Gaussian coherent-field propagation in the Markov approximation ($m=e^{-\gamma (x-x_0)}{E}_{\mathrm{free}}$)

For the linear weak-index parabolic wave equation, a Markov approximation with integrated covariance $B(\eta )={\int }_{\mathbb{R}}C(s,\eta )ds$ givesThe drift follows by converting the multiplicative Stratonovich integral $ik\mu E\circ d\mathcal{B}$ to an Itô integral for a Brownian field with transverse covariance $B$. For a Gaussian beam with one transverse coordinate entering the random region at $x_0$, $m(x,z)=e^{-\gamma (x-x_0)}{E}_{\mathrm{free}}(x,z)$. The coherent field retains transverse beam structure. The white-noise closure needs short longitudinal correlations and the corresponding separation of propagation scales; Gaussian one-point statistics alone are insufficient.