Hitting Probabilities and the Hausdorff Dimension of the Inverse Images of a Class of Anisotropic Random Fields  

作　　者：Zhen Long CHEN Quan ZHOU 

机构地区：[1]School of Statistics and Mathematics,Zhejiang Gongshang University

出　　处：《Acta Mathematica Sinica,English Series》2015年第12期1895-1922,共28页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant No.11371321)

摘　　要：Let X = {X（t）：t ∈ R^N} be an anisotropic random field with values in R^d.Under certain conditions on X,we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel-Riesz capacity.We also obtain the Hausdorff dimension of its inverse image,and the Hausdorff and packing dimensions of its level sets.These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.Let X = {X（t）：t ∈ R^N} be an anisotropic random field with values in R^d.Under certain conditions on X,we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel-Riesz capacity.We also obtain the Hausdorff dimension of its inverse image,and the Hausdorff and packing dimensions of its level sets.These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.

关 键 词：Anisotropic random field non-linear stochastic heat equations spatially homogeneous Gaussian noise hitting probabilities Hausdorff dimension inverse image 

分 类 号：O211.63[理学—概率论与数理统计]