On Spectral Theory of Random Fields in the Ball

TL;DR

This paper develops a spectral theory framework for various types of random fields in the ball, providing new representations and applications relevant to cosmology, geosciences, and embryology.

Contribution

It introduces new spectral representations for scalar, spin, and vector random fields in the ball, expanding theoretical understanding and practical modeling capabilities.

Findings

Derived spectral representations for different random field types

Applications demonstrated using Matérn models

Potential for improved simulation and analysis of ball data

Abstract

The paper investigates random fields in the ball. It studies three types of such fields: restrictions of scalar random fields in the ball to the sphere, spin, and vector random fields. The review of the existing results and new spectral theory for each of these classes of random fields are given. Examples of applications to classical and new models of these three types are presented. In particular, the Mat\'{e}rn model is used for illustrative examples. The derived spectral representations can be utilised to further study theoretical properties of such fields and to simulate their realisations. The obtained results can also find various applications for modelling and investigating ball data in cosmology, geosciences and embryology.