Asymptotics and statistical stability

We are intererested in the high-frequency asymptotics of waves propagating in random media within the paraxial approximation. Depending on the physical setting, several models can be derived. The ones we deal with here are described by either a Itô-Schrödinger equation or by a Schrödinger equation with time-independent random potential. The asymptotic analysis is performed by means of Wigner transforms that may be seen as a phase space decomposition of the energy density. It is well known that the ensemble average of the Wigner function satisfies at the limit a radiative transfer equation with a scattering cross-section depending on some statistical properties of the random potential. Much less is known about the limit of the whole process related to the Wigner transform and not only its average. We present here some results of statistical stability, that is situations where the whole process converges to its average. An appropriate tool for the analysis is the scintillation function, which is the covariance function of the Wigner transform and whose convergence to zero implies the convergence in probability. In particular:

Olivier Pinaud 2010-07-30