Intermittency and multifractality: A case study via parabolic stochastic PDEs

Abstract

Let $\xi $ denote space–time white noise, and consider the following stochastic partial differential equations on $\mathbb{R}_{+}\times \mathbb{R}$: (i) $\dot{u}=\frac{1}{2}u"+u\xi $, started identically at one; and (ii) $\dot{Z}=\frac{1}{2}Z"+\xi $, 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 $\mathbb{R}_{+}\times \mathbb{R}^{d}$ with $d\ge 2$. Gregory 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 $\mathbb{R}$ are multifractal.

Throughout, we make extensive use of the macroscopic fractal theory of Barlow and Taylor [J. Phys. A 22 (1989) 2621–2628; Proc. Lond. Math. Soc. (3) 64 (1992) 125–152]. We expand on aspects of the Barlow–Taylor theory, as well.