Stochastic heat equation with rough dependence in space
Probability
2015-05-20 v1
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 1/4\textless{}H\textless{}1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, we derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the moments of the solution.
Cite
@article{arxiv.1505.04924,
title = {Stochastic heat equation with rough dependence in space},
author = {Yaozhong Hu and Jingyu Huang and Khoa Lê and David Nualart and Samy Tindel},
journal= {arXiv preprint arXiv:1505.04924},
year = {2015}
}