  Compactness is one of the most versatile tools in the analysis of nonlinear
PDEs and systems. Usually, compactness is established by means of some
embedding theorem between functional spaces. Such theorems, in turn, rely on
appropriate estimates for a function and its derivatives. While a similar
result based on simultaneous estimates for the Malliavin and weak Sobolev
derivatives is available for the Wiener-Sobolev spaces, it seems that it has
not yet been widely used in the analysis of highly nonlinear parabolic problems
with stochasticity. In the present work we apply this result in order to study
compactness, existence of global solutions, and, as a by-product, the
convergence of a semi-discretisation scheme for a prototypical degenerate
PDE-SDE coupling.
