= Solution
\b[No: the first moment generally depends on the transverse coordinate.] At entry to the <random medium>, the initial data are deterministic, so
$$
m_1(x_0,z)=E_{\mathrm{free}}(x_0,z),
$$
which is a nonconstant <Gaussian beam> profile. <Statistical homogeneity> of the medium says that shifting both the medium and the incident data gives correspondingly shifted field statistics. It does not make the response to fixed, localized incident data translation invariant.
In particular, <Gaussian coherent-field propagation in the Markov approximation>, derived in part (d), gives
$$
m_1(x,z)=e^{-\gamma(x-x_0)}E_{\mathrm{free}}(x,z)
$$
for the linear weak-index model. A homogeneous attenuation factor multiplies the varying beam profile. Thus spatially homogeneous medium statistics can coexist with a transversely inhomogeneous <coherent field>. The first moment is not the <mean wave intensity>, and disappearance of the coherent component is not itself disappearance of the total <wave energy>.
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