On multiplication in q-Wiener chaoses

. We pursue the investigations initiated by Donati-Martin [9] and Eﬀros-Popa [10] regarding the multiplication issue in the chaoses generated by the q -Brownian motion ( q ∈ ( − 1 , 1)), along two directions: ( i ) We provide a fully-stochastic approach to the problem and thus make a clear link with the standard Brownian setting; ( ii ) We elaborate on the situation where the kernels are given by symmetric functions, with application to the study of the q -Brownian martingales.


Introduction: the q-Brownian motion
The object at the center of this study is the so-called q-Brownian motion (q ∈ (−1, 1)), which stands for one of the most celebrated family of non-commutative processes. For the sake of the clarify, let us thus start our presentation with a very short reminder on the setting of non-commutative probability theory.
First, we call a non-commutative process any path with values in a non-commutative probability space, that is a von Neumann algebra A equipped with a weakly continuous, positive and faithful trace ϕ playing the role of the "expectation" in this framework. Based on this specific structure, we can then define "L p (ϕ)"spaces along the expected procedure, that is by considering the norm When p = 2, this naturally gives rise to a Hilbert space (which will be the main setting of our study), with scalar product given by X, Y L 2 (ϕ) := ϕ XY * . Now recall that, according to a fundamental spectral result, any self-adjoint operator X in A can be associated with a probability measure µ X on R satisfying the following property: for any real polynomial P , R P (x)dµ X (x) = ϕ(P (X)). In other words, µ X and X share the same "moments". With this property in mind, self-adjoint operators in A are called (non-commutative) random variables, and the above measure µ X is called the law (or distribution) of X. By extension, we define the law of a given family {X (i) } i∈I of random variables in (A, ϕ) as the set of all of its joint moments ϕ X (i1) · · · X (ir) , i 1 , . . . , i r ∈ I, r ∈ N.
The definition of the q-Brownian can now be introduced along the following combinatorial description: 2. When π ∈ P 2 ({1, . . . , r}), we call a crossing in π any set of the form {{x 1 , y 1 }, {x 2 , y 2 }} with {x i , y i } ∈ π and x 1 < x 2 < y 1 < y 2 . The number of such crossings is denoted by Cr(π).

Definition 1.2.
For any fixed q ∈ (−1, 1), we call a q-Brownian motion in a non-commutative probability space (A, ϕ) any collection {X t } t≥0 of random variables in (A, ϕ) such that, for every integer r ≥ 1 and all t 1 , . . . , t r ≥ 0, one has ϕ X t1 · · · X tr = π∈P2 ({1,...,r}) q Cr(π) {p,q}∈π The existence issue and the basic stochastic properties of the q-Brownian motion were investigated in a series of remarkable papers by Bożejko, Kümmerer and Speicher [4,5,6]. Using (1), it can be checked in particular that for all t 1 , . . . , t r and Y := X t3 · · · X tr , ϕ X t1 X t2 Y − ϕ X t2 X t1 Y = (1 − q) P t1,...,tr (q), for some polynomial P t1,...,tr (q), which, to some extent, illustrates the following idea: the further q to 1, the less "commutative" the process. This presentation of the q-Brownian model as a possible way to quantify some non-commutativity phenomenon can actually be traced back to a paper by Frisch and Bourret [11].
Let us now consider the two following particular situations: • When letting q tend to 1 in the right-hand side of (1), one recovers the classical Wick description of the joint moments of the standard (commutative) Brownian motion; • Taking q = 0, we get that the only terms with non-zero contribution in the right-hand side of (1) are those for which Cr(π) = 0, i.e. the non-crossing pairings. The 0-Brownian motion thus corresponds to a so-called semicircular family (see [13,Lecture 8]), and the process in this case actually coincides with the celebrated free Brownian motion, whose freely-independent increments are known to be closely related to the asymptotic behaviour of large random matrices (see [16]).
At the level of its distribution, the q-Brownian model can thus be seen as a natural extension of two of the most central processes in probability theory. It is then natural to examine whether the classical stochastic properties satisfied by each of these two processes can also be extended to every q ∈ (−1, 1). A first example in this direction is given by the result of [4, Theorem 1.10] about the marginal-distribution of the process, that is the law of X t for fixed t ≥ 0. More interestingly, we can study the robustness of this interpolation with respect to the general stochastic integration issue. For instance, it has been shown by Donati-Martin in [9] that one could also define a natural "Itô integral" with respect to the q-Brownian motion, which extends both the usual Itô integral associated with the standard Brownian and the free Itô integral exhibited by Biane and Speicher in [3] when q = 0 (see also [8] for a rough-path-type approach to this stochastic integration issue).
In our study, we will more specifically be interested in the analysis of the so-called q-Wiener chaos, that is the "q-analog" of the usual Wiener chaos generated by the standard Brownian motion. The construction of the chaoses is also due to Donati-Martin (in the aforementionned paper), and it extends the procedure initiated by Biane and Speicher in the free case (for the sake of completeness, we shall briefly review these arguments in Section 2.2). Some first results about the multiplication properties prevailing in q-Wiener chaoses can be found in [9,10] (let us also quote the q-extension of the fourth-moment phenomenon in [7], at least for q ∈ [0, 1)). Our aim in these notes is to go deeper into this analysis, with a twofold objective in mind: (i) We intend to emphasize the rich combinatorial machinery governing the behaviour of q-Wiener chaoses, and in the same time offer a fully-stochastic approach to the multiplication issue. Our main result here will be the exhibition of a full q-Wick product formula for multiple integrals (Theorem 3.1). Although similar formulas can already be found in the specific q-Fock-space framework (see for instance [10,Theorem 3.3]), our proof of the result will only rely on probabilistic arguments, i.e. it will only depend on the law of the process (as given by formula (1)) and not on its representation (just as classical probability theory builds upon the law of the Brownian motion and not upon its representation). In brief, our strategy can be regarded as a q-extension of the procedure displayed in [14, Section 1.1.2] for the classical commutative case, which hopefully will make the proof (and accordingly the result) very accessible to a probabilist audience.
To achieve our purpose, we will be led to rephrase the central concept of contraction in terms of qcontraction along possibly "incomplete" pairings (see Section 2.1). Thanks to this representation, many of our arguments can then be conveniently illustrated through basic pictures, and in that sense, our formulation is somehow related to the nice combinatorial approach developed by Nica and Speicher in the free case [13].
(ii) Our second line of investigations regarding q-Wiener chaoses will consist of a close examination of the fully-symmetric situation, that is the situation where the integrals involve fully-symmetric kernels. In this case, it turns out that the coefficients in the product formula can be expressed in terms of classical q-combinatorial numbers, making the link with the classical commutative case even more clear (Theorem 4.2). We will also provide an explicit description of the stochastic dynamics governing the so-called q-Hermite martingales (Corollary 4.4), a central family of processes whose definition is based on fully-symmetric kernels. These procedures will again give us the opportunity to offer a glimpse on the rich combinatorics associated with the q-Brownian process.
The study is organized as follows. In Section 2, we introduce the main combinatorial ingredients at the core of our analysis and recall the definition of the q-Wiener chaos. Section 3 is then devoted to the presentation of our stochastic proof of the general multiplication formula. The specificities of the fully-symmetric situation are finally highlighted in Section 4.

Acknowledgements.
We are grateful to an anonymous referee for his/her careful reading of these notes.

Generalities on q-Wiener chaoses
As a first step of our analysis, let us introduce an alternative and as-graphical-as-possible approach to the notion of q-contraction, in comparison with the definition provided in [9].
2.1. Arrangement along a partition and q-contraction. We have seen through formula (1) that the law of a q-Brownian motion can be conveniently described as a sum over pairings. In order to extend such formulas at the level of the processes (and not only their laws), we will need to involve more general "incomplete pairings" into the procedure: . . , n} with blocks of one or two elements, and for every 0 ≤ k ≤ n 2 , we denote P k ≤2 the set of partitions π ∈ P ≤2 containing exactly k pairs.
c) ∈ π and a < c < b. We will denote by Cr(π) the number of crossings in π. Now observe that any partition π ∈ P k ≤2 (n) can be described in a unique way as With this notation in mind, and for all s = (s 1 , . . . , s k ) ∈ R k + and t = (t 1 , . . . , t n−2k ) ∈ R n−2k we define the arrangement π(s, t) ∈ {s, t} n as follows: for l = 1, . . . , n, When k = 0, respectively 2k = n, we naturally extend the latter definition by setting π(t) := t for every t ∈ R n + , respectively π(s) l : Example 2.2. Consider the partition π ∈ P 4 ≤2 (14) drawn below, with singletons represented by dashed lines "extending to infinity" and pairs by continuous lines. Such a representation implies in particular that each pair (a i , b i ) ∈ π satisfying a i < c j < b i , for some singleton (c j ) ∈ π, necessarily crosses the "line" (c j ). It is then easy to visualize that in this case π(s, t) = (t 1 , s 4 , t 2 , s 3 , t 3 , s 2 , s 4 , t 4 , s 1 , t 5 , s 1 , t 6 , s 3 , s 2 ) and Cr(π) = 13.
The suitable definition of a contraction in our setting now reads as follows: and π ∈ P k ≤2 (n), we call integral of f along π, and denote by π f , the function defined for every t ∈ R n−2k where π(s, t) stands for the above-defined arrangement along π.
(ii) For all f ∈ L 2 (R m + ), g ∈ L 2 (R n + ) and π ∈ P ≤2 (m ⊗ n), we define the contraction of f and g along π by the formula f ⊗ π g := π f ⊗ g. Finally, for all q ∈ [0, 1) and k ∈ {0, . . . , m ∧ n}, we define the q-contraction of f and g of order k as the sum This definition can be seen as a natural q-extension of the standard contracting procedure along a Feynman diagram (as defined in [12,Section 7.2]). It actually coincides with the notion of a q-contraction introduced in [9], which can be recovered by comparing [9, Proposition 4.1] with the subsequent Proposition 3.4 and applying the isometry property (4).
we define the mirror-symmetrized π * of π as the element of P k ≤2 (n ⊗ m) obtained by taking the symmetric of π along a vertical axis between the two blocks {1, . . . , m} and {m + 1, . . . , m + n}. Otherwise stated, if π is given by (2), then For further use, let us also label the following readily-checked relations between mirror-symmetry and q-contractions: Besides, Cr(π * ) = Cr(π) and accordingly (f ⊗ q k g) * = g * ⊗ q k f * for every k ∈ {0, . . . , m ∧ n}. 2.2. Construction of the q-Wiener chaos. With the above "graphical" approach in mind, and also for the sake of completeness, let us briefly go back here to the definition of the q-Wiener chaos.
From now on and for the rest of the paper, we fix q ∈ (−1, 1) and consider a q-Brownian motion (X t ) t≥0 = (X (q) ) t≥0 in some non-commutative probability space (A, ϕ). With the notations of Section 2.1, observe that the joint moments of (X t ) t≥0 can also be expressed as The construction of the multiple integrals of X, which give birth to the q-Wiener chaoses, can then be made along a similar procedure as in the classical commutative case (see [14]). Consider first the set E m of simple functions, that is the set of functions f of the form

using the convention that for any interval
Then, just as in the commutative case, the extension of I q m (f ) to any real-valued function f ∈ L 2 (R m + ) relies on two ingredients: the density of E m within L 2 (R m + ) on the one hand, an isometry property on the other, which, in this setting, reads as follows: Proposition 2.6. For f ∈ E m and g ∈ E n , it holds that where P q is the q-symmetrization operator defined for all f ∈ L 2 (R m + ) and s ∈ R m + by q Cr(π) f (π(s) m+1::2m ) , with π(s) m+1::2m := (π(s) m+1 , . . . , π(s) 2m ) .

Multiplication formulas in q-Wiener chaoses
With the formalism of Section 2 in hand, we can now state our main result about q-Wiener chaos, namely the extension, to every q ∈ (−1, 1), of the full Wick product formula: Theorem 3.1. Let n 1 , . . . , n r ≥ 1 and for every i ∈ {1, . . . , r}, let f i ∈ L 2 (R ni + ). Then, with the notations of Section 2.1, it holds that By letting q tend to 1 in (6) (at least at some heuristic level), we indeed recover the exact expression of the classical full Wick product formula for the standard Bm (see [12,Theorem 7.33]). Observe also that, by combining (6) The rest of this section is devoted to the proof of Theorem 3.1, which will consist in a natural three-step procedure: we first show the formula when (r = 2, n 1 = 1, n 2 ≥ 1), then extend the result to the case where (r = 2, n 1 , n 2 ≥ 1), and finally turn to the general situation. At each step, our strategy will actually be based on the (non-trivial) q-extension of the classical arguments used in the commutative framework.
Proof. For any interval A, let us write X(A) for X(1 A ). Using bilinearity and a density argument, it is readily checked that we can focus on the following situation: where k ∈ {1, . . . , n}, the intervals A i are disjoint and A, B are both intervals disjoint from the A i 's such that A = B or A ∩ B = ∅.
If A ∩ B = ∅, then clearly f ⊗ q 1 g = 0 and the formula is trivially satisfied, so we assume from now on that A = B. Then, denoting by µ(A) the length of A, it is easy to see (as in Figure 1 where we have set h ε := i =j 1 Bi ⊗ 1 A1 ⊗ · · · ⊗ 1 A k−1 ⊗ 1 Bj ⊗ 1 A k ⊗ · · · ⊗ 1 An−1 . At this point, observe that due to (4), one has which tends to 0 as ε → 0. Therefore, it remains us to check that l i=1 R ε,i L 2 (ϕ) → 0 as ε → 0, and to this end, decompose When i = j, it turns out that ϕ R * ε,i R ε,j = 0, as a consequence of the following readily-checked relations (see Figure 2 for an illustration of the first one): and ϕ [X(A n−1 ) · · · X(A 1 )] · [X(A 1 ) · · · X(A n−1 )] = µ(A 1 ) · · · µ(A n−1 ). As for the second summand in (8), it is easy to see that ϕ R , which, by letting ε tend to zero, completes the proof of our statement. Figure 2. The only partition with non-zero contribution in (9) (here, k = 4, n = 6).

Proof of Proposition 3.4.
Assume that formula (10) holds true for all m ≤ n, f ∈ L 2 (R m + ) and g ∈ L 2 (R n + ). Then for m ≥ n, it holds that where we have used Lemma 2.5 to derive the last equality.
Therefore, we can stick to an induction procedure on m ≥ 1 for n ≥ m. If m = 1, then (10) is nothing but the result of Proposition 3.3. Assume that the decomposition holds true for some m ≥ 1 and every n ≥ m, and let n ≥ m + 1. By a density argument, we can take f ∈ L 2 (R m+1 and f 2 = 1 A2 ⊗ · · · ⊗ 1 Am+1 , for disjoint intervals (A i ). Then I q m+1 (f ) = I q 1 (f 1 )I q m (f 2 ) and by the induction hypothesis which, by Proposition 3.3, yields that The conclusion immediately follows from the two identities exhibited in Lemma 3.5.
We can finally turn to the proof of the general formula.

Multiplication in the fully-symmetric case
Let us now elaborate on the specific situation where the kernels involved within the multiple integrals under consideration are given by fully-symmetric functions. Recall that a function f : R n + → R is said to be fully-symmetric if for all times t 1 , . . . , t n ∈ R + and every permutation σ of {1, . . . , n}, one has f (t 1 , . . . , t n ) = f (t σ (1) , . . . , t σ(n) ).
It is important to notice here that, contrary to the classical commutative case (with multiple integrals generated by a standard Bm), the fully-symmetric assumption must be regarded as highly restrictive in the non-commutative situation. Otherwise stated, considering any generic function f ∈ L 2 (R n + ) (n ≥ 2) and its symmetrizationf (t 1 , . . . , t n ) := 1 n σ∈Sn f (t σ (1) , . . . , t σ(n) ), there is no reason in general for the two multiple integrals I q n (f ) and I q n (f ) to be equal. Figure 4. Construction of π (third line) from π 1 (second line) and π 2 (first line).

Multiplication formula.
Given two fully-symmetric functions f 1 ∈ L 2 (R n1 + ), f 2 ∈ L 2 (R n2 + ) and a parameter k ∈ {1, . . . , n 1 ∧ n 2 }, it is readily checked that the contraction f 1 ⊗ π f 2 (see Definition 2.3) does not depend on the choice of π ∈ P k ≤2 (n 1 ⊗ n 2 ): we denote by f 1 ⊗ k f 2 this common value. For q ∈ (−1, 1), the q-contraction of f 1 and f 2 (see again Definition 2.3) is then equal to We propose to show that the latter coefficients can be conveniently expressed in terms of standard qbinomial coefficients. To this end, let us first recall the definition of the classical coefficients associated with q-combinatorics: [n] q := 1 + q + · · · + q n−1 , [n] q ! : for all n 1 , . . . , n p ≥ 0 such that n 1 + · · · + n p = n.
Injecting (14) into (10) immediately gives rise to the following combinatorial description of the product of any two multiple integrals build upon fully-symmetric kernels. Note that, when compared for instance with [14,Proposition 1.1.3], this formulation makes the transition with the classical commutative case even more clear.
Setting i := (i 1 , . . . , i k ) and j := (j 1 , . . . , j k ), the difference Cr(i, j) := Cr(π) − Cr(π ′ ) is then given by the number of crossings between a pair and a singleton in π. We can easily compute this quantity as follows (see again Figure 5): which in fact reduces to Cr(i, j) = k l=1 (i l + j l ) − k(k + 1). As a consequence, It now remains us to apply Corollary 1.
where S n refers to the group of permutations of {1, . . . , n} and inv(σ) to the number of inversions in σ. Figure 5. A partition π ∈ P 3 ≤2 (6×7) (first line) with its associated partition π ′ ∈ P 2 (3⊗3) (second line). With the notations of the proof, the positions of the pairs in π correspond to i 1 = 2, i 2 = 3, i 3 = 5, and j 1 = 2, j 2 = 4, j 3 = 7. We propose here to use the results of the previous sections (and especially identity (14)) so as to complete the program initiated in [9, Section 4.2] regarding the stochastic dynamics of the sequence (M while in the free situation, that is when q = 0, the following relation can be found in [2] M (0) In order to express our interpolation result between these two formulas, let us consider the family of kernels defined for ℓ, m ≥ 1 and t, s 1 , . . . , s ℓ ≥ 0 as where the integral in the right-hand side is understood in Itô's sense (see [9,Corollary 3.2]).
To see that identity (17) indeed provides us with the desired interpolation between (15) and (16), it suffices to observe that, on the one hand, and then applying (17) to each summand, we immediately recover (16) for q = 0, and more generally: where the integral in the right-hand side is understood in Itô's sense.
Proof of Proposition 4.3. For the sake of clarity, we fix q and t for the whole proof and drop the dependence on these two parameters in the notation, that is we write C