Hausdorff Measure and Uniform Dimension for Space-Time Anisotropic Gaussian Random Fields

Let \(X=\{ X(t), t\in \mathbb {R}^{N}\} \) $X=\{X(t),t\in {\mathbb{R}}^N\}$ be a centered space-time anisotropic Gaussian random field in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ with stationary increments, where the components \(X_{i}(i=1,\ldots ,d)\) $X_i(i=1,\dots ,d)$ are independent but distributed differently. Under certain conditions, we not only give the Hausdorff dimension of the graph sets of X in the asymmetric metric in the recurrent case, but also determine the exact Hausdorff measure functions of the graph sets of X in the transient and recurrent cases, respectively. Moreover, we establish a uniform Hausdorff dimension result for the image sets of X. Our results extend the corresponding results on fractional Brownian motion and space or time anisotropic Gaussian random fields.