  A theory for the characterization of the fourth moment of electromagnetic
wave beams is presented in the case when the source is partially coherent. A
Gaussian-Schell model is used for the partially coherent random source. The
white-noise paraxial regime is considered, which holds when the wavelength is
much smaller than the correlation radius of the source, the beam radius of the
source, and the correlation length of the medium, which are themselves much
smaller than the propagation distance. The complex wave amplitude field can
then be described by the It\^o-Schr\"odinger equation. This equation gives
closed evolution equations for the wave field moments at all orders and here
the fourth order equations are considered. The general fourth moment equations
are solved explicitly in the scintillation regime (when the correlation radius
of the source is of the same order as the correlation radius of the medium, but
the beam radius is much larger) and the result gives a characterization of the
intensity covariance function. The form of the intensity covariance function
derives from the solution of the transport equation for the Wigner distribution
associated with the second wave moment. The fourth moment results for polarized
waves is used in an application to imaging of partially coherent sources.
