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).

Infinite dimensional Poisson structures play a big role in the theory of infinite dimensional Lie algebras [

Let us recall what a Lie algebroid is [

A Lie bracket structure

A smooth fiberwise linear map

Let us recall the definition of a Poisson structure on

If

The Bracket is defined by

The anchor map

Infinite dimensional symplectic structures and their related Poisson structure were introduced by Dito and Léandre [

The infinite dimensional Poisson structure is a tool in the theory of integrable system [

Another simple Poisson structure of Dubrovin and Novikov [

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 [

The functional integral side is given by the Malliavin Calculus [

The operator algebra side is given by white noise analysis and quantum probability [

Let us recall basically the objects of these Calculi.

The main object of white noise analysis and quantum probability is given by the Bosonic Fock space

The main object of the Malliavin Calculus is the

If we consider the Brownian motion

An element

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

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

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

Léandre [

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

The goal of this paper is to define a Lie algebroid associated to the Poisson structure (

We consider the set of continuous paths

Let us recall how we construct

It should be tempting to represent

Let us consider a functional

The Sobolev norms of the Malliavin Calculus are defined by the following formula. If

If we consider the same dyadic subdivision as before, we can introduce the polygonal approximation

We consider another set of Sobolev norms [

The Nualart-Pardoux test algebra

Let us recall that

We can consider a random element of

Let

In such a case,

Let us recall the notion of a Poisson bracket

Let us motivate (

Smooth vector fields in the Nualart-Pardoux sense on the Wiener space are functions

Smooth 1-forms in the Nualart-Pardoux sense on the Wiener space are functions

A generalized vector field according to our theory [

We can define a pairing between smooth 1-form and generalized vector fields by using the formula

This allows us to put the following definition.

If

Since

This allows us to show the following theorem.

We remark that

Moreover,

Let

Let one have

Let us consider the finite dimensional Gaussian space

We can define

Let us give the scheme of the proof of this last result. When we write

But

Therefore

We get by classical results in finite dimension

We can summarize that