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On approximation for time-fractional stochastic diffusion equations on the unit sphere

T. Alodat, Q. T. Le Gia, I. H. Sloan

Abstract

This paper develops a two-stage stochastic model to investigate evolution of random fields on the unit sphere $\bS^2$ in $\R^3$. The model is defined by a time-fractional stochastic diffusion equation on $\bS^2$ governed by a diffusion operator with the time-fractional derivative defined in the Riemann-Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on $\bS^2$ as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on $\bS^2$. The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree $L\geq1$. The rate of convergence of the truncation errors as a function of $L$ and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic Hölder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.

On approximation for time-fractional stochastic diffusion equations on the unit sphere

Abstract

This paper develops a two-stage stochastic model to investigate evolution of random fields on the unit sphere in . The model is defined by a time-fractional stochastic diffusion equation on governed by a diffusion operator with the time-fractional derivative defined in the Riemann-Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on . The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree . The rate of convergence of the truncation errors as a function of and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic Hölder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.
Paper Structure (15 sections, 24 theorems, 236 equations, 6 figures)

This paper contains 15 sections, 24 theorems, 236 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functions on the sphere
  4. Isotropic random fields on the sphere
  5. Brownian motion
  6. Stochastic integrals of Mittag-Leffler functions
  7. Solution of the homogeneous problem
  8. Solution of the inhomogeneous problem
  9. Solution of the time-fractional diffusion equation
  10. Approximation to the solution
  11. Temporal increments and sample Hölder continuity of solution
  12. Numerical studies
  13. Evolution of the solution
  14. Simulations of truncation errors of solution
  15. Simulations of temporal increments

Key Result

Proposition 2.1

LanSch15 Let $\{\mathcal{C}_{\ell}: \ell\in\mathbb{N}_{0}\}$ be the angular power spectrum of a centered, $2$-weakly isotropic Gaussian random random field $\xi$ on $\mathbb{S}^2$. Then $\xi$ admits the expansion which is convergent in the $L_{2}(\Omega\times\mathbb{S}^2)$ sense, where $Y_{\ell,m},\ \ell\in\mathbb{N}_{0}, m=-\ell,\dots,\ell$, are spherical harmonic functions and the elements of t

Figures (6)

  • Figure 1: A realization of the truncated initial condition $U_{600}(0)$ with $L=600$, $\alpha=0.5$ and $\kappa_1=2.30$.
  • Figure 2: Truncated homogeneous solutions $U_{600}(\tau)=U^H_{600}(\tau)$ and $U^H_{600}(10\tau)$, using the initial realization in Figure \ref{['fig:U400 at 0']}, with $\alpha=0.5$, $\kappa_1=2.3$, and $\kappa_2=2.5$.
  • Figure 3: (a) The truncated inhomogeneous solution $U_{L}^I(10\tau)$ with $\tau=10^{-5}$, $\widetilde{A}=10^4$, $\alpha=0.5$ and $\kappa_2=2.5$, (b) the truncated combined solution $U_{600}(10\tau)$, using the truncated homogeneous and inhomogeneous solutions in Figures \ref{['HOM2']} and \ref{['inHOM1']}.
  • Figure 4: Numerical errors for the combined solution $U(t)$ with $\kappa_1=2.3$, $\kappa_2=2.5$, $\alpha=0.5$.
  • Figure 5: Numerical errors for the combined solution $U(t)$ with $\kappa_1=2.3$, $\kappa_2=2.5$, $\alpha=0.75$.
  • ...and 1 more figures

Theorems & Definitions (56)

  • Proposition 2.1
  • Remark 2.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Definition 2.2
  • Proposition 2.4
  • proof
  • Remark 2.2
  • ...and 46 more