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Abstract and Applied Analysis
Volume 2014, Article ID 947171, 18 pages
http://dx.doi.org/10.1155/2014/947171
Research Article

Polar Functions for Anisotropic Gaussian Random Fields

School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

Received 13 December 2013; Accepted 2 January 2014; Published 24 March 2014

Academic Editor: Litan Yan

Copyright © 2014 Zhenlong Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let X be an (N, d)-anisotropic Gaussian random field. Under some general conditions on X, we establish a relationship between a class of continuous functions satisfying the Lipschitz condition and a class of polar functions of X. We prove upper and lower bounds for the intersection probability for a nonpolar function and X in terms of Hausdorff measure and capacity, respectively. We also determine the Hausdorff and packing dimensions of the times set for a nonpolar function intersecting X. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise, and the operator-scaling Gaussian random field with stationary increments.