Science in China Series A: Mathematics

, Volume 52, Issue 7, pp 1478-1496

First online:

Uniform dimension results for Gaussian random fields

  • DongSheng WuAffiliated withDepartment of Mathematical Sciences, University of Alabama in Huntsville Email author 
  • , YiMin XiaoAffiliated withDepartment of Statistics and Probability, Michigan State UniversityCollege of Mathematics and Computer Science, Anhui Normal University

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Let X = {X(t), t ∈ ℝ N } be a Gaussian random field with values in ℝ d defined by
$$ X(t) = (X_1 (t),...,X_d (t)),\forall t \in \mathbb{R}^N . $$
. 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.


anisotropic Gaussian random fields sectorial local nondeterminism image Hausdorff dimension


60G15 60G17 60G60 42B10 43A46 28A80