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Mehler hemigroups and embedding of discrete skew convolution semigroups on simply connected nilpotent Lie groups. (English) Zbl 1187.43006

Hilgert, Joachim (ed.) et al., Proceedings of the fourth German-Japanese symposium on infinite dimensional harmonic analysis IV. On the interplay between representation theory, random matrices, special functions, and probability, Tokyo, Japan, September 10–14, 2007. Hackensack, NJ: World Scientific (ISBN 978-981-283-281-8/hbk). 32-46 (2009).
Some concepts and results for selfsimilar stochastic processes on groups are extended to additive processes on a homogeneous l.c.group \(G\) which are semi-selfsimilar on a discrete scale. The reconstruction of distributions of increments, forming a semistable hemigroup in \(\mathcal{M}^1(G)\), from a discrete skew convolution semigroup is treated. In passing to a duly enlarged space-time group, the notions of Mehler hemigroups and corresponding background driving processes are introduced by analogy with the case of stable hemigroups. The established generalized Lie-Trotter formulas for convolutions provide appropriate analogs of known weak sense random integral representations for driving processes. The proofs involve arguments of E. Siebert in [Probability measures on groups, Proc. 6th Conf., Oberwolfach 1981, Lect. Notes Math. 928, 362–402 (1982; Zbl 0491.60013)].
For the entire collection see [Zbl 1158.43003].

MSC:

43A80 Analysis on other specific Lie groups
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
22E25 Nilpotent and solvable Lie groups

Citations:

Zbl 0491.60013
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