Fluctuation results for Hastings–Levitov planar growth

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• 作者：Vittoria Silvestri
• 关键词：Mathematics Subject Classification60F17 ; 60G60 ; 30C85
• 刊名：Probability Theory and Related Fields
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：167
• 期：1-2
• 页码：417-460
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathem
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-2064
• 卷排序：167

文摘

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 \(\mathcal {F}\) taking values in the space of holomorphic functions on \(\{ |z|>1 \}\), of which we provide an explicit construction. The boundary values \(\mathcal {W}\) of \(\mathcal {F}\) are shown to perform an Ornstein–Uhlenbeck process on the space of distributions on the unit circle \(\mathbb {T}\), which can be described as the solution to the stochastic fractional heat equation $$\begin{aligned} \frac{\partial }{\partial t} \mathcal {W} (t,\vartheta ) = - (-\Delta )^{1/2} \mathcal {W} (t,\vartheta ) + \sqrt{2}\, \xi (t, \vartheta ), \end{aligned}$$where \(\Delta \) denotes the Laplace operator acting on the spatial component, and \(\xi (t,\vartheta )\) is a space-time white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process \(\mathcal {W}\) converges to a log-correlated fractional Gaussian field, which can be realised as \((-\Delta )^{-1/4}W\), for W complex white noise on \(\mathbb {T}\).Mathematics Subject Classification60F1760G6030C85Research supported by EPSRC Grant EP/H023348/1 for the Cambridge Centre for Analysis.