A Lie Algebroid on the Wiener Space

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 𝐸 on a smooth finite dimensional manifold 𝑀. 𝑇𝑀 is the tangent bundle of 𝑀. Γ^∞(𝐸) and Γ^∞(𝑇𝑀) denote the space of smooth section of 𝐸 and TM. A Lie algebroid on 𝐸 is given by the following data.

(i)A Lie bracket structure [⋅,⋅]_𝐸 on Γ^∞(𝐸) has in particular to satisfy the Jacobi relation [[𝑋₁,𝑋₂]_𝐸,𝑋₃]_𝐸+[[𝑋₂,𝑋₃]_𝐸,𝑋₁]_𝐸+[[𝑋₃,𝑋₁]_𝐸,𝑋₂]_𝐸=0. (1.1)(ii)A smooth fiberwise linear map 𝜌_𝐸, called the anchor map, from 𝐸 into 𝑇𝑀 satisfies the relation [𝑋,𝑓𝑌]_𝐸=𝑓[𝑋,𝑌]_𝐸+⟨𝑑𝑓,𝜌_𝐸(𝑋)⟩𝑌, (1.2) for any smooth sections 𝑋, 𝑌 of 𝐸 and any element 𝑓 of 𝐶^∞(𝑀), the space of smooth functions on 𝑀.

Let us recall the definition of a Poisson structure on 𝑀. It is an antisymmetric ℝ-bilinear map {⋅,⋅} from 𝐶^∞(𝑀)×𝐶^∞(𝑀) into 𝐶^∞(𝑀), which is a derivation on each components, vanishes on the constant and satisfies the Jacobi relation {{𝑓₁,𝑓₂},𝑓₃}+{{𝑓₂,𝑓₃},𝑓₁}+{{𝑓₃,𝑓₁},𝑓₂}=0. (1.3) A Poisson structure is given by a bivector 𝜋, that is, an element of Γ^∞(Λ²𝑇𝑀) as {𝑓₁,𝑓₂}=⟨𝜋,𝑑𝑓₁,𝑑𝑓₂⟩=𝑖_𝑑𝑓2𝑖_𝑑𝑓1𝜋, (1.4) where 𝑑𝑓₁ and 𝑑𝑓₂ can be seen as 1-forms and the dual of 𝑇^∗_𝑥(𝑀) is 𝑇_𝑥(𝑀). 𝑖_𝛼𝜋 denotes the interior product of the bivector 𝜋 by the 1-form 𝛼. If 𝜋 is a bivector, then we can define a fiberwise linear smooth map from 𝑇^∗𝑀, the cotangent bundle of 𝑀, into 𝑇𝑀, called 𝜋.

If 𝛼 is a smooth section of 𝑇^∗𝑀, then 𝜋𝛼=𝑖_𝛼𝜋. (1.5) This allows us to define a Lie algebroid structure on 𝑇^∗𝑀 [11, 14–18] as follows.

(i)The Bracket is defined by [𝛼,𝛽]_𝑇∗𝑀=𝐿_𝜋𝛼𝛽−𝐿_𝜋𝛽𝛼−𝑑(𝜋(𝛼,𝛽)), (1.6) where 𝐿 is the usual Lie derivative of a 1-form: if 𝑋 is a vector field and 𝛼 a 𝑘-form, then 𝐿_𝑋𝛼 is given by the Cartan formula 𝐿_𝑋𝛼=𝑖_𝑋𝑑𝛼+𝑑𝑖_𝑋𝛼. (1.7)(ii)The anchor map 𝜌_𝑇∗𝑀 is the map 𝜋.

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]) {𝑥(𝑠),𝑥(𝑡)}=𝛿^′₀(𝑠−𝑡) (1.8) 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 𝑡→(𝑥_𝑖(𝑡)) into (ℝ^𝑚)^∗, the dual of a Lie algebra structure on ℝ^𝑚 with structural constants 𝑐^𝑘_𝑖,𝑗. In such a case, {𝑥_𝑖(𝑠),𝑥_𝑗(𝑡)}=_𝑘𝛿₀(𝑠−𝑡)𝑐^𝑘_𝑖,𝑗𝑥_𝑘(𝑡). (1.9) 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 Fock(ℍ₂) associated to the Hilbert space ℍ₂ of 𝐿² maps from [0,1] into ℝ. Fock(ℍ₂) is constituted of series 𝜎=∑ℎ^𝑛 where ℎ^𝑛 belongs to ℍ⊗_𝑛2, the symmetric 𝑛-tensor product of ℍ₂ such that ‖𝜎‖²=𝑛!‖ℎ^𝑛‖²_ℍ⊗_𝑛2<∞. (1.10) The operator algebra is the algebra of annihilation and creation operator on the Fock space submitted to the canonical commutation relations =0,[𝑎^∗(𝑠),𝑎^∗(𝑡)]=0,[𝑎(𝑠),𝑎^∗(𝑡)]=𝛿₀(𝑠−𝑡), (1.11) where 𝑎(𝑠) is an elementary annihilation operator. 𝑎^∗(𝑡) 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. 𝑎(𝑠)+𝑎^∗(𝑠) is called the white noise and can be interpreted in the measure theory.(ii)The main object of the Malliavin Calculus is the 𝐿^𝑝 space of the Wiener measure.

If we consider the Brownian motion 𝑡→𝐵(𝑡) on ℝ (see part 2 of this work), then we can introduce the Brownian functional associated to 𝜎 in the Bosonic Fock space: 𝜓(𝜎)=_[0,1]𝑛ℎ^𝑛(𝑠₁,…,𝑠_𝑛)𝛿𝐵(𝑠₁)⋯𝛿𝐵(𝑠_𝑛). (1.12)

An element ℎ^𝑛 of ℍ⊗_𝑛2 can be realized as a symmetric map from [0,1]^𝑛 into ℝ and ∫_[0,1]𝑛ℎ^𝑛(𝑠₁,…,𝑠_𝑛)𝛿𝐵(𝑠₁)⋯𝛿𝐵(𝑠_𝑛) denotes a Wiener chaos [28]. This map 𝜓 realizes an isometry between the Fock space and the 𝐿² 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 𝑡→∫^𝑡₀ℎ(𝑠); ℎ∈ℍ₂. An element 𝑡→∫^𝑡₀ℎ(𝑠)𝑑𝑠 is called an element of the Cameron-Martin space ℍ. This operation can be interpreted as a “nonelementary” annihilation operator on the Fock space. Through this isomorphism, 𝑎(𝑡)+𝑎^∗(𝑡) can be interpreted as 𝑑/𝑑𝑡𝐵(𝑡), 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.

(i)In the case of the Malliavin Calculus, there is a natural way to choose an algebra starting from the considerations of measure theory. With the intersection of all the 𝐿^𝑝, 𝑝<∞ is indeed an algebra through the Hoelder inequality. We consider the Wiener product on the Wiener space which is the classical product of functionals.(ii)In white noise analysis, there is on the Fock space another product called the standard Wick product. The traditional product of a Wiener chaos of length 𝑛 and of length 𝑚 is not a chaos of length 𝑛+𝑚 by the help of the Itô formula. It is an infinite dimensional generalization of the fact that the product of two Hermite polynomials in finite dimension is not a Hermite polynomial. The classical Wick product consists to keep in the product of these two chaoses the chaoses of length 𝑛+𝑚. Reference[29] has defined another Wick product (called the normalized Wick product), which fits well with Stratonovitch chaos. We consider now 𝜓^𝑠𝑡(𝜎)=_[0,1]𝑛ℎ(𝑠₁,…,𝑠_𝑛)𝑑𝐵(𝑠₁)⋯𝑑𝐵(𝑠_𝑛). (1.13) We consider this time multiple Stratonovitch integrals. They are the limit when 𝑘→∞ of the classical random multiple integrals ∫_[0,1]𝑛ℎ(𝑠₁,…,𝑠_𝑛)𝑑𝐵^𝑘(𝑠₁)⋯𝑑𝐵^𝑘(𝑠_𝑛) where 𝑡→𝐵^𝑘(𝑡) is the polygonal approximation of the Brownian motion. The Itô-Stratonovitch integral is the classical one. The normalized Wick product of Léandre and Rogers [29] ∶𝜎₁⋅𝜎₂∶ of 𝜎₁ and 𝜎₂ belonging to the Hida Fock space is done in order 𝜓^𝑠𝑡(∶𝜎₁⋅𝜎₂∶)=𝜓^𝑠𝑡(𝜎₁)𝜓^𝑠𝑡(𝜎₂), (1.14) using the Itô-Stratonovitch formula [23]. This reflects in an infinite dimensional sense the classical fact that the product of two monomials is still a monomial in a finite dimensional polynomial 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 𝑠→ℎ(𝑠) of ℍ₂ which belongs to all the Sobolev spaces of the Malliavin Calculus, then the anticipative Stratonovitch integral ¹₀ℎ(𝑠)𝑑𝐵(𝑠)=lim_𝑘→∞¹₀ℎ(𝑠)𝑑𝐵^𝑘(𝑠) (1.15) 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 ℎ∈ℍ₂ 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 ∫¹₀ℎ(𝑠)𝛿𝐵(𝑠) without assuming a lot of regularity on the nonanticipative element ℎ of ℍ₂. But in the case of a nonanticipative Stratonovitch integral, ∫¹₀ℎ(𝑠)𝑑𝐵(𝑠), we have to assume that ℎ is a semi-martingale; this means some regularity on ℎ! 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 𝐿² 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 𝜋 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 is𝐻(𝑡)=∫^𝑡₀ℎ(𝑠)𝑑𝐵(𝑠)+∫^𝑡₀ℎ¹(𝑠)𝑑𝑠 where ℎ(𝑠) is chosen well such that we can define the anticipative Stratonovitch integral ∫^𝑡₀ℎ(𝑠)𝑑𝐵(𝑠). 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 𝐶([0,1];ℝ^𝑚) from [0,1] into ℝ^𝑚 endowed with the uniform topology. A typical path is denoted by 𝑡→𝐵(𝑡)=(𝐵_𝑖(𝑡)), on which we consider the Brownian motion measure 𝑑𝑃 [38].

Let us recall how we construct 𝑑𝑃. We consider the Cameron-Martin Hilbert space ℍ [39] of maps from [0,1] into ℝ^𝑚 such that ¹₀|ℎ^′_𝑠|²𝑑𝑠=‖ℎ‖²<∞, (2.1) and 𝑑𝑃 is formally the Gaussian probability measure1𝑍exp[−‖ℎ‖²2]𝑑𝐷(ℎ), (2.2) and 𝑑𝐷 is the formal Lebesgue measure on ℍ 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 ℍ₁ be a finite dimensional real Hilbert space (ℎ₁∈ℍ₁) with Hilbert norm ‖⋅‖₁. Let us consider the centered normalized Gaussian measure on ℍ₁. It is classically represented by ∑𝑒_𝑖𝑁_𝑖 where 𝑁_𝑖 are centered normalized independent one-dimensional Gaussian variables and the system of 𝑒_𝑖 constitutes an orthonormal basis of the real Hilbert space ℍ₁.

It should be tempting to represent 𝑑𝑃 by using the same procedure. We consider an orthonormal basis 𝑒_𝑖 of ℍ. The law of the Brownian motion is represented by the series ∑𝑒_𝑖𝑁_𝑖 where the 𝑁_𝑖 is a collection of independent centered one-dimensional Gaussian variables. This series does not converge in ℍ but in 𝐶([0,1];ℝ^𝑚) [45]. We refer to the textbook of Kuo [46] for the theory of infinite dimensional Gaussian measures.

Let us consider a functional 𝐹 on 𝐶([0,1];ℝ^𝑚). Its 𝑟th stochastic derivative ∇^𝑟𝐹, according to the framework of the Malliavin Calculus [26, 47, 48], is defined if it exists by ⟨∇^𝑟𝐹,ℎ₁,…,ℎ_𝑟⟩=_[0,1]𝑟⟨∇^𝑟𝐹(𝑠₁,…,𝑠_𝑟),ℎ^′_1,𝑠1,…,ℎ^′_{𝑟,𝑠𝑟}⟩𝑑𝑠₁⋯𝑑𝑠_𝑟, (2.3) where ℎ_𝑖 belongs to the Hilbert space of paths ℍ from [0,1] into ℝ^𝑚 satisfying ¹₀|ℎ^′_𝑠|²𝑑𝑠=‖ℎ‖²<∞. (2.4)

The Sobolev norms of the Malliavin Calculus are defined by the following formula. If 𝐹 is a Brownian functional, then {𝐸[(_[0,1]𝑟|∇^𝑟𝐹(𝑠₁,…,𝑠_𝑟)|²𝑑𝑠₁⋯𝑑𝑠_𝑟)^𝑝/2]}^1/𝑝=‖𝐹‖_𝑟,𝑝. (2.5) The Malliavin test algebra consists of Brownian functionals 𝐹 all of whose Sobolev norms ‖𝐹‖_𝑟,𝑝 are finite 𝑟,𝑝<∞. Let us recall how we construct these Sobolev spaces. Let 𝑓 be a smooth function from (ℝ^𝑚)^𝑑 into ℝ with compact support and some times 0<𝑡₁<⋯<𝑡_𝑑≤1. We introduce the cylindrical functional 𝐹(𝐵(⋅))=𝑓(𝐵(𝑡₁),…,𝐵(𝑡_𝑑)). We consider the Gateaux derivative of 𝐹 along a deterministic direction ℎ of ℍ: 𝜕⟨∇𝐹,ℎ⟩=⟨𝜕𝑥_𝑖𝑓(𝐵(𝑡₁),…,𝐵(𝑡_𝑑)),ℎ_𝑡𝑖⟩. (2.6) There is absolutely no problem to define it. We use the integration by parts formula for a cylindrical functional 𝐸[⟨∇𝐹,ℎ⟩]=𝐸[𝐹¹₀⟨ℎ^′_𝑠,𝛿𝐵(𝑠)⟩], (2.7) where 𝛿𝐵(𝑠) is the Itô differential. The Itô integral is the limit in all the 𝐿^𝑝(𝑑𝑃), 𝑝<∞ of the sum ∑⟨ℎ^′_𝑠𝑖,𝐵(𝑠_𝑖+1)−𝐵(𝑠_𝑖)⟩ where 0<𝑠₁<⋯<𝑠_𝑖<𝑠_𝑖+1<⋯<𝑠₂^𝑛₋₁<1=𝑠₂^𝑛 is a dyadic subdivision of [0,1] of length 2^𝑛. The convergence does not pose any problem because ℎ 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 𝐹 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 𝐵^𝑛(𝑡) of 𝐵(𝑡). Let us consider a “nondeterministic!” map from [0,1] into ℝ^𝑚𝑡→𝛽_𝑡 which belongs to 𝐿²([0,1];ℝ^𝑚). We can consider the random ordinary integral ∫¹₀⟨𝛽_𝑡,𝑑𝐵^𝑛(𝑡)⟩. It can also be easily defined. In general, the limit may not exist when 𝑛 tends to infinity, because the Brownian motion is only continuous. If we can pass to the limit, we say that the limit ∫¹₀⟨𝛽_𝑡,𝑑𝐵(𝑡)⟩ 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 [0,1]^𝑟{𝐸[|∇^𝑟𝐹(𝑠₁,…,𝑠_𝑟)−∇^𝑟𝐹(𝑠^′₁,…,𝑠^′_𝑟)|^𝑝]}^1/𝑝≤𝐶_𝑟,𝑝|𝑠_𝑖−𝑠^′_𝑖|^1/2. (2.8) The smallest 𝐶_𝑟,𝑝 such that the previous inequality is satisfied is called the first Nualart-Pardoux Sobolev norm. The second Nualart-Pardoux Sobolev norm is the smallest 𝐶¹_𝑟,𝑝 such that, for all (𝑠_𝑖)∈[0,1]^𝑟, {𝐸[|∇^𝑟𝐹(𝑠₁,…,𝑠_𝑟)|^𝑝]}^1/𝑝≤𝐶¹_𝑟,𝑝. (2.9)

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

Let us recall that 𝑁.𝑃_∞− is an algebra [31].

We can consider a random element of 𝐿²([0,1];ℝ^𝑚), 𝑡→𝛽_𝑡. We can consider its 𝑟th stochastic derivative ⟨∇^𝑟𝛽_𝑡,ℎ₁,…,ℎ_𝑟⟩=_[0,1]𝑟⟨∇^𝑟𝛽_𝑡(𝑠₁,…,𝑠_𝑟),ℎ^′_1,𝑠1,…,ℎ^′_{𝑟,𝑠𝑟}⟩𝑑𝑠₁⋯𝑑𝑠_𝑟. (2.10) Its first Nualart-Pardoux Sobolev norm 𝐶_𝑟,𝑝 is the smallest number such that outside the diagonals of [0,1]×[0,1]^𝑟{𝐸[|∇^𝑟𝛽_𝑡(𝑠₁,…,𝑠_𝑟)−∇^𝑟𝛽_𝑡′(𝑠^′₁,…,𝑠^′_𝑟)|^𝑝]}^1/𝑝≤𝐶_𝑟,𝑝(|𝑡−𝑡^′|^1/2+|𝑠_𝑖−𝑠^′_𝑖|^1/2). (2.11) The second type of Nualart-Pardoux Sobolev norm 𝐶¹_𝑟,𝑝 of 𝛽_(⋅) is the smallest number such that for all (𝑡,𝑠₁,…,𝑠_𝑟)∈[0,1]×[0,1]^𝑟{𝐸[|∇^𝑟𝛽_𝑡(𝑠₁,…,𝑠_𝑟)|^𝑝]}^1/𝑝≤𝐶¹_𝑟,𝑝. (2.12) Let us recall the theorem of Léandre [31].

Theorem 2.2. Let 𝛽 be a random element of 𝐿²([0,1];ℝ^𝑚) such that all its Nualart-Pardoux Sobolev norms are finite. Then the anticipative Stratonovitch integral ¹₀⟨𝛽_𝑡,𝑑𝐵(𝑡)⟩ (2.13) is smooth in the Nualart-Pardoux sense and its Nualart-Pardoux Sobolev norms can be estimated in terms of the Nualart-Pardoux norms of 𝛽.

In such a case, ∫¹₀⟨𝛽_𝑡,𝑑𝐵(𝑡)⟩ is the limit in all the 𝐿^𝑝(𝑑𝑃), 𝑝<∞ of ∫¹₀⟨𝛽_𝑡,𝑑𝐵^𝑛(𝑡)⟩. Moreover, ⟨∇(¹₀⟨𝛽_𝑡,𝑑𝐵(𝑡)⟩),ℎ⟩=¹₀⟨⟨∇𝛽_𝑡,ℎ⟩,𝑑𝐵(𝑡)⟩+¹₀⟨𝛽_𝑡,𝑑ℎ𝑑𝑡^𝑡⟩𝑑𝑡. (2.14) This means that the kernel of the stochastic derivative of ∫¹₀⟨𝛽_𝑡,𝑑𝐵(𝑡)⟩ is ∫¹₀⟨∇𝛽_𝑡(𝑠),𝑑𝐵(𝑡)⟩+𝛽_𝑠. Let us explain this formula; in order to take the stochastic derivative of ∫¹₀⟨𝛽_𝑡,𝑑𝐵(𝑡)⟩, we do the same formal computations as if the anticipative Stratonovitch integral had been a classical integral; we take first of all derivatives of 𝛽_𝑡 which lead to the term ⟨∇𝛽_𝑡,ℎ⟩ and derivatives of 𝑑𝐵_𝑡 which lead to 𝑑/𝑑𝑡ℎ_𝑡𝑑𝑡.

Let us recall the notion of a Poisson bracket {⋅,⋅}. We consider a commutative Frechet unital real algebra endowed with a family of Banach norms ‖⋅‖_𝑝. This means that for all 𝑝, there exists 𝑝¹ such that for all 𝐹¹,𝐹² in 𝐴‖𝐹¹𝐹²‖_𝑝≤𝐶_𝑝‖𝐹¹‖_𝑝1‖𝐹²‖_𝑝1. (2.15) A Poisson Bracket is a bilinear map from 𝐴×𝐴 into 𝐴, which is a derivation in each argument, vanishes on the unit. The derivation property means that for all 𝐹¹,𝐹²,𝐹³ in 𝐴{𝐹¹𝐹²,𝐹³}=𝐹¹{𝐹²,𝐹³}+{𝐹¹,𝐹³}𝐹². (2.16) Moreover, it satisfies the following properties: if 𝐹¹,𝐹²,𝐹³ belong to 𝐴, then {𝐹¹,𝐹²}=−{𝐹²,𝐹¹},{{𝐹¹,𝐹²},𝐹³}+{{𝐹²,𝐹³},𝐹¹}+{{𝐹³,𝐹¹},𝐹²}=0. (2.17) Moreover, for all 𝑝, there exist 𝑝^′ and 𝐶_𝑝 such that ‖{𝐹¹,𝐹²}‖_𝑝≤𝐶_𝑝‖𝐹¹‖_𝑝′‖𝐹²‖_𝑝′. (2.18) In the sequel, we will choose 𝐴=𝑁.𝑃_∞−. We consider the structural constants 𝑐^𝑘_𝑖,𝑗 of a Lie algebra structure on (ℝ^𝑚)^∗. The stochastic gradient ∇𝐹 of a functional 𝐹 can be written ∇𝐹=(∇𝐹_𝑖). Formula (1.9) reads in this framework as {𝐹¹,𝐹²}=_{𝑖,𝑗,𝑘}¹₀∇𝐹¹_𝑖(𝑠)∇𝐹²_𝑗(𝑠)𝑐^𝑘_𝑖,𝑗𝑑𝐵_𝑘(𝑠), (2.19) where we consider a Stratonovitch anticipative integral. This defines a Poisson structure on 𝑁.𝑃_∞− in our framework [4, Theorem  1].

Remark 2.3. Let us motivate (2.19). Let us consider the Hilbert space ℍ₂ of 𝐿² maps from [0,1] into ℝ^𝑚. The 𝑐^𝑘_𝑖,𝑗 define a structure of Lie algebra on ℝ^𝑚∗, and therefore, on ℍ₂. Let us consider two functionals 𝐹¹ and 𝐹² Frechet smooth on ℍ^∗₂. Their derivatives are given by kernels ⟨∇𝐹^𝑖,ℎ⟩=¹₀⟨∇𝐹^𝑖(𝑠),ℎ(𝑠)⟩𝑑𝑠. (2.20) The Lie bracket [∇𝐹¹,∇𝐹²] is given by [∇𝐹¹(𝑠),∇𝐹²(𝑠)] and the classical Lie-Poisson structure is given by {𝐹¹,𝐹²}=¹₀⟨ℎ(𝑠),[∇𝐹¹(𝑠),∇𝐹²(𝑠)]⟩𝑑𝑠. (2.21) These considerations are heuristic because the product of two elements of 𝐿² is not an element of 𝐿². If we replace 𝑑𝐵(𝑠) by ℎ(𝑠)𝑑𝑠 and if we consider the white-noise measure on ℍ₂1/𝑍exp[−‖ℎ‖²_ℍ2]𝑑𝐷(ℎ) 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 𝛽_𝑡 from [0,1] into ℝ^𝑚 such that {𝐸[|∇^𝑟𝛽_𝑡(𝑠₁,…,𝑠_𝑟)−∇^𝑟𝛽_𝑡′(𝑠^′₁,…,𝑠^′_𝑟)|^𝑝]}^1/𝑝≤𝐶_𝑟,𝑝(|𝑡−𝑡^′|^1/2+|𝑠_𝑖−𝑠^′_𝑖|^1/2) (3.1) on the connected complements of [0,1]×[0,1]^𝑟 where we have removed the diagonals and such that {𝐸[|∇^𝑟𝛽_𝑡(𝑠₁,…,𝑠_𝑟)|^𝑝]}^1/𝑝≤𝐶¹_𝑟,𝑝. (3.2) The infimums of 𝐶_𝑟,𝑝 and of 𝐶¹_𝑟,𝑝 in the previous formula are called the Nualart-Pardoux Sobolev norms of the vector field 𝛽_(⋅).

Smooth 1-forms in the Nualart-Pardoux sense on the Wiener space are functions 𝛼_(⋅) from [0,1] into ℝ^𝑚∗ such that (𝐸[|∇^𝑟𝛼_𝑡(𝑠₁,…,𝑠_𝑟)−∇^𝑟𝛼_𝑡′(𝑠^′₁,…,𝑠^′_𝑟)|^𝑝])^1/𝑝≤𝐶_𝑟,𝑝(|𝑡−𝑡^′|^1/2+|𝑠_𝑖−𝑠^′_𝑖|^1/2) (3.3) on the connected complements of [0,1]×[0,1]^𝑟 where we have removed the diagonals and such that {𝐸[|∇^𝑟𝛼_𝑡(𝑠₁,…,𝑠_𝑟)|^𝑝]}^1/𝑝≤𝐶¹_𝑟,𝑝. (3.4) The infimums of 𝐶_𝑟,𝑝 and of 𝐶¹_𝑟,𝑝 in the previous formula are called the Sobolev norms of the 1-form 𝛼_(⋅). The pairing between a 1-form 𝛼_(⋅) and a vector field 𝛽_(⋅) is realized via the formula ⟨𝛼_(⋅),𝛽_(⋅)⟩=¹₀⟨𝛼_𝑡,𝛽_𝑡⟩𝑑𝑡. (3.5) If 𝛼¹,𝛼² are two smooth 1-forms in the Nualart-Pardoux sense on the Wiener space, then the bivector 𝜋 associated to the stochastic Poisson structure is given by 𝜋(𝛼¹,𝛼²)=_{𝑖,𝑗,𝑘}¹₀𝛼¹_𝑖,𝑠𝛼²_𝑗,𝑠𝑐^𝑘_𝑖,𝑗𝑑𝐵_𝑘(𝑠). (3.6) The stochastic bivector 𝜋 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 [0,1] into ℝ^𝑚  𝛽^𝑔_(⋅) of the form 𝛽^𝑔_𝑡=_𝑖,𝑗^𝑡₀𝛽_{(𝑖,𝑗,𝑠)}𝑑𝐵_𝑗(𝑠)𝑒_𝑖+^𝑡₀𝛽_𝑠𝑑𝑠, (3.7) where 𝛽_{(𝑖,𝑗,.)} and 𝛽_(⋅) are smooth in the Nualart-Pardoux sense. The Nualart-Pardoux Sobolev norms of a generalized vector field 𝛽^𝑔_(⋅) are the collection of Nualart-Pardoux norms of 𝛽_{(𝑖,𝑗,.)} and 𝛽_(⋅).

We can define a pairing between smooth 1-form and generalized vector fields by using the formula ⟨𝛼,𝛽^𝑔⟩=_𝑖,𝑗¹₀𝛼_(𝑖,𝑠)𝛽_{(𝑖,𝑗,𝑠)}𝑑𝐵_𝑗(𝑠)+¹₀⟨𝛽_𝑠,𝛼_𝑠⟩𝑑𝑠. (3.8) This allows us to define 𝜋𝛼=𝑖_𝛼𝜋 for a smooth 1-form in the Nualart-Pardoux sense as the generalized vector field 𝜋𝛼_𝑡=_{𝑖,𝑗,𝑘}^𝑡₀𝛼_𝑖(𝑠)𝑐^𝑘_𝑖,𝑗𝑑𝐵_𝑘(𝑠)𝑒_𝑗. (3.9)

This allows us to put the following definition.

Definition 3.1. If 𝛼 and 𝛽 are smooth 1-forms in the Nualart-Pardoux sense, then we define [𝛼,𝛽]=𝐿_𝜋𝛼𝛽−𝐿_𝜋𝛽𝛼−𝑑𝜋(𝛼,𝛽), (3.10) where the Lie derivative is defined as usual by the formula 𝐿_𝛼𝛽=𝑖_𝜋𝛼𝑑𝛽+𝑑𝑖_𝜋𝛼𝛽, (3.11) and 𝑑 is the exterior derivative.

Since 𝜋𝛼 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 𝛼_{(𝑡1,𝑡2)} be a map from [0,1]×[0,1] into ℝ^𝑚∗ or later into (ℝ^𝑚∗)^⊗2 such that {𝐸[|∇^𝑟𝛼_{(𝑡1,𝑡2)}(𝑠₁,…,𝑠_𝑟)−∇^𝑟𝛼_{(𝑡′1,𝑡′2)}(𝑠^′₁,…,𝑠^′_𝑟)|^𝑝]}^1/𝑝≤𝐶_𝑟,𝑝(|𝑡₁−𝑡^′₁|^1/2+|𝑡₂−𝑡^′₂|+|𝑠_𝑖−𝑠^′_𝑖|^1/2) (3.12) on the connected complements of [0,1]×[0,1]×[0,1]^𝑟 where we have removed the diagonals and such that {𝐸[|∇^𝑟𝛼_𝑡1,𝑡2(𝑠₁,…,𝑠_𝑟)|^𝑝]}^1/𝑝≤𝐶¹_𝑟,𝑝. (3.13) The infimums of 𝐶_𝑟,𝑝 and of 𝐶¹_𝑟,𝑝 in the previous formula are called the Nualart-Pardoux Sobolev norms of the 𝛼_(⋅). In such case ∫¹₀𝛼_{(𝑡1,𝑡2)}𝑑𝐵(𝑡₁) is still smooth in the Nualart-Pardoux sense, with 𝑡₁ being included as well as ∬¹₀𝛼_𝑡1,𝑡2𝑑𝐵(𝑡₁)𝑑𝐵(𝑡₂).

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 𝜋(𝛼¹,𝛼²)=_{𝑖,𝑗,𝑘}¹₀𝛼¹_𝑖(𝑠)𝛼²_𝑗(𝑠)𝑐^𝑘_𝑖,𝑗𝑑𝐵_𝑘(𝑠), (3.14) which is smooth and its Sobolev norms can be estimated in terms of the Sobolev norms of 𝛼^𝑖 by Theorem 2.2.
Moreover, 𝑖_𝜋𝛼1𝛼²=_{𝑖,𝑗,𝑘}¹₀𝛼¹_𝑖(𝑠)𝛼²_𝑗(𝑠)𝑐^𝑘_𝑗,𝑖𝑑𝐵_𝑘(𝑠), (3.15) which is still smooth. Moreover, 𝑑𝛼²(𝑠,𝑡)=_𝑖(∇(𝛼²_𝑖)_𝑡(𝑠)−∇(𝛼²_𝑖)_𝑠(𝑡))𝑒_𝑖. (3.16) Moreover, 𝑖_𝜋𝛼1𝑑𝛼²(𝑡)=_{𝑖,𝑗,𝑘}(¹₀∇(𝛼²_𝑖)_𝑡(𝑠)𝛼¹_𝑗(𝑠)𝑐^𝑘_𝑗,𝑖𝑑𝐵_𝑘(𝑠)−¹₀∇(𝛼²_𝑖)_𝑠(𝑡)𝛼¹_𝑗(𝑠)𝑐^𝑘_𝑗,𝑖𝑑𝐵_𝑘(𝑠)), (3.17) which is smooth by the remark preceding the theorem.

Theorem 3.3. [⋅,⋅] defines a Lie bracket.

Let 0<𝑡₁<⋯<𝑡_𝑟=1 be a dyadic subdivision of [0,1]. We deduce a partition of [0,1]^𝑟 in cubes 𝐼_𝑛,𝑟 of volume 𝑉_𝑛. If 𝛽 is a function from [0,1]^𝑟 into some linear space, then we put 𝜒_𝑛𝛽(𝑡₁,…,𝑡_𝑟)=𝑉_𝑛⁻¹1_{𝐼𝑛,𝑟}(𝑡₁,…,𝑡_𝑟)_{𝐼𝑛,𝑟}𝛽(𝑠₁,…,𝑠_𝑟)𝑑𝑠₁⋯𝑑𝑠_𝑟. (3.18) If we consider the polygonal approximation 𝐵^𝑛 of 𝐵 and 𝐹_𝑛 being the 𝜎-algebra associated to 𝐵^𝑛, We denote by Π_𝑛 the operation of taking the conditional expectation of a functional 𝑓 by 𝐹_𝑛. The results of Léandre [31, Appendix], allow to state the proposition.

Proposition 3.4. Let one have ∇^𝑟Π_𝑛𝐹=Π_𝑛𝜒_𝑛∇^𝑟. (3.19) If 𝛼(𝑡₁) with values in ℝ^𝑚∗ is smooth in the Nualart-Pardoux sense, 𝑡₁ being included, then the random ordinary integral ∫¹₀⟨𝜒_𝑛Π_𝑛𝛼(𝑡),𝑑𝐵^𝑛_𝑡⟩ tends in all the Sobolev spaces of the Malliavin Calculus to the anticipative Stratonovitch integral ∫¹₀⟨𝛼(𝑡),𝑑𝐵_𝑡⟩. If 𝛼(𝑡₁,𝑡₂) which takes its values in (ℝ^𝑚∗)^⊗2 is smooth in the Nualart-Pardoux sense, then the double random ordinary integral ∬¹₀⟨𝜒_𝑛Π_𝑛𝛼(𝑡₁,𝑡₂),𝑑𝐵^𝑛_𝑡1,𝑑𝐵^𝑛_𝑡2⟩ tends in all the Sobolev spaces of the Malliavin Calculus to the double anticipative Stratonovitch integral ∬¹₀⟨𝛼(𝑡₁,𝑡₂),𝑑𝐵_𝑡1,𝑑𝐵_𝑡2⟩.

Proof of Theorem 3.3. Let us consider the finite dimensional Gaussian space 𝐵^𝑛_𝑡. A 1-form 𝛼^𝑛_𝑡 is piecewise constant as well as a vector field ℎ_𝑡. If 𝐹^𝑛 is a functional which depends on 𝐵^𝑛 only, then ∇^𝑟𝐹^𝑛 is constant on each 𝐼_𝑛,𝑟. We put {𝐹^1,𝑛,𝐹^2,𝑛}_𝑛=_{𝑖,𝑗,𝑘}¹₀∇_𝑖𝐹_𝑖^1,𝑛(𝑠)∇_𝑗𝐹^2,𝑛(𝑠)𝑐^𝑘_𝑖,𝑗𝑑𝐵^𝑛_𝑘(𝑠). (3.20) This defines a Poisson structure on the finite dimensional Gaussian space.
We can define 𝜋^𝑛, [⋅,⋅]_𝑛, and 𝜋^𝑛 according to the line of the introduction. To a 1-form smooth in the Nualart-Pardoux sense 𝛼 on the total Wiener space, we consider the 1-form Π^𝑛𝜒_𝑛𝛼=𝛼^𝑛 on the finite dimensional Gaussian space. We get [[𝛼^1,𝑛,𝛼^2,𝑛]_𝑛,𝛼^3,𝑛]_𝑛+[[𝛼^2,𝑛,𝛼^3,𝑛]_𝑛,𝛼^1,𝑛]_𝑛+[[𝛼^3,𝑛,𝛼^1,𝑛]_𝑛,𝛼^2,𝑛]_𝑛=0, (3.21) by doing as in finite dimension. By the previous proposition [[𝛼^1,𝑛,𝛼^2,𝑛]_𝑛,𝛼^3,𝑛]_𝑛 tends in all this attained 𝐿^𝑝 to [[𝛼¹,𝛼²],𝛼³]. Therefore the result.
Let us give the scheme of the proof of this last result. When we write [[𝛼^1,𝑛,𝛼^2,𝑛]_𝑛,𝛼^3,𝑛]_𝑛, there are a lot of terms which will appear. All these terms will tends separately to the corresponding term in [[𝛼¹,𝛼²],𝛼³]. Let us treat one of them, which will lead to double anticipative Stratonovitch integral. The other terms will be treated identically. For instance 𝑖_{𝜋[𝛼1,𝑛,𝛼2,𝑛]𝑛}𝑑𝛼^3,𝑛 will lead to double Stratonovitch integral which will tend in all the Sobolev spaces of the Malliavin Calculus to 𝑖_{𝜋[𝛼1,𝛼2]}𝑑𝛼³. We can consider in these expressions the term 𝑖𝜋^𝑛_[𝑖𝜋^𝑛_)(𝛼^{1,𝑛𝑑𝛼2,𝑛]}𝑑𝛼^3,𝑛 which will lead to a double stochastic integral and which will tend to 𝑖_{𝜋[𝑖1)𝜋(𝛼𝑑𝛼2]}𝑑𝛼³. But there are two parts in 𝑖_𝜋(𝛼1)𝑑𝛼² and 𝑖𝜋^𝑛_(𝛼^1,𝑛₎𝑑𝛼^2,𝑛. We will consider the parts ⟨∇𝛼²,𝜋(𝛼¹)⟩ and ⟨∇𝛼^2,𝑛,𝜋^𝑛(𝛼^1,𝑛)⟩ where we take the covariant derivative of the 1-form 𝛼² in the direction of the generalized vector field 𝜋(𝛼¹). We will show that 𝑖𝜋^𝑛_[⟨∇𝛼^2,𝑛_,𝜋^𝑛_(𝛼^1,𝑛_)⟩]𝑑𝛼^3,𝑛 tends to 𝑖_{𝜋[⟨∇𝛼2,}𝜋_(𝛼¹_)⟩]𝑑𝛼³. But in these expressions there are still two parts which can be treated similarly. We will show that the expression ⟨∇𝛼^3,𝑛,𝜋^𝑛[⟨∇𝛼^2,𝑛,𝜋^𝑛(𝛼^1,𝑛)⟩]⟩ tends to ⟨∇𝛼³,𝜋[⟨∇𝛼²,𝜋(𝛼¹)⟩]⟩ (we consider the covariant derivative of 𝛼^3,𝑛 in the direction of 𝜋^𝑛[⟨∇𝛼^2,𝑛,𝜋^𝑛(𝛼^1,𝑛)⟩]).
But 𝜋^𝑛(𝛼^1,𝑛)_𝑡=_{𝑖,𝑗,𝑘}^𝑡₀𝛼^1,𝑛_𝑠,𝑖𝑐^𝑘_𝑖,𝑗𝑑𝐵^𝑛_𝑘(𝑠)𝑒_𝑗. (3.22) Therefore ⟨∇𝛼_𝑡^2,𝑛,𝜋^𝑛(𝛼^1,𝑛)⟩=_{𝑖,𝑗,𝑘}¹₀⟨∇𝛼_𝑡^2,𝑛(𝑠),𝛼^1,𝑛_𝑠,𝑖𝑐^𝑘_𝑖,𝑗𝑑𝐵^𝑛_𝑘(𝑠)𝑒_𝑗⟩. (3.23) This implies that 𝜋^𝑛[⟨∇𝛼^2,𝑛(⋅),𝜋^𝑛(𝛼^1,𝑛)⟩]_𝑡=_{𝑖,𝑗,𝑘,𝑖′,𝑗′,𝑘′}^𝑡₀⟨¹₀⟨∇𝛼^2,𝑛_𝑢,𝑖′(𝑠),𝛼^1,𝑛_𝑠,𝑖𝑐^𝑘_𝑖,𝑗𝑑𝐵^𝑛_𝑘(𝑠)𝑒_𝑗⟩,𝑐^𝑘′_{𝑖′,𝑗′}𝑑𝐵^𝑛_𝑘′(𝑢)𝑒_𝑗′⟩, (3.24) where 𝛼_𝑡^2,𝑛(𝑠)=∑_𝑖′𝛼^2,𝑛_𝑡,𝑖′𝑒^∗_𝑖′ ( we consider a 1-form in the 𝑡 variable).
Therefore⟨∇𝛼^3,𝑛,𝜋^𝑛[⟨∇𝛼^2,𝑛,𝜋^𝑛(𝛼^1,𝑛=)⟩]⟩_{𝑖,𝑗,𝑘,𝑖′,𝑗′,𝑘′10}⟨∇𝛼^3,𝑛_{(⋅),𝑗′}(𝑢),⟨¹₀⟨∇𝛼^2,𝑛_𝑢,𝑖′(𝑠),𝛼^1,𝑛_𝑠,𝑖𝑐^𝑘_𝑖,𝑗𝑑𝐵^𝑛_𝑘(𝑠)𝑒_𝑗⟩,𝑐^𝑘′_{𝑖′,𝑗′}𝑑𝐵^𝑛_𝑘′(𝑢)𝑒_𝑗′⟩⟩. (3.25) By Proposition 3.4, this tends in all the Sobolev spaces of the Malliavin Calculus to _{𝑖,𝑗,𝑘,𝑖′,𝑗′,𝑘′}¹₀⟨∇𝛼³_{(⋅),𝑗′}(𝑢),⟨¹₀⟨∇𝛼²_𝑢,𝑖′(𝑠),𝛼¹_𝑠,𝑖𝑐^𝑘_𝑖,𝑗𝑑𝐵_𝑘(𝑠)𝑒_𝑗⟩,𝑐^𝑘′_{𝑖′,𝑗′}𝑑𝐵_𝑘′(𝑢)𝑒_𝑗′⟩⟩. (3.26) We recognize in this quantity ⟨∇𝛼³,𝜋[⟨∇𝛼²,𝜋(𝛼¹)⟩]⟩ where we take the covariant derivative of 𝛼³ in the direction of the generalized vector field 𝜋[⟨∇𝛼²,𝜋(𝛼¹)⟩] (we had taken the covariant derivative of 𝛼² in the direction of the generalized vector field 𝜋(𝛼²)).

Theorem 3.5. 𝜋 is an anchor map. This means that for all 1-form 𝛼, 𝛽 which are smooth in the Nualart-Pardoux sense all functional 𝐹 are smooth in the Nualart-Pardoux sense; one has the relation: [𝛼,𝐹𝛽]=𝐹[𝛼,𝛽]+⟨∇𝐹,𝜋(𝛼)⟩𝛽. (3.27)

Proof of Theorem 3.5. We get by classical results in finite dimension [𝛼^𝑛,𝐹^𝑛𝛽^𝑛]_𝑛=𝐹^𝑛[𝛼_𝑛,𝛽_𝑛]_𝑛+⟨∇𝐹^𝑛,𝜋^𝑛(𝛼^𝑛)⟩𝛽^𝑛. (3.28) By the results of Proposition 3.4, this tends when 𝑛→∞ to the formula [𝛼,𝐹𝛽]=𝐹[𝛼,𝛽]+⟨∇𝐹,𝜋(𝛼)⟩𝛽. (3.29) Therefore the result is attained.

4. Conclusion

We can summarize that ([⋅,⋅],𝜋) 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. 𝜋 takes its values in the space of generalized vector fields.