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Isotropic Q-fractional Brownian motion on the sphere: regularity and fast simulation

Annika Lang, Björn Müller

Abstract

As an extension of isotropic Gaussian random fields and Q-Wiener processes on d-dimensional spheres, isotropic Q-fractional Brownian motion is introduced and sample Hölder regularity in space-time is shown depending on the regularity of the spatial covariance operator Q and the Hurst parameter H. The processes are approximated by a spectral method in space for which strong and almost sure convergence are shown. The underlying sample paths of fractional Brownian motion are simulated by circulant embedding or conditionalized random midpoint displacement. Temporal accuracy and computational complexity are numerically tested, the latter matching the complexity of simulating a Q-Wiener process if allowing for a temporal error.

Isotropic Q-fractional Brownian motion on the sphere: regularity and fast simulation

Abstract

As an extension of isotropic Gaussian random fields and Q-Wiener processes on d-dimensional spheres, isotropic Q-fractional Brownian motion is introduced and sample Hölder regularity in space-time is shown depending on the regularity of the spatial covariance operator Q and the Hurst parameter H. The processes are approximated by a spectral method in space for which strong and almost sure convergence are shown. The underlying sample paths of fractional Brownian motion are simulated by circulant embedding or conditionalized random midpoint displacement. Temporal accuracy and computational complexity are numerically tested, the latter matching the complexity of simulating a Q-Wiener process if allowing for a temporal error.

Paper Structure

This paper contains 11 sections, 8 theorems, 31 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Real-valued stochastic processes and spherical Gaussian random fields
  3. Q-fractional Brownian motion on the sphere
  4. $Q$-fractional Brownian motion on the d-dimensional hypersphere
  5. Efficient simulation of Q-fractional Brownian motion
  6. Spectral approximation in space
  7. Simulation of real-valued fractional Brownian motion
  8. Circulant embedding
  9. CRMD
  10. Comparison of computational performance
  11. Proof of Theorem \ref{['thm:KC_kratschmer_Gaussian']}

Key Result

Theorem 3.2

Let $Q$ satisfy Assumption assump:summability and $H \in (0, 1)$. Then, $Q$-fBm exists with basis expansion where $(\beta_{\ell , m}^H, {\ell \in \mathbb{N}_0, m=-\ell,\ldots,\ell})$ is a sequence of independent real-valued fBms with Hurst parameter $H$. Furthermore, $B_{Q}^{H} \in C^{H^-}(\mathbb{T}; L^2(\mathbb{S}^2))$.

Figures (4)

  • Figure 1: Samples of $Q$-fBm for $H=0.1, 0.5, 0.9$ at time $T= 1, 2, 3$.
  • Figure 2: Visual representation of the CRMD method ($\mu=2, \nu=1$). $X_{3,5}$ is simulatedconditional on $X_{2,3}$, $X_{3,3}$, and $X_{3,4}$,ignoring its dependence on $X_{2,4}$, $X_{3,1}$, and $X_{3,2}$.
  • Figure 3: Empirical decay of the error of CRMD.
  • Figure 4: Performance comparison of CE and CRMD.

Theorems & Definitions (13)

  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Corollary 3.5
  • proof
  • Lemma 3.6
  • proof
  • Remark 3.7
  • proof : Proof of Theorem \ref{['thm:fBm_space_time_reg']}
  • ...and 3 more