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MULTIPLE INTERSECTIONS OF SPACE-TIME ANISOTROPIC GAUSSIAN FIELDS*
Zhenlong Chen, Weijie Yuan
Acta mathematica scientia,Series B. 2024, 44 (1):
275-294.
DOI: 10.1007/s10473-024-0115-1
Let $X=\{X(t)\in\mathbb{R}^{d},t\in\mathbb{R}^{N}\}$ be a centered space-time anisotropic Gaussian field with indices $H=(H_{1},\cdots ,H_{N})\in(0,1)^{N}$, where the components $X_{i}\ (i=1,\cdots ,d)$ of $X$ are independent, and the canonical metric $\sqrt{\mathbb{E}(X_{i}(t)-X_{i}(s))^{2}}\ (i=1,\cdots ,d)$ is commensurate with $\gamma^{\alpha_{i}}(\sum\limits_{j=1}^{N}|t_{j}-s_{j}|^{H_{j}})$ for $s=(s_{1},\cdots ,s_{N}), t=(t_{1},\cdots ,t_{N})\in\mathbb{R}^{N}$, $\alpha_{i}\in(0,1]$, and with the continuous function $\gamma(\cdot)$ satisfying certain conditions. First, the upper and lower bounds of the hitting probabilities of $X$ can be derived from the corresponding generalized Hausdorff measure and capacity, which are based on the kernel functions depending explicitly on $\gamma(\cdot)$. Furthermore, the multiple intersections of the sample paths of two independent centered space-time anisotropic Gaussian fields with different distributions are considered. Our results extend the corresponding results for anisotropic Gaussian fields to a large class of space-time anisotropic Gaussian fields
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