Spherically Restricted Random Hyperbolic Diffusion

Abstract

This paper investigates solutions of hyperbolic diffusion equations in R 3 with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere S 2 are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the meansquare convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the Hölder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings.Stochastic partial differential equations; Hyperbolic diffusion equation; Spherical random field; Hölder continuity; Long-range dependence; Approximation errors; Cosmic microwave background arXiv:1912.08378v1 [math.PR]

On Multifractionality of Spherical Random Fields with Cosmological Applications

“…-Develop the distribution theory for the estimators of H(t); -Investigate changes of the Hölder exponents depending on evolutions of random fields driven by SPDEs on the sphere, see [27,46,50,51];…”

On Spectral Theory of Random Fields in the Ball

“…-Study rates of convergence in these spectral series representations; -Extend the developed spectral theory to spatio-temporal fields; -Apply the obtained series expansions to investigate evolutions of random fields in the ball driven by SPDEs , see the corresponding results for spherical random fields in [3,4,10]; -Apply the developed methodology to real data, in particular, to new highresolution cosmological data from future CMB-S4 and Euclid mission surveys.…”

“…Professor Yadrenko was one of pioneering researchers and leading figures in developing this theory. Later on, it was demonstrated that the behaviour of the power angular spectrum determines various properties of these fields and evolutions of their spatio-temporal counterparts, see [3,4,10]. However, the known results about spherical fields are not directly translatable to the random fields defined in the ball.…”

On approximation for fractional stochastic partial differential equations on the sphere

“…Various fractional versions of SPDEs have also been constructed in diverse applications [6,7,3,9,18,20,26,27,28,32,34]. A distinct merit of these fractional models is that they can be used to maintain long-range dependence in the evolution of complex systems, such as models of climate change and density fluctuations in the primordial universe as inferred from the cosmic microwave background (CMB) [2,13,14,15,39,40,41].…”