On intersections of independent anisotropic Gaussian random fields

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X^H = \{ X^H (s),s \in \mathbb{R}^{N_1 } \} $\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X^K = \{ X^K (t),t \in \mathbb{R}^{N_2 } \} $\end{document} be two independent anisotropic Gaussian random fields with values in ℝd with indices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H = (H_1 ,...,H_{N_1 } ) \in (0,1)^{N_1 } ,K = (K_1 ,...,K_{N_2 } ) \in (0,1)^{N_2 } $\end{document}, respectively. Existence of intersections of the sample paths of XH and XK is studied. More generally, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_1 \subseteq \mathbb{R}^{N_1 } ,E_2 \subseteq \mathbb{R}^{N_2 } $\end{document} and F ⊂ ℝd be Borel sets. A necessary condition and a sufficient condition for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{P}\{ (X^H (E_1 ) \cap X^K (E_2 )) \cap F \ne \not 0\} > 0$\end{document} in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathbb{R}^{N_1 + N_2 + d} ,\tilde \rho )$\end{document} are proved, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde \rho $\end{document} is a metric defined in terms of H and K. These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets.