The Hausdorff measure of the range and level sets of Gaussian random fields with sectorial local nondeterminism

TL;DR

This paper precisely characterizes the Hausdorff measure of the range and level sets of certain Gaussian fields with sectorial local nondeterminism, and establishes a Chung-type law of the iterated logarithm, with applications to Brownian sheets and stochastic wave equations.

Contribution

It provides exact Hausdorff measure functions for Gaussian fields with sectorial local nondeterminism and proves a Chung-type law of the iterated logarithm, extending understanding of these stochastic processes.

Findings

Exact Hausdorff measure functions for Gaussian fields with sectorial local nondeterminism.

Chung-type law of the iterated logarithm established for these fields.

Applications to Brownian sheets and stochastic wave equations.

Abstract

We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space-time white noise or colored noise.