Stable hemigroups and mixing of generating functionals (Q1866680)

The author studies convolution hemigroups (distributions of not necessarily stationary independent increment processes) on a nilpotent Lie group \(G\). Many objects, usually studied in connection with convolution semigroups of measures, are investigated for the hemigroup case. In particular, some subgroups of \(\text{Aut}(G)\) associated with a hemigroup (like the invariance group and the decomposability group), the set of exponents, and the notion of a stable hemigroup are considered. A hemigroup can be obtained from a family of convolution semigroups via the procedure of mixing their generating functionals with the help of a certain measure. Particular classes of mixing include the use of random integrals of group-valued Lévy processes. The mixing properties of generating functionals are employed for obtaining characterizations of stable and semi-stable convolution semigroups.