  We present the variational multiscale (VMS) method for partial differential
equations (PDEs) with stochastic coefficients and source terms. We use it as a
method for generating accurate coarse-scale solutions while accounting for the
effect of the unresolved fine scales through a model term that contains a
fine-scale stochastic Green's function. For a natural choice of an "optimal"
coarse-scale solution and L^2-orthogonal stochastic basis functions, we
demonstrate that the fine-scale stochastic Green's function is intimately
linked to its deterministic counterpart. In particular, (i) we demonstrate that
whenever the deterministic fine-scale function vanishes, the stochastic
fine-scale function satisfies a weaker, and discrete notion of vanishing
stochastic coefficients, and (ii) derive an explicit formula for the fine-scale
stochastic Green's function that only involves quantities needed to evaluate
the fine-scale deterministic Green's function. We present numerical results
that support our claims about the physical support of the stochastic fine-scale
function, and demonstrate the benefit of using the VMS method when the
fine-scale Green's function is approximated by an easier to implement, element
Green's function.
