Exact moduli of continuity for operator-scaling Gaussian random fields

TL;DR

This paper establishes the exact uniform and local moduli of continuity for operator-scaling Gaussian random fields, using strong local nondeterminism and quasi-metrics related to their scaling properties.

Contribution

It provides the first precise characterization of the regularity properties of operator-scaling Gaussian fields through exact moduli of continuity.

Findings

Proves strong local nondeterminism for the fields.

Derives exact uniform and local moduli of continuity.

Illustrates regularity changes with examples.

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 Bierm\'{e}, 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.