## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 6B (2017), 4561-4616.

### Stochastic heat equation with rough dependence in space

Yaozhong Hu, Jingyu Huang, Khoa Lê, David Nualart, and Samy Tindel

#### Abstract

This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable. The existence and uniqueness of the solution $u$ are proved assuming the nonlinear coefficient $\sigma(u)$ is differentiable with a Lipschitz derivative and $\sigma(0)=0$.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 6B (2017), 4561-4616.

**Dates**

Received: May 2015

Revised: December 2016

First available in Project Euclid: 12 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1513069267

**Digital Object Identifier**

doi:10.1214/16-AOP1172

**Mathematical Reviews number (MathSciNet)**

MR3737918

**Zentralblatt MATH identifier**

06838127

**Subjects**

Primary: 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 65C30: Stochastic differential and integral equations

**Keywords**

Stochastic heat equation fractional Brownian motion Feynman–Kac formula Wiener chaos expansion intermittency

#### Citation

Hu, Yaozhong; Huang, Jingyu; Lê, Khoa; Nualart, David; Tindel, Samy. Stochastic heat equation with rough dependence in space. Ann. Probab. 45 (2017), no. 6B, 4561--4616. doi:10.1214/16-AOP1172. https://projecteuclid.org/euclid.aop/1513069267