  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.
