Is there a stochastic analog for the GreenGauss 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.

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 twopoint correlation function of the random vector field $E(r)$ (with components $E_i$). 


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 GreenGauss 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. 

