On the Self-Averaging of Wave Energy in Random Media

We consider the stabilization (self-averaging) and destabilization of the energy of waves propagating in random media. Propagation is modeled here by an Itô Schrödinger equation. The explicit structure of the resulting transport equations for arbitrary statistical moments of the wave field is used to show that wave energy density may be stable in the high frequency regime, in the sense that it only depends on the statistics of the random medium and not on the specific realization. Stability is conditional on having sufficiently smooth initial energy distributions. We show that wave energy is not stable, and instead scintillation is created by the wave dynamics, when the initial energy distribution is sufficiently singular. Application to time reversal of high frequency waves is also considered.