Anisotropic Gaussian random fields: Criteria for hitting probabilities and applications

TL;DR

This paper establishes new criteria for the hitting probabilities of anisotropic Gaussian random fields, providing bounds using capacity and Hausdorff measure, and applies these to stochastic PDEs with fractional and colored noises.

Contribution

It introduces generalized criteria for hitting probabilities of anisotropic Gaussian fields, extending classical estimates with capacity and Hausdorff measure, and applies these to complex stochastic PDE systems.

Findings

Derived bounds for hitting probabilities using capacity and Hausdorff measure.

Extended classical estimates to anisotropic Gaussian fields.

Applied criteria to stochastic PDEs with fractional and colored noises.

Abstract

We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel-Riesz capacity and the $\gamma$-dimensional Hausdorff measure. We apply the criteria to a system of linear stochastic partial differential equations driven by space-time noises that are fractional in time and either white or colored in space.