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

Abstract

Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d defined bywhere 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.

Sample paths of the solution to the fractional-colored stochastic heat equation

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L-Kuramoto–Sivashinsky SPDEs vs. time-fractional SPIDEs: Exact continuity and gradient moduli, 1/2-derivative criticality, and laws

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“…Proof. For any fixed ε > 0, let δ ε (·, ·) be defined as in (14), with the corresponding coefficients b ℓm (ε) and κ ℓm (ε) such that conditions (15), (16), (17), (19), (20) and (21) hold. Now we consider…”

“…The analysis of sample path properties of random fields has been considered by many authors, see, for instance, [4,7,14,15,21,22,25,26,30,31] and their combined references. These papers have covered a wide variety of circumstances, including scalar and vector valued random fields, isotropic and anisotropic behaviour, analytic and geometric properties.…”