WEAK AND STRONG ERROR ANALYSIS FOR MEAN-FIELD RANK-BASED PARTICLE APPROXIMATIONS OF ONE-DIMENSIONAL VISCOUS SCALAR CONSERVATION LAWS

• 成果类型: Article
• 署名作者: Bencheikh, Oumaima; Jourdain, Benjamin
• 署名单位: Inria; Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1776
• 发表日期: 2022
• 页码: 4143-4185
• 关键词: probabilistic approximation; convergence rate; propagation; chaos; limit

摘要

In this paper, we analyse the rate of convergence of a system of N in-teracting particles with mean-field rank-based interaction in the drift coef-ficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikov (Ann. Probab. 46 (2018) 1042-1069) to check trajectorial propagation of chaos with optimal rate N-1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy (Math. Comp. 73 (2004) 777-812) to check the convergence in L1(R) with rate O ( root N1 + h)of the empirical cumulative dis-tribution function of the Euler discretization with step h of the particle system to the solution of a one-dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as O(N1 + h). We provide numerical results which confirm our theoretical estimates.