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:

• We obtain optimal rates of convergence according to some physical parameters of the problem, as the domain of measurements or the regularity of the initial conditions. For instance, if $\eta \ll 1$ is the dimensionless transversal wavelength, statistical stability is observed in the Itô-Schrödinger regime for smooth initial conditions as soon as the Wigner function is averaged over a spatial domain of typical size $\eta^{1-s_1}$, for $s_1>0$ [8].
  

• We characterize the statistical instabilities by computing the first-order corrector for the Itô-Schrödinger regime scintillation, and obtain for instance that the instabilities come from either simple of double scattering, that is the fraction of the wave that has interacted once or twice with the medium, but not from higher scattering events [2].
  

• For the Schrödinger equation with time-independent random potentials with long-range correlations, we consider the simple scattering contribution to the scintillation and show by computing the first-order scintillation corrector that statistical stability occurs for any dimension and any decay of the correlation function provided the power spectrum of the fluctuations is integrable [1].