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Convolution semigroups of stable distributions over a nilpotent Lie group. (English) Zbl 0824.60007

Let \(G\) be a simply connected nilpotent Lie group, \({\mathcal G}\) its Lie algebra and \(\mu_ t\) be a weakly continuous convolution semigroup of probability distributions on \(G\) with infinitesimal generator \(L\). The first result is a representation theorem of \(L\) by means of a symmetric nonnegative matrix \(A\) on \({\mathcal G}\), a measure \(M\) on \({\mathcal G} - \{0\}\) with \(\int {| X |^ 2 \over 1 + | X |^ 2} dM(X) < \infty\) and a vector \(b \in{\mathcal G}\). More precisely, for any \(f \in C^ 2\) one has: \[ Lf(g) = {1 \over 2} \sum_{j, k} a_{j,k} X_ jX_ kf(g) + \sum_ j b_ j X_ jf(x) + \int \left( f(g \exp X) - f(g) - {Xf(g) \over 1 + | X |^ 2} \right) dM(X), \] and the triple \((A,M,b)\) uniquely determines the semigroup \(\mu_ t\). Furthermore, the author gives necessary and sufficient conditions on the triple \((A,M,B)\) in order that the semigroup \(\mu_ t\) be stable under a dilation \((\gamma_ r)_{r > 0}\).
Reviewer: J.Lacroix (Paris)

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A10 Measure algebras on groups, semigroups, etc.
60E07 Infinitely divisible distributions; stable distributions
60E10 Characteristic functions; other transforms
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