Fluctuation results for Hastings-Levitov planar growth

• 成果类型: Article
• 署名作者: Silvestri, Vittoria
• 署名单位: University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0688-7
• 发表日期: 2017
• 页码: 417-460
• 关键词: WEAK-CONVERGENCE; internal dla; diffusion; LIMITS

摘要

We study the fluctuations of the outer domain of Hastings-Levitov clusters in the small particle limit. These are shown to be given by a continuous Gaussian process taking values in the space of holomorphic functions on , of which we provide an explicit construction. The boundary values of are shown to perform an Ornstein-Uhlenbeck process on the space of distributions on the unit circle , which can be described as the solution to the stochastic fractional heat equation Equation ID=Equ34 MediaObject MediaObject Equation>where denotes the Laplace operator acting on the spatial component, and is a space-time white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process converges to a log-correlated fractional Gaussian field, which can be realised as , for W complex white noise on .