Joint Hölder Continuity of Parabolic Anderson Model

We obtain the Holder continuity and joint Hölder continuity in space and time for the random field solution to the parabolic Anderson equation (∂t−12Δ)u=u⋄W˙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\partial_t-\frac{1}{2}\Delta)u=u\diamond\dot{W}$$\end{document} in d-dimensional space, where Ẇ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density µ(ξ). We assume that γ0(t)≤c|t|α0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma_0(t)\leq{c}|t|^{\alpha_0}$$\end{document} and |μ(ξ)|≤c∏i=1d|ξi|−αior|μ(ξ)|≤c|ξ|−α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mu(\xi)|\leq{c}\prod_{i=1}^d|\xi_i|^{-\alpha_i}\;{\rm{or}}\;|\mu(\xi)|\leq{c}|\xi|^{-\alpha}$$\end{document}, where αi, i = 1, …, d (or α) can take negative value.