Abstract
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 \(\sqrt {{{\mathbb{E}({X_i}(t) - {X_i}(s))}^2}} \,(i = 1, \cdots ,d)\) is commensurate with \({\gamma ^{{\alpha _i}}}(\sum\limits_{j = 1}^N {|{t_j} - {s_j}{|^{{H_j}}})} \) for s = (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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This research was supported by the National Natural Science Foundation of China (12371150, 11971432), the Natural Science Foundation of Zhejiang Province (LY21G010003), the Management Project of “Digital+” Discipline Construction of Zhejiang Gongshang University (SZJ2022A012, SZJ2022B017), the Characteristic & Preponderant Discipline of Key Construction Universities in Zhejiang Province (Zhejiang Gongshang University-Statistics) and the Scientific Research Projects of Universities in Anhui Province (2022AH050955).
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Chen, Z., Yuan, W. Multiple intersections of space-time anisotropic Gaussian fields. Acta Math Sci 44, 275–294 (2024). https://doi.org/10.1007/s10473-024-0115-1
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DOI: https://doi.org/10.1007/s10473-024-0115-1