Convolution semigroups and resolvent families of measures on hypergroups
The authors continue the studies on probabilities on hypergroups [cf. Rend. Mat. Appl., VII. Ser. 2, 315-334 (1982; Zbl 0501.60016); ibid. 2, 547-563 (1982; Zbl 0507.60004); see also the second author, Probability measures on groups VII, Proc. Conf., Oberwolfach 1983, Lect. Notes Math. 1064, 481-550 (1984; Zbl 0543.60013)]. The underlying space K is a commutative hypergroup with dual \(K{\hat{\;}}\). At first notions of positive definite and negative definite functions on \(K{\hat{\;}}\) are introduced in order to prove a Schoenberg-type-theorem (theorem 3.7), i.e. continuous convolution semigroups \((\mu_ t)_{t\geq 0}\) on K correspond to functions of the type \(e^{-t\psi}\), \(\psi\) negative definite. In {\S} 4-{\S} 6 resolvents, potentials and positive definite measures are studied, in order to discuss in {\S} 7 transience and recurrence properties of convolution semigroups resp. of the corresponding Markov processes. Especially a sufficient condition for transcience, analogous to the wellknown results for Abelian l.c. groups, is established (Thm. 7.10): a continuous convolution semigroup \((\mu_ t)_{t\geq 0}\) with \({\hat \mu}{}_ t=e^{-t\psi}\) is transient if \(1/\psi\) is locally \(\omega_{K{\hat{\;}}}\)-integrable.
positive definite and negative definite functions
scientific article; zbMATH DE number 3890420
scientific article