Multiple Self-decomposable Laws on Vector Spaces and on Groups: The Existence of Background Driving Processes


Following K. Urbanik, we define for simply connected nilpotent Lie groups G multiple self-decomposable laws as follows: For a fixed continuous one-parameter group (T t ) of automorphisms put \(L^{(0)} : = M^1 \left( G \right)\,and\,L^{(m + 1)} : = \{ \mu \in M^1 \left( G \right):\forall t > 0\ \exists\ v(t) \in L^{(m)} :\mu = T_t (\mu )*v(t)\} \,for \,m \ge 0.\)

Under suitable commutativity assumptions it is shown that also for m > 0 there exists a background driving Lévy process with corresponding continuous convolution semigroup (v s)s≥0 determining μ and vice versa. Precisely, μ and v s are related by iterated Lie Trotter formulae.


Convolution Semigroup Multivariate Anal Weak Operator Topology Convolution Structure Logarithmic Moment 
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© Physica-Verlag Heidelberg 2009

Authors and Affiliations

  1. 1.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

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