Science in China Series A: Mathematics

, Volume 52, Issue 7, pp 1478–1496

Uniform dimension results for Gaussian random fields



Let X = {X(t), t ∈ ℝN} be a Gaussian random field with values in ℝd 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 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.


anisotropic Gaussian random fields sectorial local nondeterminism image Hausdorff dimension 


60G15 60G17 60G60 42B10 43A46 28A80 


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© Science in China Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Alabama in HuntsvilleHuntsvilleUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA
  3. 3.College of Mathematics and Computer ScienceAnhui Normal UniversityWuhuChina

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