Skorohod Integral at Vacuum State on Guichardet-Fock Spaces

In this paper, we define expectation of f F ∈ , i.e. E f f ( ) ( ) = ∅ , according to Wiener-Ito-Segal isomorphic relation between Guichardet-Fock space F and Wienerspace W. Meanwhile, we derive a formula for the expectation of random Hermite polynomial in Skorohod integral on GuichardetFock spaces. In particular, we prove that the anticipative Girsanov identities under the condition n E H x x n 2 ( ( ( ) ) , 0 1 ) , ≥ = δ ‖ ‖ on Guichardet-Fock spaces.


Introduction
The quantum stochastic calculus developed by Hudson and Parthasarathy [1] is essentially a noncommutative extension of classical Ito stochastic calculus.In thistheory, annihilation, creation, and number operator processes in boson Fock space play the role of "quantum noises" [2] [3], which are in continuous time.In 2002, Attal [4] discussed and extended quantum stochastic calculus by means of the Skorohod integral of anticipation processes and the related gradient operator on Guichardet-Fock spaces.Usually, Fock spaces as the models of the Particle Systems are widely used in quantumphysics.Meanwhile, vacuum states described by empty set on Guichardet-Fockspaces play very important role at quantum physics.
Recently Privault [5] [6] developed a Malliavin-type theory of stochastic calculus on Wiener spaces and showed its several interesting applications.In his article, Privault surveyed the moment identities for Skorohod integral and derived a formula for the expectation of random Hermit polynomials in Skorohod integral on Wiener spaces.It is well known that Guichardet-Fock space F and Wiener space W are Wiener-Ito-Segal isomorphic.Motivated by the above, we would like to study the expectation of random Hermit polynomials in Skorohod integral on Guichardet-Fock spaces.However, how to define the expectation on Guichardet-Fock spaces is the primary problem.
In this argument, we define expectation of f F ∈ according to isomorphic relation, i.e. ( ) ( ) E f f = ∅ .Meanwhile, we prove a moment identity for the Skorohod integrals and derive a formula for the expectation of random Hermite polynomial in Skorohod integral on Guichardet-Fock spaces.Particularly, under the condition 2 ( ( ( ), )) 0, 1 , we prove the anticipative Girsanov identities on Guichardet-Fock spaces.This paper is organized as follows.Section 2, we fix some necessarynotations and recall main notions and facts about Skorohod integral in Guichardet-Fock spaces.Section 3 and Section 4 state our main results.

Notations
In this section, we fix some necessary notations and recall mainnotions in Guichardet-Fock spaces.For detail formulation of Skorohod integrals, we refer reader to [4].
Let R + be the set of all nonnegative real numbers and Γ the finite power set of R + , namely where σ ♯ denotes the cardinality of σ as a set.Particularly, let ∅ ∈ Γ be an atom of measure 1.We denote by 2 ( ) L Γ the usual space of square integral real-valued functions on Γ .Fixing a complex separable Hilbert space η , Guichardet-Fock space tensor product 2 ( ) L η ⊗ Γ , which we identify with the space of square-integrable functions 2 ( ; ) L η Γ , is denoted by F. For a Hilbert space-valued map : x R denotes the Skorohod integral operator.For a vector space-valued map :   1 respectively denote the stochastic gradient operator of f and the adapted gradient operator of f.Moreover, we write Dom∇ for the domain of the stochastic gradient as anunbounded Hilbert apace operator:  E x x x is square integrable and the function ( , , ) ( ), ( ) ∈ , we have where

Random Hermit Polynomials
In Theorem 3.1 below, we compute the expectation of the random Hermit polynomial x R Especially, for x and , (( ) ) 0, 0 2, Proof We divide two steps to prove the stability result.
Step 1.We first prove that for any Step 2. For f F ∈ , and 0 i l ≤ ≤ , we have .
Hence, replacing 1 above with l i − , we get

Definition 2 . 1
The value of f F ∈ at empty set is called the expectation of f on Guichardet-Fock space and is denoted by ( ). . .

2 (
result will be applied in Section 4 to anticipate Girsanov identities on Guichardet-Fock spaces.Theorem 3.1 For any 0 n ≥ and : 1 k f x x and l n k = 〈 〉 = − − above, and use (2.3) in step 1, we get 1