LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA

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1 Communications on Stochastic Analysis Vol. 6, No Serials Publications LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI* Dedicated to Professor K. R. Parthasarathy on the occasion of his 75th birthday Abstract. We extend the Lie algebra time shift technique, introduced in [2], from the usual Weyl algebra associated to the additive group of a Hilbert space H to the generalized Weyl algebra oscillator algebra, associated to the semi direct product of the additive group H with the unitary group on H the Euclidean group of H, in the terminology of [14]. While in the usual Weyl algebra the possible quantum extensions of the time shift are essentially reduced to isomorphic copies of the Wiener process, in the case of the oscillator algebra a larger class of Lévy process arises.our main result is the proof of the fact that the generators of the quantum Markov semigroups, associated to these time shifts, share with the quantum extensions of the Laplacian the important property that the generalized Weyl operators are eigenoperators for them and the corresponding eigenvalues are explicitly computed in terms of the Lévy Khintchin factor of the underlying classical Lévy process. 1. Introduction The classical heat semigroup on R is the Markov semigroup canonically associated to the classical, real valued Brownian motion W t via the basic formula of Markov processes: E 0] v t fw 0 = E 0] fw t = P t f = e t f ; t 0, where E 0] denotes the W -conditional expectation onto the past σ-algebra of time 0, v t is the usual time shift in the Wiener space and f is any Borel measurable function. In the quantum formulation of the classical, real valued Brownian motion there is also a time shift u t and E 0] denotes the restriction of the vacuum conditional expectation. However u t acts trivially on the initial algebra and therefore the generator of the corresponding semigroup is zero, in particular it cannot coincide with the time shift of the classical Brownian motion. P.A. Meyer noticed this discrepancy and, in the Oberwolfach 1987 quantum probability workshop, posed the question if there exists a quantum extension of the classical time shift in Wiener space. For any such extension the generator of the associated quantum Received ; Communicated by K. B. Sinha Mathematics Subject Classification. Primary 60J65; Secondary 60J45, 60H40. Key words and phrases. Generalized Weyl operator, white noise, Lie algebra time shift, oscillator algebra, generator of Lévy process. * This research is supported by the Centro Vito Volterra, Università di Roma II, Tor Vergata. 125

2 126 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI Markov semigroup would provide a quantum extension of the classical laplacian. A solution of Meyer s problem was given by Accardi in the same workshop see [2] and it was based on the idea that the Fock space time shift u t is the time shift of the increment process associated to the classical Brownian motion, i.e., the integrated white noise, thus its action on the algebra of measurable functionals of the process is characterized by the property of being the unique continuous endomorphism satisfying: u t W s W r := W s+t W r+t On the other hand the usual time shift vt in Wiener space is the unique continuous endomorphism of the associated algebra of measurable functions given by the property: vt W s := W s+t Therefore, denoting ĵ t the restriction of the Wiener time shift on the time zero algebra, vt is uniquely determined by the pair ĵ t, u t through the identity v t W s := W s+t = W t + W s+t W t = ĵ t W 0 + u t W s W 0. But it is known that, in the quantum formulation of the classical Wiener process, the initial random variable is identified to position operator q 0 and the increment noise is identified to the momentum process This leads to this identification W t W 0 = P 0,t]. So that W s = W 0 + W s W 0 = q P 0,s] v t W s = v t q P 0,s] = ĵ t q u t P 0,s] = ĵ t q P t,t+s] On the other hand v t W s = W s+t = W 0 + W s+t W 0 = q P 0,s+t] From this it follows that = q P 0,t] P t,s+t] ĵ t q 0 = q P 0,t] is a possible answer to our problem. In [2] it was shown that this is indeed the case and that the generator of the associated quantum Markov semigroup is indeed a quantum generalization of the classical laplacian, in the sense that its restriction to the operators of multiplication by smooth functions coincides with the usual laplacian. The systematic investigation of all possible solutions to this problem when the initial algebra is a general not necessarily finite dimensional Weyl algebra, begun in paper [3] see also [4], led in particular to the unexpected identification of the Lévy laplacian with a usual Volterra Gross laplacian, corresponding to a usual Brownian motion with values in a special Hilbert space the Cesaro Hilbert space, and opened the way to a multiplicity of new developments on the structure of the heat semigroups associated to the whole hierarchy of exotic laplacians, to which

3 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 127 the previously developed analytical techniques did not apply see for example [11], [6], [7]. In the present paper, we extend the results of [2], [3], [4] from the usual Weyl algebra over an Hilbert space H, which gives a projective representation of the additive group H acting on itself by translations, to the oscillator algebra, which gives a projective representation of the semi direct product of the additive group H with the unitary group on H, denoted UH. We combine this extension of the quantum time shifts techniques with the techniques developed by Araki, Woods, Parthasarathy and Schmidt see [15] for a systematic exposition and bibliography, to construct quantum Markov processes of random walk type, i.e., obtained by adding, to an initial operator process, the increments of a quantum independent increment operator process. This simple additive picture at Lie algebra level produces, after exponentiation, a projective representation of Lie groups. While in the usual Weyl algebra the possible quantum extensions of the classical time shift are essentially reduced to isomorphic copies of the Wiener process, in the case of the oscillator algebra a larger class of Lévy process arises as possible candidates for quantum extensions of the time shift. We determine the explicit form of these time shifts and construct the associated Markov cocycles hence, via the quantum Feynman Kac technique [1], the associated quantum Markov semigroup. From this we deduce the main result of the present paper which can be described as follows. The quantum Markov semigroup associated to the usual quantum Brownian motion is the quantum heat semigroup and usual Weyl operators are eigenoperators of its generator the quantum laplacian. Analogously the generalized Weyl operators, associated to the oscillator algebra, are eigenoperators of the generator of the quantum Markov semigroups canonically associated to the Lie algebra time shifts of the oscillator algebra which are given by classical Lévy process. Furthermore the corresponding eigenvalues are explicitly computed in terms of the Lévy Khintchin factor of the underlying classical Lévy process. The standard GKSL form of these generators can be easily computed using stochastic calculus. Generally these generators are unbounded even in the case of finite dimensional Brownian motion. The fact that they have a total, self-adjoint, set of linearly independent eigenoperators singles out an interesting new class of quantum Markov semigroups up to now the only non trivial known example in this class was the quantum laplacian and can be probably exploited to achieve a deeper understanding of the analytical structure of this class. 2. Notations and Preliminaries 2.1. Boson Fock space. In the following all Hilbert spaces are assumed to be complex and separable with inner product linear in the second variable denoted,,. For any Hilbert space H, we denote: BH the algebra of all bounded linear operators on H ΓH the symmetric boson Fock space over H ψ u u H the exponential vector associated with u: ψ u := n 0 u n n! ΓH ; ψ 0 = Φ vacuum vector.

4 128 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI For any dense linear subspace S of H the family {ψ u, u S} is total and linearly independent in ΓH. We denote by ES the vector space algebraically generated by it. If S is as above a linear operator may be defined densely on ΓH by giving arbitrarily its action on the family {ψ u, u S}. We simply use the notation E when S = H. The annihilation, creation and Weyl operators are defined respectively by: A vψ u := v, u ψ u ; A + vψ u := d ds s=0 ψ u+sv, W vψ u = e 1 2 v 2 v,u ψ u+v ; v H. The boson creation and annihilation operators satisfy the canonical commutations relations CCR: [A u, A + v] = < u, v > [A u, A v] = [A + u, A + v] = 0, 2.2 for any u, v H, where [x, y] := xy yx is the commutator and 1 0 is the identity operator on ΓH. The second quantized ΓT of a self-adjoint bounded operator T on H is given by the relation: ΓT ψ u := ψ T u. The differential second quantization operator ΛT or the number operator of T is defined via the Stone theorem by: Its action on ES is given by: Γe itt =: e itλt, t R. ΛT ψ u = 1 i d ds s=0ψ e ist u. If T is a bounded but not necessarily self-adjoint operator on H, then by writing T as sum of two self-adjoint operators then we can define ΛT by: T = T 1 + it 2 = T + T ΛT := ΛT 1 + iλt i T T, 2i Hence ΛT is linear in T and the following canonical commutations relations hold weakly on the set of the exponential vectors [ΛT, A + u] = A + T u, [ΛT, Au] = AT u 2.3 and [ΛT 1, ΛT 2 ] = Λ[T 1, T 2 ]. 2.4

5 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA Markov flows on white noise spaces. In the notations of section 2.1, if the space H has the form L 2 R +, H 0 L 2 R + H 0 where H 0 is an Hilbert space, the associated Fock space is called a white noise space and the space H 0 a multiplicity or polarization space. Usually an initial or system space is added to the white noise space and in the following we will fix the choice H w := ΓH 0 Γ L 2 R +, H 0. Notice that we choose the initial space to be the Fock space over the multiplicity space. This is a common feature in the theory of quantum time shifts see [3] whose motivation will be clear from the following development. This space has two natural Hilbert space filtrations past and future defined asymmetrically by H t] := ΓH 0 Γ L 2 [0, t], H 0 ; H [t = Γ L 2 [t, + [, H 0 ; t 0 where, here and in the following, H t] resp. H [t will be identified to the subspace Φ 0 H t] Φ [t resp. Φ t] H [t, Φ [t resp. Φ 0, Φ t] being the vacuum vector in H [t resp. H 0, H t]. With these notations the following factorization property holds H w = ΓH 0 Γ L 2 R +, H 0 = H t] H [t. Similarly we define the Von Neumann algebra pure noise filtration: B t] = B Γ L 2 [0, t], H 0 B Γ L 2 [0, t], H 0 1 [t, B [t = B Γ L 2 [t, + [, H 0 1 t] BH [t and B = B ΓL 2 R +, H 0 B t] B [t, where 1 t] and 1 [t are respectively the identities on the spaces Γ L 2 [0, t], H 0 and Γ L 2 [t, + [, H 0. If A 0 is a C subalgebra of BΓH 0, we define the filtrations: A t] := A 0 B t], A [t := B [t, A = A t] A [t = A 0 B and denote by 1 0 resp. 1 the identity on ΓH 0 resp. Γ L 2 R +, H 0. The noise creation, annihilation and conservation increment processes, acting on the space Γ L 2 R +, H 0, will be denoted respectively A + s,tξ := A + χ [s,t] ξ, A s,tξ := A χ [s,t] ξ, Λ s,t ξ := ΛM χ[s,t] T. 2.5 If s = 0, we simply write A + t ξ := A + 0,t ξ, A t ξ := A 0,t ξ, Λ tt := Λ 0,t T 2.6 and, when no ambiguity is possible, the same notation will be used for their natural action on ΓH 0 Γ L 2 R +, H 0. Denote by θ t, the right shift on L 2 R +, H 0, so that t 0 { fs t, if s t 0; θ t fs = 2.7 0, if 0 s t.

6 130 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI The operator θ t is isometric with θ t fs = fs + t. The white noise time shift is the 1 parameter endomorphism semigroup of B ΓH 0 B Γ L 2 R +, H 0 characterized by the property that, for all b 0 BΓH 0, b B Γ L 2 R +, H 0 and t 0 one has: u t b 0 b = b 0 Γθ t bγθ t. 2.8 The shift u t is a normal, injective endomorphism and satisfies the following: 1 For s, t 0, u su t = u s+t. 2 For s 0, u sa = A 0 1 s] B [s. The vacuum conditional expectations, defined on the algebra A and with values in the past filtration algebra are defined by: E t] b t] b [t := Φ [0, b [t Φ [0 b t], b t] A t], b [t A [t. 2.9 Definition 2.1. A stochastic process on A 0 is a family {j t : A 0 A t], t 0} of homomorphisms with j 0 x = x 1. A stochastic process is called normal if for each t 0, j t is σ-weakly continuous. Let j t t 0 be a normal stochastic process on A 0. We define j t as the unique normal homomorphism j t : A 0 1 [0,t] B [t A characterized by j t x 1 = j t x, j t 1 b [t = j t 1 b [t Each j t can be extended in an obvious way to the algebraic linear span of the elements of the form x 1 [0,t] Y [t, where x A 0, Y [t is an operator on H [t. Definition 2.2. A stochastic process on A 0 is said to be a Markov cocycle if, in the notations of 2.10, it satisfies the cocycle equation: for all s, t 0 and x A 0 j 0 x = x 1 ; j s+t x = j s u s j t x It is said to be σ-weakly continuous if the map t, x j t x is continuous w.r.t the σ-weak topology of the Von Neumann algebras involved. According to the Feynman-Kac formula to every Markov cocycle j t on A 0, one can associate Markov semigroup P t on A 0, characterized by the identity: P t x = E 0] j t x 2.12 for all t 0, x A 0. Moreover the cocycle identity 2.11 and condition 2.12 imply that for each s, t 0 E s] j s+t = j s P t In the following we will take the initial space ΓH 0 to be ΓH. 3. The Generalized Weyl Operator In this section we introduce the generalized Weyl operator associated to the Fock representation of the oscillator Weyl algebra. We use the notations of section 2.1. Definition 3.1. For all unitary operator U, on H and u, v H, z C, define the exponential operator on the set of the exponential vectors by: Γu, U, v, z := e A+ u ΓUe A v e z. 3.1

7 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 131 In the following we need some known results which we sum up in Lemmas 3.2 and 3.3. Lemma 3.2. For all unitary operator U on H and u, v H, one has Proof. Γu, U, v, zψ x = e z+ v,x ψ u+ux ; x H. 3.2 Γu, U, v, zψ x = e z e A+ u ΓUe A v ψ x = e z e A+ u ΓU e v,x ψ x = e z+ v,x e A+ u ψ Ux = e z+ v,x ψ u+ux. Lemma 3.3. For all unitary operators U j, on H and u j, v j H, z j C; j = 1, 2, we have the following relation: where Γu 1, U 1, v 1, z 1 Γu 2, U 2, v 2, z 2 = Γu, U, v, z, 3.3 u = u 1 + U 1 u 2 ; U = U 1 U 2 ; v = v 2 + U 2 v 1 ; z = z 1 + z 2 + v 1, u Proof. Let x H. Using Lemma 3.2 one finds Γ 1 Γ 2 ψ x := Γu 1, U 1, v 1, z 1 Γu 2, U 2, v 2, z 2 ψ x = Γu 1, U 1, v 1, z 1 e z 2+ v 2,x ψ u2+u 2x = e z 2+ v 2,x Γu 1, U 1, v 1, z 1 ψ u2 +U 2 x = e z 2+ v 2,x e z 1+ v 1,u 2 +U 2 x ψ u1+u 1u 2+U 2x = e z 1+z 2 + v 1,u 2 + v 2 +U 2 v 1,x ψ u1+u 1u 2+U 1U 2x = Γu 1 + U 1 u 2, U 1 U 2, v 2 + U 2 v 1, z 1 + z 2 + v 1, u 2 ψ x. Lemma 3.4. Let fz = n 0 a n z n, be an entire function. Denoting fz := n 0 a n z n, f z := n 0 -i Let T be a bounded operator on H, then the operator ft := n 0 a n T n a n z n is well defined and bounded on H with norm less than f T. Moreover, the adjoint of ft is ft. -ii If f and g are two analytic functions then the operator fgt := ft gt is well defined and it is bounded on H. -iii If e 1 and e 2 are the analytic functions defined respectively by: e 1 z := + n=1 z n 1 n! = ez 1, 3.5 z

8 132 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI e 2 z := + n=2 z n 2 n! = ez z 1 z 2, 3.6 then the operators e 1 T and e 2 T enjoy the following properties: e 1 T = e 1 T, 3.7 e 2 T = e 2 T, 3.8 e T e 1 T = e 1 T, 3.9 e 1 T e 1 T = e 2 T + e 2 T, 3.10 e 1 T e 1 T ; e 2 T e 2 T Proof. The statements i and ii are clear. 3.7, 3.8 and 3.11 follow from i.3.9 and 3.10 follow from the identities e x e 1 x = e 1 x ; e 1 xe 1 x = e 2 x + e 2 x, x C. In the following, we denote by B s H, the set of all bounded self-adjoint operators on H. Definition 3.5. For ξ H and T B s H, denote u ξ,t := ie 1 it ξ ; v ξ,t = ie 1 it ξ ; z ξ,t := ξ, e 2 it ξ The operator W ξ, T defined by W ξ, T := Γ u ξ,t, e it, v ξ,t, z ξ,t is called generalized Weyl operator over H Proposition 3.6. For all pair ξ, T H B s H, W ξ, T is a unitary operator on ΓH whose action on the exponential vectors is given, in the notation 3.12, by W ξ, T ψ x = e z ξ,t + v ξ,t,x ψ uξ,t +e it x Proof follows from 3.2. Since the domain E is dense in ΓH, unitarity is equivalent to W ξ, T ϕ, W ξ, T ψ = ϕ, ψ ϕ, ψ E Let x, y H, then 3.14 gives W ξ, T ψ x, W ξ, T ψ y = e z ξ,t + v ξ,t,x +z ξ,t + v ξ,t,y where ψ uξ,t +e it x, ψ uξ,t +e it y = e z ξ,t +z ξ,t + x,v ξ,t + v ξ,t,y + u ξ,t +e it x,u ξ,t +e it y = e h ξ,t x,y, 3.16 h ξ,t x, y = z ξ,t + z ξ,t + x, v ξ,t + v ξ,t, y + u ξ,t + e it x, u ξ,t + e it y.

9 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 133 Using the fact that v ξ,t = e it u ξ,t and the properties 3.7, 3.8 and 3.10, one has h ξ,t x, y = ξ, e 2 it ξ ξ, e 2 it ξ e it x, u ξ,t u ξ,t, e it y + u ξ,t, e it y + e it x, u ξ,t + u ξ,t, u ξ,t + e it x, e it y = ξ, e 2 it + e 2 it ξ + ξ, e 1 it e 1 it ξ + x, y = x, y. From the expression of h ξ,t and from equation 3.16, one obtains W ξ, T ψ x, W ξ, T ψ y = e x,y = ψ x, ψ y which can be extended by linearity to whole of E. Remark 3.7. We use Lie algebra notations. The standard notation for the generalized Weyl operator, here denoted W ξ, T, is W ξ, e it see [14]. Moreover, as shown in Theorem 3.8 below, our definition of generalized Weyl operator differs by a phase from the standard one. Our choice has the notational advantage that it is better suited for the transition between the Lie algebra and the Lie group language. For example, while in the standard notation the inverse adjoint of W u, U is given by W u, U = W U u, U, in our notation it becomes W u, U = W u, U or, in Lie algebra notations, identity 3.18 below. In the following, we will use the notations Rz and Iz, respectively, for real and imaginary parts of a such complex number z. Theorem 3.8. In the notations of Definition 3.5, and denoting W E the standard generalized Weyl operator, one has W ξ, T = e iiz ξ,t W E u ξ,t, e it Moreover, W ξ, T = W ξ, T Proof. We have W E u ξ,t, e it = Γ u ξ,t, e it, e it u ξ,t, 1 2 u ξ,t 2. But e it u ξ,t = e it u ξ,t = v ξ,t

10 134 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI and 1 2 u ξ,t 2 = 1 2 ie 1iT ξ, ie 1 it ξ = 1 2 ξ, e 1 it e 1 it ξ Therefore = 1 2 ξ, e 2 it + e 2 it ξ = 1 ξ, e 2 it ξ + ξ, e 2 it ξ 2 = 1 z ξ,t + z ξ,t 2 = Rz ξ,t W E u ξ,t, e it = Γ u ξ,t, e it, v ξ,t, Rz ξ,t = Γ u ξ,t, e it, v ξ,t, z ξ,t e iiz ξ,t = e iiz ξ,t W ξ, T 3.20 which proves To prove Eq.3.18 notice that, from Eq and using the fact that z ξ, T = z ξ,t, v ξ,t = u ξ, T, one deduces that W ξ, T = e iiz ξ,t W E u ξ,t, e it = e iiz ξ,t W E e it u ξ,t, e it = e iiz ξ,t W E e it u ξ,t, e it = e iiz ξ,t W E v ξ,t, e it = e iiz ξ, T W E u ξ, T, e it = W ξ, T. Remark 3.9. A more direct proof of 3.17 can be obtained, noting that the action of the standard Weyl operator is given by and that W E u ξ,t, e it ψ x = e 1 2 u ξ,t 2 u ξ,t,e it x ψ uξ,t +e it x u ξ,t, e it x = e it u ξ,t, x = v ξ,t, x. Therefore, from Eq and 3.14, one obtains W E u ξ,t, e it ψ x = e Rz ξ,t + v ξ,t,x ψ uξ,t +e it x = e iiz ξ,t W ξ, T ψ x.

11 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 135 Theorem The operator valued function t W tξ, tt is a strongly continuous one-parameter unitary group with generator Gξ, T which is the closure of Hξ, T := A + ξ + A ξ + ΛT To prove the above theorem, we need the following lemmata. Lemma In the notation 3.12, let u t := u tξ,tt ; v t := v tξ,tt = e itt u t ; z t := z tξ,tt, t R, 3.22 then the following relations hold Proof. Define Then f s is analytic in z and u s + e ist u t = u s+t, 3.23 v t + e itt v s = v s+t, 3.24 z s + z t + v s, u t = z s+t f s z := se 1 isz. f s z + e isz f t z = s eisz 1 + te isz eitz 1 = s + te 1 is + tz = f s+t z. isz itz While u t = if t T ξ for all t R, then u s + e ist u t = i f s T + e ist f t T ξ = if s+t T ξ = u s+t 3.26 which proves From the definition of v t in 3.22 and from 3.23, one has v t + e itt v s = e itt u t + e itt e ist u s = e is+tt e ist u t + u s = e is+tt u t+s = v s+t then 3.24 is proved. To prove 3.25, denote g s,t z = s 2 e 2 isz + t 2 e 2 itz + ste 1 isze 1 itz. Clearly g s,t is analytic and it is not difficult to see that then g s,t z = s + t 2 e 2 is + tz z s + z t + v s, u t = sξ, e 2 ist sξ tξ, e 2 itt tξ + ie 1 ist sξ, ie 1 itt tξ = ξ, s 2 e 2 ist ξ ξ, t 2 e 2 itt ξ ξ, ste 1 ist e 1 itt ξ = ξ, s 2 e 2 ist + t 2 e 2 itt + ste 1 ist e 1 itt ξ = ξ, g s,t T ξ = ξ, s + t 2 e 2 is + tt ξ = s + tξ, e 2 is + tt s + tξ hence 3.25 is proved. = z s+t

12 136 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI Lemma Let u t, v t, z t are as in Lemma 3.11, then and, for all x, y H, the function is derivable at t = 0 and Proof. We have and and lim u t = lim v t = lim z t = t 0 t 0 t 0 t R ht := z t + v t, x + y, u t + y, e itt x h 0 = i ξ, x + y, ξ + T y, x u t = t e 1 itt ξ t ξ e 1 t T 0, as t 0 v t = e itt u t = u t 0, as t 0 t 2 ξ, e 2 itt ξ t 2 ξ 2 e 2 t T 0, as t 0. For the second part, note that h0 = y, x, then ht h0 t = 1 t t 2 ξ, e 2 itt ξ + it e 1 itt ξ, x + it y, e 1 itt ξ + y, e itt x y, x = t ξ, e 2 itt ξ + i e 1 itt ξ, x + i y, e 1 itt ξ + y, eitt 1 x. t Taking the limit as t 0, one obtains h is derivable at t = 0 and Eq.3.28 holds. Proof. of Theorem We have already shown that W ξ, T is unitary, for all ξ H, T B s H, so is for W tξ, tt, t R. In the following we prove that: -a W sξ, st W tξ, tt = W s + tξ, s + tt, s, t R ; -b the map t W tξ, tt is strongly continuous. For simplicity we use the notation: W t := W tξ, tt = Γ u t, e itt, v t, z t, t R 3.29 then a follows from Lemma 3.11 because W sw t = Γ u s, e ist, v s, z s Γ u t, e itt, v t, z t = Γ u s + e ist u t, e ist e itt, v t + e itt v s, z s + z t + v s, u t = Γ u s + e ist u t, e is+tt, v t + e itt v s, z s + z t + v s, u t = Γ u s+t, e is+tt, v s+t, z s+t = W s + t.

13 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 137 By the group property a, the strong continuity is reduced to time zero. We prove that, for all ψ ΓH, Let ψ = ψ x, x H, then lim W tψ ψ = 0. t 0 W tψ ψ 2 = W tψ x ψ x, W tψ x ψ x = W tψ x 2 2R ψ x, W tψ x + ψ x 2 = 2 ψ x 2 2R e z t+ v t,x ψ x, ψ ut+e itt x = 2e x 2 2R e zt+ vt,x e x,ut + x,eitt x. Using Lemma 3.11, one has then lim v t, x = lim x, u t = lim z t = 0 ; lim e itt x = x t 0 t 0 t 0 t 0 lim W tψ t 0 ψ 2 = 0. This limit can be extended by linearity to all ψ E. Finally an arbitrary element ψ of ΓH is a limit of a sequence ψ n n E. It follows that, for all ɛ > 0, there is n 0 N such that This gives ψ n0 ψ ɛ/4. W tψ ψ = W tψ ψ n0 + W tψ n0 ψ n0 + ψ n0 ψ W tψ ψ n0 + W tψ n0 ψ n0 + ψ n0 ψ = 2 ψ n0 ψ + W tψ n0 ψ n0 ɛ 2 + W tψ n 0 ψ n0. But we have already shown that lim t 0 W tψ n0 ψ n0 = 0. Therefore for t sufficiently small we have This gives W tψ n0 ψ n0 ɛ 2. W tψ ψ ɛ hence lim t 0 W tψ = ψ for all ψ ΓH which proves the property b. By Stone s theorem, there is a self-adjoint operator Gξ, T, such that Let us prove that, in the notation 3.21 W t = e itgξ,t. Hξ, T Gξ, T. 3.30

14 138 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI Let x, y H, then ψ y, W tψ x = e z t+ v t,x ψ y, ψ ut +e itt x = e z t+ v t,x e y,u t+e itt x = e z t+ v t,x + y,u t + y,e itt x = e ht, 3.31 where ht is as in Lemma Taking the derivative of the left hand side of 3.31 at time zero, one obtains d dt t=0 ψ y, W tψ x = ψ y, d dt t=0 W tψ x = ψ y, d dt t=0 e itgξ,t ψ x = ψ y, igξ, T ψ x On the other hand the derivative of the right hand side of 3.31 gives d dt t=0 e ht = h 0e h0 = i ξ, x + y, ξ + T y, x e y,x But we have ψ y, ihξ, T ψ x = ψ y, ia + ξ + A ξ + ΛT ψ x Combining Eqs , one obtains this gives 3.30 which ends the proof. = i A ξψ y, ψ x + ψ y, A ξψ x + A T xψ y, ψ x = i y, ξ + ξ, x + y, T x ψ y, ψ x = i y, ξ + ξ, x + y, T x e y,x ψ y, ihξ, T ψ x = ψ y, igξ, T ψ x 3.35 Lemma Let D be the set of self-adjoint bounded operators on H with spectrum in [ π, π[ D := {T B s H; σt [ π, π[} Then for all unitary operator U UH there exists a unique operator T π D such that U = e it π. Proof. Existence. Let U be a unitary operator. Since U is normal, the Von Neumann algebra generated by U is abelian hence, by a theorem of Von Neumann see [17], it is generated by a single self-adjoint operator T. Denote F : R [ π, π[ the map defined by F x := x mod 2π ; x R. Clearly F is a measurable, bounded, real valued function on R. Therefore T π := F T, defined by the spectral theorem, is bounded, self-adjoint and by construction satisfies e it = e it π = U.

15 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 139 Moreover, denoting T = xe T dx the spectral decomposition of T, one has T π 2 = F x 2 de T x R π 2 de T x = π 2. Uniqueness. Let T 1 D be an operator such that e it 1 = e it π = U. Then the spectra of T 1 and T π can differ only by multiples of 2π. Since both T 1 and T π are in D, this implies that T 1 = F T 1 = F T π = T π. Theorem For all ξ j, T j H B s H; j = 1, 2, the generalized Weyl relations hold where the pair ξ, T H D and γ ξ,t relations R W ξ 1, T 1 W ξ 2, T 2 = e iγ ξ,t W ξ, T 3.36 R are uniquely determined by the e it 1 e it 2 = e it, 3.37 e 1 it ξ = e 1 it 1 ξ 1 + e it1 e 1 it 2 ξ 2, 3.38 γ ξ,t = i ξ 1, e 2 it 1 ξ 1 + ξ 2, e 2 it 2 ξ 2 ξ, e 2 it ξ + e 1 it 1 ξ 1, e 1 it 2 ξ Remark The reality of γ ξ,t, given by Eq. 3.39, i.e. S := iγ ξ,t + iγ ξ,t = 0 can be directly checked as follows: From Eqs. 3.7 and 3.8, one has, for all self-adjoint bounded operator T on H, e1 it = e1 it ; e2 it = e2 it. Then ξ, e 2 it ξ = e 2 it ξ, ξ = ξ, e 2 it ξ T B s H, ξ H. From Eq. 3.39, one has S = ξ 1, e 2 it 1 ξ 1 + ξ 2, e 2 it 2 ξ 2 ξ, e 2 it ξ + e 1 it 1 ξ 1, e 1 it 2 ξ 2 + ξ 1, e 2 it 1 ξ 1 + ξ 2, e 2 it 2 ξ 2 ξ, e 2 it ξ + e 1 it 2 ξ 2, e 1 it 1 ξ 1 = ξ 1, e 2 it 1 + e 2 it 1 ξ 1 + ξ 2, e 2 it 2 + e 2 it 2 ξ 2 ξ, e 2 it + e 2 it ξ + e 1 it 1 ξ 1, e 1 it 2 ξ 2 + e 1 it 2 ξ 2, e 1 it 1 ξ

16 140 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI Using Eqs. 3.9 and 3.10, Eq becomes S = ξ 1, e 1 it 1 e 1 it 1 ξ 1 + ξ 2, e 1 it 2 e 1 it 2 ξ 2 ξ, e 1 it e 1 it ξ + e it 1 e 1 it 1 ξ 1, e 1 it 2 ξ 2 + e 1 it 2 ξ 2, e it 1 e 1 it 1 ξ 1 = e 1 it 1 ξ 1, e 1 it 1 ξ 1 + e 1 it 2 ξ 2, e 1 it 2 ξ 2 e 1 it ξ, e 1 it ξ + e 1 it 1 ξ 1, e it 1 e 1 it 2 ξ 2 + e it 1 e 1 it 2 ξ 2, e 1 it 1 ξ Denote η = e 1 it ξ, η j = e 1 it j ξ j ; j = 1, Then Eq.3.38 becomes η = η 1 + e it 1 η and from equation 3.41, one obtains S = η η 2 2 η 2 + η 1, e it 1 η 2 + e it 1 η 2, η 1 = η η 2 2 η 1 + e it1 η η 1, e it1 η 2 + e it1 η 2, η 1 = η η 2 2 η e it 1 η η 1, e it 1 η 2 + e it 1 η 2, η 1 + η 1, e it 1 η 2 + e it 1 η 2, η 1 = 0. Proof. of Theorem Let ξ j, T j H B s H; j = 1, 2. Using the notations of Eq in Lemma 3.6, one obtains W ξ 1, T 1 W ξ 2, T 2 = Γu ξ1,t 1, e it1, v ξ1,t 1, z ξ1,t 1 Γu ξ2,t 2, e it2, v ξ2,t 2, z ξ2,t Denote u j = u ξj,t j ; v j = v ξj,t j ; z j = z ξj,t j ; j = 1, 2. Using 3.3, Eq becomes W ξ 1, T 1 W ξ 2, T 2 = Γ u 1 + e it 1 u 2, e it 1 e it 2, v 2 + e it 2 v 1, z 1 + z 2 + v 1, u While e it 1 e it 2 is unitary then by Lemma 3.13 there exists a unique operator T D such that Eq holds. But it is well known see [16] that the Bernoulli series b n n! xn is convergent with radius equal to 2π and with sum n 0 e 1 x := x e x 1 = 1 e 1 x ; x < 2π. It follows that for all operator T BH with norm T < 2π, e 1 T is invertible in BH with inverse equal to e 1 T. Since σit i[ π, π[, then it is of norm less than 2π. From these prescriptions e 1 it is invertible with inverse e 1 it. Denote ξ, the unique vector of H defined by ξ := e 1 it e 1 it 1 ξ 1 + e it 1 e 1 it 2 ξ 2 so that Eq holds and u 1 + e it 1 u 2 = i e 1 it 1 ξ 1 + e it 1 e 1 it 2 ξ 2 = ie 1 it ξ =: u ξ,t. 3.46

17 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 141 On the other hand, we have v 2 + e it 2 v 1 = i e 1 it 2 ξ 2 + e it 2 e 1 it 1 ξ 1 = i e it 2 e 1 it 2 ξ 2 + e it 2 e it 1 e 1 it 1 ξ 1 = ie it 2 e it 1 e it 1 e 1 it 2 ξ 2 + e 1 it 1 ξ 1 Let γ ξ,t C defined by = ie it 1 e it 2 1 e 1 it ξ = ie it 1 e 1 it ξ = ie it e 1 it ξ = ie 1 it ξ = : v ξ,t iγ ξ,t := z 1 + z 2 + v 1, u 2 z ξ,t, 3.48 where z ξ,t be as in Combining 3.45 with Eqs , we deduce that W ξ 1, T 1 W ξ 2, T 2 = Γu ξ,t, e it, v ξ,t, z ξ,t e iγ ξ,t Remark The set of all generalized Weyl operators W := {W ξ, T = e iγ ξ,t W ξ, T ; ξ H, T B s H} is not linearly independent. In fact, for example, W 0, π1 = W 0, π1. In the following we will look for a domain D, of pairs of the form ξ, T with the property that the associated Weyl operators W ξ, T are linearly independent and which is maximal with respect to this property. To this goal define the equivalence relation, on H B s H, by ξ 1, T 1 ξ 2, T 2 there exists α C ; such that W ξ 2, T 2 = αw ξ 1, T 1 in this case necessarily α = 1. Denote by H B s H /, the set of equivalence classes for this relation. For all equivalence class, we can choose only one representant ξ, T in some domain D H B s H, hence we construct a bijection between D and H B s H /. Note that several choices of domain D are possible. Intuitively there is a some domain D 0 which is chosen in a natural way. This domain will be called the principal domain of generalized Weyl operators, but here we will not discuss this choice. see [9] for an example of principal domain of quadratic Weyl operator. In the spirit to deal with some properties of generalized Weyl operators, such as independence linearity, we assume, in the following, that a such domain D is fixed. Clearly, from above prescriptions, that the linear span of W, coincides with the linear span of the set of W ξ, T, of which ξ, T in D. Note also that this space is in fact a algebra and this comes from theorems 3.8 and 3.14.

18 142 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI Definition The C subalgebra of B ΓH, generated by W is called oscillator Weyl algebra over H, we denote it by W g H. Lemma For all ξ j, T j D; j = 1, 2, ξ 1, T 1 = ξ 2, T 2 Proof. Let ξ j, T j D; j = 1, 2 such that Then u ξ1,t 1 = u ξ2,t 2 and e it 1 = e it 2 u ξ1,t 1 = u ξ2,t 2 ; e it 1 = e it 2. v ξ2,t 2 : = ie 1 it 2 ξ 2 = e it 2 ie 1 it 2 ξ 2 = e it 2 u ξ2,t 2 = e it 1 u ξ1,t 1 this gives = v ξ1,t 1 W ξ 2, T 2 : = Γu ξ2,t 2, e it 2, v ξ2,t 2, z ξ2,t 2 = Γu ξ1,t 1, e it 1, v ξ1,t 1, z ξ2,t 2 = e z ξ 2,T 2 z ξ1,t 1 Γuξ1,T 1, e it1, v ξ1,t 1, z ξ1,t 1 = e z ξ 2,T 2 z ξ1,t 1 W ξ1, T 1 which implies ξ 2, T 2 ξ 1, T 1 but ξ 1, T 1, ξ 2, T 2 D, then ξ 1, T 1 = ξ 2, T 2. Lemma Let j = 1,..., m, m N and ξ j, T j D satisfying Denote ξ j, T j ξ k, T k j, k = 1,..., m, j k f j : H y f j y = u ξj,t j + e itj y then, there exists x H separating maps f j, i.e. f j x f k x j, k = 1,..., m, j k. Proof. From condition 3.50 and Lemma 3.18, we deduce that for j, k = 1,..., m, j k, u ξj,t j u ξk,t k or e itj e it k Denoting J = {j, k {1,..., m} 2, E j,k := e it j e itk 0} -a If J =, then for all j, k = 1,..., m, one has e itj = e it k then from 3.51, one has u ξj,t j u ξk,t k j k, then f j x f k x j k x H. Hence all x of H separates maps f j. -b If J, we consider F = kere j,k. j,k J -b.i Let us prove, in a first step, that F H. By contradiction, assuming that F = H. While the reunion of finite subspaces is a vectorial space only if, one among them contains all the others, then there exists some j 0, k 0 J such that H = kere j,k = kere j0,k 0 j,k J.

19 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 143 This gives then by definition of J, contradiction, that e it j 0 = e it k0 j 0, k 0 / J absurd. -b.ii In a second step, assuming by x H\F, j, k J, f j x = f k x. Fix x H\F and consider x n = x n H\F, n 1. it is easily checked that x n H\F. By hypothesis H, there exists for x n a pair j n, k n J such that H f jn x n = f kn x n While J is a finite set then, the sequence j n, k n has a stationary subsequence, what we take the same notation. Then for n sufficiently big, j n, k n = j 0, k 0 J and Eq. 3.52, becomes f j0 x n = f k0 x n which gives Taking the limit as n +, one has and Eq becomes u ξj0,t j0 u ξk0,t k0 = e it k 0 e it j0 xn u ξj0,t j0 u ξk0,t k0 = 0 E j0,k 0 x n := e it k 0 e it j0 xn = 0. Then x n kere j0,k 0 and also x = nx n kere j0,k 0 F which is not true by definition of x, i.e. absurd. From above prescriptions, we conclude that there exists x 0 H\F, separating maps f j corresponding to all pair j, k J. The case for which the pair j, k / J, is similar to the one for which J =. In Conclusion, x 0 separates maps corresponding to all pair j, k, j, k = 1,..., m, j k. Proposition The set of generalized Weyl operators is linearly independent. WD := {W ξ, T, ξ, T D} Proof. Let ξ j, T j D, α j C, j = 1,..., m, m 1, satisfying conditions 3.50 of Lemma 3.19 and m α j W ξ j, T j = j=1 By Lemma 3.19, there exists x 0 H separating maps f j. Applying Eq to exponential vector ψ x0, one obtains m α j e z ξ j,t j +<v ξj,t j,x 0 > ψ fj x 0 = j=1 Since the exponential vectors are linearly independent, then which implies α j = 0 for all j = 1,..., m. α j e z ξ j,t j +<v ξj,t j,x 0> = 0

20 144 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI 4. Lie Algebra Time Shift 4.1. Lie algebra time shift as homomorphism of Lie algebras. Let L be a complex Lie algebra. Let {X + α, X α, X 0 β ; α F ; β F 0 } where F, F 0 are disjoint sets, be set of generators of L satisfying the following conditions: X 0 β = X 0 β β F 0, X + α = X α α F. We assume that there exists a single central element, denoted 1 0, among the generators and that it is of X 0 -type i.e. self-adjoint. We will denote C γ α,β ε, ε, ε the structure constants of L with respect to the generators Xα, ε i.e. with α, β F F 0, ε, ε, ε {+,, 0}, and assuming summation over repeated indices: [Xα, ε Xβ ε ] = C γ α,β ε, ε, 0Xγ 0 + C γ α,β ε, ε, +X γ + + C γ α,β ε, ε, Xγ. γ F 0 γ F γ F In the following we will consider only locally finite Lie algebra, i.e. such that, for any pair α, β F F 0 only a finite number of structure constants C γ α,β ε, ε, ε are different from zero. Definition 4.1. Let be given: - a Lie algebra L with canonical set of generators {X ε α, ε {+,, 0}, α F F 0 } with Lie-bracket as in the above definition - a measurable space S, BS - a sub-algebra C L C S, BS. The current algebra over S of {L, Xα}, ε with test function algebra C, is the Lie algebra LS, C defined as follows: as a vector space LS, C is the algebraic linear span of the family {X ε αf, f C, ε {+,, 0}, α F 0 F } and the generators are independent in the sense that Xαf ε ε,α = 0 ; f ε,α = 0 ε, α ε,α the map f X ε αf is linear for ε = +, 0 and anti linear for ε = the Lie brackets are defined by [X ε αf, X ε β g] = γ,ε C γ α,β ε, ε, ε X ε γ f εε g ε ε where, for a test function f, we use the notation and operations εε = ε ε = f ε = f ; if ε = +, 0 ; f ε = f ; if ε = { +, if ε, ε {+, 0, +, +, 0, 0,, } ;, if ε, ε {+,, 0, }

21 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 145 the involution is defined by with X ε αf := X ε α f ε α + := ; := + ; 0 := 0 ; ε α = +, if α F ; ε α =, if α F 0. Example 4.2. The one-mode Heisenberg Lie algebra heis1, is the complex Lie algebra generated as vector space by a +, a, 1 with involution 1 = 1 ; a + = a and Lie bracket [a, a + ] = 1 ; [a ±, 1] = 0. The current algebra of heis1 over R is the Lie algebra generated by the set {a + f, a f, 1 ; f C := L 2 R L R} which are linearly independent in the sense that, f, g C, λ C a + f + a g + λ1 = 0 f = g = 0, λ = 0. The involution and the Lie bracket are given by and a + f = a f [a f, a + g] = f, g 1 ; [a + f, a + g] = [a f, a g] = [a ± f, 1] = 0. Definition 4.3. A unitary representation of a Lie algebra L is a triple: {H, π, D}, where: D H is a total subset of L which is a core for each πa a L π : L L a D adjointable linear maps from lin-spand H πa +, i.e. the H adjoint of πa is defined on D and πa + = πa, π[a, b] = [πa, πb] a, b L, where the identity is meant weakly on D. Definition 4.4. Let L be a real or complex Lie algebra with center CentreL. A central decomposition of L is a direct sum of vector spaces where L 0 is a sub Lie algebra of L. L = CentreL L 0, Remark 4.5. Let L and G be a real or complex Lie algebras and Φ : L G a isomorphism of Lie algebras. Clearly if L has a central decomposition then G has the central decomposition L = CentreL L 0, G = ΦCentreL ΦL 0.

22 146 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI In the following all Lie algebras are identified with their images under the corresponding unitary representations and these are omitted from the notations. We assume that all Lie algebras have a scalar center, i.e. with central decomposition L = C1 0 L 0. If given a Lie algebra, denote by L s the real subspace of L, of self-adjoint non central elements, i.e. such that L s L 0 and a = a for all a L s. For I R, denote by C I, the space of test functions in C with support in I. Definition 4.6. Let be given: - a complex Lie algebra L with scalar center C1 0, - a space of test functions C, - a current algebra LR, C of L over R. A Lie algebra time shift is a family of homomorphisms of Lie algebras with the following structure: ĵ t : L L L[0, t], C [0,t] 4.1 ĵ t X := T X T [0,t] X 4.2 with the property that the exponential map exists and the map j t, defined by j t e ix := e iĵ t X, X L s 4.3 is well defined and extends to a homomorphism of the C algebra generated by the set {e ix, X L s }. Note that in the above definition the map T must be a Lie algebra homomorphism of L and T [0,t] is a Lie algebra homomorphism from L to its current algebra over [0, t]. In the following we take T = Id, i.e. T X = X, for all X L An example of Lie algebra time shift of L osc H. In this section we give an example of Lie algebra time shift of the oscillator Lie algebra. Definition 4.7. The oscillator Lie algebra algebra over an Hilbert space H, is the complex Lie algebra L osc H generated as vector space by A + u, A u, ΛT, 1 0 the identity on H, u H \ {0}, T BH \ {0} which are linearly independent and satisfying the commutation relations from 2.1 to 2.4. In the definition 4.6, taking L = L osc H and sets of indices F = H \ {0}, F 0 = BH. L osc H, is generated as vector space by the set {X ε α ; α F F 0 ; ε = +,, 0}, where: If α = u F, X α ± = A ± u and if α = T F 0 \ {0}, Xα 0 = ΛT and X0 0 = 1 0. The current algebra, L osc R, BR, of L osc H over R is given by the following -the space of test functions is C = L 2 R L R, -for all f C, Xαf ε = A ± f u, if ε = ± and α = u F ; ΛM f T, if ε = 0 and α = T F 0 \ {0}; fxdx1 0, if ε = 0 and α = 0,

23 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 147 where the operator M f, is the multiplication by f. Since f u H := L 2 R H L 2 R, H and M f T acts on H, the representation space of this current algebra is ΓL 2 R H ΓL 2 R, H. If f = χ [0,t], denote X ε t α := X ε αχ [0,t], then we have A + t u = A + χ [0,t] u ; A t u = A χ [0,t] u ; Λ t T = ΛM χ[0,t] T. 4.4 This shows clearly, with these notations, that the operators A ± t u and Λ t T are in the current algebra L osc [0, t], C [0,t]. The Lie algebra time shift can be defined by its action on generators of L osc H which takes the form: ĵ t X ε α = X ε α γ,ε X ε t γ. 4.5 Several choices of homomorphisms of Lie algebras are possible. In our case, we consider only a class of Lie algebra homomorphisms and we study in which cases they are a Lie algebra time shifts. To this goal, let us fix: - i a continuous unital homomorphism ρ : BH BH, - ii a surjective ρ 1 cocycle, i.e. a linear map δ : BH H with the property δt T = ρt δt T, T BH. 4.6 Then the cocycle property implies that the linear functional L : BH C, defined by is hermitian and satisfies: In fact we have and LT := δ1, δt T BH 4.7 LT T = δt, δt T, T BH. 4.8 LT = δt, δ1 = ρt δ1, δ1 = δ1, ρt δ1 = δ1, δt = LT LT T = δ1, δt T = δ1, ρt δt = ρt δ1, δt = δt, δt. Example 4.8. Let u be a fixed vector in H. Taking ρt := T and δt = ρt u, one finds LT = u, ρt u.

24 148 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI With the following choices: ĵ t A + ξ := A + ξ A + t T 1 ξ + A t T 2 ξ, 4.9 ĵ t A ξ := A ξ A + t T 2 ξ + A t T 1 ξ, 4.10 ĵ t ΛT := ΛT A + t δt + A t δt + Λ t ρt + tlt 1, 4.11 ĵ t 1 0 = R t, 4.12 ĵ t will be a R linear map whenever T 1, T 2 are R linear operators on H and R t is a self-adjoint operator acting on ΓL 2 R, H. Proposition 4.9. The map ĵ t, defined by the equations from 4.9 to 4.12, is a Lie algebra homomorphism if and only if T 1 = T 2 = 0 and R t = 0 for all t 0. In this case, denoting H t ξ, T := A + t ξ + A t ξ + Λ t T 4.13 one has ĵ t Hξ, T = Hξ, T H t δt, ρt + tlt 1 ξ, T H BH Proof. The property is easily verified from the construction. The map ĵ t is a Lie algebra morphism if and only if for all X, Y {A + ξ, A ξ, ΛT, 1 0 }, one has ĵ t [X, Y ] = [ĵ t X, ĵ t Y ], this is equivalent to the following conditions: ξ, η R t = t T 1 ξ, T 1 η T 2 η, T 2 ξ, 4.15 T 2 ξ, T 1 η = T 2 η, T 1 ξ, 4.16 ρt T 1 ξ = T 1 ξ, 4.17 ρt T 2 ξ = T 2 ξ, 4.18 δt, T 1 ξ = T 2 ξ, δt 4.19 for all ξ, η H and T BH. Taking T = 1 the identity of BH in 4.18, since ρ1 = 1, it follows that T 2 = 0. Therefore 4.19 implies that also T 1 = 0 because δ is surjective. From this we see that 4.15 implies that R t = 0 for all t 0. Given the above, from the real linearity of ĵ t and the notation 4.13, one sees that Eq is the sum of 4.9, 4.10 and Then Defining a map j t on the set WD by the following: j t W ξ, T j t W ξ, T = jt e ihξ,t := e iĵ t Hξ,T = e i Hξ,T 1 +i 1 0 H tδt,ρt +tlt 1 = e ihξ,t e ihtδt,ρt e itlt = W ξ, T W t δt, ρt e itlt, 4.21

25 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 149 where we have used the notation W t ξ, T := e ihtξ,t. Note that from definition of D, the map j t is well defined by Eq Since WD is a linearly independent set, then j t can be extended by linearity to whole the linear span of WD and we have the following theorem. Theorem The action of j t given by 4.21 holds also for all ξ, T H B s H. Moreover, j t extends to a isomorphism from the C algebra W g H onto its image. Furthermore the map ĵ t is a Lie algebra time shift. For the proof of this theorem we need the following lemma. Lemma Let fz = n 0 a nz n be an entire function and ρ and δ as in 4.6. Then we have: i For all T BH, fρt = ρft ii For all T BH, ξ H, fχ [0,t] T χ [0,t] ξ = χ [0,t] ft ξ Proof. Note that from Lemma 3.4, ρft and fρt are well-defined. Moreover, since ρ is continuous, -i fρt = n a n ρt = a n ρt n n 0 n 0 continuity = ρ a n T n = ρft, -ii n 0 fχ [0,t] T χ [0,t] ξ = n 0 a n χ[0,t] T n χ[0,t] ξ Proof. of Theorem = a n χ [0,t] T n ξ = χ [0,t] a n T n ξ n 0 n 0 = χ [0,t] ft ξ. Step I. In this Step we will prove the first part of Theorem. Let ξ, T D then j t W ξ, T = W ξ, T W t δt, ρt e itlt where and = W ξ, T ΓU t, e itt, V t, Z t e itlt, 4.24 T t = χ [0,t] ρt, U t = ie 1 it t χ [0,t] δt, V t = e it t U t Z t = χ [0,t] δt, e 2 it t χ [0,t] δt.

26 150 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI Using Eqs and 4.23 in Lemma 4.11, we get e it t = χ [0,t] ρe it 1, 4.25 U t = e 1 χ [0,t] ρit χ [0,t] δit = χ [0,t] e 1 ρit δit = χ [0,t] ρ e 1 it δit = χ [0,t] δ e 1 it it = χ [0,t] δe it 1, 4.26 V t = e χ [0,t] ρ it χ [0,t] δe it 1 = χ [0,t] ρ e it δe it 1 = χ [0,t] δ e it e it 1 = χ [0,t] δe it 1, 4.27 Z t = χ [0,t] δt, χ [0,t] ρ e 2 it δt = t δt, δ e 2 it T = tl e 2 it it 2 = tle it it Injecting right-hand sides of Eqs in 4.24, one obtains j t W ξ, T = W ξ, T Γ χ [0,t] δe it 1, e it t, χ [0,t] δe it 1, tle it Note that from 4.25, arguments of exponential operator Γ in the right-hand side of the above equation dependent of e it. Now taking ξ, T H B s H then there exists a pair ξ, T D such that W ξ, T = e iα W ξ, T, α R. But from definition of generalized Weyl operator given in Eq we get e it = e it. Then by linearity of j t we obtain j t W ξ, T = e iα j t W ξ, T = e iα W ξ, T Γ χ [0,t] δe it 1, e it t, χ [0,t] δe it 1, tle it 1 = W ξ, T Γ χ [0,t] δe it 1, e it t, χ[0,t] δe it 1, tle it 1 = W ξ, T W t δt, ρt = e iĵ t Hξ,T which proves that we can take the same expression of j t for all W ξ, T Step II. Assuming, provisionally, that for all ξ, ξ 1, ξ 2 H and T, T 1, T 2 B s H, the following relations hold j t W ξ, T = jt W ξ, T 4.31 and j t W ξ 1, T 1 W ξ 2, T 2 = j t W ξ 1, T 1 j t W ξ 2, T 2, 4.32

27 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 151 then j t is a homomorphism from the algebra generated by the set of generalized Weyl operators onto A t]. In fact if w = α j W j, then j t w = j t α j W j = 1 n = j t 1 j n 1 j n 1 j n ᾱ j jt W j = 1 j n ᾱ j j t W j ᾱ j Wj = j t w On the other hand, let w 1 = 1 j n α jw j ; w 2 = 1 j n β jw j, then j t w 1 w 2 = j t α j β k W j W k = α j β k j t Wj W k = 1 j,k n 1 j,k n α j β k j t W j j t W k = 1 j,k n 1 j n α j j t W j 1 k n β k j t W k = j t w 1 j t w Eqs and 4.34 imply that j t is a homomorphism. From the expression of j t in equation 4.21, we see that it is an injective map. Since j t is a homomorphism of algebra and identity preserving then it is automatically continuous. Hence it can be extended to the C algebra generated by the set of Weyl operators W g H. Hence it is a isomorphism from W g H onto its image. Step III. Here, we prove the property We have j t W ξ, T = e iĵ t Hξ,T = e i ĵt Hξ,T = e iĵ t Hξ,T = e iĵ t Hξ,T = j t e ihξ,t W = j t ξ, T. Step IV. In this step we investigate to prove the multiplicative property From Theorem 3.14 and linearity of j t, we get j t W ξ 1, T 1 W ξ 2, T 2 = j t e iγ ξ,t W ξ, T = e iγ ξ,t j t W ξ, T = e iγ ξ,t W ξ, T Γ U t, e itt, V t, ζ t, 4.35 where e itt, U t and V t are as in Eqs and ζ t = tle it 1. Similarly we obtain j t W ξ 1, T 1 j t W ξ 2, T 2 = W ξ 1, T 1 Γ U 1 1 it t, e t, V 1 t, ζ 1 t W ξ 2, T 2 Γ U 2 2 it t, e t, V 2 t, ζ 2 t [ = e iγ ξ,t W ξ, T Γ U 1 1 it t, e t, V 1 t, ζ 1 t ] Γ U 2 2 it t, e t, V 2 t, ζ 2 t,

28 152 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI where U j j it t, e t, V j t and ζ j ; j = 1, 2 are as U t, e it t, V t and ζ t respectively. Using Lemma 3.3, we get j t W ξ 1, T 1 j t W ξ 2, T 2 = e iγ ξ,t W ξ, T Γ U, e it, V, ζ, 4.36 where and Then U = U 1 1 it t + e t U 2 t ; e it 1 2 it = e t it e t ; V = V 2 2 it t + e t V 1 t ζ = ζ 1 t + ζ 2 t + V 1 t, U 2 t. U = χ [0,t] δe it e χ [0,t] ρit 1 χ [0,t] δe it 2 1 = χ [0,t] δe it1 1 + χ [0,t] e ρit1 δe it2 1 = χ [0,t] δe it ρe it 1 δe it 2 1 = χ [0,t] δe it1 1 + δe it1 e it2 1 = χ [0,t] δe it1 e it2 1 = χ [0,t] δe it 1 = U t. The same computations give e it = e it t, V = V t and ζ = ζ t. Finally comparing Eqs and 4.36, one has The Generator of the Quantum Lévy Process In this section, we use the notations of Section 2.2 and we consider the C algebra A 0 = W g H and the stochastic process j t : A 0 : A t] given on the generalized Weyl operators by Eq We have the following theorem: Theorem 5.1. The stochastic process j t is a Markov cocycle. Proof. Clearly that j 0 W ξ, T = W ξ, T 1 for all ξ, T H Bs H. Then by homomorphism properties we obtain j 0 a = a 1 for all a A 0. Let j t be as in Eq of Section 2.2. Then from 4.21, one has j t W ξ, T = W ξ, T e X [0,t], where, in the notation 4.13: X [0,t] = i H t δt, ρt + tlt 1.

29 LÉVY PROCESSES THROUGH TIME SHIFT ON OSCILLATOR WEYL ALGEBRA 153 Using Eqs. 2.7 and 2.8, one has j s u s j t W ξ, T = j s u s W ξ, T e X [0,t] = j s W ξ, T Γθ s e X [0,t] Γθs = j s W ξ, T 1 [0,s] e X [s,s+t] W = j s ξ, T [0,s] e X [s,s+t] = j s W ξ, T 1 j s [0,s] e X [s,s+t]. From Eq. 2.10, we get j s u s j t W ξ, T = j s W ξ, T = W ξ, T e X [0,s] j s 1 0 e X [s,s+t] s] e X [s,s+t] which proves the cocycle identity = W ξ, T e X [0,s] e X [s,s+t] = W ξ, T e X [0,s]+X [s,s+t] = W ξ, T e X [0,s+t] = j s+t W ξ, T Theorem 5.2. Let P t be the Markov semigroup associated to the process j t via Eq Then its action on the generalized Weyl operators is given by P t W ξ, T = e tleit 1 W ξ, T. 5.1 Moreover, W ξ, T are eigenoperators of its generator G 0 so that G 0 W ξ, T = Le it 1W ξ, T. 5.2 Proof. With help of Eqs. 2.12, 4.21, 2.9, 4.28 respectively, we get P t W ξ, T = E 0] j t W ξ, T = itlt E0] W ξ, T W t δt, ρt e = Φ [0, W t δt, ρt e itlt Φ [0 W ξ, T = e itlt Φ [0, W t δt, ρt Φ [0 W ξ, T = e itlt Φ [0, e Zt Φ [0 W ξ, T = e itlt +tleit it 1 W ξ, T = e tleit 1 W ξ, T. Remark 5.3. Taking the state L such that LT = u, T u, where u is a vector of H with norm equal to 1, Eq. 5.1 becomes G 0 W ξ, T = u, e it 1u W ξ, T = e ix 1µ T,u dx W ξ, T, 5.3 σt

30 154 LUIGI ACCARDI, HABIB OUERDIANE, AND HABIB REBEI where µ T,u is the spectral measure of T associated to u. In the above equation, the eigenvalues of G 0 η := e ix 1µ T,u dx, σt are explicitly computed in terms of the Lévy-Khintchin factor of the underlying classical Lévy process and the generalized Weyl operators, associated to the oscillator Lie algebra, are eigenoperators of the generator of the quantum Markov semigroups canonically associated to the quantum extension of the classical Lévy process induced by these Lie algebra shifts. Acknowledgment. Habib Rebei is grateful to Professor Luigi Accardi for kind hospitality during his visit at the Centro Vito Volterra. He would like to thanks his supervisor Habib Ouerdiane for many helpful discussions during the preparation of this paper. References 1. Accardi, L.: On the Quantum Feynman-Kac Formula, Rendiconti del Seminario Mathematico e Fisico di Milano, Vol. XLVIII Accardi, L.: A note on Meyer s note, Quantum Probability and Applications III, Oberwolfach 1987, Lecture Notes in Mathematics Vol. 1303, Springer 1988, Accardi, L, Barhoumi, A and Ouerdiane, H.: A quantum approach to Laplace operators, Infin. Dimens. Anal. Quantum Probab. Relat. Top. Vol. 9, no , Accardi, L, Barhoumi, A, Ouerdiane, H and Rebei, H.: White noise quantum time shifts, Random Operators/ Stochastic Eqs , Accardi, L, Franz, U and Skeide, M.: Renormalized squares of white noise and other non- Gaussian as Lévy processes on real Lie algebras, Comm. Math. Phys , Accardi, L, Ji, U.C and Saitô, K.: Exotic Laplacians and associated stochastic processes, Infin. Dimen. Anal. Quantum Probab. Rel. Top , Accardi, L, Ji, U.C and Saitô, K.: Exotic Laplacians and Derivatives of White Noise, Infin. Dimen. Anal. Quantum Probab. Rel. Top Accardi, L and Mohari, A.: On the Structure of Classical and Quantum Flows, J. Funct. Anal , Accardi, L, Ouerdiane, H and Rebei, H.: On the quadratic Heisenberg group, Infin. Dimens. Anal.Quantum Probab. Relat. Top. Vol: 13 no pp Fagnola, F.: On the realisation of classical Markov processes as quantum flows in Fock space, Probability theory and mathematical statistics Vilnius, 1993, , TEV, Vilnius, Ji, U.C and Saitô K.: A similarity between the Gross Laplacian and the Lévy Laplacian, Infin. Dimen. Anal. Quantum Probab. Rel. Top , Meyer, P.A.: A note on Shifts and Cocycles, Quantum Probability and Applications III, Oberwolfach 1987, Lecture Notes in Mathematics Vol. 1303, Springer 1988, Meyer, P.A.: Quantum Probability for Probabilists, Lectur Notes in Math., Vol. 1538, Springer-Verlag, 1995, 2nd edn. 14. Parthasarathy, K.R.: An introduction to quantum stochastic calculus, Birkhaüser Verlag, Basel Parthasarathy, K.R and Schmidt, K.: Positive definite kernels continuous tensor products and central limit theorems of probability theory, Springer Lecture Notes in Mathematics no Torossian, C.: Formule de Campbell-Baker-Hausdorff, Isomorphisme de Duflo Conjecture de Kashiwara-Vergne, Cours de Master, Tunis Fev Yosida, K.: Functional analysis, Springer Verlag 1965.