Convolution semigroups of local type on a commutative hypergroup (Q749984)

Let K be a commutative hypergroup with neutral element e. This paper continues the investigation of a convolution semigroup \((\mu_ t)_{t\geq 0}\) of sub-probability measures on K in terms of its (infinitesimal) generator. It is based on previous work of \textit{W. R. Bloom} and the author [e.g. Probability measures on groups IX, Proc. 9th Conf., Oberwolfach/FRG 1988, Lect. Notes Math. 1379, 21-35 (1989; Zbl 0693.60009); J. Theor. Probab. 1, No.3, 271-286 (1988; Zbl 0648.60011)] and of \textit{R. Lasser} [Pac. J. Math. 127, No.2, 353-371 (1987; Zbl 0652.43001)]. The method of proof is similar to the Abelian group case as developed by \textit{C. Berg} and \textit{G. Forst} [Potential theory on locally compact Abelian groups (1975; Zbl 0308.31001)], i.e. potential theoretical ideas are applied. `There are, however, significant limitations of the translation procedure.' Throughout the paper it is assumed that K admits a hypergroup dual \(K^\wedge\), this comprises large subclasses of commutative hypergroups (cf. 5.6 for an extensive list of examples). Let us formulate two major results: 1. There exists a positive measure \(\eta\) on \(K\setminus \{e\}\) such that \(\lim_{t\downarrow 0}t^{- 1}\int f d\mu_ t=\int f d\eta\) for all bounded continuous functions f on K vanishing in a neighbourhood of e (Theorem 4.1). As usual \(\eta\) is termed the Lévy measure of \((\mu_ t)_{t\geq 0}\). This proposition is slightly more general than a result of Lasser. 2. The convolution semigroup \((\mu_ t)_{t\geq 0}\) is of local type (i.e. its generator is a local operator) iff \(\lim_{t\downarrow 0}t^{-1}\mu_ t(\complement W)=0\) for all open neighbourhoods W of e or, equivalently, iff its Lévy measure \(\eta\) is zero (Theorem 5.3).