Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 76 / 1 / c / Solution

{c}
{title2=$m=e^{-\gamma(x-x_0)}E_{\mathrm{free}}$}

For the linear weak-index <parabolic wave equation>, a <Markov approximation> with integrated covariance $B(\eta)=\int_{\mathbb R}C(s,\eta)ds$ gives
$$
 m_x=\frac{i}{2k}m_{zz}-\gamma m,\qquad \gamma=\frac{k^2\mu^2B(0)}2.
$$
The 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.

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 Incoming links: Gaussian coherent-field propagation in the Markov approximation