Derivatives of Local Times and Hausdorff Dimension of Inverse Images of Space-time Anisotropic Gaussian Random Fields

Let X = {X(t), t ∈ ∝N} be a centered space-time anisotropic Gaussian random field with values in ℝd. Under some general conditions, the existence, joint continuity and Hölder conditions of higher-order derivative of local times of X are studied. Moreover, we obtain the uniform Hausdorff dimension of the inverse images of X. The existing results of Gaussian random fields are extended to space-time anisotropic Gaussian random fields with approximate independent components.