PATHWISE CONVERGENCE OF THE HARD SPHERES KAC PROCESS

• 成果类型: Article
• 署名作者: Heydecker, Daniel
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1475
• 发表日期: 2019
• 页码: 3062-3127
• 关键词: boltzmann-equation; homogeneous boltzmann; moment production; spectral gap; chaos; inequalities; propagation; equilibrium; entropy; MODEL

摘要

We derive two estimates for the deviation of the N-particle, hard-spheres Kac process from the corresponding Boltzmann equation, measured in expected Wasserstein distance. Particular care is paid to the long-time properties of our estimates, exploiting the stability properties of the limiting Boltzmann equation at the level of realisations of the interacting particle system. As a consequence, we obtain an estimate for the propagation of chaos, uniformly in time and with polynomial rates, as soon as the initial data has a kth moment, k > 2. Our approach is similar to Kac's proposal of relating the long-time behaviour of the particle system to that of the limit equation. Along the way, we prove a new estimate for the continuity of the Boltzmann flow measured in Wasserstein distance.