独立随机场的多重相交性与Hausdorff维数  

Multiple intersections of independent random fields and Hausdorff dimension

作　　者：陈振龙[1] 

机构地区：[1]浙江工商大学统计与数学学院,杭州310018

出　　处：《中国科学：数学》2016年第9期1279-1304,共26页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11371321)资助项目

摘　　要：设X^H={X^H(s),s∈R^(N1)}和X^K={X^K(t),t∈R^(N2)}是指数分别为H∈(0,1)^(N1)和K∈(0,1)^(N2)、取值于R^d的两个独立的各向异性随机场,它们可以不具有Gauss性.在某些一般条件下,本文研究了X^H和X^K的样本轨道相交的存在性.更一般地,对任意的Borel集F?R^d,本文给出了F包含X^H和X^K相交点的一些充分和必要条件,确定了X^H与X^K在F中的相交时集的Hausdorff维数.这些结果适用于带有乘性噪声的非线性随机热方程的非Gauss解和双分数Brown单.Let XH = {XH（s）,s C RN1} and XK = {xK （t）,t C RN2} be two independent anisotropic random fields, which may not possess the Gaussianity, with values in Rd with indices H C （0, 1）N1, K E （0, 1）N2, respectively. Under certain general conditions, the existence of intersections of the sample paths of XH and Xn is investigated. More generally, for any Borel set F Rd, some necessary conditions and sufficient conditions for F to contain intersection points of Xg and XK are established, and the Hausdorff dimension of intersection times such that F contains intersection points of XH and XK is determined. These results are applicable to non- Gaussian solutions of non-linear stochastic heat equations with multiplicative noise and bifractional Brownian sheets.

关 键 词：多重相交性 各向异性随机场 容度 HAUSDORFF维数 随机热方程 双分数Brown单 

分 类 号：O211.6[理学—概率论与数理统计]