Spectral Conditions for Strong Local Nondeterminism and Exact Hausdorff Measure of Ranges of Gaussian Random Fields

Abstract

Let X={X(t),t∈ℝN} be a Gaussian random field with values in ℝd defined by $$X(t) = (X_1(t), \ldots, X_d(t)),\quad t \in {\mathbb{R}}^N,$$ where X 1,…,X d are independent copies of a real-valued, centered, anisotropic Gaussian random field X 0 which has stationary increments and the property of strong local nondeterminism. In this paper we determine the exact Hausdorff measure function for the range X([0,1]N).We also provide a sufficient condition for a Gaussian random field with stationary increments to be strongly locally nondeterministic. This condition is given in terms of the spectral measures of the Gaussian random fields which may contain either an absolutely continuous or discrete part. This result strengthens and extends significantly the related theorems of Berman (Indiana Univ. Math. J. 23:69–94, 1973, Stochast. Process. Appl. 27:73–84, 1988), Pitt (Indiana Univ. Math. J. 27:309–330, 1978) and Xiao (Asymptotic Theory in Probability and Statistics with Applications, pp. 136–176, 2007, A Minicourse on Stochastic Partial Differential Equations, Lecture Notes in Math, vol. 1962, pp. 145–212, 2009), and will have wider applicability beyond the scope of the present paper.