WEAK CONVERGENCE OF THE LOCALIZED DISTURBANCE FLOW TO THE COALESCING BROWNIAN FLOW

• 成果类型: Article
• 署名作者: Norris, James; Turner, Amanda
• 署名单位: University of Cambridge; Lancaster University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP845
• 发表日期: 2015
• 页码: 935-970
• 关键词: model; web

摘要

We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and having a certain weak flow property. The space is equipped with a complete separable metric. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric, also complete and separable. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps, each jump arising from a small localized disturbance of the circle. A local version of this result is also obtained, in which the weak limit law is that of the coalescing Brownian flow on the line. Our set-up is well adapted to time-reversal and our weak limit result provides a new proof of time-reversibility of the coalescing Brownian flow. We also identify a martingale associated with the coalescing Brownian flow on the circle and use this to make a direct calculation of the Laplace transform of the time to complete coalescence.