Continuous automorphism groups on a locally compact group contracting modulo a compact subgroup and applications to stable convolution semigroups
The following observation in the theory of probability measures on groups was the starting point for our investigations. Let \((\mu_ t)_{t\geq 0}\) be a continuous convolution semigroup of probability measures on a locally compact group G. Then \(\mu_ 0\) is the normed Haar measure \(\omega_ K\) on a compact subgroup K of G. In addition let \(T=(\tau_ t)_{t>0}\) be a continuous automorphism group on G. Then the semigroup \((\mu_ t)_{t\geq 0}\) is said to be T-stable (in the strict sense) if \(\tau_ t(\mu_ 1)=\mu_ t\) for all \(t>0\). Obviously K is T-invariant. Moreover all the measures \(\mu_ t\) are concentrated on the Borel measurable subgroup \(C_ K(T)\) of G where T acts contracting modulo K (cf. Proposition 4.2). Consequently, in order to determine the class of groups G on which there exist T-stable convolution semigroups, it is necessary to study the structure of \(C_ K(T)\). This investigation will be performed in the first part of this paper. Turning matters around we are then going to show that the structure of \(C_ K(T)\) has a strong impact on the structure of the T-stable convolution semigroup \((\mu_ t)_{t\geq 0}\).
scientific article; zbMATH DE number 3931300
scientific article