  Navier-Stokes equations in the whole space R^3 subject to an anisotropic
viscosity and a random perturbation of multiplicative type is described. By
adding a term of Brinkman-Forchheimer type to the model, existence and
uniqueness of global weak solutions in the PDE sense are proved. These are
strong solutions in the probability sense. The convective term given in terms
of the Brinkman-Forchheirmer provides some extra regularity in the space
L^{2$\alpha$+2} (R^3), with $\alpha$ > 1. As a consequence, the nonlinear term
has better properties which allows to prove uniqueness. The proof of existence
is performed through a control method. A Large Deviations Principle is given
and proven at the end of the paper.
