  Using the Stochastic Adhesion Model (SAM) as a simple toy model for cosmic
structure formation, we study renormalization and the removal of the cutoff
dependence from loop integrals in perturbative calculations. SAM shares the
same symmetry with the full system of continuity+Euler equations and includes a
viscosity term and a stochastic noise term, similar to the effective theories
recently put forward to model CDM clustering. We show in this context that if
the viscosity and noise terms are treated as perturbative corrections to the
standard eulerian perturbation theory, they are necessarily non-local in time.
To ensure Galilean Invariance higher order vertices related to the viscosity
and the noise must then be added and we explicitly show at one-loop that these
terms act as counter terms for vertex diagrams. The Ward Identities ensure that
the non-local-in-time theory can be renormalized consistently. Another
possibility is to include the viscosity in the linear propagator, resulting in
exponential damping at high wavenumber. The resulting local-in-time theory is
then renormalizable to one loop, requiring less free parameters for its
renormalization.
