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Fractional Stochastic Partial Differential Equation for Random Tangent Fields on the Sphere

Vo V. Anh, Andriy Olenko, Yu Guang Wang

Abstract

This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the Lévy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Loève expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. Approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion.

Fractional Stochastic Partial Differential Equation for Random Tangent Fields on the Sphere

Abstract

This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the Lévy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Loève expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. Approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion.

Paper Structure

This paper contains 11 sections, 9 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Vector Spherical Harmonics
  3. Function Spaces and Vector Spherical Harmonics
  4. Random Tangent Fields and Fractional Brownian Motion
  5. Random Tangent Fields
  6. Vector-valued Fractional Brownian Motion
  7. Fractional Stochastic Partial Differential Equations for Tangent Fields
  8. Fractional Diffusion Operator for Random Tangent Fields
  9. Fractional SPDE and Solution
  10. Covariance Function
  11. Analysis of temporal increments

Key Result

Theorem 4.3

Let $\alpha \in (0,2]$, $\alpha +\gamma \in \lbrack 0,2]$, $\beta \in (0,1]$ and $H \in \lbrack 1/2,1)$. Let Eq. eq:vfSDE.S2 be defined with the fractional diffusion operator eq:fDiffu, fractional Brownian motion eq:fBm, Hurst index $H$ and variances $A_{\ell }^{1}$ and and $A_{\ell }^{2}$ at $t=1$ respectively for $\tau :=\max \{\frac{2}{\beta }(1-\beta - H ),0\}$. Then, the solution to eq:vfSDE

Theorems & Definitions (14)

  • Remark 4.1
  • Remark 4.2
  • Theorem 4.3
  • Corollary 4.4
  • Theorem 4.5
  • Remark 4.6
  • Remark 4.7
  • Corollary 4.8
  • Corollary 4.9
  • Proposition 4.10
  • ...and 4 more