间歇性与多重分形：基于抛物随机偏微分方程的案例研究
Intermittency and multifractality: A case study via parabolic stochastic PDEs
摘要：
令$ξ$表示时空白噪声，并考虑以下随机偏微分方程：（i）$\dot{u}=\frac{1}{2} u" + uξ$，相同地从1开始;和（ii）$\dot{Z}=\frac12 Z" + ξ$，相同地从0开始。众所周知，（i）的解决方案是间歇性的，而（ii）的解决方案则不是。这两个方程属于不同的普适类。我们证明了这两个系统的高峰是在一个自然的大尺度意义上的多重分形。这项工作的一些扩展也建立多重分形行为的随机偏微分方程的峰值$\mathbf{R}_+\times\mathbf{R}^d$与$d\ge 2$。G.劳勒问我们，不连续性是否等同于多重分形。本文对这一问题给出了否定的回答。作为我们的方法的副产品，我们还证明了布朗运动的峰形成一个大规模的单分形，而$\mathbf{R}$上的Ornstein-Uhlenbeck过程的峰是多重分形。在整个过程中，我们广泛使用的宏观分形理论的M. T。Barlow和S. J. Taylor（1989，1992）。我们还扩展了巴洛-泰勒理论的各个方面。
英文摘要：
Let $ξ$ denote space-time white noise, and consider the following stochastic partial differential equations: (i) $\dot{u}=\frac{1}{2} u" + uξ$, started identically at one; and (ii) $\dot{Z}=\frac12 Z" + ξ$, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on $\mathbf{R}_+\times\mathbf{R}^d$ with $d\ge 2$. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein--Uhlenbeck process on $\mathbf{R}$ are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S. J. Taylor (1989, 1992). We expand on aspects of the Barlow-Taylor theory, as well.
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