Diffusive limits of Lipschitz functionals of Poisson measures

2021/07/12 by Eustache Besançon, Besançon, Eustache, Laurent Decreusefond +4 · 1 citation
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications

paper · doi:10.48550/arxiv.2107.05339

openalex publication_date 2021/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Continuous Time Markov Chains, Hawkes processes and many other interesting processes can be described as solution of stochastic differential equations driven by Poisson measures. Previous works, using the Stein's method, give the convergence rate of a sequence of renormalized Poisson measures towards the Brownian motion in several distances, constructed on the model of the Kantorovitch-Rubinstein (or Wasserstein-1) distance. We show that many operations (like time change, convolution) on continuous functions are Lipschitz continuous to extend these quantified convergences to diffuse limits of Markov processes and long-time behavior of Hawkes processes.

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