Exact moduli of continuity for operator-scaling Gaussian random fields

Abstract

Let $X=\{X(t),t\in\mathrm{R}^{N}\}$ be a centered real-valued operator-scaling Gaussian random field with stationary increments, introduced by Biermé, Meerschaert and Scheffler (Stochastic Process. Appl. 117 (2007) 312–332). We prove that $X$ satisfies a form of strong local nondeterminism and establish its exact uniform and local moduli of continuity. The main results are expressed in terms of the quasi-metric $\tau_{E}$ associated with the scaling exponent of $X$. Examples are provided to illustrate the subtle changes of the regularity properties.