Strong Local Nondeterminism and Exact Modulus of Continuity for Spherical Gaussian Fields

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on ${\mathrm{\S }}^2$. 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.