Semistable convolution semigroups on measurable and topological groups.

*(English)*Zbl 0544.60021Semistable probability measures on the real line have been first considered by P. Lévy. The extension to operator-semistable laws on Euclidean spaces is due to R. Jajte. Far reaching generalizations to topological and measurable groups have been studied by A. Tortrat [see e.g., Probability measures on groups, Proc. 6th Conf., Oberwolfach 1981, Lect. Notes Math. 928, 452-466 (1982; Zbl 0514.60014)]. Following this approach we adopt in this paper the following definition: A convolution semigroup (c.s.g.) \((\mu_ t)_{t>0}\) of probability measures on a measurable group G is said to be (strictly) semistable with respect to a measurable homomorphism \(\delta\) of G and with coefficient \(c\in(0,1)\cup(1,\infty)\) if \(\delta(\mu_ t)=\mu_{ct}\) for all \(t>0.\)

Several examples from the literature that fit into this framework are presented in section 1. In section 2 a rather general result on purity of semistable c.s.g. is proved (theorem 1): These semigroups respect certain band decompositions of the Banach lattice of bounded measures on G. On the one hand side this leads to the well known zero-one laws for semistable measures and on the other hand side to the result that semistable c.s.g. on locally compact groups are of pure Lebesgue type. In section 3 we show that under rather weak conditions a semistable c.s.g. is holomorphic (theorem 5). This extends a well known property of stable c.s.g. also due to the author [Holomorphic convolution semigroups on topological groups. Probability measures on groups VII. Proc. Conf., Oberwolfach 1983. Lect. Notes Math. 1064, 421-449 (1984)]. In the semistable case the proof is more complicated; one first has to establish the quasianalyticity of the c.s.g. (theorem 4). In section 4 the results of section 3 are specialized to Euclidean spaces. In particular the support structure of a semistable c.s.g. is investigated (theorem 7).

Remark: It has been incorrectly stated in our paper that a topological Hausdorff group G is also a measurable group (cf. p. 150). (This holds if G has a countable basis of its topology.) Nevertheless all of our results hold true: The product measure \(\mu \otimes \nu\) can be uniquely extended to a \(\tau\)-regular signed measure \(\mu {\hat \otimes}\nu\) on \({\mathcal B}(G\times G)\), and Fubini’s theorem remains valid for \(\mu {\hat \otimes}\nu\) [cf. I. Csiszár, Probab. Struct. algébriques, Actes Colloque internat. 186, Clermont-Ferrand 1969, 75-97 (1970; Zbl 0229.60004)].

Several examples from the literature that fit into this framework are presented in section 1. In section 2 a rather general result on purity of semistable c.s.g. is proved (theorem 1): These semigroups respect certain band decompositions of the Banach lattice of bounded measures on G. On the one hand side this leads to the well known zero-one laws for semistable measures and on the other hand side to the result that semistable c.s.g. on locally compact groups are of pure Lebesgue type. In section 3 we show that under rather weak conditions a semistable c.s.g. is holomorphic (theorem 5). This extends a well known property of stable c.s.g. also due to the author [Holomorphic convolution semigroups on topological groups. Probability measures on groups VII. Proc. Conf., Oberwolfach 1983. Lect. Notes Math. 1064, 421-449 (1984)]. In the semistable case the proof is more complicated; one first has to establish the quasianalyticity of the c.s.g. (theorem 4). In section 4 the results of section 3 are specialized to Euclidean spaces. In particular the support structure of a semistable c.s.g. is investigated (theorem 7).

Remark: It has been incorrectly stated in our paper that a topological Hausdorff group G is also a measurable group (cf. p. 150). (This holds if G has a countable basis of its topology.) Nevertheless all of our results hold true: The product measure \(\mu \otimes \nu\) can be uniquely extended to a \(\tau\)-regular signed measure \(\mu {\hat \otimes}\nu\) on \({\mathcal B}(G\times G)\), and Fubini’s theorem remains valid for \(\mu {\hat \otimes}\nu\) [cf. I. Csiszár, Probab. Struct. algébriques, Actes Colloque internat. 186, Clermont-Ferrand 1969, 75-97 (1970; Zbl 0229.60004)].

##### MSC:

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

43A05 | Measures on groups and semigroups, etc. |