Random Spherical Hyperbolic Diffusion

Abstract

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 Hölder 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.

Spectral Analysis of Fractional Hyperbolic Diffusion Equations with Random Data

“…The case α = 1 is related to the random spherical hyperbolic diffusion equation, see [4] . In this case the isotropic spherical random field has the following spectral representation in the Hilbert space L 2 ( × S 2 , sin θ dθ dϕ) :…”

“…All plots were made using Mathematica 12. In order to compare our results with the ones of [4], let us consider the values c = 1, D = 1 and k = 0.01. Then the quantity A defined in eq.…”

“…(20) equals A = ( √ 10001 − 1)/2. In [4] the authors have used the time variable t = c 2 t/2D, but because of the presence of the parameter α in t α in our expressions, we prefer to use the variable t, so that, for α = 1, we have that t = t/2. Moreover, like in [4], we consider l ≤ 2500.…”

“…1 we have the plots of the multiplication factor F l,α (t) as a function of l for four different times and for three different values of α. The case α = 1 corresponds to the case studied in [4]. In Fig.…”

“…One such model was recently provided by [4], where stochastic hyperbolic diffusion equations were used for the spherical random fields. The hyperbolic heat equation is formally identical to the linear telegraph equation introduced by Heaviside in his study of transmission lines, and it was introduced by Cattaneo [7] in order to impose a bounded speed of propagation for the temperature disturbances, in contrast therefore with the classical parabolic heat equation, that has an unbounded propagation speed.…”