Local times of anisotropic Gaussian random fields and stochastic heat equation

TL;DR

This paper investigates the local times of anisotropic Gaussian random fields and stochastic heat equations, establishing moment estimates, path properties, and geometric characteristics of level sets using advanced probabilistic and geometric tools.

Contribution

It introduces new moment estimates and geometric methods for analyzing local times of anisotropic Gaussian fields and applies these to stochastic heat equations to characterize their level sets.

Findings

Established moment estimates for local times.

Derived sample path properties related to Chung's law.

Determined Hausdorff measure functions for solution level sets.

Abstract

We study the local times of a large class of Gaussian random fields satisfying strong local nondeterminism with respect to an anisotropic metric. We establish moment estimates and H\"{o}lder conditions for the local times of the Gaussian random fields. Our key estimates rely on geometric properties of Voronoi partitions with respect to an anisotropic metric and the use of Besicovitch's covering theorem. As a consequence, we deduce sample path properties of the Gaussian random fields that are related to Chung's law of the iterated logarithm and modulus of non-differentiability. Moreover, we apply our results to systems of stochastic heat equations with additive Gaussian noise and determine the exact Hausdorff measure function with respect to the parabolic metric for the level sets of the solutions.