Hemigroups of probability measures on a locally compact group
The author considers hemigroups of probability measures on a locally compact group G, i.e. families \((\mu (s,t))_{0\leq s\leq t}\) of probability measures with \[ \mu (r,s)*\mu (s,t)=\mu (r,t)\quad for\quad 0\leq r\leq s\leq t. \] The corresponding probability operators constitute evolution families on the space \(C_ 0(G)\) of continuous real valued functions on G, vanishing at infinity. He discusses the characterization and generation of hemigroups resp. their evolution families by families of linear operators on \(C_ 0(G)\), which indeed are generators of continuous convolution semigroups of probability measures on G. Two different approaches are presented. The first approach is based on ideas of T. Kato and leads to a self contained presentation of the generation of hemigroups. For the other approach he applies a Hille- Yosida theory for evolutions developed by \textit{J. V. Herod} and \textit{R. W. McKelvey} [Isr. J. Math. 36, 13-40 (1980; Zbl 0452.47044)]. This supplies with a bijective mapping between generator families and hemigroups of bounded variation.
characterization and generation of hemigroups
scientific article; zbMATH DE number 4124685
scientific article