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.
 A: 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 $E_i$).
A: 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.
