Polar sets of anisotropic Gaussian random fields

TL;DR

This paper investigates the conditions under which anisotropic Gaussian random fields almost surely do not hit certain sets, establishing bounds on hitting probabilities and characterizing polar sets via Hausdorff dimension.

Contribution

It provides new bounds on hitting probabilities for anisotropic Gaussian fields and extends results to translated fields with bounded Hölder norm.

Findings

Sets with small Hausdorff dimension are polar.

Upper bounds for hitting probabilities are established.

Results apply to translated Gaussian fields with independent bounded Hölder norm fields.

Abstract

This paper studies polar sets of anisotropic Gaussian random fields, i.e. sets which a Gaussian random field does not hit almost surely. The main assumptions are that the eigenvalues of the covariance matrix are bounded from below and that the canonical metric associated with the Gaussian random field is dominated by an anisotropic metric. We deduce an upper bound for the hitting probabilities and conclude that sets with small Hausdorff dimension are polar. Moreover, the results allow for a translation of the Gaussian random field by a random field, that is independent of the Gaussian random field and whose sample functions are of bounded H\"older norm.