Multiple intersections of space-time anisotropic Gaussian fields

Let X = {X(t) is an element of R-d, t is an element of R-N} be a centered space-time anisotropic Gaussian field with indices H = (H-1,center dot center dot center dot, H-N) is an element of (0, 1)(N), where the components X-i (i = 1, center dot center dot center dot, d) of X are independent, and the canonical metric root E(Xi(t)-Xi(s))(2) (i=1,center dot center dot center dot,d) is commensurate with gamma alpha i (Sigma(N)(j=1) |t(j) - s(j)|(Hj)) for s = (s(1), center dot center dot center dot, s(N)), t = (t(1), center dot center dot center dot, t(N)) is an element of R-N, alpha(i) is an element of (0, 1], and with the continuous function gamma(center dot) 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 gamma (center dot). 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.