NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH

August 25, 2008 10:5 WSPC/102-IDAQPRT

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Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 11, No. 3 (2008) 351–375 c World Scientific Publishing Company

NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH KESTEN DISTRIBUTIONS AND RELATED POISSON PROCESSES

ROMUALD LENCZEWSKI∗ and RAFAL SALAPATA† Instytut Matematyki i Informatyki, Politechnika Wroclawska, Wybrze˙ze Wyspia´ nskiego 27, 50-370 Wroclaw, Poland ∗ [email protected] † [email protected] Received 25 October 2007 Communicated by Y. G. Lu We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, 0 in which to each such partition P we assign the weight w(P ) = pe(P ) q e (P ) , where e(P ) 0 and e (P ) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney’s numbers (related to inner blocks in non-crossing partitions) and generalized Euler’s numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes. Keywords: Monotone independence; free independence; free Brownian motion; monotone Brownian motion; Kesten laws; Poisson process; Fock space. AMS Subject Classification: 46L53, 46L54, 60F05

1. Introduction In noncommutative probability several noncommutative Brownian motions have been introduced and studied. In particular, different notions of noncommutative independence (either abstract, or restricted to the Fock space level) lead to different noncommutative (i) central limit theorems, (ii) analogues of the classical Brownian motion, (iii) Poisson-type processes. 351


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R. Lenczewski & R. Salapata

In this context let us mention here the well-known examples of the boson Brownian motion on the symmetric Fock space of Hudson and Parthasarathy,10 the fermion Brownian motion on the antisymmetric Fock space of Applebaum and Hudson,1 the free Brownian motion on the free Fock space of Speicher21 and the monotone Brownian motion of Muraki18 (see also Lu17 ) on the monotone Fock space. An important feature of the free Brownian motion is that it can be obtained as the limit in distribution (as the dimension becomes infinite) of a sequence of Brownian motions in the finite-dimensional Hermitian matrices, as shown by Biane.4 Interpolations between these examples have also been studied. For instance, an interpolation between the boson, fermion and free Brownian motions, called the q-Brownian motion, was studied by Bo˙zejko and Speicher.6 In the bialgebra and Hopf algebra context, respectively, two different q-central limit theorems and related Brownian motions were studied by Sch¨ urmann20 and the author.14,15 In this paper, we introduce and study an interpolation between noncommutative Brownian motions in monotone probability of Muraki19 and free probability of Voiculescu.22,23 Our interpolation depends on two continuous non-negative parameters p, q. The most important reason why we find this interpolation interesting is that it is based on the combinatorics of ordered non-crossing partitions ON C and thus gives a very natural interpolation between the combinatorics of non-crossing partitions N C in free probability21 and the combinatorics of monotone non-crossing partitions MN C in monotone probability.19 By a monotone non-crossing partition of the set [n] := {1, 2, . . . , n} we understand a sequence P = (P1 , P2 , . . . , Pr ) of blocks, in which the fact that block Pj is inner with respect to block Pi implies that i < j. In other words, the natural partial order on the set of blocks of P implemented by the relation of being inner or outer is respected by the formal order defined by positions of blocks in P . The term “monotone non-crossing partition” comes from our work,16 but it can be traced back to the partitions studied by Muraki.19 The key observation is that the class of non-crossing partitions as well as the class of monotone non-crossing partitions can be included in the scheme of ordered non-crossing partitions if one assigns the weight pe(P ) to each P ∈ ON C, where p ≥ 0 and e(P ) is the number of disorders, or Euler inversions. By a disorder in P = (P1 , P2 , . . . , Pr ) we understand a pair of blocks {Pi , Pj } such that i < j, Pi is inner with respect to Pj and lies immediately under Pj , i.e. with no intermediate blocks between. Then, for p = 0 we obtain monotone non-crossing partitions with the same weight and the case p = 1 corresponds to all ordered non-crossing partitions with the same weight, which reduces to N C. In a similar way, we can introduce the second parameter q ≥ 0 and assign to 0 each P the weight q e (P ) , where e0 (P ) is equal to the number of orders in P (by an order we understand any pair {Pi , Pj } such that i < j and Pi is outer with respect to Pj ). The second parameter is not necessary to give all Kesten laws as such (in particular, the Wigner law), but it is needed when we want to reproduce all mixed


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moments of the free Brownian motion. Besides, it provides a natural symmetry between orders and disorders in our combinatorics. As a consequence, we obtain a nice relation between Delaney’s numbers D(n, k + j), which give the numbers of partitions π ∈ N C 22n which have k +j inner blocks, and generalized Euler’s numbers E(n, k, j), by which we understand the numbers of ordered partitions P ∈ ON C 22n which have k disorders and j orders. Using a weight function on L2 (R+ ) × L2 (R+ ) related to w(P ), we define (p, q)creation and annihilation processes on the free Fock space, (at )t≥0 and (a∗t )t≥0 , respectively, and the corresponding canonical position process (ωt )t≥0 , or Brownian motion, where ωt = at + a∗t (parameters p, q are suppressed in the notation) (in a similar way we can define the canonical momentum process ηt = i(at − a∗t )). In particular, we obtain the combinatorial formula X w(P )tn (1.1) ϕ(ωt2n ) = b(P )! 2 P ∈ON C 2n

for the even moments of the position process in the vacuum state ϕ on F(R+ ) (the odd moments vanish), where the weight is given by 0

w(P ) = pe(P ) q e (P )

(1.2) ON C n (ON C 2n )

and b(P ) denotes the number of blocks of P . By we denote the set of ordered non-crossing partitions (pair-partitions) of [n]. Similarly, the moments of suitably defined processes of Poisson type (γt )t≥0 (dependence on p, q is suppressed again) can be expressed as ϕ(γtn ) =

X

P ∈ON C n

w(P )tb(P ) . b(P )!

(1.3)

It is easy to see that for (p, q) = (0, 1) and (p, q) = (1, 1) the above formulas give the moments of canonical position processes and Poisson processes in monotone probability and free probability, respectively. The case (p, q) = (1, 0) corresponds to the anti-monotone processes. Finally, the case (p, q) = (0, 0) corresponds to Boolean processes. Moreover, we show that our canonical position processes are associated with Kesten distributions. For instance, for t = 1, the moments of ω1 agree with the moments of Kesten measures with densities p p p 2(p + q) − x2 1 fp,q (x) = , x ∈ [− 2(p + q), 2(p + q)] (1.4) π 2 − (2 − p − q)x2 and atoms at x = ± √

1 1−(p+q)/2

for p + q < 1. In particular, for (p, q) = (0, 1) (as

well as for (p, q) = (1, 0)) we obtain the standard arcsine law and for (p, q) = (1, 1) — the standard Wigner law. Let us point out that our model interpolates not only between the moments of the canonical free and monotone position processes, but also between the corresponding mixed moments of creation and annihilation processes. Moreover, it


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reproduces independence on both Fock spaces. Therefore, at least on the Fock space level, it may be viewed as an interpolation between monotone independence and free independence. This feature is absent in the t-interpolation of Bo˙zejko and Wysocza´ nski,7 which also reproduces the moments of Kesten laws,11 but does not reproduce monotone independence. Namely, neither the mixed moments of monotone independent creation and annihilation processes nor the mixed moments of position processes (with arcsine distributions) associated with disjoint intervals agree with the corresponding moments of t-deformed processes for the right value of t. To put it in the general framework of interacting Fock spaces, let us notice that one of the main points of the t-interpolation and of the “gaussianization of probability measures”2 is that one can reproduce the moments of classical probability measures by means of noncommutative Gaussian operators on the one-mode interacting Fock spaces (see also Refs. 8 and 9 for a related result on symmetric measures). Of course, one-mode interacting Fock spaces are examples of interacting Fock spaces3 in which deformations of the inner product on the free Fock space are very simple (and are related to the Jacobi parameters of probability measures). Our deformations of the free Fock spaces (or, of the corresponding creation and annihilation operators) are more complicated and, roughly speaking, they might be viewed as examples of two-mode interacting Fock spaces. Although our motivation is of combinatorial nature rather than related to the interacting Fock space structure, it seems that this is the reason why we can also reproduce noncommutative independence apart from the classical properties like “gaussianization” in the one-mode case. Let us mention here that a discrete interpolation between monotone probability and free probability, called the monotone hierarchy of freeness, was studied in Ref. 16. In particular, we obtained a combinatorial formula for the mixed moments of the hierarchy of m-monotone Gaussians, based on the combinatorics in which one counts blocks which are inner with respect to each block (as in the case of Poisson operators studied by Muraki19 ) rather than on the combinatorics based on blocks’ depths,2 or levels of Catalan paths,8,9 which is equivalent in the case of symmetric measures. Moreover, this also allowed us to reproduce the mixed moments of monotone independent creation and annihilation operators, which are not reproduced by the formulas given in Ref. 2. 2. Combinatorics of Kesten Laws The set of all partitions of the set [n] will be denoted by Pn . We say that π ∈ Pn is non-crossing if there are no pairs {k, k 0 } ⊂ πi and {m, m0 } ⊂ πj with i 6= j and such that k < m < k 0 < m0 . The set of all non-crossing partitions of the set [n] will be denoted by N C n . By Pn2 we will denote the set of all pair-partitions of [n] and N C 2n := N C n ∩ Pn2 . On the set of blocks of π ∈ N C n we can introduce a natural partial order. Namely, we will say that πi is inner with respect to πj for i 6= j if there exist a,


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b ∈ πj such that for all c ∈ πi it holds that a < c < b, in which case we shall write πj < πi . Moreover, we set πj ≤ πi iff πj < πi or πj = πi . Equivalently, we will say that πj is outer w.r.t. πi . We will say that blocks πj , πi are neighboring if they are comparable in the above sense and there are no other “intermediate” blocks between them, i.e. if πj < πi and πj ≤ πk ≤ πi implies that k = i or k = j. If πi , πj are neighboring blocks and πj < πi , we will write πj ≺ πi . A block πi is called outer in π if π has no blocks which are outer with respect to πi . The pair P = (π, σ), where π = {π1 , π2 , . . . , πk } ∈ Pn and σ is a permutation from the symmetric group Sk , will be called an ordered partition of the set [n] and will be identified with the sequence P = (P1 , P2 , . . . , Pk ), where Pi = πσ(i) . In particular, we will write Pj ≺ Pi if πσ(j) ≺ πσ(i) . The set of all ordered (pair, non-crossing, non-crossing pair) partitions of [n] will be denoted OP n (respectively, OP 2n , ON C n , ON C 2n ). Let us observe that in each ordered partition P = (π, σ), the permutation σ defines a linear order on the set of blocks of π. Comparing this order with the partial order given by the relation of being inner (outer) for P ∈ ON C n , we can introduce “disorders” betwen blocks. Definition 2.1. If P = (P1 , P2 , . . . , Pk ) ∈ ON C n , we will say that the pair {Pi , Pj } forms a disorder or Euler inversion if i < j and Pj ≺ Pi . If i < j and Pi ≺ Pj , we will say that the pair {Pi , Pj } forms an order. The numbers of all disorders and orders in P will be denoted e(P ) and e0 (P ), respectively. In a similar way we define orders and disorders in the permutation σ ∈ Sn associated with each index i ∈ {1, 2, . . . , n − 1} for which σ(i) < σ(i + 1) and σ(i) > σ(i + 1), respectively. The numbers of all disorders and orders in σ will be denoted e(σ) and e0 (σ), respectively. Remark 2.1. Well-known Euler numbers, denoted mutations of the set [n] which have k disorders.

n k

, give the numbers of per-

Example 2.1. Consider ordered non-crossing partitions P ∈ ON C 28 and Q ∈ ON C 8 given in Fig. 1. Partition P has 3 pairs of neighboring blocks: P4 ≺ P2 , P4 ≺ P1 and P2 ≺ P3 . Only the first two give disorders, thus e(P ) = 2 and e0 (P ) = 1. Pairs of neighboring blocks in Q are the following: Q1 ≺ Q2 , Q1 ≺ Q3 and Q2 ≺ Q4 . None of them gives a disorder, thus e(Q) = 0 and e0 (Q) = 3. Noncrossing ordered partitions which do not have disorders are called monotone.16 4

1 2 3

1

2 3 4 q q q q q q q q 1 2 3 4 5 6 7 8

q q q q q q q q 1 2 3 4 5 6 7 8

P

2, 3 , 4, 7 , 5, 6 , 1, 8 Fig. 1.

Q

1, 8 , 3, 4, 7 , 2 , 5, 6

Examples of ordered non-crossing partitions.


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By ON CC n (ON CC 2n ) we denote the set of those ordered non-crossing partitions (pair-partitions) of [n], in which the numbers 1 and n belong to the same block. Such partitions will be called covered. Let us introduce numbers X X 1 1 rn = w(P ) , sn = w(P ) , n! n! 2 2 P ∈ON C 2n

P ∈ON CC 2n

for n ≥ 1 and set r0 = 1, s0 = 0, where w(P ) is given by (1.2). P

r r r r Q 1 Q 2 Fig. 2.

r r Q m

P ∈ ON C constructed from Q(1) , . . . , Q(m) ∈ ON CC.

∞ Proposition 2.1. The following relation between sequences (rn )∞ n=1 and (sn )n=1 holds: n X X n ≥ 1. rn = s k1 s k2 · · · s km , m=1 k1 +k2 +···+km =n

where the second summation runs over positive indices k1 , . . . , km . (i)

(i)

(i)

Proof. Let Q(i) = (Q1 , Q2 , . . . , Qki ), where i = 1, 2, . . . , m, be arbitrary pairpartitions from the sets ON CC 22ki , respectively, such that k1 + · · · + km = n. From the blocks of all these partitions we construct ordered pair partitions of [n] with the (i) shape given by Fig. 2. We order all blocks Qj of the subpartition Q(i) in such a way that the order between blocks from the same partition Q(i) is preserved. There are 2 n! k1 !···km ! such orderings and each of them defines exactly one P ∈ ON C 2n . Moreover, each partition from ON C 22n can be obtained in this fashion by an appropriate choice of covered partitions Q(i) . From the above reasoning we obtain |ON C 22n | =

n X

X

m=1 k1 +k2 +···+km =n

n! |ON CC 22k1 | · · · |ON CC 22km | . k1 ! · · · km !

(2.1)

Clearly, between blocks which belong to different partitions Q(i) and Q(j) , i 6= j, there are no orders or disorders. Therefore, e(P ) = e(Q(1) ) + e(Q(2) ) + · · · + e(Q(r) ) ,

e0 (P ) = e0 (Q(1) ) + e0 (Q(2) ) + · · · + e0 (Q(r) ) , which gives multiplicativity of the weights


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w(P ) = w(Q(1) )w(Q(2) ) · · · w(Q(r) ) and that, together with (2.1), gives the assertion. Corollary 2.1. Let R(z) = series. Then

P∞

n=0 rn z

n

R(z) =

and S(z) =

1 . 1 − S(z)

P∞

n=0 sn z

n

be formal power

Proof. Using Proposition 2.1 and simple algebraic computations, we get ∞ ∞ X n X X X R(z) − 1 = rn z n = s k1 s k2 · · · s km z n n=1

=

n=1 m=1 k1 +k2 +···+km =n

∞ X ∞ X

X

m=1 n=m k1 +k2 +···+km =n

=

∞ X

(S(z))m =

m=1

s k1 s k2 · · · s km z n

S(z) 1 − S(z)

from which our assertion follows.

Pn

r

1

r r r r Q1

r r

Q2

Fig. 3.

r

Qr

Partition covered by the last block Pn+1 .

In order to find R(z), we introduce another sequence of numbers, denoted (an )∞ n=0 and defined by the combinatorial formula an =

n X

X

m=1 k1 +k2 +···+km =n

p m s k1 s k2 · · · s km

(2.2)

for n ≥ 1 and we set a0 = 1. Let us observe that an is the sum of contributions to rn+1 of these pair partitions P = (P1 , . . . , Pn+1 ) ∈ ON CC 22n+2 which are covered by the block of highest color, namely Pn+1 . Therefore, X 1 an = w(P ) (2.3) n! 2 P ∈ON CC 2n+2 Pn+1 ={1,2n+2}

for n ≥ 0. In fact, each neighboring block of Pn+1 corresponds to a certain covered partition Qi ∈ ON CC 22ki for i = 1, . . . , m, as Fig. 3 shows. Of course, k1 +· · ·+km =


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n. Besides, let us observe that each neighboring block of Pn+1 forms a disorder with block Pn+1 . This justifies equivalence of (2.2) and (2.3). P∞ Corollary 2.2. Let A(z) = n=0 an z n be a formal power series. Then 1 . A(z) = 1 − pS(z) Proof. Using algebraic calculations and Eq. (2.2), we obtain A(z) − 1 = =

∞ X

an z n =

n=1 ∞ X ∞ X

X

n=1 m=1 k1 +k2 +···+km =n

pm

∞ X

X

n=m k1 +k2 +···+km =n

m=1

=

∞ X n X

(pS(z))m =

m=1

p m s k1 s k2 · · · s km z n

s k1 s k2 · · · s km z n

pS(z) 1 − pS(z)

from which we get the assertion. To find a relation between R(z) and A(z), we shall need some additional notations. Let ON CC n (r) (ON CC 2n (r)) denote the sets of ordered partitions (pair partitions) of [n] with r outer blocks, and, for r = 1, . . . , n, introduce sequences X 1 w(P ) (2.4) s(r) n = n! 2 Q∈ON CC 2n (r)

for n ≥ 1, and set

(r) s0

(1)

(r)

= 0. Of course, sn = sn and sr = 1. P∞ (r) Proposition 2.2. Let S (r) (z) = n=r sn z n , where r ≥ 1. Then S (r) (z) = (S(z))r .

Proof. Notice that we have the following relation between sequences (sn )n≥1 and (r) (sn )n≥1 : X s(r) n ≥ r, s k1 · · · s kr , n = k1 +k2 +···+kr =n

where k1 , k2 , . . . , kr are assumed to be positive integers. This gives !r ∞ ∞ X X X (r) n n S (z) = s k1 · · · s kr z = sn z = (S(z))r . n=r k1 +k2 +···+kr =n

n=1

which proves our assertion.

Lemma 2.1. For n ≥ r + 1 it holds that s(r) n

n−r+1 n−r r X q X (r−1) (r) = ak−1 sn−k + (2n − 2k − r)ak−1 sn−k . n n k=1

k=1


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Proof. Let us split (2.4) into two sums: the first one running over those partitions from ON CC 22n (r) in which the block of highest color is outer, and the second one — over the remaining partitions. Then X X 1 1 w(P ) + w(P ) s(r) n = n! n! 2 2 P ∈ON CC (r) 2n Pn −outer

=

n−r+1 1 X n!

X

n−r 1 X n! k=1

=

X

T ∈ON CC 2 2k Tk ={1,2k}

k=1

+

P ∈ON CC (r) 2n Pn −inner

r

Q∈ON CC 22n−2k (r−1)

X

X

T ∈ON CC 2 2k Tk ={1,2k}

n−1 w(T )w(Q) k−1

(2n − 2k − r)

Q∈ON CC 22n−2k (r)

n−1 qw(T )w(Q) k−1

n−r+1 n−r r X q X (r−1) (r) ak−1 sn−k + (2n − 2k − r)ak−1 sn−k . n n k=1

k=1

which completes the proof. From the above lemma we get a relation between the S (r) (z)’s and A(z) in the form of a differential equation. Corollary 2.3. The functions S (r) (z) and A(z) satisfy the differential recurrence (S (r) (z))0 = rS (r−1) (z)A(z) + 2qz(S (r) (z))0 A(z) − qrA(z)S (r) (z) , with initial conditions S (r) (0) = 0, A(0) = 1. Proof. In view of Lemma 2.1, we have (S (r) (z))0 =

∞ X

n−1 ns(r) n z

n=r

= rz r−1 + r

∞ X

n=r+1 ∞ n−r X X

+q

n=r+1 k=1

=r

∞ n−r+1 X X

n=r

− qr

n−r+1 X

(r−1)

ak−1 sn−k z n−1

k=1

(r)

(2n − 2k − r)ak−1 sn−k z n−1 (r−1)

ak−1 sn−k z n−1 + 2q

n=r+1 k=1

k=1

∞ n−r X X

n=r+1 k=1

∞ n−r X X

(r)

ak−1 sn−k z n−1

(r)

(n − k)ak−1 sn−k z n−1


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=r

∞ X

n−(r−1)

n=r−1

− qr

X

(r−1)

ak sn−k z n + 2qz

n=r k=0

k=0

∞ n−r X X

∞ n−r X X

(r)

ak (n − k)sn−k z n−1

(r)

ak sn−k z n

n=r k=0

= rS (r−1) (z)A(z) + 2qz(S (r) (z))0 A(z) − qrA(z)S (r) (z) . Of course, S (r) (0) = 0 and A(0) = a0 = 1, which completes the proof. Using the relations bewteen functions R(z), A(z) and S (r) (z), r ≥ 1, we can now derive the explicit form of R(z), which will turn out to be related to the moment generating function of Kesten laws. Theorem 2.1. For each p ≥ 0, q ≥ 0, p + q ≥ 0, the sequence (mn )n≥0 , where ( rk if n = 2k , mn = 0 if n is odd is the sequence of moments of the Kesten measure (1.4). The probability measure determined by the moments is unique. Proof. Using Corollary 2.3 and Proposition 2.2, we obtain a differential equation for S(z), namely rS 0 (z) = rA(z) + 2qrzS 0 (z)A(z) − qrS(z)A(z) which, in view of Corollary 2.1, leads to the differential equation for R(z) of the form R2 (z)((1 − q)R(z) + q) R0 (z) = R(z)(1 − p − 2qz) + p

with the initial condition R(0) = 1. Let us observe now that the function R(z) must be symmetric with respect to p and q (this easily follows from the definition of R(z) if we reverse the order in all ordered non-crossing pair partitions over which the summation is taken). Thus, if we denote R(z) = Rp,q (z), then Rp,q (z) = Rq,p (z). Therefore, we obtain another differential equation for R(z) with p and q interchanged. This allows us then to reduce the above equation to the (algebraic) quadratic equation, namely AR2 (z) + BR(z) + C = 0 , where A = (1 − q)2 − (1 − p)2 − 2pz + 2qz , B = 2q(1 − q) − 2p(1 − p) , C = q 2 − p2 ,


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which has two solutions, p p + q − 1 ± 1 − 2(p + q)z R(z) = p + q − 2 + 2z

but only the one corresponding to the minus sign satisfies the initial condition R(0) = 1. If we define the corresponding moment generating function by taking mn = rk for n = 2k with odd moments equal to zero, we obtain the function P n 2 M (z) = ∞ n=0 mn z = R(z ) and the corresponding Cauchy transform p (p + q − 1)z − z 2 − 2(p + q) 1 1 = G(z) = M z z 2 − (2 − p − q)z 2 turns out to be the Cauchy transform of the (uniquely determined) Kesten distribution µp,q (with the absolutely continuous part given by (1.4)). Remark 2.2. The continued fraction representation of G(z) takes the form 1

G(z) = z−

,

1 z−

t z−

t . z − ..

where t = (p + q)/2. Such Cauchy transforms were obtained in the context of the so-called t-transformation of measures.7 However, we will show later that our combinatorics is different and, when carried over to the Fock space level, gives a different Brownian motion (although it also has Kesten distributions). 3. Noncommutative Brownian Motions We will now construct new types of noncommutative Brownian motions on the free Fock space which have Kesten distributions and are parametrized by two nonnegative real numbers p, q. They can be viewed as an interpolation between the free Brownian motion obtained for (p, q) = (1, 1) and the monotone Brownian motion corresponding to (p, q) = (0, 1) (the anti-monotone and Boolean Brownian motions are also obtained, for (p, q) = (1, 0) and (p, q) = (0, 0), respectively). Moreover, the mixed moments of the associated creation and annihilation operators in the vacuum state agree with their counterparts in free probability and monotone probability, a feature absent in other interpolations. By the free Fock space over a Hilbert space H = L2 (R+ ) we understand the direct sum ∞ ∞ M M L2 (Rn+ ) (3.1) H⊗n ∼ F(H) = CΩ ⊕ = CΩ ⊕ n=1

with the canonical inner product.

n=1


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Let w be the weight function on R+ × R+ given by    p if 0 < s < t , w(s, t) = q if 0 < t < s ,   1 otherwise ,

where p > 0, q ≥ 0. Now, introduce special vectors in F(H) denoted by the “tensorlike” symbol f1 ~ f2 ~ · · · ~ fn , where p p (f1 ~ f2 ~ · · · ~ fn )(t1 , t2 , . . . , tn ) := f1 (t1 ) w(t1 , t2 )f2 (t2 ) · · · w(tn−1 , tn )fn (tn ) .

These vectors remind simple tensors, but they have the “nearest neighbor coupling”. Note that the set of such vectors is dense in F(H) if p > 0 and q > 0. If (p, q) = (0, 1), it is dense in the monotone Fock space M(H) = CΩ ⊕

∞ M

L2 (∆(n) ) ,

n=1

where ∆(n) = {(t1 , . . . , tn ) ∈ Rn+ ; t1 ≤ · · · ≤ tn }. In turn, if (p, q) = (1, 0), it is dense in the anti-monotone Fock space (similar to the monotone Fock space, but with the reversed order of coordinates). If (p, q) = (0, 0), we obtain in turn CΩ ⊕ L2 (R+ ). Let us define suitable creation, gauge and annihilation operators which are (p, q)deformations of their free counterparts. Definition 3.1. Define the (p, q)-creation operator a(f ) : F(H) → F(H) associated with f ∈ H as the bounded linear extension of a(f )Ω = f , a(f )(f1 ~ f2 ~ · · · ~ fn ) = f ~ f1 ~ · · · ~ fn ,

(3.2) (3.3)

where f , f1 , . . . , fn ∈ H. Definition 3.2. For any f ∈ L∞ (R+ ) with |f (0)| < ∞, by the (p, q)-gauge operator on F(H) associated with f we understand the bounded linear extension of M (f )Ω = f (0)Ω , M (f )(f1 ~ f2 ~ · · · ~ fn ) = (f f1 ) ~ f2 ~ · · · ~ fn , for any f1 , f2 , . . . , fn ∈ H. In particular, by M (f, g) we will denote the gauge operator associated with the function defined by the weighted inner product on H, namely Z hhf, gii(t) := f (s)g(s)w(s, t)ds . R+

Note that this gauge operator multiplies the vacuum vector by the value of the inner product of f and g, i.e. M (f, g)Ω = hf, giΩ

(3.4)


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thanks to our assumption that w(s, 0) = 1 for any s ≥ 0. The (p, q)-gauge operator allows us to write a simple formula for the action of the annihilation operators. Proposition 3.1. The action of the (p, q)-annihilation operator a∗ (f ) associated with f ∈ H, adjoint with respect to the creation operator a(f ), is given by the bounded linear extension of a∗ (f )Ω = 0 , a∗ (f )(f1 ~ f2 ~ · · · ~ fn ) = M (f1 , f )(f2 ~ f3 ~ · · · ~ fn , ) ,

(3.5)

where f1 , f2 , . . . , fn ∈ H. Thus, in particular, a∗ (f )f1 = hf1 , f iΩ. Remark 3.1. Equivalently, we can use a deformed inner product on the free Fock space and define the creation operators to be the free creation operators, whereas the annihilation operators to be their adjoints with respect to the (p, q)-deformed inner product given by Z X hF, Gi = δn,m F (t1 , . . . , tn )G(t1 , . . . , tn )dt1 · · · dtn , w(σ −1 ) σ∈Sn

∆σ

for any p, q > 0, where F ∈ L2 (Rn+ ), G ∈ L2 (Rm + ), and hΩ, Ωi = 1, hΩ, F i = 0. This definition can be extended to p, q ≥ 0 except that one has to divide the above vector space by the corresponding kernel of the sesquilinear form. The gauge operators (which commute among themselves) allow us to write relations between creation, annihilation and gauge operators in a simple form as the proposition given below shows (we omit the elementary proof). Proposition 3.2. The following relations hold: a∗ (g)a(f ) = M (f, g) ,

M (h)a(f ) = a(hf ) ,

¯ = a∗ (f h) , a∗ (f )M (h)

(3.6)

where f, g ∈ H and h ∈ L∞ (R+ ) with |h(0)| < ∞. By the (p, q)-Gaussian operator associated with the function f ∈ H we will understand the self-adjoint position operator given by the sum ω(f ) = a∗ (f ) + a(f ). In turn, by the associated Brownian motion we will understand the process (ωt )t≥0 , where ωt = ω(χ[0,t) ). Below we will find a formula for the mixed moments ϕ(ω(f1 )ω(f2 ) · · · ω(fn )), where ϕ is the vacuum state on F(H) and f1 , f2 , . . . , fn ∈ Θ, where Θ := {χ[s,t) ; 0 ≤ s < t < ∞}

(3.7)

is the set of characteristic functions of intervals. We shall assume that supports of these functions are pairwise disjoint and are ordered by the partial order I1 < I2 whenever t1 < t2 for all t1 ∈ I1 , t2 ∈ I2 . We then set I1 ≤ I2 whenever I1 < I2 or I1 = I2 . The same notation will be used for the corresponding characteristic functions f = χI1 , g = χI2 , i.e. f < g and f ≤ g.


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Note that, as in the monotone case, it is not possible to obtain a similar formula for arbitrary functions, or even for characteristic functions with arbitrary supports. Example 3.1. Let f := f1 = f2 = f5 = f6 = χ[1,2) and g := f3 = f4 = χ[0,1) . Besides, to simplify notation, we set a (fi ) = ai for i = 1, . . . , 6 and = 1, ∗. Then ϕ(ω(f1 )ω(f2 ) · · · ω(f6 )) = ϕ(a∗1 a2 a∗3 a4 a∗5 a6 ) + ϕ(a∗1 a∗2 a3 a4 a∗5 a6 )

+ ϕ(a∗1 a2 a∗3 a∗4 a5 a6 ) + ϕ(a∗1 a∗2 a3 a∗4 a5 a6 ) + ϕ(a∗1 a∗2 a∗3 a4 a5 a6 ) ,

(3.8)

since the other mixed moments certainly vanish. However, the second, third and fourth summands also give zero contribution to (3.8) since in each of them a creation operator associated with g is paired with an annihilation operator associated with f or vice versa and these functions have disjoint supports. Therefore, it is enough to compute the contribution from the first and last summands: ϕ(a∗1 a∗2 a∗3 a4 a5 a6 ) = ha∗1 a∗2 a∗3 (g ~ f ~ f ), Ωi = pha∗1 a∗2 (f ~ f ), Ωi Z 2Z 2 pq + p =p w(t1 , t2 )dt1 dt2 = 2 1 1

ϕ(a∗1 a2 a∗3 a4 a∗5 a6 ) = 1 . Thus

pq + p + 2 . 2 Before we find a formula for all moments, let us introduce some additional notations (most of them are taken from Ref. 16). ϕ(ω(f1 )ω(f2 ) · · · ω(f6 )) =

Definition 3.3. Let f1 , f2 , . . . , fn ∈ Θ have pairwise identical or disjoint supports. We will say that P = (P1 , . . . , Pm ) ∈ OP n is adapted to (f1 , f2 , . . . , fn ), which we denote P ∼ (f1 , f2 , . . . , fn ), if and only if it satisfies two conditions: (1) i, j ∈ Pk =⇒ fi = fj , (2) i ∈ Pk , j ∈ Pl and k < l =⇒ fi ≤ fj . In turn, if π = {π1 , . . . , πm } ∈ Pn , then we will say that π is adapted to (f1 , f2 , . . . , fn ) if and only if its blocks satisfy only the first condition, which we denote π ∼ (f1 , . . . , fn ). Definition 3.4. If π ∼ (f1 , f2 , . . . , fn ), then the support of block πi is supp πi = supp fj , for any j ∈ πi . In turn, the support of the partition π ∈ Pn is supp π = {(t1 , . . . , tm ); tk ∈ supp πk , k = 1, . . . , m} .

Finally, for π ∈ N C 22k and any f1 , f2 , . . . , f2k ∈ Θ we will also use a simplified notation aπ (f1 , f2 , . . . , f2k ) = a1 (f1 )a2 (f2 ) · · · a2k (f2k ) ,

(3.9)


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where we understand that i = ∗ and j = 1 whenever {i, j} is a block of π and i < j. Lemma 3.1. If f1 , f2 , . . . , f2n ∈ Θ have pairwise identical or disjoint supports and π ∈ N C 22n is not adapted to (f1 , f2 , . . . , f2n ), then ϕ(aπ (f1 , f2 , . . . , f2n )) = 0. Proof. Since π is not adapted to (f1 , . . . , f2n ), there exists a block {i, j} ∈ π, such that fi , fj have disjoint supports. Assuming that i < j, we obtain ϕ(aπ (f1 , f2 , . . . , f2n )) = ha1 (f1 ) · · · a∗ (fi ) · · · a(fj ) · · · a2n (f2n )Ω, Ωi = ha1 (f1 ) · · · a∗ (fi ) · · · a(fj )(g1 ~ · · · ~ gk ), Ωi = ha1 (f1 ) · · · a∗ (fi ) · · · aj−1 (fj−1 )(fj ~ g1 ~ · · · ~ gk ), Ωi = cha1 (f1 ) · · · a∗ (fi )(fj ~ g1 ~ · · · ~ gk ), Ωi = cha1 (f1 ) · · · ai−1 (fi−1 )M (fj , fi )g1 ~ · · · ~ gk , Ωi = 0, where we used the fact that M (fj , fi ) = 0 since fj and fi have disjoint supports. Let π = {π1 , . . . , πk } ∼ (f1 , . . . , fn ), where f1 , f2 , . . . , fn ∈ Θ have pairwise identical or disjoint supports and let σ ∈ Sk . Then the pair (π, σ) can be identified with an ordered partition, for which we can define the set ∆(π,σ) = supp π ∩ ∆σ , i.e. ∆(π,σ) = {(t1 , . . . , tn ) ∈ Rn ; tσ(1) ≤ · · · ≤ tσ(n) , ti ∈ supp πi , i = 1, . . . , n} .

(3.10)

Then the following proposition holds. Proposition 3.3. Suppose f1 , . . . , f2n ∈ Θ have pairwise identical or disjoint supports and let π = {π1 , . . . , πn } be a pair partition which is adapted to (f1 , . . . , f2n ). Then, for any σ ∈ Sn such that (π, σ) (f1 , . . . , f2n ) it holds that ∆(π,σ) = ∅. Proof. We know that π ∼ (f1 , . . . , f2n ) and (π, σ) (f1 , . . . , f2n ). Therefore, (π, σ) deos not satisfy condition 2 of Definition 3.3, i.e. ∃ k, l ∈ {1, . . . , n} ∀ i ∈ πσ(k) , j ∈ πσ(l)

k < l and fi > fj .

Let us suppose that (t1 , . . . , tn ) ∈ ∆(π,σ) . Then it must hold that tσ(k) ≤ tσ(l) ,

(3.11)

since k < l. On the other hand, fi > fj and tσ(k) ∈ supp πσ(k) = supp fi as well as tσ(l) ∈ supp πσ(l) = supp fj , thus tσ(k) > tσ(l) , which contradicts (3.11).


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In the sequel we will have to collect functions with the same supports in a suitable way. Suppose that f1 , . . . , f2n ∈ Θ have pairwise identical or disjoint supports and let g1 , . . . , gr ∈ Θ be such that {f1 , . . . , f2n } = {g1 , . . . , gr } and g1 < · · · < gr

(3.12)

and the corresponding (ordered) intervals by I1 < I2 < · · · < Ir . Then we can introduce numbers 1 i = 1, . . . , r . (3.13) bi = |{j; supp fj = supp g (i) }| , 2 Of course, b1 + b2 + · · · + br = n. If there exists a pair partition which is adapted to (f1 , . . . , f2n ), then numbers bi are integers and the following easy proposition holds. Proposition 3.4. If f1 , . . . , f2n ∈ Θ have pairwise identical or disjoint supports and (π, σ) = (πσ(1) , πσ(2) , . . . , πσ(n) ) ∈ ON C 22n is adapted to (f1 , . . . , f2n ), then λ(∆(π,σ) ) =

r Y (λ(Ii ))bi

i=1

bi !

,

where λ denotes the (one-dimensional as well as n-dimensional ) Lebesgue measure. Proof. Since the volume of ∆(π,σ) does not depend on σ, we have Z r Y (λ(Ii ))bi λ(∆(π,σ) ) = dt1 · · · dtn = bi ! ∆(π,id) i=1 which gives the assertion. Theorem 3.1. If f1 , f2 , . . . , fn ∈ Θ have pairwise identical or disjoint supports, then r Y X (λ(Ii ))bi w(P ) . (3.14) ϕ(ω(f1 )ω(f2 ) · · · ω(fn )) = bi ! i=1 P ∈ON C 2 n P ∼(f1 ,...,fn )

Proof. Of course, if n is odd, the above moments vanish, which gives the assertion since (3.14) is a sum over the empty set. Assume therefore that n = 2k. First, observe that Lemma 3.1 gives X ϕ(ω(f1 )ω(f2 ) · · · ω(f2k )) = ϕ(aπ (f1 , f2 , . . . , f2k )) π∈N C 22k

=

X

ϕ(aπ (f1 , f2 , . . . , f2k )) .

π∈N C 2 2k π∼(f1 ,...,f2k )

If π = {π1 , . . . , πk } ∈ N C 22k , then to each number i ∈ {1, . . . , k} we can assign the number li ∈ {0, 1, . . . , k}, in such a way that block πli is a neighboring outer block


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of πi whenever πi has any outer blocks, and otherwise we set li = 0 and t0 = 0. Notice that Z ϕ(aπ (f1 , f2 , . . . , f2k )) = w(t1 , tl1 ) · · · w(tk , tlk )dt1 · · · dtk . supp π

Therefore, we have ϕ(ω(f1 )ω(f2 ) · · · ω(f2k )) =

X

π∈N C 2 2k π∼(f1 ,...,f2k )

Z

X

=

supp π

(π,ρ)∈ON C 2 2k (π,ρ)∼(f1 ,...,f2k )

=

r Y (λ(Ii ))bi

i=1

bi !

Z

w(t1 , tl1 ) · · · w(tk , tlk )dt1 · · · dtk

∆(π,ρ)

X

w(t1 , tl1 ) · · · w(tk , tlk )dt1 · · · dtk

w(P ) ,

P ∈ON C 2 2k P ∼(f1 ,...,f2k )

using Propositions 3.3 and 3.4. Remark 3.2. Note that there are two reasons why the mixed moments (3.14) depend on the supports of fi ’s. The first one is the presence of the lengths of intervals I1 , . . . , Ir (in fact, in order that the R.H.S. not be zero, each fi must appear an even number of times and therefore each length can be replaced by an inner product). The second one is the fact that the summation runs only over these P which are adapted to (f1 , f2 , . . . , fn ). For instance, if f1 < f2 < · · · < fm and g1 > g2 > · · · > gm , we obtain factorizations ϕ(ω(f1 ) · · · ω(fm )ω(fm ) · · · ω(f1 )) = q n−1 ϕ(ω 2 (f1 )) · · · ϕ(ω 2 (fn )) , ϕ(ω(g1 ) · · · ω(gm )ω(gm ) · · · ω(g1 )) = pn−1 ϕ(ω 2 (g1 )) · · · ϕ(ω 2 (gn )) . In particular, if q = 1, the first property reflects the so-called “pyramidal factorization” of the mixed moments.13 In fact, one can show that for q = 1 our processes satisfy all three conditions of the so-called “generalized Brownian motion” given in Ref. 6 (pyramidal factorization, stationarity and gaussianity). Example 3.2. In this example we will evaluate the sixth moment of ω(f ) for f = χ[0,1) by computing the contributions associated with all partitions P ∈ ON C 26 corresponding to products of creation and annihilation operators. These contributions will be compared with their counterparts obtained for the moments of t-deformed operators studied in Ref. 7. We have ϕ(ω 6 (f )) = ϕ(a∗ aa∗ aa∗ a) + ϕ(a∗ a∗ aaa∗ a) + ϕ(a∗ aa∗ a∗ aa) + ϕ(a∗ a∗ aa∗ aa) + ϕ(a∗ a∗ a∗ aaa) .


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R. Lenczewski & R. Salapata Table 1.

Comparison of mixed moments.

(p, q)-interpolation

t-deformation

t → (p + q)/2

1

1

1

a∗ aa∗ aa∗ a a∗ a∗ aaa∗ a

(p + q)/2

t

(p + q)/2

a∗ aa∗ a∗ aa

(p + q)/2

t

(p + q)/2

a∗ a∗ aa∗ aa

(p2 + pq + q 2 )/3

t2

(p2 + 2pq + q 2 )/4

a∗ a∗ a∗ aaa

(p2

t2

(p2 + 2pq + q 2 )/4

2t2 + 2t + 1

((p + q)2 + 2p + 2q + 2)/2

P

+ 4pq +

q 2 )/6

((p + q)2 + 2p + 2q + 2)/2

These mixed moments of creation and annihilation operators are given in Table 1. Analogous computations can be done for t-deformed creation and annihilation operators. In Table 1 we compare the values of the summands in the above equation for the (p, q)-interpolation and for the t-deformation. The main observation is that, in general, the mixed moments of (p, q)-free creation and annihilation operators are different than the corresponding mixed moments of t-deformed operators. In particular, for p = 0 (arcsine law), the t-deformed moments do not reproduce the moments in the monotone case. Moreover, we can compare two combinatorial formulas for the moments. It follows from Ref. 7 that the even moments of Kesten laws (1.4) satisfy the equation µ2n =

X

tin(π) =

π∈N C 22n

n−1 X k=0

D(n, k)tk

(3.15)

for n ≥ 1, where t = (p + q)/2 and in(π) is the number of inner blocks in π and the D(n, k) are the so-called Delaney’s numbers, which give the numbers of pair partitions in N C 22n which have exactly k inner blocks. They are given by the explicit formula n+k−1 n+k−1 D(n, k) = − , k k−1 where k ∈ {0, 1, . . . , n − 1}, n ≥ 1 and

n −1

= 0.

Definition 3.5. For n ≥ 1, the numbers of partitions from the set ON C 22n which have exactly k disorders and j orders will be called generalized Euler’s numbers and will be denoted E(n, k, j), i.e. E(n, k, j) = |{P ∈ ON C 22n ; e(P ) = k where 0 ≤ j, k ≤ n − 1.

and e0 (P ) = j}|


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Using Ref. 7 and the results of this section, we can find a relation between Delaney’s numbers and generalized Euler’s numbers. Proposition 3.5. For n ≥ 1 and k ∈ {0, 1, . . . , n − 1}, it holds that n! k + j D(n, k + j) . E(n, k, j) = k+j k 2 Proof. Substituting t = (p + q)/2 in (3.15) and performing elementary algebraic calculations, we obtain n−1 X n−1 X D(n, l) l µ2n = (p + q)k , 2l k k=0 l=k

for n ≥ 1. On the other hand, we know that µ2n =

1 n!

X

P ∈ON C 22n

pe(P ) =

n−1 X

k,j=0

E(n, k, j) (p + q)k n!

Comparing the coefficients of these two polynomials, we get the desired relation. 4. Central Limit Theorem on Discrete Free Fock Space In this section we will define discrete free creation and annihilation operators on the free Fock space F(H) over a Hilbert space H with a fixed orthonormal basis {e i }∞ i=1 (for a discussion of the discrete free Fock space, see Ref. 23). Using them, we will formulate an elementary version of the central limit theorem for “(p, q)-independent” random variables. An abstract treatment of the notion of independence involved here goes beyond the scope of this paper and will be given in a separate paper. Using the infinite matrix    p if i < j , wi,j = q if i > j ,   1 if i = j , we define a rescaled orthogonal basis in F(H), √ ei1 ~ ei2 ~ · · · ~ ein = wi1 ,i2 wi2 ,i3 · · · win−1 ,in ei1 ⊗ ei2 ⊗ · · · ⊗ ein , where i1 , i2 , . . . , in ∈ N. It is easy to see that hei1 ~ei2 ~· · ·~ein , ej1 ~ej2 ~· · ·~ejn i = δi1 ,j1 δi2 ,j2 · · · δin ,jn wi1 ,i2 wi2 ,i3 · · · win−1 ,in . Using this basis, we define the creation operators Ai , i ∈ N, by equations Ai (Ω) = ei , Ai (ei1 ~ ei2 ~ · · · ~ ein ) = ei ~ ei1 ~ · · · ~ ein


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with their adjoints, annihilation operators, acting as follows: A∗i (Ω) = 0 , A∗i (ej ) = δi,j Ω , A∗i (ei1 ~ ei2 ~ · · · ~ ein ) = wi1 ,i2 δi,i1 (ei2 ~ ei3 ~ · · · ~ ein ) . We will study the asymptotic behavior of normalized sums N 1 X ωi , SN = √ N i=1

(4.1)

where ωi = Ai + A∗i for i ∈ N. It is easy to see that the position operators ωi have mean zero and variance one with respect to the vacuum state ϕ on F(H). They will play the role of “independent” random variables in the central limit theorem. In the propositions given below we state their properties which are rather standard in the central limit context. Proposition 4.1. If, among the indices i1 , i2 , . . . , in ∈ N, there exists ij , j ∈ [n], such that ij 6= ik for any k 6= j, then ϕ(ωi1 · · · ωin ) = 0. If, in turn, (i1 , i2 , . . . , i2n ) is a sequence of indices associated with a partition P ∈ OP 22n , then ( w(P ) if P ∈ ON C 22n , ϕ(ωi1 · · · ωi2n ) = 0 if P ∈ / ON C 22n . Proof. The first assertion is the usual singleton condition, which clearly holds in our case. The second assertion is obvious if P ∈ / ON C 22n (it easily follows from the ∗ definition of Ai and Ai ). Suppose that P ∈ ON C 22n . Then there exists r such that ir = ir+1 6= ir+2 6= · · · 6= in . Therefore ϕ(ωi1 · · · ωin ) = hωi1 · · · ωir−1 A∗ir Air ωir+2 · · · ωin Ω, Ωi = wir+1 ,ir+2 hωi1 · · · ωir−1 ωir+2 · · · ωin Ω, Ωi = wir+1 ,ir+2 ϕ(ωi1 · · · ωir−1 ωir+2 · · · ωin ) , where wir+1 ,ir+2 is equal to p or q, depending on whether ir+1 < ir+2 (then block {r, r + 1} forms a disorder with its neighboring outer block) or ir+1 > ir+2 (then block {r, r + 1} forms an order with its neighboring outer block), respectively, and otherwise wir+1 ,ir+2 = 1 (then block {r, r + 1} does not have outer blocks). The assertion follows by induction. Theorem 4.1. For any n ∈ N it holds that 2n lim ϕ(SN )=

N →∞

1 n!

and the odd moments vanish in the limit.

X

P ∈ON C 22n

w(P )


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Proof. The proof is standard since the mixed moments of the ωi are invariant under order preserving injections, which gives 2n X X 1 1 X N 2n ϕ(SN ) = n ϕ(ωi1 ωi2 · · · ωi2n ) = n ϕ(ωP ) , N i ,...,i N r=1 r 1

P ∈OP 2n (r)

2n

where OP 2n (r) is the set of ordered partitions of the set [n] which have r blocks and ϕ(ωP ) denotes ϕ(ωi1 ωi2 · · · ωin ) for any sequence (i1 , i2 , . . . , in ) associated with P . Using Proposition 4.1 and standard arguments, we obtain the assertion. 5. Poisson Processes In this section we shall introduce processes of Poisson type, denoted (γt )t≥0 , which correspond to the (p, q)-Brownian motion studied in the previous section. The moments of γt in the vacuum state on F(R+ ) are given by the combinatorial formula ϕ(γtn ) =

X

P ∈ON C n

tb(P ) w(P ) , b(P )!

where t > 0 (p, q are supressed in the notation). Again, as in the case of position processes, for (p, q) = (1, 1) we obtain the moments of the free Poisson process 21 which are given by X ϕ(γtn ) = tb(π) , π∈N C n

where b(π) denotes the number of blocks of π. In turn, for (p, q) = (0, 1) we get the moments of the monotone Poisson process ϕ(γtn ) =

X

P ∈MON n

tb(P ) , b(P )!

Therefore, the process (γt )t≥0 plays the role of a natural interpolation between these two processes. To construct γt we shall use the gauge operator (Definition 3.2) associated with the characteristic function χ[0,t) , namely mt = M (χ[0,t) ), which is given by the explicit formula mt Ω = 0 , mt (f1 ~ f2 ~ · · · ~ fn ) = (χ[0,t) f1 ) ~ f2 ~ · · · ~ fn . Such operators were also used in the free case in a similar context,21 where the Poisson process was defined as lt + lt∗ + lt∗ lt + mt , where lt denotes the free creation operator associated with the function χ[0,t) . An analogous form of the Poisson process will be adopted for the (p, q)interpolation, namely γt = at + a∗t + a∗t at + mt

(5.1)


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and will be called the (p, q)-Poisson process. It follows from Proposition 3.2 that a∗t at = nt = M (χ[0,t) , χ[0,t) ) is another gauge operator given by the explicit formula nt Ω = tΩ ,

where W (s) =

Rt

nt (f1 ~ f2 ~ · · · ~ fn ) = (W f1 ) ~ f2 ~ · · · ~ fn ,

0

w(u, s)du = ((p − q)s + qt).

Example 5.1. Let us compute low-order moments of the (p, q)-Poisson process. hγt1 Ω, Ωi = t , hγt2 Ω, Ωi = t + t2 , hγt3 Ω, Ωi = t +

p+q+4 2 t + t3 , 2

hγt4 Ω, Ωi = t +

3p + 3q + 6 2 p2 + pq + q 2 + 3p + 3q 3 t + t + t4 , 2 3

hγt5 Ω, Ωi = t + (3p + 3q + 4)t2 +

11p2 + 11pq + 11q 2 + 24p + 24q + 36 3 t 6

3p3 + 3p2 q + 3pq 2 + 3q 2 + 8p2 + 8q 2 + 18p + 18q + 48 4 t + t5 . 12 For instance, we get +

hγt4 Ω, Ωi = thγt3 Ω, Ωi + hγt3 (χ(t1 )), Ωi = thγt3 Ω, Ωi + hγt2 (χ(t2 )χ(t1 ) + (W (t1 ) + 1)χ(t1 ) + tΩ), Ωi = thγt3 Ω, Ωi + thγt2 Ω, Ωi p+1 2 + γt (W 2 (t1 ) + 3W (t1 ) + 1)χ(t1 ) + t + t Ω ,Ω 2 = t2 + +

Z

=t+

p+q+4 3 t + t4 + t2 + t3 2 t

(W 2 (t1 ) + 3W (t1 ) + 1)dt1 + t2 +

0

p+q 3 t 2

3p + 3q + 6 2 p2 + pq + q 2 + 3p + 3q + 9 3 t + t + t4 , 2 3

where, for simplicity, we denote χ = χ[0,t) . Before we compute all moments of γt , we introduce some notations. Let π = {π1 , . . . , πb } ∈ N C n be a partition of [n]. Let us divide the set [n] into the following disjoint subsets: aπ = {i ∈ [n] : ∃r∈{1,...,b} |πr | > 1, i = max πr } ,


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a∗π = {i ∈ [n] : ∃r∈{1,...,b} |πr | > 1, i = min πr } , nπ = {i ∈ [n] : ∃r∈{1,...,b} πr = {i}} , mπ = [n]\(nπ ∪ aπ ∪ a∗π ) . In other words, the sets a∗π and aπ consist of left and right legs of the blocks of π, respectively, mπ corresponds to the “middle” legs of the blocks π1 , . . . , πb , and the set nπ corresponds to singletons in π. Using these sets, we can assign to each π = {π1 , . . . , πb } ∈ N C n an operator cπ = c1 · · · cn , where  at if i ∈ aπ ,      a∗ if i ∈ a∗π , t ci =   mt if i ∈ mπ ,    nt if i ∈ nπ .

Of course, the mapping π → cπ is one-to-one, but it is not onto. However, the products of operators at , a∗t , nt , mt which do not correspond to any non-crossing partition π will turn out to be irrelevant. In Fig. 4 we show a non-crossing partition π ∈ ON C 10 (5) with the corresponding operator cπ . In turn, examples of products which do not correspond to any non-crossing partitions are, for instance, given by mt mt mt , a∗t at at , at a∗t a∗t at . Lemma 5.1. Let c1 , . . . , cn ∈ {at , a∗t , mt , nt }, be a sequence of operators, for which there exists no partition π ∈ N C n such that cπ = c1 · · · cn . Then hc1 · · · cn Ω, Ωi = 0. Proof. Observe that if cπ 6= c1 · · · cn for all π ∈ N C n , then one of the following three cases must hold: (1) The number of creation operators at in the sequence c1 , . . . , cn must be different from the number of annihilation operators a∗t . Then the assertion is certainly true. (2) There exists i ∈ [n] such that among ci , ci+1 , . . . , cn there are more annihilation oparators a∗t than creation operators at . In that case, the assertion certainly holds. (3) There exists i ∈ [n] such that ci = mt and in the sequence ci+1 , . . . , cn there are as many annihilation operators a∗t as creation operators at . Then we have

cπ

at at nt mt mt at at at at nt

q q q q q q q q q q 1 2 3 4 5 6 7 8 9 10 Fig. 4.

Partition π and the corresponding operator cπ .


374

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hc1 · · · cn Ω, Ωi = hc1 · · · ci−1 mt CΩ, Ωi = 0 , for a certain constant C. Theorem 5.1. The moments of the Poisson process (γt )t≥0 in the vacuum state are given by ϕ(γtn ) =

X

P ∈ON C n

tb(P ) w(P ) b(P )!

for any n ∈ N and ϕ(γt0 ) = 1. Proof. First, notice that from Lemma 5.1 it follows that X ϕ(γtn ) = hcπ Ω, Ωi , π∈N C n

and therefore it suffices to show that if π ∈ N C n has b(π) = b blocks, then it holds that hcπ Ω, Ωi =

tb X w(π, σ) , b! σ∈Sb

where w(π, σ) = w(P ) for P = (π, σ). For that purpose, let us construct a pairpartition π 00 ∈ N C 22b such that hcπ Ω, Ωi = haπ00 Ω, Ωi and e(π, σ) = e(π 00 , σ), e0 (π, σ) = e0 (π 00 , σ) for any permutation σ ∈ Sb , where aπ00 is the abbreviated notation for aπ00 (χ[0,t) , . . . , χ[0,t) ). Let us notice that in the product cπ we can omit all occurrences of mt since they correspond to the “middle” legs of blocks of π, and therefore their action will not change the value of hcπ Ω, Ωi. In other words,  + * → Y ci  Ω, Ω = hcπ0 Ω, Ωi , hcπ Ω, Ωi =  ∗ i∈a∗ π ∪aπ ∪nπ

where π 0 is a certain non-crossing partition with b blocks, each consisting of one or two elements. Substituting the operator a∗t at for each mt in the product cπ0 corresponds to replacing singletons by two-element blocks {i, i + 1}. Therefore, cπ0 = aπ00 for a certain pair-partition π 00 ∈ N C 22b . Moreover, from the construction of π 00 it follows that e(π, σ) = e(π 00 , σ) and e0 (π, σ) = e0 (π 00 , σ) for any permutation σ ∈ Sb . Now, from the proof of Theorem 3.1 we have hcπ Ω, Ωi = haπ00 Ω, Ωi =

which completes the proof.

tb X tb X w(π 00 , σ) = w(π, σ) , b! b! σ∈Sb

σ∈Sb


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375

References 1. D. B. Applebaum and R. L. Hudson, Fermion Ito’s formula and stochastic evolution, Commun. Math. Phys. 96 (1986) 473–496. 2. L. Accardi and M. Bo˙zejko, Interacting Fock space and gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998) 663–670. 3. L. Accardi, Y. G. Lu and I. Volovich, Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations, Publ. IIAS (Kyoto), 1997. 4. Ph. Biane, Free Brownian motion, free stochastic calculus and random matrices, in Free Probability Theory (Waterloo, 1995), Fields Inst. Commun., Vol. 12 (Amer. Math. Soc., 1997). 5. M. Bo˙zejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996) 357–388. 6. M. Bo˙zejko and R. Speicher, An example of a generalized Brownian motion, Commun. Math. Phys. 137 (1991) 519–531. 7. M. Bo˙zejko and J. Wysocza´ nski, Remarks on t-transformations of measures and convolutions, Ann. I. H. Poincaré-PR 37 (2001) 737–761. 8. T. Cabanal-Duvillard, Probabilités libres et calcul stochastique. Application aux grandes matrices aléeatoires, Ph.D. Thesis, l’Université Pierre et Marie Curie, 1998. 9. T. Cabanal-Duvillard and V. Ionescu, Un théorème central limite pour des variables aléatoires non-commutatives, C. R. Acad. Sci. Paris 325 (1997) 1117–1120. 10. R. L. Hudson and K. R. Parthasarathy, Quantum Ito’s formula and stochastic evolution, Commun. Math. Phys. 93 (1984) 301–323. 11. H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959) 336–354. 12. A. Krystek and H. Yoshida, Generalized t-transformations of probability measures and deformed convolution, Probab. Math. Statist. 24 (2004) 97–119. 13. B. Kummerer, Markov dilations and non-commutative Poisson processes, preprint. 14. R. Lenczewski, On sums of q-independent SUq (2) quantum variables, Commun. Math. Phys. 154 (1993) 127–134. 15. R. Lenczewski, Quantum random walk for Uq (su(2)) and a new example of quantum noise, J. Math. Phys. 37 (1996) 2260–2278. 16. R. Lenczewski and R. Salapata, Discrete interpolation between monotone probability and free probability, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 77–106. 17. Y. G. Lu, On the interacting Fock space and the deformed Wigner law, Nagoya Math. J. 145 (1997) 1–28. 18. N. Muraki, Noncommutative Brownian motion in monotone Fock space, Commun. Math. Phys. 183 (1997) 557–570. 19. N. Muraki, Monotonic independence, monotonic central limit theorem and monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001) 39–58. 20. M. Sch¨ urmann, Quantum q-white noise and a q-central limit theorem, Commun. Math. Phys. 140 (1991) 589–615. 21. R. Speicher, A new example of “Independence” and “White noise”, Probab. Theor. Related Fields 84 (1990) 141–159. 22. D. Voiculescu, K. Dykema and A. Nica, Free Random Variables, CRM Monograph Series, No. 1 (Amer. Math. Soc., 1992). 23. D. Voiculescu, Lectures on free probability theory, in Lectures on Probability Theory and Statistics (Saint-Flour, 1998), Lecture Notes in Math., Vol. 1738 (Springer, 2000), pp. 279–349.

NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH

NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH

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