The time horizon for stochastic homogenization of the one-dimensional wave equation
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Date
2023-07
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Abstract
The wave equation with stochastic 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 ε, classical
homogenization fails for times of the order ε−2. We consider the
one-dimensional wave equation and are interested in the critical
time scale ε−β 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 ε−1
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Keywords
wave