Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Abstract

Let (μ t ) t⪖0 be a τ-semistable convolution semigroup of probability measures on a Lie groupG whose idempotentμ 0 is the Haar measure on some compact subgroupK. Then all the measuresμ 1 are supported by theK-contraction groupC K(τ) of the topological automorphism τ ofG. We prove here the structure theoremC K(τ)=C(τ)K, whereC(τ) is the contraction group of τ. 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.