# 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

## 2 Answers

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 $E_i$).

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Carlo, that is exactly what I wanted to hear. So how is it related? Let's say I have a three dimensional Gaussian random field. How do I determine the mean flux across its boundary? I can brute force calculate this by generating many, many realizations of a 3D Gaussian random field on a computer, using the conventional Green-Gauss theorem on each deterministic realization and then taking an average over all realizations. Does stochastic calculus provide any tool to analytically solve this problem? Thank you.

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Hi Dave Richards, if you register a MathOverflow account, you will be able to add comments to your original answer / edit your post to include clarifications. When you do so, flag your question for moderator attention and comment that you'd like your accounts merged. –  j.c. Sep 23 '11 at 14:59