Strong local nondeterminism and exact modulus of continuity for spherical Gaussian fields

Abstract

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on S 2 . In particular, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the high-frequency behaviour of its angular power spectrum; we then exploit this result to establish an exact uniform modulus of continuity for its sample paths. We also discuss the range of values of the spectral index for which the sample functions exhibit fractal or smooth behaviour.

Strong local nondeterminism of spherical fractional Brownian motion

“…This paper provides an important step towards this direction. More specifically, we demonstrate that the coefficients {d ℓ } play the same role as the angular power spectrum of an isotropic Gaussian field on S 2 in [16] and their high frequency behavior determines the property of strong local nondeterminism of SFBM. For this reason, we will also call the sequence {d ℓ , ℓ = 0, 1, .…”

“…Recently, Lan, Marinucci and Xiao [16] have studied the SLND property of a class of Gaussian random fields indexed by the unit sphere S 2 , which are also called spherical Gaussian random fields. The main difference between [16] and the aforementioned work for Gaussian fields indexed by the Euclidean space is that [16] takes the spherical geometry of S 2 into full consideration and its method relies on harmonic analysis on the sphere. More specifically, Lan, Marinucci and Xiao have considered a centered isotropic Gaussian random field…”

“…See [20] for a systematic account on random fields on S 2 . By applying harmonic analytic tools on the sphere, Lan, Marinucci and Xiao [16] have proved that the SLND property of an isotropic Gaussian field T on S 2 is determined by the high-frequency behavior of its angular power spectrum. Moreover, by applying SLND, they have established exact uniform modulus of continuity for a class of isotropic Gaussian fields on S 2 .…”

“…Moreover, by applying SLND, they have established exact uniform modulus of continuity for a class of isotropic Gaussian fields on S 2 . Since SFBM B = B (x) , x ∈ S 2 is not isotropic in the sense of (1), the results on SLND in [16] are not directly applicable. In our approach we will make use of the Karhunen-Loève expansion of the spherical fractional Brownian motion obtained by Istas [13] and derive optimal upper and lower bounds for the coefficients {d ℓ , ℓ = 0, 1, .…”

“…of SFBM. In Section 3, we combine the high frequency behavior of {d ℓ } and Proposition 7 in [16] to establish the property of strong local nondeterminism for SFBM.…”