A generalisation of the fractional Brownian field based on non-Euclidean norms

TL;DR

This paper generalizes the fractional Brownian field by replacing the Euclidean norm with arbitrary norms, characterizing all self-similar Gaussian fields with stationary increments and exploring non-Euclidean variants of fractional Poisson fields.

Contribution

It provides a complete description of self-similar Gaussian fields with stationary increments based on non-Euclidean norms and introduces non-Euclidean fractional Poisson fields sharing covariance structures.

Findings

Characterization of all admissible norms for self-similar Gaussian fields

Integral representations of the generalized fields

Construction of non-Euclidean fractional Poisson fields with shared covariance

Abstract

We explore a generalisation of the L\'evy fractional Brownian field on the Euclidean space based on replacing the Euclidean norm with another norm. A characterisation result for admissible norms yields a complete description of all self-similar Gaussian random fields with stationary increments. Several integral representations of the introduced random fields are derived. In a similar vein, several non-Euclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameters of the Poisson and Brownian variants are related by convex geometry transforms, namely the radial $p$th mean body and the polar projection transforms.