Moment Bounds for a Generalized Anderson Model with Gaussian Noise Rough in Space

In this article, we study a generalized Anderson model driven by Gaussian noise which is white/colored in time and has the covariance of a fractional Brownian motion with Hurst index H<12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<\frac{1}{2}$$\end{document} in space. We prove the existence of the solution in the Skorohod sense and obtain upper and lower bounds for the pth moments for all p=2,3,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2,3,\ldots $$\end{document}. Then we can prove that solution of this equation in the Skorohod sense is weakly intermittent. Hölder continuity of the solution with respect to the time and space variables is also deduced.