The Annals of Probability

Stochastic heat equation with rough dependence in space

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

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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

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

Received: May 2015
Revised: December 2016
First available in Project Euclid: 12 December 2017

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Zentralblatt MATH identifier

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

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


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.

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