×

Convolution semigroups and resolvent families of measures on hypergroups. (English) Zbl 0558.60008

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 § 4-§ 6 resolvents, potentials and positive definite measures are studied, in order to discuss in § 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.
Reviewer: W.Hazod

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A05 Measures on groups and semigroups, etc.
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A35 Positive definite functions on groups, semigroups, etc.
43A40 Character groups and dual objects
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Berg, C.: Sur les semi-groups de convolution. Théorie du potentiel et analyse harmonique, Exposés des Journées de la Soc. Math. France, Inst. Recherche Math. Avancée, Strasbourg, 1973, pp. 1-26. Lecture Notes in Math., vol. 404., Berlin-Heidelberg-New York: Springer 1974
[2] Berg, C., Forst, G.: Potential theory on locally compact abelian groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 87. Berlin-Heidelberg-New York: Springer 1975 · Zbl 0308.31001
[3] Bloom, W.R., Heyer, H.: The Fourier transform for probability measures on hypergroups. Rend. Mat.2, 315-334 (1982) · Zbl 0501.60016
[4] Bloom, W.R., Heyer, H.:Convergence of convolution products of probability measures on hypergroups. Rend. Mat.2, 547-563 (1982) · Zbl 0507.60004
[5] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, Vol. 1. Die Grundlehren der Mathematischen Wissenschaften, Band 115. Berlin-Göttingen-Heidelberg: Springer 1963 · Zbl 0115.10603
[6] Heyer, H.: Probability theory on hypergroups: A survey. Probability Measures on Groups, Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach, 1983, pp. 481-550. Lecture Notes in Math., Vol. 1064. Berlin-Heidelberg-New York: Springer 1984
[7] Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. in Math.18, 1-101 (1975) · Zbl 0325.42017 · doi:10.1016/0001-8708(75)90002-X
[8] Lasser, R.: Orthogonal polynomials and hypergroups: Contributions to analysis and probability theory. Technische Universität München, Institut für Mathematik, 1981
[9] Lasser, R.: Bochner theorems for hypergroups and their applications to orthogonal polynomial expansions. J. Approx. Theory37, 311-325 (1983) · Zbl 0524.43004 · doi:10.1016/0021-9045(83)90040-0
[10] Lasser, R.: Orthogonal polynomials and hypergroups. Rend. Mat.3, 185-209 (1983) · Zbl 0538.33010
[11] Lasser, R.: On the Lévy-Hin?in formula for commutative hypergroups. Probability Measures on Groups, Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach 1983, pp. 298-308. Lecture Notes in Math., Vol. 1064. Berlin-Heidelberg-New York: Springer 1984
[12] Port, S.C., Stone, C.J.: Infinitely divisible processes and their potential theory I, II. Ann. Inst. Fourier (Grenoble)21 1, 157-275 (1971);21 2, 179-265 (1971) · Zbl 0195.47601
[13] Spector, R.: Mesures invariantes sur les hypergroupes Trans. Amer. Math. Soc.239, 147-165 · Zbl 0428.43001
[14] Vrem, R.C.: Harmonic analysis on compact hypergroups. Pacific J. Math.85, 239-251 (1979) · Zbl 0458.43002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.