On intersections of independent anisotropic Gaussian random fields  

作　　者：CHEN ZhenLong XIAO YiMin 

机构地区：[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

出　　处：《Science China Mathematics》2012年第11期2217-2232,共16页中国科学：数学（英文版）

基　　金：supported by Zhejiang Provincial Natural Science Foundation of China(Grant No. Y6100663);National Science Foundation of US (Grant No. DMS-1006903)

摘　　要：Let X^H = {X^H（8）,8∈ R^N1} and XK = {X^K（t）,t ∈R^2} 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 XH and XK is studied. More generally, let E1 R^N1, E2 R^N2 and F R^d be Borel sets. A necessary condition and a sufficient condition for P{（X^H（E1） ∩ X^K（E2）） ∩ F ≠ Ф} 〉 0 in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 x E2 x F in the metric space （R^N1＋N2＋d, ρ） are proved, whereρ 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.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.

关 键 词：INTERSECTION anisotropic Gaussian fields hitting probability Hausdorff dimension stochastic heatequation fractional Brownian sheet 

分 类 号：O343.8[理学—固体力学] TN911.23[理学—力学]