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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 Guichardet- Fock spaces. In particular, we prove that the anticipative Girsanov identities under the condition *E*(*H*_{x}(*δ*(*x*),‖*x*‖^{2})),n≥1 on Guichardet-Fock spaces.

The quantum stochastic calculus developed by Hudson and Parthasarathy [

Recently Privault [

In this argument, we define expectation of

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

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.

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 [

Let

where

Fixing a complex separable Hilbert space

For a Hilbert space-valued map

denotes the Skorohod integral operator. For a vector space-valued map

respectively denote the stochastic gradient operator of f and the adapted gradient operator of f. Moreover, we write

Definition 2.1 The value of

Definition 2.2 For the map

expectation of

Lemma 2.1 Let x be a map

we denote

Lemma 2.2 Let

is integrable, then

Lemma 2.3 Let

where

Theorem 2.1 For any

where

Lemma 2.4 Let

In Theorem 3.1 below, we compute the expectation of the random Hermit polynomial

Theorem 3.1 For any

Especially, for

then we have

Proof We divide two steps to prove the stability result.

Step 1. We first prove that for any

For

replace 1 above with

Hence, taking

Step 2. For

Hence, replacing 1 above with

thus letting

Corollary 4.1 Assume that

have

Proof We have

hence

By Theorem 3.1 and Fubini theorem, we have

This shows that

we have

i.e.

The authors are extremely grateful to the referees for their valuable comments and suggestions on improvement of the first version of the present paper. The authors are supported by National Natural Science Foundation of China (No. 11261027 and No. 11461061), supported by scientific research projects in Colleges and Universities in gansu province (No. 2015A-122) and supported by doctoral research start-up fund project of Lanzhou City Universities (No. LZCU-BS2015-02) and SRPNWNU (No. NWNU-LKQW-14-2).

Jihong Zhang,Yongjun Li,Xiaochun Sun, (2016) Skorohod Integral at Vacuum State on Guichardet-Fock Spaces. Journal of Applied Mathematics and Physics,04,1321-1326. doi: 10.4236/jamp.2016.47141