Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups
Let \((\mu_ t)_{t\geq 0}\) be a \(\tau\)-semistable convolution semigroup of probability measures on a Lie group G whose idempotent \(\mu_ 0\) is the Haar measure on some compact subgroup K. Then all the measures \(\mu_ t\) are supported by the K-contraction group \(C_ K(\tau)\) of the topological automorphism \(\tau\) of G. We prove here the structure theorem \(C_ K(\tau)=C(\tau)K\), where C(\(\tau)\) is the contraction group of \(\tau\). Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.
convolution semigroup of probability measures
scientific article; zbMATH DE number 4078353
scientific article