## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 6A (2017), 3697-3751.

### Intermittency and multifractality: A case study via parabolic stochastic PDEs

Davar Khoshnevisan, Kunwoo Kim, and Yimin Xiao

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 6A (2017), 3697-3751.

**Dates**

Received: March 2015

Revised: July 2016

First available in Project Euclid: 27 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1511773662

**Digital Object Identifier**

doi:10.1214/16-AOP1147

**Mathematical Reviews number (MathSciNet)**

MR3729613

**Zentralblatt MATH identifier**

06838105

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60K37: Processes in random environments

**Keywords**

Intermittency multifractality macroscopic/large-scale Hausdorff dimension stochastic partial differential equations

#### Citation

Khoshnevisan, Davar; Kim, Kunwoo; Xiao, Yimin. Intermittency and multifractality: A case study via parabolic stochastic PDEs. Ann. Probab. 45 (2017), no. 6A, 3697--3751. doi:10.1214/16-AOP1147. https://projecteuclid.org/euclid.aop/1511773662