A Minicourse on Stochastic Partial Differential Equations

Volume 1962 of the series Lecture Notes in Mathematics pp 145-212

Sample Path Properties of Anisotropic Gaussian Random Fields

  • Yimin XiaoAffiliated withDepartment of Statistics and Probability, Michigan State University Email author 

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Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation.

This paper is concerned with sample path properties of anisotropic Gaussian random fields in general. Let \(X = \left\{ {X\left( t \right),t \in {\rm{R}}^N } \right\}\) be a Gaussian random field with values in Rd and with parameters H1,…,HN. Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly.

Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random fields. An important tool for our study is the various forms of strong local nondeterminism.