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MULTIPLE INTERSECTIONS OF SPACE-TIME ANISOTROPIC GAUSSIAN FIELDS*
Zhenlong Chen, Weijie Yuan
数学物理学报(英文版). 2024 (1):
275-294.
DOI: 10.1007/s10473-024-0115-1
Let X={X(t)∈Rd,t∈RN} be a centered space-time anisotropic Gaussian field with indices H=(H1,⋯,HN)∈(0,1)N, where the components Xi (i=1,⋯,d) of X are independent, and the canonical metric √E(Xi(t)−Xi(s))2 (i=1,⋯,d) is commensurate with γαi(N∑j=1|tj−sj|Hj) for s=(s1,⋯,sN),t=(t1,⋯,tN)∈RN, αi∈(0,1], and with the continuous function γ(⋅) 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 γ(⋅). 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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