Domain and range symmetries of operator fractional Brownian fields

TL;DR

This paper characterizes the symmetries of operator fractional Brownian fields, revealing their structure via group theory and a new anisotropic representation, and describes possible symmetry pairs in specific dimensions.

Contribution

It introduces a novel anisotropic polar-harmonizable representation of OFBF and characterizes their domain and range symmetries as maximal groups and intersections of centralizers.

Findings

Characterization of domain and range symmetries as maximal groups.

New anisotropic polar-harmonizable representation of OFBF.

Description of possible symmetry pairs in (m,1) and (2,2) dimensions.

Abstract

An operator fractional Brownian field (OFBF) is a Gaussian, stationary increment R^n-valued random field on R^m that satisfies the operator self-similarity property {X(c^E t)}_{t in R^m} L= {c^H X(t)}_{t in R^m}, c > 0, for two matrix exponents (E,H). In this paper, we characterize the domain and range symmetries of OFBF, respectively, as maximal groups with respect to equivalence classes generated by orbits and, based on a new anisotropic polar-harmonizable representation of OFBF, as intersections of centralizers. We also describe the sets of possible pairs of domain and range symmetry groups in dimensions (m,1) and (2,2).