Time dependent random fields on spherical non-homogeneous surfaces

TL;DR

This paper introduces a novel class of isotropic, time-dependent random fields on non-homogeneous spheres using time-changed spherical Brownian motion, capturing anisotropies relevant in cosmology.

Contribution

It presents a new mathematical framework for modeling isotropic random fields on non-homogeneous spheres via time-changed spherical Brownian motion and fractional derivatives.

Findings

Model captures anisotropies in cosmology.

Defines a stochastic process on non-homogeneous spheres.

Provides a new approach to isotropic random fields.

Abstract

We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere represented by a time-changed spherical Brownian motion of order \nu \in (0,1] with which some anisotrophies can be captured in Cosmology. This process is a time-changed rotational diffusion (TRD) or the stochastic solution to the equation involving the spherical Laplace operator and a time-fractional derivative of order \nu. TRD is a diffusion on the non-homogeneous sphere and therefore, the spherical coordinates given by TRD represent the coordinates of a non-homogeneous sphere by means of which an isotopic random field is indexed. The time dependent random fields we present in this work is therefore realized through composition and can be viewed as isotropic random field on randomly varying sphere.