The time horizon for stochastic homogenization of the one-dimensional wave equation

Abstract

The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity $\epsilon$ , classical homogenization fails for times of the order ${\epsilon }^{-2}$ . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale $\epsilon$ and are interested in the critical time scale ${\epsilon }^{-\beta }$ from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ${\epsilon }^{-1}$ .