Central limit theorem for a class of one-dimensional kinetic equations

• 成果类型: Article
• 署名作者: Bassetti, Federico; Ladelli, Lucia; Matthes, Daniel
• 署名单位: Polytechnic University of Milan; University of Pavia; Technische Universitat Wien
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0269-8
• 发表日期: 2011
• 页码: 77-109
• 关键词: fixed-points; equilibrium; CONVERGENCE; speed

摘要

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law nu (alpha), then the limit is a scale mixture of nu (alpha). Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.