Polar Functions for Anisotropic Gaussian Random Fields

Zhenlong Chen

doi:10.1155/2014/947171

Abstract

LetXbe an (N,d)-anisotropic Gaussian random field. Under some general conditions onX, we establish a relationship between a class of continuous functions satisfying the Lipschitz condition and a class of polar functions ofX. We prove upper and lower bounds for the intersection probability for a nonpolar function andXin terms of Hausdorff measure and capacity, respectively. We also determine the Hausdorff and packing dimensions of the times set for a nonpolar function intersectingX. 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.

Highlights

Gaussian random fields have been extensively studied in probability theory and applied in a wide range of scientific areas including physics, engineering, hydrology, biology, economics, and finance
It has been known that the sample path properties such as fractal dimensions of these anisotropic Gaussian random fields can be very different from those of isotropic ones such as Levy’s fractional Brownian motion; see, for example, [3,4,5,6,7]
These results in this paper are applicable to solutions of SPDEs such as the linear string process considered by Mueller and Tribe [6], linear hyperbolic SPDEs considered by Dalang and Nualart [10], and nonlinear stochastic heat equations considered by Dalang et al [8]

Summary

Introduction

Gaussian random fields have been extensively studied in probability theory and applied in a wide range of scientific areas including physics, engineering, hydrology, biology, economics, and finance. Le Gall [13] made a further discussion for the d-dimensional Brownian motion and proposed an open problem about the existence of its no-polar continuous function satisfying the Holder condition. Some of these results have been extended partially to fractional Brownian motion with stationary increments by Xiao [14], to the Brownian sheet with independent increments by Chen [15], and recently to the fractional Brownian sheets with anisotropy by Chen [4]. More specific constants in Section i are numbered as ci., ci.2,

Some Preliminary Estimates

Characteristics of Polar Functions

Applications to SPDEs