Uniform dimension results for Gaussian random fields (Q1042980)

Let \(X = \{ X(t), t \in \mathbb{R}^N \}\) be a Gaussian random field with values in \(\mathbb{R}^d\), and let \(X(t) = (X_1(t),\dotsc,X_d(t))\) be the coordinate processes. The properties of space and time anisotropy of \(X\) and their connections to uniform Hausdorff dimension results are discussed. An example of a space-anisotropic Gaussian random field for which the uniform Hausdorff dimension result does not hold is given. For a time anisotropic Gaussian random field \(X\), where the coordinates \(X_1,\dotsc,X_d\) are independent and indentically distributed, uniform Hausdorff dimension results are established for the image sets of \(X\). These results extend the corresponding results for the one-dimensional Brownian motion, fractional Brownian motion and the Brownian sheet. An important class of space-anisotropic random fields is operator-self-similar processes in the space variable, i.e.\ \(\{X(ct), t \in \mathbb{R}^N \} =^{\text{dist}} \{c^D X(t), t \in \mathbb{R}^N\}\) for some \(D \in \mathbb{R}^{d \times d}\). An important class of time-anisotropic random fields is operator-self-similar processes in the time variable, i.e.\ \(\{X(c^A t), t \in \mathbb{R}^N \} =^{\text{dist}} \{c X(t), t \in \mathbb{R}^N\}\) for some \(A \in \mathbb{R}^{N \times N}\).