Uniform dimension results for Gaussian random fields  

作　　者：WU DongSheng XIAO YiMin 

机构地区：[1]Department of Mathematical Sciences,University of Alabama in Huntsville,Huntsville,AL 35899,USA [2]Department of Statistics and Probability,Michigan State University,East Lansing,MI 48824,USA [3]College of Mathematics and Computer Science,Anhui Normal University,Wuhu 241000,China

出　　处：《Science China Mathematics》2009年第7期1478-1496,共19页中国科学：数学（英文版）

基　　金：supported by National Science Foundation of the United States (Grant No.DMS-0706728)

摘　　要：Let X = {X(t), t ∈ ? N } be a Gaussian random field with values in ? d defined by (1) $$ X(t) = (X_1 (t),...,X_d (t)),\forall t \in \mathbb{R}^N . $$ . The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed. It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X.When X is an (N, d)-Gaussian random field as in (1), where X 1,...,X d are independent copies of a real valued, centered Gaussian random field X 0 which is anisotropic in the time variable. We establish uniform Hausdorff dimension results for the image sets of X. These results extend the corresponding results on one-dimensional Brownian motion, fractional Brownian motion and the Brownian sheet.Let X = {X(t),t ∈ RN} be a Gaussian random field with values in Rd defined by X(t) =(X1(t),...,Xd(t)), t ∈ RN.(1) The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed.It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X.When X is an(N,d)-Gaussian random field as in(1),where X1,...,Xd are independent copies of a real valued,centered Gaussian random field X0 which is anisotropic in the time variable.We establish uniform Hausdorff dimension results for the image sets of X.These results extend the corresponding results on one-dimensional Brownian motion,fractional Brownian motion and the Brownian sheet.

关 键 词：anisotropic Gaussian random fields sectorial local nondeterminism IMAGE Hausdorff dimension 60G15 60G17 60G60 42B10 43A46 28A80 

分 类 号：O211.6[理学—概率论与数理统计]