  The paper starts by giving a motivation for this research and justifying the
considered stochastic diffusion models for cosmic microwave background
radiation studies. Then it derives the exact solution in terms of a series
expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy
problem with random initial conditions is studied. All assumptions are stated
in terms of the angular power spectrum of the initial conditions. An
approximation to the solution is given and analysed by finitely truncating the
series expansion. The upper bounds for the convergence rates of the
approximation errors are derived. Smoothness properties of the solution and its
approximation are investigated. It is demonstrated that the sample H\"older
continuity of these spherical fields is related to the decay of the angular
power spectrum. Numerical studies of approximations to the solution and
applications to cosmic microwave background data are presented to illustrate
the theoretical results.
