  The evolution Stokes equation in a perforated domain subject to Fourier
boundary condition on the boundaries of the holes is considered. We assume that
the dynamic is driven by a stochastic perturbation on the interior of the
domain and another stochastic perturbation on the boundaries of the holes. The
macroscopic (homogenized) equation is derived as another stochastic partial
differential equation, defined in the whole non perforated domain. Here, the
initial stochastic perturbation on the boundary becomes part of the homogenized
equation as another stochastic force. We use the two-scale convergence method
after extending the solution with 0 in the wholes to pass to the limit. By
It\^o stochastic calculus, we get uniform estimates on the solution in
appropriate spaces. In order to pass to the limit on the boundary integrals, we
rewrite them in terms of integrals in the whole domain. In particular, for the
stochastic integral on the boundary, we combine the previous idea of rewriting
it on the whole domain with the assumption that the Brownian motion is of trace
class. Due to the particular boundary condition dealt with, we get that the
solution of the stochastic homogenized equation is not divergence free.
However, it is coupled with the cell problem that has a divergence free
solution. This paper represents an extension of the results from of Duan and
Wang (Comm. Math. Phys. 275:1508--1527, 2007), where a reaction diffusion
equation with a dynamical boundary condition with a noise source term on both
the interior of the domain and on the boundary was studied, and through a
tightness argument and a pointwise two scale convergence method the homogenized
equation was derived.
