Sample Path Properties of Operator-Slef-Similar Gaussian Random Fields

Abstract

We study the Hausdorff dimension of the image and graph set, hitting probabilities, transience, and other sample path properties of certain isotropic operator-self-similar Gaussian random fields $X = \{X(t),\ t \in {\bf R}^N\}$ with stationary increments, including multiparameter operator fractional Brownian motion. Our results show that if $X({\bf 1})$, where $ {\bf 1} =(1, 0, \ldots, 0) \in {\bf R}^N$, is full, then many of such sample path properties are completely determined by the real parts of the eigenvalues of the self-similarity exponent~D.