  This paper aims to show constructions of scale dependence and interaction on
some probabilistic models which may be revelant for renormalization theory in
Quantum Field Theory. We begin with a review of the convolution product's use
in the Kreimer-Connes formalism of perturbative renormalization. We show that
the Wilson effective action can be obtained from a convolution product
propriety of regularized Gaussian measures on the space of fields. Then, we
propose a natural C*-algebraic framework for scale dependent field theories
which may enhance the conceptual approach to renormalization theory. In the
same spirit, we introduce a probabilistic construction of interacting theories
for simple models and apply it for quantum field theory by defining a partition
function in this setting.
