= Solution
Write $\psi=e^{ikx}E(x,\mathbf s)$, $\mathbf s=(y,z)$. In the weak-contrast approximation $n^2\simeq1+2\mu W$, neglecting $E_{xx}$ relative to $kE_x$ gives the <parabolic wave equation>
$$
2ikE_x+\Delta_\perp E+2k^2\mu WE=0,
\qquad E_x=\frac{i}{2k}\Delta_\perp E+ik\mu WE.
$$
For the <mutual coherence> $\Gamma(x;\mathbf s_1,\mathbf s_2)=\mathbb E[E_1E_2^*]$, differentiating gives the exact moment equation within this wave approximation,
$$
\boxed{\Gamma_x=\frac{i}{2k}(\Delta_1-\Delta_2)\Gamma
+ik\mu\,\mathbb E[(W_1-W_2)E_1E_2^*].}
$$
This is an evolution equation but not a closed one. Even a jointly <Gaussian random field> does not make $W$ independent of $E$, which depends on its history. Transverse <statistical homogeneity> alone therefore does not close the second moment. A covariance or longitudinal Markov approximation is additional information.
Under the white-in-$x$ assumption introduced in the next part, interpret $W\,dx=d\mathcal B_x$, a <longitudinal Brownian field with transverse covariance> $C$, and use the <Stratonovich integral> as the smooth-medium limit. Conversion to the <Itô integral> yields
$$
dE=\left(\frac{i}{2k}\Delta_\perp E-\frac12k^2\mu^2C(0)E\right)dx
+ik\mu E\,d\mathcal B_x.
$$
The product rule with quadratic variation gives the closed <mutual coherence in a white-noise parabolic medium> equation,
$$
\boxed{\Gamma_x=\frac{i}{2k}(\Delta_1-\Delta_2)\Gamma
-k^2\mu^2[C(0)-C(\mathbf s_1-\mathbf s_2)]\Gamma.}
$$
The two attenuation drifts contribute $-k^2\mu^2C(0)$ and the cross variation contributes $+k^2\mu^2C(\mathbf s_1-\mathbf s_2)$. The coefficient $C$ is the white-noise covariance intensity; it replaces, rather than literally retains, the original finite pointwise variance of $W$.
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