  We consider semigroups of operators for hierarchies of evolution equations of
large particle systems, namely, of the dual BBGKY hierarchy for marginal
observables and the BBGKY hierarchy for marginal distribution functions. We
establish that the generating operators of the expansions for one-parametric
families of operators of these hierarchies are the corresponding order
cumulants (semi-invariants) of semigroups for the Liouville equations. We also
apply constructed semigroups to the description of the kinetic evolution of
interacting stochastic Markovian processes, modeling the microscopic evolution
of soft active matter. For this purpose we consider the mean field asymptotic
behavior of the semigroup generated by the dual BBGKY hierarchy for marginal
observables. The constructed scaling limit is governed by the set of recurrence
evolution equations, namely, by the Vlasov-type dual hierarchy. Moreover, the
relationships of this hierarchy of evolution equations with the Vlasov-type
kinetic equation with initial correlations are established.
