#### Abstract

We define a Lie algebroid on the space of smooth 1-forms in the Nualart-Pardoux sense on the Wiener space associated to the stochastic linear Poisson structure on the Wiener space defined Léandre (2009).

#### 1. Introduction

Infinite dimensional Poisson structures play a big role in the theory of infinite dimensional Lie algebras [1], in the theory of integrable system [2], and in field theory [3]. But for instance, in [2], the test functional space where the hydrodynamic Poisson structure acts continuously is not conveniently defined. In [4, 5] we have defined such a test functional space in the case of a linear Poisson bracket of hydrodynamic type. On the other hand, it is very well known [6] that the theories of Lie groupoids and Lie algebroids play a key role in Poisson geometry. It is interesting to study a Lie algebroid for the Poisson structure [4] defined analytically in the framework of [4]. We postpone until later the study the Lie groupoid associated to the same Poisson structure but in the algebraic framework of [5]. The definition of this Lie groupoid in the framework of [4] presents, namely, some difficulties. Moreover some deformation quantizations for symplectic structures in infinite dimensional analysis were recently performed (see the review of Léandre [7] on that). The theory of groupoids is related [8] to Kontsevich deformation quantization [9].

Let us recall what a Lie algebroid is [6, 10–13]. We consider a bundle on a smooth finite dimensional manifold . is the tangent bundle of . and denote the space of smooth section of and . A Lie algebroid on is given by the following data.

(i)A Lie bracket structure on has in particular to satisfy the Jacobi relation (ii)A smooth fiberwise linear map , called the anchor map, from into satisfies the relation for any smooth sections , of and any element of , the space of smooth functions on .Let us recall the definition of a Poisson structure on . It is an antisymmetric -bilinear map from into , which is a derivation on each components, vanishes on the constant and satisfies the Jacobi relation A Poisson structure is given by a bivector , that is, an element of as where and can be seen as 1-forms and the dual of is . denotes the interior product of the bivector by the 1-form . If is a bivector, then we can define a fiberwise linear smooth map from , the cotangent bundle of , into , called .

If is a smooth section of , then This allows us to define a Lie algebroid structure on [11, 14–18] as follows.

(i)The Bracket is defined by where is the usual Lie derivative of a 1-form: if is a vector field and a -form, then is given by the Cartan formula (ii)The anchor map is the map .Infinite dimensional symplectic structures and their related Poisson structure were introduced by Dito and Léandre [19], Léandre [7, 20–22], and Léandre and Obame [23] in the infinite dimensional analysis, motivated by the theory of deformation quantization in infinite dimension. We refer to the review of Léandre on that in [7].

The infinite dimensional Poisson structure is a tool in the theory of integrable system [24]. We refer to the review of Mokhov in [3] and Dubrovin and Novikov in [2] on that. In particular, some partial differential equations of the theory of integrable systems are described as Hamiltonian systems associated to some Poisson structure. For instance, the Gardner-Zakharov-Faddeev (ultralocal) bracket (see [2, pages 52-53]) is used to describe the KdV equation as a Hamiltonian system.

Another simple Poisson structure of Dubrovin and Novikov [2] is given as follows. We consider the set of smooth paths into , the dual of a Lie algebra structure on with structural constants . In such a case, It is useful to avoid the presence of the Dirac mass, and Léandre [4] has given an appropriate definition of the Poisson structure of the previous formula in the framework of Malliavin Calculus.

The goal of infinite dimensional analysis is to give a rigorous meaning to some formal considerations of mathematical physics. The formal operations of mathematical physics are defined consistently on some functional spaces. It is very well known, for instance, that the vacuum expectation of some operator algebras [25] is given by formal path integrals on the fields. Infinite dimensional analysis deals in the simplest case where these objects are mathematically well established.

(i)The functional integral side is given by the Malliavin Calculus [26].(ii)The operator algebra side is given by white noise analysis and quantum probability [27, 28].Let us recall basically the objects of these Calculi.

(i)The main object of white noise analysis and quantum probability is given by the Bosonic Fock space associated to the Hilbert space of maps from into . is constituted of series where belongs to , the symmetric -tensor product of such that The operator algebra is the algebra of annihilation and creation operator on the Fock space submitted to the canonical commutation relations where is an elementary annihilation operator. is an elementary creation operator. The presence of a Dirac mass leads to the same difficulties as in (1.8) and (1.9) and leads white noise analysis to consider an improvement of the Fock space (called the Hida Fock space) such that these operators act continuously on it. is called the white noise and can be interpreted in the measure theory.(ii)The main object of the Malliavin Calculus is the space of the Wiener measure.If we consider the Brownian motion on (see part 2 of this work), then we can introduce the Brownian functional associated to in the Bosonic Fock space:

An element of can be realized as a symmetric map from into and denotes a Wiener chaos [28]. This map realizes an isometry between the Fock space and the of the Wiener measure. The main ingredient of the Malliavin Calculus is to take the derivative (*almost surely defined!*) in the direction of an element ; . An element is called an element of the Cameron-Martin space . This operation can be interpreted as a “nonelementary” annihilation operator on the Fock space. Through this isomorphism, can be interpreted as , the white noise associated to the Brownian motion, which does not exist in the traditional sense because the Brownian motion is *only continuous!* Since there are integrations by parts associated to a derivativation along an element of the Cameron-Martin space of a cylindrical functional, this operation is closable. It is the generalization in infinite dimension of the traditional definition of Sobolev spaces on finite dimensional spaces. But in infinite dimension, we consider Gaussian measures and not Lebesgue measure, which does not exist as a measure in infinite dimension! But since there is no Sobolev imbedding in infinite dimension, functionals which belong to all the Sobolev spaces of the Malliavin Calculus (these functionals are said to be smooth in the Malliavin sense) are in general only *almost surely defined*!

The study of Poisson structures requests that the test functional space where this Poisson structure acts is an *algebra*.

Léandre [4] has given an appropriate definition to the Poisson structure of the previous formula on an algebra of functional on the Wiener space of the Malliavin type. The main difficulty to overcome is that the good understanding of this Poisson structure leads to the study of some anticipative Stratonovitch integrals. The traditional Malliavin Calculus [26] is not suitable to study some anticipative Stratonovitch integrals. This means that if we consider a random element of which belongs to all the Sobolev spaces of the Malliavin Calculus, then the anticipative Stratonovitch integral does not exist. This pathology is not true if we consider a refinement of the Itô integral called the Hitsuda-Skorokhod integral [30]. Let us recall that the first authors who have studied anticipative Stratonovitch integrals are Nualart and Pardoux [30]. Léandre has defined conveniently some Sobolev spaces on the Wiener space such that the map anticipative Stratonovitch integral acts continuously on them [31–37]. Thisat means that if belongs to all the Sobolev spaces in the Nualart-Pardoux sense, then the anticipative Stratonovitch integral (1.15) exists. The main difference between the classical definition of the Sobolev space in the Malliavin sense and the Sobolev spaces in the Nualart-Pardoux sense is that some regularity on the kernels on the derivatives on the considered functionals is requested. This is a generalization of the following fact: we can define a nonanticipative Itô integral without assuming a lot of regularity on the nonanticipative element of . But in the case of a nonanticipative Stratonovitch integral, , we have to assume that is a semi-martingale; this means some regularity on ! The Sobolev spaces of Nualart-Pardoux type were introduced by Léandre in [31–36] in order to study some Sobolev cohomology theories of some loop space endowed with the Brownian bridge measure on a compact Riemannian manifold. So the Poisson structure (1.9) can be defined consistently on the Nualart-Pardoux test algebra [4].

Let us recall that in white noise analysis, the algebraic counterpart of the Malliavin Calculus, the main tool is the Fock space and the algebra of creation and annihilation operators on the Fock space. The Bosonic Fock space is transformed into the of an infinite dimensional Gaussian measure by the help of the map Wiener chaoses. The Poisson structure (1.9) was defined by Léandre in [5] on the Hida test algebra endowed with the normalized Wick product.

The goal of this paper is to define a Lie algebroid associated to the Poisson structure (1.9) on the Nualart-Pardoux test algebra. The main remark is that the map transforms a 1-form on the Wiener space smooth in the Nualart-Pardoux sense in a generalized vector field on the Wiener space, whose theory was done by Léandre in [32, 35], and not in an ordinary vector field on the Wiener space! Classical vector field on the Wiener space are random elements of the Cameron-Martin space which belongs to all the Sobolev spaces of the Malliavin Calculus. Generalized vectorfield is where is chosen well such that we can define the anticipative Stratonovitch integral . In general, we cannot define the derivative of a functional which belongs to all the Sobolev spaces of the Malliavin Calculus along a generalized vector field. But we can do that if the functional belongs to all the Sobolev spaces in the Nualart-Pardoux sense. In this paper, since we consider smooth 1-forms in the Nualart-Pardoux sense, we can define still their interior product by a generalized vector field through our theory of anticipative Stratonovitch integral. So the formulas (1.1) and (1.2) are still true, but almost surely!

#### 2. The Linear Stochastic Poisson Structure

We consider the set of continuous paths from into endowed with the uniform topology. A typical path is denoted by , on which we consider the Brownian motion measure [38].

Let us recall how we construct . We consider the Cameron-Martin Hilbert space [39] of maps from into such that and is formally the Gaussian probability measure and is the formal Lebesgue measure on which does not exist as a measure (we refer to the works of Léandre [40–42], Asada [43], and Pickrell [44] for various approaches to the Lebesgue measure in infinite dimension). Let be a finite dimensional real Hilbert space () with Hilbert norm . Let us consider the centered normalized Gaussian measure on . It is classically represented by where are centered normalized independent one-dimensional Gaussian variables and the system of constitutes an orthonormal basis of the real Hilbert space .

It should be tempting to represent by using the same procedure. We consider an orthonormal basis of . The law of the Brownian motion is represented by the *series * where the is a collection of independent centered one-dimensional Gaussian variables. This series *does not converge* in but in [45]. We refer to the textbook of Kuo [46] for the theory of infinite dimensional Gaussian measures.

Let us consider a functional on . Its th stochastic derivative , according to the framework of the Malliavin Calculus [26, 47, 48], is defined if it exists by where belongs to the Hilbert space of paths from into satisfying

The Sobolev norms of the Malliavin Calculus are defined by the following formula. If is a Brownian functional, then
The Malliavin test algebra consists of Brownian functionals all of whose Sobolev norms are finite . Let us recall how we construct these Sobolev spaces. Let be a smooth function from into with compact support and some times . We introduce the cylindrical functional . We consider the Gateaux derivative of along a deterministic direction of :
There is absolutely no problem to define it. We use the integration by parts formula for a cylindrical functional
where is the Itô differential. The Itô integral is the limit in all the , of the sum where is a dyadic subdivision of of length . The convergence does not pose any problem because is deterministic. Since we have the integration by parts formula (2.7), we can extend the operation of taking the stochastic derivative of a Brownian functional consistently, as we establish classically the definition of Sobolev spaces in finite dimension. The main novelty of the Malliavin Calculus with respects of [49–52] motivated by mathematical physics is that the *algebra* of functionals which belong to all the Sobolev spaces of the Malliavin Calculus (these functionals are said to be smooth in the Malliavin sense) is constituted of functionals almost surely defined. The reader interested in the Malliavin Calculus can see the books of [26, 47].

If we consider the same dyadic subdivision as before, we can introduce the polygonal approximation of . Let us consider a “nondeterministic!” map from into which belongs to . We can consider the random ordinary integral . It can also be easily defined. In general, the limit may not exist when tends to infinity, because the Brownian motion is only continuous. If we can pass to the limit, we say that the limit is an anticipative Stratonovitch integral. Nualart and Pardoux [30] are the first authors who have defined some anticipative Stratonovitch integrals. An appropriate theory was established by Léandre [31–36] in order to understand some Sobolev cohomology theories on the loop space. Let us recall it quickly.

We consider another set of Sobolev norms [31]. We suppose that outside the diagonals of The smallest such that the previous inequality is satisfied is called the first Nualart-Pardoux Sobolev norm. The second Nualart-Pardoux Sobolev norm is the smallest such that, for all ,

*Definition 2.1. *The Nualart-Pardoux test algebra consists of functionals whose all Nualart-Pardoux Sobolev norms of first type and second type are finite. The elements of are said to be smooth in the Nualart-Pardoux sense.

Let us recall that is an algebra [31].

We can consider a random element of , . We can consider its th stochastic derivative Its first Nualart-Pardoux Sobolev norm is the smallest number such that outside the diagonals of The second type of Nualart-Pardoux Sobolev norm of is the smallest number such that for all Let us recall the theorem of Léandre [31].

Theorem 2.2. *Let be a random element of such that all its Nualart-Pardoux Sobolev norms are finite. Then the anticipative Stratonovitch integral
**
is smooth in the Nualart-Pardoux sense and its Nualart-Pardoux Sobolev norms can be estimated in terms of the Nualart-Pardoux norms of .*

In such a case, is the limit in all the , of . Moreover, This means that the kernel of the stochastic derivative of is . Let us explain this formula; in order to take the stochastic derivative of , we do the same formal computations as if the anticipative Stratonovitch integral had been a classical integral; we take first of all derivatives of which lead to the term and derivatives of which lead to .

Let us recall the notion of a Poisson bracket . We consider a commutative Frechet unital real algebra endowed with a family of Banach norms . This means that for all , there exists such that for all in A Poisson Bracket is a bilinear map from into , which is a derivation in each argument, vanishes on the unit. The derivation property means that for all in Moreover, it satisfies the following properties: if belong to , then Moreover, for all , there exist and such that In the sequel, we will choose . We consider the structural constants of a Lie algebra structure on . The stochastic gradient of a functional can be written . Formula (1.9) reads in this framework as where we consider a Stratonovitch anticipative integral. This defines a Poisson structure on in our framework [4, Theorem 1].

*Remark 2.3. *Let us motivate (2.19). Let us consider the Hilbert space of maps from into . The define a structure of Lie algebra on , and therefore, on . Let us consider two functionals and Frechet smooth on . Their derivatives are given by kernels
The Lie bracket is given by and the classical Lie-Poisson structure is given by
These considerations are heuristic because the product of two elements of is not an element of . If we replace by and if we consider the white-noise measure on instead of the Brownian measure (2.2), then this heuristic formula gives the formula (2.19). This is relevant of the so-called Malliavin transfer principle: a formula becomes almost surely true through the theory of Stratonovitch integrals.

#### 3. The Stochastic Lie Algebroid

Smooth vector fields in the Nualart-Pardoux sense on the Wiener space are functions from into such that on the connected complements of where we have removed the diagonals and such that The infimums of and of in the previous formula are called the Nualart-Pardoux Sobolev norms of the vector field .

Smooth 1-forms in the Nualart-Pardoux sense on the Wiener space are functions from into such that on the connected complements of where we have removed the diagonals and such that The infimums of and of in the previous formula are called the Sobolev norms of the 1-form . The pairing between a 1-form and a vector field is realized via the formula If are two smooth 1-forms in the Nualart-Pardoux sense on the Wiener space, then the bivector associated to the stochastic Poisson structure is given by The stochastic bivector realizes a continuous bilinear map on the space of smooth 1-forms smooth into the space of smooth functionals.

A generalized vector field according to our theory [32, 35] is a a random application from into of the form where and are smooth in the Nualart-Pardoux sense. The Nualart-Pardoux Sobolev norms of a generalized vector field are the collection of Nualart-Pardoux norms of and .

We can define a pairing between smooth 1-form and generalized vector fields by using the formula This allows us to define for a smooth 1-form in the Nualart-Pardoux sense as the generalized vector field

This allows us to put the following definition.

*Definition 3.1. *If and are smooth 1-forms in the Nualart-Pardoux sense, then we define
where the Lie derivative is defined as usual by the formula
and is the exterior derivative.

Since is a generalized vector field, the introduction of the Lie derivative leads to stochastic integral (we refer to [32, 33, 35] for similar constructions). Let us recall some results of [31, 32]. Let be a map from into or later into such that on the connected complements of where we have removed the diagonals and such that The infimums of and of in the previous formula are called the Nualart-Pardoux Sobolev norms of the . In such case is still smooth in the Nualart-Pardoux sense, with being included as well as .

This allows us to show the following theorem.

Theorem 3.2. *
is a continuous antisymmetric bilinear application acting on the space of smooth 1-forms in the Nualart-Pardoux sense with values in the set of smooth 1-forms in the Nualart-Pardoux sense.*

*Proof of Theorem 3.2. *We remark that
which is smooth and its Sobolev norms can be estimated in terms of the Sobolev norms of by Theorem 2.2.

Moreover,
which is still smooth. Moreover,
Moreover,
which is smooth by the remark preceding the theorem.

Theorem 3.3. *
defines a Lie bracket.*

Let be a dyadic subdivision of . We deduce a partition of in cubes of volume . If is a function from into some linear space, then we put If we consider the polygonal approximation of and being the -algebra associated to We denote by the operation of taking the conditional expectation of a functional by . The results of Léandre [31, Appendix], allow to state the proposition.

Proposition 3.4. *Let one have
**
If with values in is smooth in the Nualart-Pardoux sense, being included, then the random ordinary integral tends in all the Sobolev spaces of the Malliavin Calculus to the anticipative Stratonovitch integral . If which takes its values in is smooth in the Nualart-Pardoux sense, then the double random ordinary integral tends in all the Sobolev spaces of the Malliavin Calculus to the double anticipative Stratonovitch integral .*

*Proof of Theorem 3.3. *Let us consider the finite dimensional Gaussian space . A 1-form is piecewise constant as well as a vector field . If is a functional which depends on only, then is constant on each . We put
This defines a Poisson structure on the finite dimensional Gaussian space.

We can define , , and according to the line of the introduction. To a 1-form smooth in the Nualart-Pardoux sense on the total Wiener space, we consider the 1-form on the finite dimensional Gaussian space. We get
by doing as in finite dimension. By the previous proposition tends in all this attained to . Therefore the result.

Let us give the scheme of the proof of this last result. When we write , there are a lot of terms which will appear. All these terms will tends separately to the corresponding term in . Let us treat one of them, which will lead to double anticipative Stratonovitch integral. The other terms will be treated identically. For instance will lead to double Stratonovitch integral which will tend in all the Sobolev spaces of the Malliavin Calculus to . We can consider in these expressions the term which will lead to a double stochastic integral and which will tend to . But there are two parts in and . We will consider the parts and where we take the covariant derivative of the 1-form in the direction of the generalized vector field . We will show that tends to . But in these expressions there are still two parts which can be treated similarly. We will show that the expression tends to (we consider the covariant derivative of in the direction of ).

But
Therefore
This implies that
where ( we consider a 1-form in the variable).

Therefore
By Proposition 3.4, this tends in all the Sobolev spaces of the Malliavin Calculus to
We recognize in this quantity where we take the covariant derivative of in the direction of the generalized vector field (we had taken the covariant derivative of in the direction of the generalized vector field ).

Theorem 3.5. * is an anchor map. This means that for all 1-form , which are smooth in the Nualart-Pardoux sense all functional are smooth in the Nualart-Pardoux sense; one has the relation:
*

*Proof of Theorem 3.5. *We get by classical results in finite dimension
By the results of Proposition 3.4, this tends when to the formula
Therefore the result is attained.

#### 4. Conclusion

We can summarize that realizes a stochastic Lie algebroid acting on the space of smooth 1-forms in the Nualart-Pardoux sense on the Wiener space and functional smooth in the Nualart-Pardoux sense on the Wiener space. takes its values in the space of generalized vector fields.