# Stochastic Green-Gauss Theorem

Is there a stochastic analog for the Green-Gauss theorem? I'm looking for an expression that relates the flux (or statistical moments of the flux) through a random surface to the divergence of the random field.

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what kind of relation are you after? the flux is linearly related to the divergence, so the p-th moment of the flux is directly related to the p-point correlation function of the divergence. –  Carlo Beenakker Sep 23 '11 at 13:09

Since the flux is linearly related to the field, if the field is Gaussian with zero mean, the mean flux vanishes. The variance of the flux $\Phi$ through a surface $\delta V$ is given by a double integral over the enclosed volume $V$,
Var $\Phi = \int dr \int dr'\;\sum_{i,j} \frac{\partial^2 M_{ij}(r,r')}{\partial r_i\partial r'_j}$
where $M_{ij}(r,r')=\langle E_i(r)E_j(r')\rangle$ is the two-point correlation function of the random vector field $E(r)$ (with components $E_i$).