Stochastic regularity of a quadratic observable of high-frequency waves

We consider high-frequency waves satisfying the scalar wave equation with highly oscillatory initial data. The wave speed, and the phase and amplitude of the initial data are assumed to be uncertain, described by a finite number of random variables with known probability distributions. We define quantities of interest (QoIs), or observables, as local averages of the squared modulus of the wave solution. We aim to quantify the regularity of these QoIs in terms of the input random parameters, and the wave length, i.e., to estimate the size of their derivatives. The regularity is important for uncertainty quantification methods based on interpolation in the stochastic space. In particular, the size of the derivatives should be bounded independently of the wave length. In this paper, we are able to show that when these QoIs are approximated by Gaussian beam superpositions, they indeed have this property, despite the highly oscillatory character of the waves.