Regularity Properties of Gaussian Random Fields and Stochastic Heat Equation on the Sphere

This talk is concerned with sample path regularities of isotropic Gaussian fields and the solution of the stochastic heat equation on the unit sphere ${\mathbb S}$. In the first part, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the high-frequency behavior of its angular power spectrum; we then apply 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 behavior. In the second part, we consider the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on ${\mathbb S}^2$ and establish the exact uniform moduli of continuity of the solution in the time and spatial variable, respectively. This talk is based on joint works with Xiaohong Lan and Domenico Marinucci.Non UBCUnreviewedAuthor affiliation: Michigan State UniversityFacult