Hausdorff dimension of set generated by exceptional oscillations of a class of N-parameter Gaussian processes

A class of N-parameter Gaussian processes are introduced, which are more general than the N-parameter Wiener process. The definition of the set generated by exceptional oscillations of a class of these processes is given, and then the Hausdorff dimension of this set is defined. The Hausdorff dimensions of these processes are studied and an exact representative for them is given, which is similar to that for the two-parameter Wiener process by Zacharie (2001). Moreover, the time set considered is a hyperrect angle which is more general than a hyper-square used by Zacharie (2001). For this more general case, a Fernique-type inequality is established and then using this inequality and the Slepian lemma, a Lévy’s continuity modulus theorem is shown. Independence of increments is required for showing the representative of the Hausdorff dimension by Zacharie (2001). This property is absent for the processes introduced here, so we have to find a different way.