Local times of anisotropic Gaussian random fields and stochastic heat equation

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 proper...

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Bibliographic Details
Published inarXiv.org
Main Authors Cheuk Yin Lee, Xiao, Yimin
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 30.10.2023
Subjects
Estimates
Fields (mathematics)
Random noise
Stochastic processes
Thermodynamics
Voronoi graphs
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Summary: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.
ISSN:2331-8422