= Mutual coherence in a white-noise parabolic medium
{title2=$\Gamma_x=\frac{i}{2k}(\Delta_1-\Delta_2)\Gamma-k^2\mu^2[C(0)-C(\mathbf s_1-\mathbf s_2)]\Gamma$}
A <parabolic wave equation> driven by longitudinal <Gaussian white noise> with transverse <covariance kernel> $C$ has the displayed closed <mutual coherence> equation. The noise is the derivative of a <longitudinal Brownian field with transverse covariance>; the physical smooth-noise limit uses the <Stratonovich integral>. Conversion to the <Itô integral> supplies the attenuation drift and the cross variation supplies the $C(\mathbf s_1-\mathbf s_2)$ term. For unit plane-wave incidence and transverse <statistical homogeneity>, $\Gamma=\exp[-k^2\mu^2x(C(0)-C(\mathbf s_1-\mathbf s_2))]$. General Gaussian colored media do not have this local closure.
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