Multiple intersections of space-time anisotropic Gaussian fields

Abstract

Let X = {X(t) ∈ ℝ^d, t ∈ℝ^N} be a centered space-time anisotropic Gaussian field with indices H = (H₁, ⋯, H_N) ∈ (0, 1)^N, where the components X_i (i = 1, ⋯, d) of X are independent, and the canonical metric \(\sqrt {{{\mathbb{E}({X_i}(t) - {X_i}(s))}^2}} \,(i = 1, \cdots ,d)\) is commensurate with \({\gamma ^{{\alpha _i}}}(\sum\limits_{j = 1}^N {|{t_j} - {s_j}{|^{{H_j}}})} \) for s = (s₁, ⋯, s_N), t = (t₁, ⋯, t_N) ∈ ℝ^N, α_i ∈ (0, 1], and with the continuous function γ(·) satisfying certain conditions. First, the upper and lower bounds of the hitting probabilities of X can be derived from the corresponding generalized Hausdorff measure and capacity, which are based on the kernel functions depending explicitly on γ (·). Furthermore, the multiple intersections of the sample paths of two independent centered space-time anisotropic Gaussian fields with different distributions are considered. Our results extend the corresponding results for anisotropic Gaussian fields to a large class of space-time anisotropic Gaussian fields.