Semigroups related to additive and multiplicative, free and Boolean convolutions

Abstract

Belinschi and Nica introduced a composition semigroup of maps on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know quantitatively if a measure is freely infinitely divisible or not.In the first half of the paper, we further investigate this indicator: we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bożejko's conjecture which says the Boolean powers µ ⊎ , t ∈ [0, 1], of a probability measure µ are freely infinitely divisible if the measure µ is freely infinitely divisible.In the other half of this paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible relative to the free multiplicative convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then the multiplicative analogue of Bożejko's conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection.

The Exponential Map in Non-commutative Probability

“…In the same manner, as pointed out in [AH13], the multiplicative analog of the commutation relation proved in [BN08]`µ 'p˘Z q , "´µ Zp 1¯' q 1 , q 1 " 1´p`pq, pq " p 1 q 1 needs some care because the multiplicative powers are multi-valued. We will solve this problem by the use of Propostion 23.…”

“…Recall that the (additive) divisibility indicator was defined in Definition 1.4 of [BN08] as sup tt ě 0 : µ P B t pPpRqqu , while the multiplicative divisibility indicator was defined in Definition 4.6 of [AH13] as…”

Limit theorems for free Lévy processes

“…The correspondence µ → η is a generalization of the multiplicative Bercovici-Pata map from Boolean to free [AH13] which is not surjective. We can show that η ∈ E, but η may not be the η-transform of a probability measure.…”

Free Subordination and Belinschi–Nica Semigroup

“…When µ = ν, the subordination map has a special meaning. [AH13b]. The measure B µ (µ) is ⊠-infinitely divisible for any µ ∈ P(R + ) \ {δ 0 }, but this map is not surjective onto the set of ⊠-infinitely divisible measures.…”