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Free Brownian motion and evolution towards boxplus-infinite divisibility for k-tuples

Let D be the space of non-commutative distributions of k-tuples of selfadjoints in a C*-probability space (for a fixed k). We introduce a semigroup of transformations B_t of D, such that every distribution in D evolves under the B_t towards infinite divisibility with respect to free additive convolution. The very good properties of B_t come from some special connections that we put into evidence between free additive convolution and the operation of Boolean convolution. On the other hand we put into evidence a relation between the transformations B_t and free Brownian motion. More precisely, we introduce a transformation Phi of D which converts the free Brownian motion started at an arbitrary distribution m in D into the process B_t (Phi(m)), t>0.


Introduction
1.1 Review of past work. The study of noncommutative forms of independence for random variables has led to several "convolution operations" that can be defined on the space M of probability distributions on R. Two such operations are the free (additive) convolution ⊞ and the Boolean convolution ⊎, which reflect the operations of addition of freely independent and respectively Boolean independent random variables.
In the paper [2] we introduced a family {B t | t ≥ 0} of transformations of M, defined by the formula B t (µ) = µ ⊞(1+t) ⊎(1/(1+t)) , ∀ t ≥ 0, ∀ µ ∈ M. (1.1) The transformations B t turn out to form a semigroup: B s • B t = B s+t , ∀ s, t ≥ 0. On the other hand for t = 1 it turns out that the range set B 1 (M) is precisely the set M inf-div of distributions in M which are infinitely divisible with respect to ⊞; and moreover, the map M ∋ µ → B 1 (µ) ∈ M inf-div coincides with a remarkable bijection discovered by Bercovici and Pata [3] in their study of relations between infinite divisibility with respect to ⊞ and to ⊎. Due to the above properties of the transformations B t , for a fixed µ ∈ M the process t → B t (µ) can be viewed as a kind of "evolution towards ⊞-infinite divisibility" (where infinite divisibility is always reached by the time t = 1). In [2] it was observed that this process is related to free Brownian motion. Recall that the free Brownian motion started at ν ∈ M is the process {ν ⊞ γ t | t ≥ 0}, where γ t ∈ M is the centered semicircular distribution of variance t. The connection between this and the transformations B t is described as follows.
For a distribution µ ∈ M let G µ and F µ denote the Cauchy transform and respectively the reciprocal Cauchy transform of µ; that is, we have , and F µ (z) = 1/G µ (z), z ∈ C \ R.
By using basic facts from the theory of the Cauchy transform, one easily sees that for every distribution ν ∈ M there exists a unique µ ∈ M such that One can thus define a map Φ : M → M by putting Φ(ν) := µ with µ and ν related as in (1.2). The map Φ turns out to be one-to-one, with image Φ(M) consisting precisely of those distributions µ ∈ M which have ∞ −∞ t 2 dµ(t) = 1 and ∞ −∞ t dµ(t) = 0. (A detailed presentation of these facts appears in Section 2 of the paper [4].) The relation between the transformations B t and free Brownian motion can be expressed by using the map Φ and the following formula: Φ( ν ⊞ γ t ) = B t ( Φ(ν) ), ∀ ν ∈ M, ∀ t > 0. (1.3) In other words the free Brownian motion started at ν corresponds exactly, via Φ, to the process {B t (µ) | t ≥ 0} started at µ = Φ(ν).
1.2 Description of results of this paper. In this paper we find multi-variable analogues for the results described above. Let k be a positive integer, and let D c (k) denote the space of non-commutative distributions of k-tuples of selfadjoint elements in a C *probability space. The convolution operations ⊞ and ⊎ can be defined on D c (k), and for every µ ∈ D c (k) it makes sense to define convolution powers µ ⊞p , ∀ p ≥ 1 and µ ⊎q , ∀ q > 0. One can thus define a family {B t | t ≥ 0} of transformations of D c (k) by exactly the same formula as in (1.1): , ∀ t ≥ 0, ∀ µ ∈ D c (k). (1.4) Brownian motion started at ν ∈ D c (k) is the process {ν ⊞ γ t | t ≥ 0} where γ t ∈ D c (k) now stands for the joint distribution of a free family x 1 , . . . , x k of selfadjoint elements in a C *probability space, such that every x i has a centered semicircular distribution of variance t.
In order to connect this to the transformations B t , we use a multi-variable analogue for the map Φ which had been defined via Equation (1.2). The multi-variable version of Equation (1.2) involves formal power series in k non-commuting indeterminates z 1 , . . . , z k (instead of complex analytic functions of one variable z). For µ ∈ D c (k) let M µ be its moment series, µ(X i 1 · · · X in ) z i 1 · · · z in ; and let us moreover denote where (1 + M µ ) −1 is the inverse of 1 + M µ under multiplication in the ring of power series C z 1 , . . . , z k . With these notations, the multi-variable analogue for (1.2) is (equality of formal power series). We prove that for every ν ∈ D c (k) there exists a unique µ ∈ D c (k) such that (1.5) holds. One can thus define a map Φ : D c (k) → D c (k) by putting Φ(ν) := µ with µ, ν as in (1.5), and it turns out that we then have the analogue of (1.3): Φ( ν ⊞ γ t ) = B t ( Φ(ν) ), ∀ ν ∈ D c (k), ∀ t > 0. (1.6) Concerning the seemingly different appearance of Equations (1.2) and (1.5), we make the following comment. Suppose that k = 1 and that µ, ν ∈ D c (1) are identified as compactly supported distributions on R. Then η µ and M ν can be viewed as analytic functions on a neighbourhood of 0, and (for z running in a suitable domain of C) we have By substituting these formulas into Equation (1.2), and by replacing z with 1/z, we bring which is exactly the 1-dimensional version of Equation (1.5).

Further remarks.
The results in [2] were proved by using complex analytic functions. The methods used in this paper are completely different, they rely on the combinatorics of non-crossing partitions. For most of the paper we use the larger algebraic framework of D alg (k), the space of all possible joint distributions of k-tuples in a (sheer algebraic) non-commutative probability space. It makes sense to define B t as a bijective transformation of D alg (k), then prove the algebraic statements made about {B t | t ≥ 0} in this larger framework; these properties of the B t are summarized in Theorem 4.11 of the paper. In the same theorem we also point out an additional property of B t , that where ⊠ (the free multiplicative convolution) is the operation on D alg (k) which corresponds to the multiplication of free k-tuples of random variables in a non-commutative probability space.
The map Φ defined via Equation (1.5) and the relation between the transformations B t and free Brownian motion given in (1.6) can be considered on D alg (k) as well. The proof of formula (1.6) is in fact done in this algebraic framework, in Theorem 6.2 of the paper.
After the algebraic results are established, what remains to be done is make sure that the transformations B t do indeed the job they are supposed to, when they are restricted to the smaller space D c (k). Some of the verifications needed here are direct consequences of things proved in our preceding paper [1]. But there is one verification that we are left with, namely that the map Φ : D alg (k) → D alg (k) carries D c (k) into itself. We prove this fact by providing an operator model for how Φ works on D c (k); this operator model is discussed in Remark 7.4 and Theorem 7.5 of the paper.
We conclude the introduction with an outline of how the paper is organized. Throughout the whole paper k is a fixed positive integer -the "number of indeterminates" we are working with. In Section 2 we review the algebraic framework of D alg (k), and the R and η transforms for distributions in D alg (k). Section 3 is a review section as well, devoted to an important bijection on power series introduced in [1], the bijection "Reta" sending R µ → η µ for every µ ∈ D alg (k)); this bijection is the workhorse for many of the computations with power series done in the present paper. In Section 4 we introduce B t as a bijective transformation of D alg (k) and we prove some of its basic properties; the results of the section are summarized in Theorem 4.11. In Section 5 we establish a formula for moments of the free Brownian motion, which is needed in the proof of the connection between free Brownian motion and the B t . The proof of this connection is then done in Section 6, Theorem 6.2. The final Section 7 of the paper deals with the framework of D c (k), the main point of the section being the operator model for Φ.

Non-commutative convolutions and transforms on D alg (k)
We denote by C X 1 , . . . , X k the algebra of non-commutative polynomials in X 1 , . . . , X k . Thus C X 1 , . . . , X k has a linear basis where the monomials in the basis are multiplied by concatenation. When needed, C X 1 , . . . , X k will be viewed as a * -algebra, with * -operation determined uniquely by the fact that each of X 1 , . . . , X k is selfadjoint. 2 o Let (A, ϕ) be a non-commutative probability space; that is, A is a unital algebra over C, and ϕ : A → C is a linear functional, normalized by the condition that ϕ(1 A ) = 1. For x 1 , . . . , x k ∈ A, the joint distribution of x 1 , . . . , x k is the linear functional µ x 1 ,...,x k : C X 1 , . . . , X k → C which acts on the linear basis (2.1) by the formula It is immediate that D alg (k) is precisely the set of linear functionals on C X 1 , . . . , X k that can arise as joint distribution for some k-tuple x 1 , . . . , x k in a non-commutative probability space.
Remark 2.2. (The operations ⊞ and ⊎ on D alg (k).) These operations are defined via the general principle that if one has a form of independence for non-commutative random variables, then the addition of independent k-tuples of random variables will induce a "convolution" operation on D alg (k). The operation ⊞ arises in this way, in connection to the concept of free independence. Given µ, ν ∈ D alg (k), one can always find random variables x 1 , . . . , x k , y 1 , . . . , y k in a non-commutative probability space (A, ϕ) such that the joint distribution of the k-tuple x 1 , . . . , x k is equal to µ, the joint distribution of y 1 , . . . , y k is equal to ν, and such that {x 1 , . . . , x k } is freely independent from {y 1 , . . . , y k } in (A, ϕ). The joint distribution of the k-tuple x 1 + y 1 , . . . , x k + y k turns out to depend only on µ and ν; and the free additive convolution µ ⊞ ν is equal, by definition, to the joint distribution of x 1 + y 1 , . . . , x k + y k .
The operation ⊎ is defined in the same way, but where we use the concept of Boolean independence: given µ, ν ∈ D alg (k), the Boolean convolution µ ⊎ ν is the (uniquely determined) distribution of x 1 + y 1 , . . . , x k + y k , where the joint distribution of x 1 , . . . , x k is equal to µ, the joint distribution of y 1 , . . . , y k is equal to ν, and {x 1 , . . . , x k } is Boolean independent from {y 1 , . . . , y k }.
A commonly used method for studying the operations ⊞ and ⊎ goes by considering cumulants for distributions in D alg (k): in relation to ⊞ one considers the free cumulants introduced in [7], while for ⊎ one uses the Boolean cumulants which go back all the way to [8]. In this paper we will work with the concepts, equivalent to cumulants, of linearizing transforms for ⊞ and ⊎. Specifically, for a distribution µ ∈ D alg (k) we will work with the R-transform R µ (a formal power series which records the free cumulants of µ) and with the the η-series η µ (which does the same job in connection to the Boolean cumulants of µ). The precise definitions of R µ and η µ will be reviewed in the next notation and remark. The meaning of the statement that R and η are "linearizing transforms" for ⊞ and respectively for ⊎ is that we have: and η µ⊎ν = η µ + η ν , ∀ µ, ν ∈ D alg (k).
(2.5) Notation 2.3. We will denote by C 0 z 1 , . . . , z k the space of power series with complex coefficients and with vanishing constant term, in k non-commuting indeterminates z 1 , . . . , z k . The general form of a series f ∈ C 0 z 1 , . . . , z k is thus where the coefficients α (i 1 ,...,in) are from C.
Remark 2.4. (review of the series M µ , R µ , η µ ). Let µ be a distribution in D alg (k). 1 o We will denote by M µ the series in C 0 z 1 , . . . , z k given by M µ is called the moment series of µ, and its coefficients (µ(X i 1 · · · X in ), with n ≥ 1 and 1 ≤ i 1 , . . . , i n ≤ k) are called the moments of µ.
2 o The η-series of µ is where (1 + M µ ) −1 is the inverse of 1 + M µ under multiplication, in the ring of series C z 1 , . . . , z k . The coefficients of η µ are called the Boolean cumulants of µ.
3 o There exists a unique series R µ ∈ C 0 z 1 , . . . , z k which satisfies the functional equation Indeed, it is easily seen that Equation (2.9) amounts to a recursion which determines uniquely the coefficients of R µ in terms of those of M µ . The series R µ is called the Rtransform of µ, and its coefficients are called the free cumulants of µ. (See the discussion in Lecture 16 of [6], and specifically Theorem 16.15 and Corollary 16.16 of that lecture.) Remark 2.5. It is immediate that for every f ∈ C 0 z 1 , . . . , z k there exists a unique distribution µ ∈ D alg (k) such that η µ = f . (This is because, as one immediately checks, the equation Likewise, the map D alg (k) ∋ µ → R µ ∈ C 0 z 1 , . . . , z k is bijective. The fact that for every g ∈ C 0 z 1 , . . . , z k there exists a unique µ ∈ D alg (k) such that R µ = g is easily seen when one writes explicitly the relations between the coefficients of R µ and M µ that are coming out of (2.9) -see Lectures 11 and 16 of [6].
Remark 2.6. (Convolution powers.) For µ ∈ D alg (k) and a positive integer n one denotes the n-fold convolution µ ⊞ · · · ⊞ µ by µ ⊞n . From the additivity (2.4) of the R-transform it follows that R µ ⊞n = n · R µ , and the latter formula can be extended to the case when n is not necessarily integer. More precisely, for every µ ∈ D alg (k) and t ∈ (0, ∞) one defines the convolution power µ ⊞t to be the unique distribution in D alg (k) which has R-transform equal to It is immediate that the ⊞-convolution powers defined in this way satisfy the usual rules for operating with exponents: µ ⊞s ⊞ µ ⊞t = µ ⊞(s+t) and µ ⊞s ⊞t = µ ⊞st , ∀ µ ∈ D alg (k), ∀ s, t > 0. (2.11) Note that, as a consequence, one has that for every fixed t ∈ (0, ∞) the map µ → µ ⊞t is a bijection from D alg (k) onto itself. A similar discussion can be made in connection to the convolution powers with respect to ⊎: for every µ ∈ D alg (k) and t ∈ (0, ∞) one defines the convolution power µ ⊎t to be the unique distribution in D alg (k) which has η-series equal to (2.12) Then the ⊎-convolution powers satisfy the usual rules of operating with exponents, and for every fixed t ∈ (0, ∞) the map µ → µ ⊎t is a bijection from D alg (k) onto itself.
3. The bijection "Reta", and its combinatorial properties Definition 3.1. We will denote where R, η : D alg (k) → C 0 z 1 , . . . , z k are the bijections µ → R µ and respectively µ → η µ that were discussed in Remark 2.5. In other words, Reta is the bijection from C 0 z 1 , . . . , z k onto itself which is uniquely determined by the requirement that Remark 3.2. The bijection Reta was introduced in our previous paper [1]. Its name was chosen by looking at Equation (3.2) (the transformation of C 0 z 1 , . . . , z k that "converts R into η"). It is very useful that one can alternatively describe Reta via an explicit formula which gives directly the coefficients of the series Reta(f ) in terms of those of f , for f ∈ C 0 z 1 , . . . , z k . This formula is reviewed (following [1]) in Proposition 3.5 below. It involves summations indexed by non-crossing partitions, and in order to present it we will start with a very concise review (intended mostly for setting notations) of the lattice N C(n) of non-crossing partitions. For a more detailed introduction to N C(n) and to how it is used in free probability, we refer to [6], Lectures 9 and 10. . . , B p are pairwise disjoint non-void sets (called the blocks of π), and B 1 ∪ · · · ∪ B p = {1, . . . , n}. We say that π is non-crossing if for every 1 ≤ i < j < k < l ≤ n such that i is in the same block with k and j is in the same block with l, it necessarily follows that all of i, j, k, l are in the same block of π. The set of all non-crossing partitions of {1, . . . , n} will be denoted by N C(n).
2 o For π ∈ N C(n), the number of blocks of π will be denoted by |π|.
3 o On N C(n) we consider the partial order given by reversed refinement: for π, ρ ∈ N C(n), we write "π ≤ ρ" to mean that every block of ρ is a union of blocks of π. The minimal and maximal element of (N C(n), ≤) are denoted by 0 n (the partition of {1, . . . , n} into n singleton blocks) and respectively 1 n (the partition of {1, . . . , n} into only one block).
4 o In the considerations about Reta, an important role is played by another partial order relation on N C(n), which was introduced in [1] and is denoted by "≪". For π, ρ ∈ N C(n) we will write "π ≪ ρ" to mean that π ≤ ρ and that, in addition, the following condition is fulfilled: For every block C of ρ there exists a block B of π such that min(C), max(C) ∈ B.
It is immediately verified that "≪" is indeed a partial order relation on N C(n). It is much coarser than the reversed refinement order. For instance, the inequality π ≪ 1 n is not holding for all π ∈ N C(n), but it rather amounts to the condition that the numbers 1 and n belong to the same block of π. At the other end of N C(n), the inequality π ≫ 0 n can only take place when π = 0 n . 1 o For n ≥ 1 and 1 ≤ i 1 , . . . , i n ≤ k we will denote by the linear functional which extracts the coefficient of z i 1 · · · z in in a series f ∈ C 0 z 1 , . . . , z k . Thus for f written as in Equation (2.6) we have Cf (i 1 ,...,in) (f ) = α (i 1 ,...,in) . 2 o Suppose we are given a positive integer n, some indices i 1 , . . . , i n ∈ {1, . . . , k}, and a partition π ∈ N C(n). We define a (generally non-linear) functional Then we define The quantities Cf (i 1 ,...,in);π (f ) will be referred to as generalized coefficients of the series f . Proposition 3.5. Let f, g be series in C 0 z 1 , . . . , z k such that Reta(f ) = g. Then for every n ≥ 1 and 1 ≤ i 1 , . . . , i n ≤ k we have More generally, we have the following formula for a generalized coefficient Cf (i 1 ,...,in);ρ (g), where ρ is an arbitrary partition in N C(n): Remark 3.6. 1 o For the proof of the above formulas (3.7) and (3.8) we refer to Proposition 3.9 of [1]. Let us mention here that the same Proposition 3.9 of [1] also gives an explicit formula for how Equation (3.7) can be inverted in order to write the coefficients of f in terms of those of g. This latter fomula says that for every n ≥ 1 and 1 ≤ i 1 , . . . , i n ≤ k we have: Note that, since (−1) 1+|π| Cf (i 1 ,...,in);π (g) can also be written as "−Cf (i 1 ,...,in);π (−g)", an equivalent way of recording the formula (3.9) is by stating that 2 o Let f, g be two series in C 0 z 1 , . . . , z k such that Reta(f ) = g. An immediate consequence of Equation (3.7) is that the linear and quadratic coefficients of g are identical with the corresponding coefficients of f : (This is because for n ≤ 2 the only partition π ∈ N C(n) which satisfies π ≪ 1 n is 1 n itself.) The first time when we see a difference between f and g is when we look at coefficients of order 3: 3 o In Section 4 of the paper we will need a formula for the iterations of Reta, which we derive in Proposition 3.8 below. The proof of this formula is based on a property of the partial order ≪ which was proved in Proposition 2.13 of [1], and goes as follows.
Lemma 3.7. Let π be a partition in N C(n) such that π ≪ 1 n . For every integer p satisfying 1 ≤ p ≤ |π|, we have that: Proposition 3.8. Let f be a series in C 0 z 1 , . . . , z k and let s be in R, s = −1. We have Proof. Fix n ≥ 1 and 1 ≤ i 1 , . . . , i n ≤ k for which we verify the equality of the coefficients of z i 1 · · · z in for the series on the two sides of Equation (3.11). We start from the left-hand side of this equation, and compute: By reversing the order of summation in the double sum that has appeared, we continue our sequence of equalities with: (3.7)).
Remark 3.9. In the case when s = −1, the expression "Reta sReta(f ) " is not treated by the preceding proposition, but rather by using Equation (3.10) of Remark 3.6.1, which gives us that Every B t is a bijection from D alg (k) onto itself (which happens because, as noticed in Remark 2.6, both the maps form in fact a semigroup under composition; this will follow from a "commutation relation", stated in the next proposition, satisfied by the convolution powers with respect to ⊞ and to ⊎. Proposition 4.2. Let p, q be two real numbers such that p ≥ 1 and q > (p − 1)/p. We have where the new exponents p ′ , q ′ > 0 are defined by Proof. If q = 1 then it follows that q ′ = 1 and p ′ = p, and both sides of Equation (4.2) are equal to µ ⊞p . For the rest of the proof we will assume that q = 1, which implies that q ′ = 1 as well. Our strategy is to prove that the distributions on the two sides of Equation (4.2) have equal R-transforms. We prove this by calculating explicitly the R-transforms in question, where we take advantage of the fact that the convolution powers with respect to ⊞ and with respect to ⊎ are scaled by the R-transform and respectively by the η-series (Equations (2.10) and (2.12) in Remark 2.6). The calculations may occasionally come to the point where we deal with the R-transform of a ⊎-power, or with the η-series of a ⊞-power; in such a situation we apply Reta (or Reta −1 ) and go on, remaining that the compositions of Reta's that arise in this way are dealt with by using Proposition 3.8. To be specific, on the left-hand side of (4.2) we calculate: On the right-hand side of (4.2) we calculate: 3.11)).
It only remains to observe that the definition of p ′ and q ′ ensures that 1 − q ′ = (1 − q)p and p ′ q ′ /(1 − q ′ ) = q/(1 − q), hence the two R-transforms calculated above are indeed equal to each other.
Remark 4.4. If in the calculation for the R-transform of µ ⊞p ⊎q that was shown in the proof of Proposition 4.2 we make p = 1 + t and q = 1/(1 + t) (for some t > 0) we obtain We leave it as an exercise to the reader to check that the similar calculation done with η-series instead of R-transforms leads to the analogous formula Remark 4.6. An intriguing property of the map B which was observed in [1] is that it is a homomorphism with respect to the operation of free multiplicative convolution ⊠ on D alg (k). This operation is defined as follows. Given µ, ν ∈ D alg (k), one can always find random variables x 1 , . . . , x k , y 1 , . . . , y k in a non-commutative probability space (A, ϕ) such that the joint distribution of the k-tuple x 1 , . . . , x k is equal to µ, the joint distribution of the k-tuple y 1 , . . . , y k is equal to ν, and such that {x 1 , . . . , x k } is freely independent from {y 1 , . . . , y k } in (A, ϕ). The joint distribution of the k-tuple x 1 y 1 , . . . , x k y k turns out to depend only on µ and ν; and the free multiplicative convolution µ ⊠ ν is equal, by definition, to the joint distribution of x 1 y 1 , . . . , x k y k .
In the remaining part of this section we will show that every B t is a homomorphism with respect to ⊠. The argument is short, because it takes advantage of what had already been proved in [1] -the essential point is to use Theorem 7.3 of that paper. We mention that in the 1-dimensional case another derivation of the ⊠-homomorphism property of B t can be obtained by using the concept of S-transform (see Section 3 of [2]).
In the proof that B t is a ⊠-homomorphism we will also use a binary operation denoted by ⋆ on C 0 z 1 , . . . , z k , which was introduced in [5], and is uniquely determined by the fact that R µ ⋆ R ν = R µ⊠ν , ∀ µ, ν ∈ D alg (k). In other words, ⋆ is the operation with formal power series which reflects the multiplication of two free k-tuples in terms of their R-transforms.
A remarkable fact proved in Theorem 7.3 of [1] is that we also have That is, ⋆ is at the same time the operation with formal power series which reflects the multiplication of two free k-tuples in terms of their η-series. It is immediate that formula (4.8) is actually just another form of stating the ⊠-multiplicativity of B. We prefer this formula which makes explicit use of ⋆, because we want to combine it with other properties that ⋆ has, in connection to dilations and scalar multiplication of power series (as reviewed in Remark 4.8 below).
2 o For f ∈ C 0 z 1 , . . . , z k and r > 0 we denote by f • D r the series in C 0 z 1 , . . . , z k determined by the condition that (4.10) f • D r is called the dilation of f by r.
Remark 4.8. It is easy to see, directly from the definitions, that all three series M µ , R µ , η µ associated to a distribution µ ∈ D alg (k) behave well with respect to dilations; that is, we have Let us also record here two formulas from [5] which involve dilations and the operation ⋆.
The first formula simply says that ⋆ behaves well with respect to dilations: The second formula puts into evidence a special connection with scalar multiplication of series. While ⋆ is highly non-linear (and doesn't generally behave well with respect to scalar multiplication), it is remarkable that we have For the proof of (4.12) and (4.13) we refer to Notation 4.1 and Lemma 4.4 of [5].
In order to prove that B t is a homomorphism with respect to ⊠ we will show in the next proposition that, in fact, each of the two kinds of convolution powers involved in the definition of B t is "only a dilation away" from being itself a ⊠-homomorphism. Proposition 4.9. For every t > 0 and every µ, ν ∈ D alg (k) we have and Proof. In order to establish the fomula (4.14) we check that the distributions appearing on the two sides of this formula have the same R-transform: = t R µ⊠ν • D t (by (4.7)) = R (µ⊠ν) ⊞t • D t (by (2.10)) = R (µ⊠ν) ⊞t •Dt (by (4.11)).
The verification of (4.15) is done in a similar way, where now we check that the distributions on the two sides of the formula have identical η-series. The calculation is virtually identical to the one shown in the verification of (4.14), only that we have to replace everywhere R-transforms by η-series, and ⊞-powers by ⊎-powers. (An important point included in this "mutatis mutandis" argument is that, right at the beginning of the calculation, we can invoke the formula (4.8) relating η-series to the operation ⋆.) Proof. This is a straightforward consequence of Proposition 4.9: the dilation factors which appear when we take succesively the powers "⊞(t + 1)" and "⊎1/(t + 1)" cancel each other, and we are left with the plain ⊠-multiplicativity stated in Equation (4.16).
The results of this section are thus summarized in the following theorem, which puts together Corollary 4.3, Remark 4.5, and Corollary 4.10.

A formula for the moments of the free Brownian motion
Our goal in this section is to prove an explicit formula via summations over non-crossing partitions for moments (ν ⊞ γ t )(X i 1 · · · X in ), where ν is an arbitrary distribution in D alg (k) and γ t is defined as follows.
Notation 5.1. For t > 0 we will denote by γ t ∈ D alg (k) the joint distribution of a k-tuple (x 1 , . . . , x k ) where x 1 , . . . , x k form a free family, and every x i has a centered semicircular distribution of variance t.
The formula for moments which is the main result of the section will be stated in Proposition 5.4. We start by introducing a few natural conventions of notations for noncrossing partitions that will be useful in Proposition 5.4. Remark 5.2. 1 o It will be convenient that instead of sticking strictly to "N C(n)", we use the more general notation "N C(M )" for an arbitrary totally ordered finite set M . Of course, N C(M ) can always be identified canonically to N C( |M | ), where one uses the unique increasing bijection from M onto {1, . . . , |M |} in order to identify partitions of M with partitions of {1, . . . , |M |}.
2 o Let M be a totally ordered finite set, and let L be a non-empty subset of M . For π ∈ N C(M ) we can consider the restricted partition π | L of L into blocks of the form A ∩ L, with A block of π such that A ∩ L = ∅. It is immediately verified that π | L ∈ N C(L) (where L is endowed with the total order inherited from M ).
3 o Let M be a totally ordered finite set, and suppose that M = L 1 ∪ L 2 , disjoint union. If π 1 is a partition of L 1 and π 2 is a partition of L 2 , then there is an obvious way of putting π 1 and π 2 together to form a partition of M ; we will denote this partition by π 1 ⊔ π 2 . It is clear that in order to have π 1 ⊔ π 2 ∈ N C(M ) it is necessary but not sufficient that π 1 ∈ N C(L 1 ) and π 2 ∈ N C(L 2 ).
4 o Let M, L 1 , L 2 be as above and let π 1 be a fixed partition in N C(L 1 ). It is easy to see that among the partitions π 2 ∈ N C(L 2 ) with the property that π 1 ⊔ π 2 ∈ N C(M ) there is one, π, which is larger than all the others with respect to reversed refinement order on N C(L 2 ). So π ∈ N C(L 2 ) is characterized by the fact that for a partition π 2 ∈ N C(L 2 ) we have the equivalence The formula for moments that will be proved in Proposition 5.4 uses the class of noncrossing partitions discussed in the following notation. 1 o We will denote by N C ≤2 (n) the set of partitions ρ ∈ N C(n) such that every block of ρ has either 1 or 2 elements.
3 o Let ρ be in N C ≤2 (n), and let us consider the partition ρ | D(ρ) ∈ N C( D(ρ) ). We will denote by ρ the maximal partition in N C( S(ρ) ) that can be combined with ρ | D(ρ) into a non-crossing partition of {1, . . . , n}, in the sense discussed in part 4 o of the preceding remark. Thus ρ is characterized by the fact that for a partition σ ∈ N C( S(ρ) ) we have the equivalence Remark 5.5. Let us comment a bit on what is achieved by the formula (5.4). An important point is, of course, that we explicitly identify a combinatorial structure -namely N C ≤2 (n) -which indexes the sum leading to (ν ⊞ γ t )(X i 1 · · · X in ). Let us moreover fix a partition ρ ∈ N C ≤2 (n) and let us examine the term indexed by ρ on the right-hand side of (5.4). First there is an issue of compatibility. Let us say that "ρ is compatible with the n-tuple (i 1 , . . . , i n )" when the following happens: whenever B = {p, q} is a 2-element block of ρ, it follows that i p = i q . If ρ is not compatible with (i 1 , . . . , i n ), then the term indexed by ρ on the right-hand side of (5.4) vanishes.
Suppose then that ρ is compatible with (i 1 , . . . , i n ). Let S(ρ) = {b 1 < b 2 < · · · < b m } be the set of singletons of ρ, and let ρ be the non-crossing partition of S(ρ) that was put into evidence in Notation 5.3.3. The term indexed by ρ on the right-hand side of (5.4) is then equal to  ) ). Thus the generalized coefficient of M ν we have to look at is Cf (i 1 ,i 3 ,i 6 ,i 7 ,i 9 );b ρ (M ν ), which is just ν(X i 1 X i 9 )ν(X i 3 X i 6 X i 7 ). Hence the term indexed by ρ in the sum on the righthand side of (5.4) is in this concrete example equal to Remark 5.6. We now move towards proving the formula stated in Proposition 5.4. In preparation of the proof, let us review the basic "moments vs. free cumulants" formula which expresses the moments of a distribution µ ∈ D alg (k) in terms of its free cumulantsthat is, in terms of the coefficients of the R-transform R µ . This formula says that Cf (i 1 ,...,in);π (R µ ), ∀ n ≥ 1, ∀ 1 ≤ i 1 , . . . , i n ≤ k; (5.6) and more generally, that for any ρ ∈ N C(n) we have Cf (i 1 ,...,in);π (R µ ) (5.7) (where Equation (5.6) corresponds to the case when ρ = 1 n ). For more details on this, see Lectures 11 and 16 of [6]. Also in preparation of the proof of Proposition 5.4 it is convenient to introduce the following elements of notation.
Notation 5.7. Let n be a positive integer.
Proof of Proposition 5.4. We fix for the whole proof a positive integer n and some indices 1 ≤ i 1 , . . . , i n ≤ k for which we will prove that Equation (5.4) holds.
We start from the left-hand side of the equation. From the moment-cumulant formula (5.6) and the fact that R ν⊞γt = R ν + R γt , we have: Now let us fix for the moment a partition π ∈ N C(n), and let us look at the term indexed by π on the right-hand side of (5.8). We write this term explicitly: Cf (i 1 ,...,in);π (R ν + R γt ) = A block of π Cf (i 1 ,...,in)|A (R ν ) + Cf (i 1 ,...,in)|A (R γt ) (5.9) and we expand the product on the right-hand side of (5.9) into a sum of 2 |π| terms. The general term of the sum is obtained by splitting the set of blocks of π into two sets of blocks S 1 and S 2 , and by forming the product But a fundamental fact about free semicircular systems is that the R-transform of γ t is just (see [6], Lectures 11 and 16). Thus the second product in (5.10) is non-zero if and only if every block B ∈ S 2 is of the form B = {p, q} with 1 ≤ p < q ≤ n such that i p = i q . When this requirement is satisfied, the set S 2 of blocks of π corresponds naturally to a partial pairing ρ ∈ N C ≤2 (n; i 1 , . . . , i n ) such that ρ ⊳ π (where Notation 5.7 is used). For our fixed π ∈ N C(n) we thus arrive to an equation of the form Cf (i 1 ,...,in);π (R ν + R γt ) = ρ∈N C ≤2 (n;i 1 ,...,in) such that ρ⊳π term ρ (5.11) where the quantities "term ρ " are further discussed in the next paragraph. So let π ∈ N C(n) be as in the preceding paragraph, and let ρ ∈ N C ≤2 (n; i 1 , . . . , i n ) be such that ρ ⊳ π. In connection to this ρ we will use the notations D(ρ), S(ρ) and ρ ∈ N C( S(ρ) ) that were introduced in Notation 5.3. The contribution "term ρ " to the sum (5.11) is of the form shown in (5.10), where S 2 is the set of blocks of π which also are 2-element blocks of ρ. The product " B∈S 2 · · · " in (5.10) is then clearly equal to t |D(ρ)|/2 . For the other product " A∈S 1 · · · " in (5.10) we note that the union of the blocks counted in S 1 is equal to S(ρ), and this gives us that A∈S 1 Cf (i 1 ,...,in)|A (R ν ) = Cf ( (i 1 ,...,in)|S(ρ) );(π|S(ρ)) (R ν ).
But the latter quantity is in turn equal to Cf ( (i 1 ,...,in)|S(ρ));b ρ (M ν ), due to the moments vs. free cumulant formula (used now in the more general form that was reviewed in (5.7)). Replacing this in (5.13) takes us precisely to the right-hand side of Equation (5.4), and this concludes the proof.

Relation between B t and the free Brownian motion
Recall from Remark 2.5 that the map µ → η µ is a bijection from D alg (k) onto the space of series C 0 z 1 , . . . , z k . It thus makes sense to define a map Φ : D alg (k) → D alg (k) via the η-series prescription described as follows. Definition 6.1. For every ν ∈ D alg (k), we let Φ(ν) be the unique distribution µ ∈ D alg (k) which has η-series given by: Our goal in the present section is to prove the following result.
Theorem 6.2. Let ν be a distribution in D alg (k). We have that where γ t ∈ D alg (k) is the distribution of the scaled free semicircular system from Notation 5.1.
A key point in the proof of Theorem 6.2 will be to use a natural combinatorial construction of "assigning singletons to doubletons" in a partial pairing, which is described next. Remark 6.3. ("Assigning singletons to doubletons for ρ ∈ N C ≤2 (n)".) Let a partition ρ ∈ N C ≤2 (n) be given. We will denote by α(ρ) the non-crossing partition of {0, 1, . . . , n, n + 1} which is obtained as follows. Start with the partial pairing of {0, 1, . . . , n, n + 1} that is obtained by adding to ρ the 2-element block {0, n + 1}. Consider the picture of this new partial pairing (drawn in the usual way -with the points 0, 1, . . . , n, n+1 on a horizontal line, and with a family of non-intersecting "hooks" drawn under that horizontal line, to represent the 2-element blocks of the partial pairing). In this picture we draw some additional vertical line segments, starting at every singleton of ρ, and going down until they meet a hook representing a doubleton. When these new vertical segments are added to the picture, we now have the picture of a non-crossing partition of {0, 1, . . . , n, n+1}, which will be denoted by α(ρ).
A concrete example: if n = 9 and ρ ∈ N C ≤2 (n) is as in ( Clearly, the definition of α(ρ) could also be stated without referring to pictures. That is, the rule for assigning the singletons of ρ to doubletons (in order to create α(ρ)) can be expressed in plain algebraic terms. Indeed, for every 1-element block {i} of ρ, exactly one the following two possibilities (1) and (2) applies: (1) Either there is no 2-element block B = {p, q} of ρ such that p < i < q. In this case i is assigned to the doubleton {0, n + 1} that was added to ρ. the "assign-singletons-to-doubletons" map. It is easily checked that the image of α is π ∈ N C( {0, 1, . . . , n + 1} ) 0 π ∼ n + 1 and π has no 1-element blocks , (6.4) where the notation "0 π ∼ n + 1" in (6.4) is a shorthand for "0 and n + 1 belong to the same block of π". It is also immediate that the map α from (6.3) is one-to-one. The map β : π ∈ N C( {0, 1, . . . , n + 1} ) 0 π ∼ n + 1 and π has no 1-element blocks → N C ≤2 (n) (6.5) which is inverse to α is described as follows. Let π = {A 1 , . . . , A p } be a partition from the set (6.4), and say that A 1 is the block of π that contains 0 and n + 1. Then Proof of Theorem 6.2. Fix t > 0 for which we will prove that (6.2) holds. We will prove this equality by showing that the distributions on its two sides have the same η-series: We first observe that on both sides of (6.7) we have series in C 0 z 1 , . . . , z k that are of the form k i=1 z 2 i + (terms of order ≥ 3). (6.8) Indeed, from the definition of Φ in Equation (6.1) it is clear that η Φ(σ) is of the form (6.8) for every σ ∈ D alg (k), and this applies in particular to the left-hand side of (6.7). On the right-hand side of (6.7) we first invoke Remark 4.4 and write then we use the fact that η Φ(ν) is of the form (6.8), combined with the observation (see Remark 3.6) that applying Reta does not change the linear and quadratic terms of a series in C 0 z 1 , . . . , z k . In order to prove (6.7), we should thus fix a monomial of length ≥ 3 in z 1 , . . . , z k , and prove that the coefficients for this monomial in η Φ(ν⊞γt) and in η Bt(Φ(ν)) are equal to each other. It will be convenient to denote our fixed monomial in z 1 , . . . , z k as z i 0 z i 1 · · · z in z i n+1 for some n ≥ 1 and i 0 , i 1 , . . . , i n+1 ≤ k. Our job for the remaining of the proof is to verify that Cf (i 0 ,i 1 ,...,i n+1 ) η Φ(ν⊞γt) = Cf (i 0 ,i 1 ,...,i n+1 ) η Bt(Φ(ν)) , (6.9) for this fixed n and i 0 , i 1 , . . . , i n+1 .
On the left-hand side of (6.9) we have The latter moment is exactly of the kind studied in Section 5 of the paper, and can be expressed (by Proposition 5.4) as a summation indexed by N C ≤2 (n). Thus for the lefthand side of (6.9) we come to where for every ρ ∈ N C ≤2 (n) the contribution term ′ ρ of ρ is as on the right-hand side of Equation (5.4) in Proposition 5.4.
Let us next observe that if the indices i 0 , i 1 , . . . , i n+1 fixed since (6.9) do not satisfy the condition i 0 = i n+1 , then the right-hand sides of both (6.10) and (6.12) vanish. This is clear for (6.10), while for (6.12) we argue as follows: if i 0 = i n+1 then the product in (6.13) is guaranteed to vanish (since one of the blocks of π contains 0 and n + 1), hence every term term ′′ π on the right-hand side of (6.12) is equal to 0. So let us then assume that i 0 = i n+1 . The equality (6.9) that we have to prove is reduced (by virtue of (6.10) and (6.12)) to π in the set f rom (6.4) term ′′ π , (6.14) where the quantities term ′ ρ and term ′′ π are described in Equations (5.4) and (6.11), respectively. But the equality (6.14) is immediately verified by using the "assign-singletons-todoubletons" construction from Remark 6.3. Indeed, in Remark 6.3 we pointed out a natural bijection β from the set in (6.4) onto N C ≤2 (n), and by using the explicit description provided there for β it is immediately seen that term ′′ π = term ′ β(π) , for every π in the set from (6.4). Thus β provides a term-by-term identification of the sums on the two sides of (6.14), and this completes the proof.

Restricting to the framework of D c (k)
In this section we show that the results from the Sections 4-6 of the paper continue to hold when we work in C * -framework.
∃ C * -probability space (A, ϕ) and selfadjoint elements x 1 , . . . , x k ∈ A such that µ x 1 ,...,x k = µ    (7.1) (where the joint distribution µ x 1 ,...,x k is defined as in Equation (2.2) from Definition 2.1). The fact that (A, ϕ) is a C * -probability space means here that A is a unital C * -algebra and that ϕ : A → C is a positive linear functional such that ϕ(1 A ) = 1. The notation "D c (k)" is chosen to remind of "distributions with compact support"indeed, in the case when k = 1 we have a natural identification between D c (1) and the set of probability distributions with compact support on R.
Remark 7.2. In the preceding sections, the operations ⊞ and ⊎ and the convolution powers with respect to them were considered in the larger framework of the space D alg (k). But by considering sums of freely independent and respectively Boolean independent k-tuples of selfadjoint elements in a C * -probability space, one sees that if µ, ν ∈ D c (k) then µ ⊞ ν and µ ⊎ ν belong to D c (k) as well. Hence ⊞ and ⊎ make sense as binary operations on D c (k). Moreover, concerning convolution powers we have that The fact stated in (7.2(a)) was proved in [5], by using compressions with free projections. The proof of (7.2(b)) is done by constructing an operator model for µ ⊎t -see Remark 4.7 and Proposition 4.8 of [1]. The following result is then an immediate consequence of (7.2) and of what was proved in algebraic framework in Theorem 4.11.
where B is the multi-variable Boolean Bercovici-Pata bijection from Theorems 1 and 1' of the paper [1].
In the remaining part of this section we will show that the above Theorem 6.2 also carries through to the C * -framework. The main point that needs to be addressed is that the map Φ : D alg (k) → D alg (k) introduced in Definition 6.1 sends D c (k) into itself. We will prove this via an "operator model" for Φ, described in the next remark and theorem. where H is a Hilbert space, a 1 , . . . , a k ∈ B(H) are selfadjoint operators, and ξ o ∈ H is a unit vector. Starting from this data, we proceed as follows: (i) We consider the Hilbert space K := C ⊕ k j=1 H , and the unit vector Ω 0 := 1 ⊕ 0 ⊕ 0 ⊕ · · · ⊕ 0 k times ∈ K. For 1 ≤ j ≤ k we let v j : H → K be the embedding defined by v j (ξ) = 0 ⊕ 0 ⊕ · · · ⊕ 0 j−1 times ⊕ξ ⊕ 0 ⊕ · · · ⊕ 0 k−j times ∈ K, ξ ∈ H.
(ii) For 1 ≤ j ≤ k we denote v j ξ 0 =: Ω j ∈ K, and we consider the rank-one partial isometry w j ∈ B(K) which carries Ω 0 to Ω j . The operator w j and its adjoint are thus described by the formulas: (iii) For 1 ≤ j ≤ k we consider the selfadjoint operators x j , y j ∈ B(K) defined by x j := 0 ⊕ a j ⊕ · · · ⊕ a j k times and y j := w j + x j + w * j .
The system (K; y 1 , . . . , y k ; Ω 0 ) will be called the output of the operator model for Φ. The terms "input" and "output" used in this construction are justified by the following theorem.
Proof. If i ′ = i ′′ then both sides of Equation (7.6) are equal to 0: the right-hand side vanishes because of how η Φ(ν) is defined (see Definition 6.1), while the vanishing on the left-hand side follows immediately from the operating rules described in (7.5). So we will assume that i ′ = i ′′ =: i, when the relation that has to be proved becomes; w * i x j 1 · · · x jm w i Ω 0 = ν(X j 1 · · · X jm )Ω 0 .
Proof of Theorem 7.5. We fix for the whole proof a positive integer n and some indices 1 ≤ i 1 , . . . , i n ≤ k, for which we verify that µ(X i 1 · · · X in ) = Φ(ν) (X i 1 · · · X in ). (7.10) By the definition of µ, the left-hand side of (7.10) is µ(X i 1 · · · X in ) = y i 1 · · · y in Ω 0 , Ω 0 = (w i 1 + x i 1 + w * i 1 ) · · · (w i 1 + x i 1 + w * i 1 )Ω 0 , Ω 0 , and the latter quantity expands as a sum of 3 n terms of the form u 1 · · · u n Ω 0 , Ω 0 , with But from the rules (7.5) for how the operators w i , x i , w * i act on the decomposition CΩ 0 ⊕ v 1 (H) ⊕ · · · ⊕ v k (H) of K it follows that many of these 3 n terms vanish. We leave it as an easy exercise to the reader to verify that we have in fact u 1 · · · u n Ω 0 , Ω 0 = 0 whenever u 1 , . . . , u n from (7.11) are not chosen according to the recipe (7.7) from Lemma 7.9. By taking Lemma 7.9 into account, we thus find that µ(X i 1 · · · X in ) = π∈Int(n), with no singletons Cf (i 1 ,...,in);π η Φ(ν) . (7.12) It remains to note that on the right-hand side of (7.12) it does not cost anything to add the terms Cf (i 1 ,...,in);π η Φ(ν) where π ∈ Int(n) has some singleton blocks; indeed, each of these added terms is in fact equal to 0, because the linear terms of the series η Φ(ν) vanish. So from (7.12) we can write: µ(X i 1 · · · X in ) = π∈Int(n) Cf (i 1 ,...,in);π η Φ(ν) = Cf (i 1 ,...,in) M Φ(ν) (by Remark 7.6) which is what we wanted to obtain. Proof. Let ν be in D c (k). By using the GNS construction one can realize ν as the joint distribution of a k-tuple a 1 , . . . , a k of selfadjoint operators on a Hilbert space H, with respect to a vector-state · ξ o , ξ o on B(H). Then Theorem 7.5 gives Φ(ν) as the joint distribution of y 1 , . . . , y k ∈ B(K) with respect to · Ω 0 , Ω 0 , where (K; y 1 , . . . , y k ; Ω 0 ) are constructed as in Remark 7.4. This implies that Φ(ν) ∈ D c (k).
It thus follows that the statement of Theorem 6.2 also holds in C * -framework: Corollary 7.11. Let ν be a distribution in D c (k). Then for every t > 0 we have Φ( ν ⊞ γ t ) = B t ( Φ(ν) ) ∈ D c (k), (7.13) where γ t is the distribution of the scaled free semicircular system from Notation 5.1.