On intersections of independent anisotropic Gaussian random fields

Abstract

Let $X^H = \{ X^H (s),s \in \mathbb{R}^{N_1 } \}$ and $X^K = \{ X^K (t),t \in \mathbb{R}^{N_2 } \}$ be two independent anisotropic Gaussian random fields with values in ℝ^d with indices $H = (H_1 ,...,H_{N_1 } ) \in (0,1)^{N_1 } ,K = (K_1 ,...,K_{N_2 } ) \in (0,1)^{N_2 }$, respectively. Existence of intersections of the sample paths of X ^H and X ^K is studied. More generally, let $E_1 \subseteq \mathbb{R}^{N_1 } ,E_2 \subseteq \mathbb{R}^{N_2 }$ and F ⊂ ℝ^d be Borel sets. A necessary condition and a sufficient condition for $\mathbb{P}\{ (X^H (E_1 ) \cap X^K (E_2 )) \cap F \ne \not 0\} > 0$ in terms of the Bessel-Riesz type capacity and Hausdorff measure of E ₁ × E ₂ × F in the metric space $(\mathbb{R}^{N_1 + N_2 + d} ,\tilde \rho )$ are proved, where $\tilde \rho$ 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.