Weakly infinitely divisible measures on some locally compact Abelian groups

Abstract

Abstract. On the torus group, on the group of p-adic integers and on the p-adic solenoid we give a construction of an arbitrary weakly infinitely divisible probability measure using a random element with values in a product of (possibly infinitely many) subgroups of R. As a special case of our results, we have a new construction of the Haar measure on the p-adic solenoid.

Explicit representation of roots on p-adic solenoids and non-uniqueness of embeddability into rational one-parameter subgroups

“…The remaining part of this Introduction is about the probabilistic impact to the group theoretic questions. In the last decade solenoids have drawn attention as relevant examples on various fields of probability theory (see [1][2][3][4]). …”

“…The same is true for the above-mentioned general group theoretic proofs, which show the equivalence of connectedness and divisibility for compact (Abelian) groups. Among other things the infinite divisibility of Dirac measures on S p shows that the characterization of weakly infinitely divisible probability measures on S p in [1] is in fact a characterization of all infinitely divisible probability measures. Beyond their existence, we will prove an explicit representation of the roots on S p in §2, which also gives their multiplicity.…”

“…In particular, for Gaussian measures γ * ω C * δ x in the sense of Parthasarathy [22], where γ is a symmetric Gaussian and ω C is the Haar probability measure on some compact subgroup C ⊆ S p , in case x ∈ S p \S arc p we do not have continuous embeddability, and in any case we have non-unique rational Gaussian embedding semigroups. These play an important role in [1][2][3][4].…”