Criteria for hitting probabilities with applications to systems of stochastic wave equations

TL;DR

This paper establishes bounds on hitting probabilities of random fields, emphasizing the influence of parameter space dimension, and applies these findings to analyze a system of stochastic wave equations driven by spatially homogeneous Gaussian noise.

Contribution

It introduces new bounds on hitting probabilities using Hausdorff measure and Bessel-Riesz capacity, specifically tailored for stochastic wave equations with spatially homogeneous noise.

Findings

Derived upper and lower bounds for hitting probabilities

Applied bounds to stochastic wave equations in various spatial dimensions

Highlighted the impact of parameter space dimension on hitting probabilities

Abstract

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel--Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension $k\ge1$ driven by a $d$-dimensional spatially homogeneous additive Gaussian noise that is white in time and colored in space.