Chung-type law of the iterated logarithm and exact moduli of continuity for a class of anisotropic Gaussian random fields

TL;DR

This paper proves a Chung-type law of the iterated logarithm and precise moduli of continuity for anisotropic Gaussian fields, extending existing results without stationary increments assumptions and applying to stochastic PDE solutions.

Contribution

It introduces new Chung-type laws and exact moduli of continuity for anisotropic Gaussian fields, relaxing previous assumptions and providing sharper bounds.

Findings

Established Chung-type law of the iterated logarithm.

Derived exact local and uniform moduli of continuity.

Applicable to solutions of stochastic heat equations driven by fractional-colored Gaussian noise.

Abstract

We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Gaussian noise, including the stochastic heat equation.