  Self-similar symmetric $\alpha$-stable, $\alpha\in(0,2)$, mixed moving
averages can be related to nonsingular flows. By using this relation and the
structure of the underlying flows, one can decompose self-similar mixed moving
averages into distinct classes and then examine the processes in each of these
classes separately. The relation between processes and flows involves
semi-additive functionals. We establish a general result about semi-additive
functionals related to cocycles, and identify the presence of a new
semi-additive functional in the relation between processes and flows. This new
functional is useful for finding the kernel function of self-similar mixed
moving averages generated by a given flow. It also sheds new light on previous
results on the subject.
