Hausdorff Measure and Uniform Dimension for Space-Time Anisotropic Gaussian Random Fields

Weijie Yuan () and Zhenlong Chen ()
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Weijie Yuan: Zhejiang Gongshang University
Zhenlong Chen: Zhejiang Gongshang University

Journal of Theoretical Probability, 2024, vol. 37, issue 3, 2304-2329

Abstract: Abstract Let $$X=\{ X(t), t\in \mathbb {R}^{N}\} $$ X = { X ( t ) , t ∈ R N } be a centered space-time anisotropic Gaussian random field in $$\mathbb {R}^d$$ R d with stationary increments, where the components $$X_{i}(i=1,\ldots ,d)$$ X i ( i = 1 , … , d ) are independent but distributed differently. Under certain conditions, we not only give the Hausdorff dimension of the graph sets of X in the asymmetric metric in the recurrent case, but also determine the exact Hausdorff measure functions of the graph sets of X in the transient and recurrent cases, respectively. Moreover, we establish a uniform Hausdorff dimension result for the image sets of X. Our results extend the corresponding results on fractional Brownian motion and space or time anisotropic Gaussian random fields.

Keywords: Hausdorff measure; Uniform dimension; Graph; Image; Anisotropic Gaussian field; Strong local nondeterminism; 60G15; 60G17; 60G60 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-024-01323-7

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