Self-averaging of kinetic models for waves in random media

TL;DR

This paper investigates the statistical stability of wave energy density in random media using kinetic models, demonstrating conditions under which the energy density self-averages and providing explicit formulas in high frequency regimes.

Contribution

It provides optimal estimates for the statistical instability of wave energy density and characterizes self-averaging behavior in the Itô-Schrödinger regime.

Findings

Energy density is asymptotically statistically stable in many configurations.

Explicit asymptotic expression for scintillation function in high frequency limit.

Quantification of kinetic model limitations via statistical instability analysis.

Abstract

Kinetic equations are often appropriate to model the energy density of high frequency waves propagating in highly heterogeneous media. The limitations of the kinetic model are quantified by the statistical instability of the wave energy density, i.e., by its sensitivity to changes in the realization of the underlying heterogeneous medium modeled as a random medium. In the simplified It\^o-Schr\"odinger regime of wave propagation, we obtain optimal estimates for the statistical instability of the wave energy density for different configurations of the source terms and the domains over which the energy density is measured. We show that the energy density is asymptotically statistically stable (self-averaging) in many configurations. In the case of highly localized source terms, we obtain an explicit asymptotic expression for the scintillation function in the high frequency limit.