On intersections of independent anisotropic Gaussian random fields |
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Authors: | ZhenLong Chen YiMin Xiao |
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Affiliation: | 1. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China 2. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA
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Abstract: | Let XH = {XH(s),s ∈RN1} and X K = {XK(t),t ∈R N2} be two independent anisotropic Gaussian random fields with values in R d with indices H =(H1,...,HN1) ∈(0,1)N1,K =(K1,...,KN2) ∈(0,1) N2,respectively.Existence of intersections of the sample paths of X H and X K is studied.More generally,let E1■RN1,E2■RN2 and FRd be Borel sets.A necessary condition and a sufficient condition for P{(XH(E1)∩XK(E2))∩F≠Ф}>0 in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1×E2×F in the metric space(RN1+N2+d,) are proved,where is a metric defined in terms of H and K.These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets. |
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Keywords: | intersection anisotropic Gaussian fields hitting probability Hausdorff dimension stochastic heat equation fractional Brownian sheet |
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