Assumptions ass1 and ass2 are satisfied. Fix $T \in (0,\infty)$ and suppose that for some normalizing constants $a^N_T,b^N_T$ the normalized maxima of the i.i.d. system eq_McKV_SDE converge weakly to a nondegenerate distribution function $\Gamma_T$ on ${\mathbb R}$: Then, the sequence $({\mathbb Q}^N\circ U_N^{-1})_{N \in \mathbb{N}}$ of probability laws on $M_p(E)$ converges to the law o

Theorems & Definitions (6)

  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Example 3.1: Gaussian particles
  • Example 3.2: Rank-based diffusions