Wick Analysis for Bernoulli Noise Functionals

A Gel’fand triple S(Ω) ⊂ L2(Ω) ⊂ S∗(Ω) is constructed of functionals of Z, where Z = (Z n ) n∈N is an appropriate Bernoulli noise on a probability space (Ω,F,P). Characterizations are given to both S(Ω) and S(Ω). It is also shown that a Wick-type product can be defined on S∗(Ω) and moreover S∗(Ω) forms a commutative algebra with the product. Finally, a transform named S-transform is defined on S∗(Ω) and its continuity as well as other properties are examined.


Introduction
In the constructive quantum field theory, there have been encountered infinite quantities, which originated (from a mathematical point of view) from the product of generalized functions (see [1,2]).To obtain useful information out of these infinite quantities, Wick [3] first introduced the now so-called Wick renormalization, which was actually a formal manipulation.Hida and Ikeda [4] first rigorously defined the Wick renormalization on functionals of Brownian motion and called it Wick product instead.Meyer and Yan [5] extended the Wick product to cover generalized functionals of Gaussian white noises (also known as Hida distributions).Now the Wick product is widely used as a tool in stochastic differential equations, stochastic partial differential equations, stochastic quantization [2], and many other fields.
Bernoulli noise functionals (also known as functionals of Bernoulli noises) have attracted much attention in recent years.In 2001 Émery [6] discussed the chaotic representation property of a class of discrete-time stochastic processes including Bernoulli noises.In 2008 Privault [7] surveyed his discrete-time chaotic calculus.In 2010 Nourdin [8] considered Rademacher functionals (a special case of Bernoulli noise functionals) by using Stein's method.The same year, Wang et al. [9] introduced a notion of quantum Bernoulli noises and defined corresponding quantum stochastic integrals, which are actually about operator processes acting on Bernoulli noise functionals.Recently Wang et al. [10] have presented an alternative approach to Privault's discretetime chaotic calculus.There are other works devoted to development of a theory of quantum Bernoulli noises (see, for instance, [11,12]).
As can be seen, most of the existing work concerning the Wick product has been done within generalized functionals of Gaussian white noises.In this paper, we aim to construct generalized functionals of Bernoulli noises and develop a Wick-type calculus on them.
The paper is organized as follows.In Section 2, we fix some necessary notation and recall main notions and facts about Bernoulli noises for our later use.
Sections 3, 4, and 5 are our main work.In Section 3, we first introduce a weight function on the finite power set of N, where N denotes the set of all nonnegative integers.And then, by using the weight function and the full Wiener integral operator J, we construct a nuclear space (Ω) of functionals of , where  = (  ) ∈N is the Bernoulli noise fixed in Section 2. With (Ω) as the testing functional space, we get the generalized functional space  * (Ω) by taking the dual.We also give characterizations to both (Ω) and  * (Ω).In Section 4, we define the Wick product on the generalized functional space  * (Ω) and prove that  * (Ω) forms a commutative algebra with the Wick product.Finally, in Section 5, we introduce a transform, called -transform, on the generalized functional space  * (Ω) by using the coherent states.We examine its continuity as well as other properties.

Preliminaries
Throughout this paper, N designates the set of all nonnegative integers.We denote by Γ the finite power set of N, namely, where ♯ means the cardinality of  as a set.We use  2 (Γ) to mean the usual Hilbert space of square summable real-valued functions on Γ.
Let (Ω, F, P) be a probability space and  = (  ) ∈N an independent sequence of real-valued random variables on (Ω, F, P) satisfying that with   = √(1 −   )/  , 0 <   < 1, and F = (  ;  ∈ N), namely,  generates F. We note that such a sequence of random variables does exist.In fact,  is a discrete-time Bernoulli stochastic process and if we put then (  ) ∈N is a martingale.Hence we may call  a Bernoulli noise.
The next lemma [9] shows that  has the chaotic representation property.
Lemma 2. There exists a unique isometric isomorphism J :  2 (Γ)  →  2 (Ω) such that where the series on the right-hand side converges in the norm ‖ ⋅ ‖.
The isometric isomorphism J is known as the full Wiener integral operator [10].Lemma 3.For each  ≥ 0, there exists a bounded operator   on  2 (Ω) such that where  \  =  \ {} and 1  () the indicator of  as a subset of N.
The operator   and its adjoint operator  *  are referred to as the annihilation operator and creation operator [10], respectively.

The Framework of Bernoulli Noise Analysis
In the present section, we mainly construct a Gel'fand triple of functionals of the Bernoulli noise , which serves as the framework where we work.
To start with, we first introduce a weight function on Γ, which will play a key role in stating and proving our main results.
Definition 9.One sets and endows it with the topology generated by the norm sequence {‖ ⋅ ‖  } ≥0 .One calls (Ω) the testing functional space and its elements testing functionals.
Proposition 10.The testing functional space (Ω) is a nuclear space, namely, for any  ≥ 0, there exists  ≥  such that the inclusion mapping   :   →  from   (Ω) to   (Ω) is a Hilbert-Schmidt operator.
Then  ∈ (Ω) if and only if  ∈ S + (Γ).In that case Proof.The first part is easy to verify.We need only to show that for each  ≥ 0, the series in (24) converges to  in norm ‖ ⋅ ‖  .To do so, we take  ≥ 0. Since { −    } ∈Γ forms an orthonormal basis of   (Ω), we have in norm ‖ ⋅ ‖  .Note that the series on the right-hand side also converges to  in the norm of  2 (Ω) since ‖ ⋅ ‖  2 (Ω) = ‖ ⋅ ‖ ≤ ‖ ⋅ ‖  .On the other hand, because of  = J(), we have Thus () = ⟨⟨ −    , ⟩⟩   − ,  ∈ Γ, which together with (25) implies Definition 12.One defines  * (Ω) as the dual of (Ω) and endows it with the strong topology.One calls  * (Ω) the generalized functional space and its elements generalized functionals.Now, by identifying  2 (Ω) with its dual, we come to a real Gel'fand triple This justifies naming  * (Ω) the generalized functional space.By convention we may call the above triple the framework of Bernoulli noise analysis.

Wick Product
In this section, we define the Wick product on the generalized functional space  * (Ω).We prove that  * (Ω) forms a commutative algebra with this product.Some other properties are also shown of the Wick product.
Definition 16.For ,  ∈ S − (Γ), one defines a function  ⬦  on Γ as and calls it the Wick product of  and .
Note that the generalized functional space  * (Ω) can be characterized in terms of S − (Γ).Thus we naturally come to the following definition.
Proof.Let ,  ∈ S − (Γ) be the Guichardet representations of Φ and Ψ, respectively.Then, by Definition 18, we find that Φ ⬦ Ψ just has  ⬦  as its Guichardet representation.Thus by using Proposition 17 we know the claim is true.
Proof.Let   ∈ S − (Γ) be the Guichardet representation of Φ  , where  = 1, 2, 3.Then, for each  ∈ Γ, we can show that where The following two propositions can be proved similarly and their proofs are omitted.
The following proposition offers a link between the Wick product and the operator   .

𝑆-Transform
It is known that the -transform on generalized functionals of Gaussian white noises plays an important role in Hida's white noise analysis [13].In the last section, we define the -transform on our generalized functional space  * (Ω) and examine its fundamental properties.