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arXiv · 1505.04924

Stochastic heat equation with rough dependence in space

Yaozhong Hu·Jingyu Huang·Khoa Lê·David Nualart·Samy Tindel
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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.

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Yaozhong Hu, Jingyu Huang, Khoa Lê, David Nualart, Samy Tindel. 2015-05-19. Stochastic heat equation with rough dependence in space. https://arxiv.org/abs/1505.04924

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