We define the Wiener product on a bosonic Connes space associated to a Bilaplacian and we introduce formal Wiener chaos on the path space. We consider the vacuum distribution on the bosonic Connes space and show that it is related to the heat semigroup associated to the Bilaplacian. We deduce a Cameron-Martin quasi-invariance formula for the heat semigroup associated to the Bilaplacian by using some convenient coherent vector. This paper enters under the Hida-Streit approach of path integral.

1. Introduction

Let us recall some basic tools of Wiener analysis. Let Bt be a one-dimensional Brownian motion starting from 0. It is classically related to the heat equation on ℝ:
(1.1)∂∂tE[f(Bt)]=12E[Δf(Bt)],
where Δ=∂2/∂x2 is the standard Laplacian and f is a smooth function with bounded derivatives at each order. Associated to the heat equation there is a convenient probability measure on a convenient path space. Almost surely, the trajectory of B is continuous. We construct by this way the Wiener measure dP on the continuous path space endowed with its Borelian σ-algebra. Let ℍ be the Hilbert space L2([0,1];ℝ). We consider the symmetric tensor product ℍ⊗^n of this Hilbert space. It is constituted of maps hn(s1,…,sn) symmetric in si such that
(1.2)∥hn∥2=∫[0,1]nhn2(s1,…,sn)ds1⋯dsn<∞.
We consider the symmetric Fock space F(ℍ) of set σ=∑n=0∞hn such that
(1.3)∥σ∥2=∑n!∥hn∥2<∞.
We consider the vacuum expectation.
(1.4)μ[σ]=h0.
With an element hn of ℍ⊗^n is associated the Wiener chaos
(1.5)Ψ(hn)=∫[0,1]nhn(s1,…,sn)dBs1⋯dBsn.
The mat Ψ realizes a isomorphism between F(ℍ) and L2(dP). On the level of the Fock space some important elements are constituted by coherent vectors:
(1.6)σ=∑h⊗nn!.
The functional associated to such a coherent vector is a so-called exponential martingale
(1.7)Ψ(σ)=exp[∫01hsdBs-∥h∥22].
We refer to the books of Hida et al. [1], to the book of Obata [2], and to the book of Meyer [3] for an extensive study on that subject. Especially on the Fock space, we can define the Wiener product:
(1.8)Ψ(σ1·σ2)=Ψ(σ1)Ψ(σ2),
where we consider the ordinary product of the two Ψ(σi). For that, we use the Itô table for the Laplacian(1.9)dBs·dBs=12ds,dBs·ds=ds·ds=0
which reflect algebraically the Itô formula for the Brownian motion. From this Itô table, we deduce classically that if σ is an exponential vector, Ψ(σ)=exp[∫01hsdBs-∥h∥2/2] and not exp[∫01hsdBs].

The law of Bt+∫0thsds is absolutely continuous with respect of the law of Bt, and the Radon-Nikodym derivative between these two laws is Ψ(σ)=exp[-∫01hsdBs-∥h∥2/2]. It is the subject of the Cameron-Martin formula.

The construction of a full path probability measure associated to a semi-group is related to Hunt theory: the generator L of the semi-group has to satisfy maximum principle. We are motivated where we take others type of generator. To simplify the computations we take the simplest of such operators L=-∂4/∂x4. We have implemented recently some stochastic tools for semi-groups whose generators do not simplify maximum principle ([4–10]). We construct in [8, 9] the Wiener distribution associated to a Bilaplacian using the Hida-Streit approach of path integrals as distribution. We refer to the works of Funaki [11], Hochberg [12], Krylov [13], and the review paper of Mazzucchi [14] for other approaches. We refer to the review paper of Albeverio [15] for various approach of path integrals.

In the Hida-Streit approach of path integral, there are basically 3 objects:

an algebraic space, generally a kind of Fock space;

a map Ψ from this algebraic space into a set of functionals on a mapping space;

the path integral is continuous on the level of the algebraic set. We say that it is an Hida-type distribution.

Generally, people were considering map Ψ as the map Wiener chaos. A breakdown was performed by Getzler [16] motivated by the works of Atiyah-Bismut-Witten relating the structure of the free loop space and the Index theory. Developments were done by Léandre in [17, 18]. Especially, in [8, 9] we were using map Ψ as related to cylindrical functional to define a path integral associated to the Bilaplacian and to state some properties related to this path integral.

In this paper, we come back to the original map Ψ of Wiener, by using Wiener chaos. But we use formal Wiener chaos. We consider a continuous path ws. We consider a map hni1,…,in(s1,…,sn)s1<s2<⋯<sn<1 with values in ℝ. We consider the formal Wiener chaos:
(1.10)Ψ(hn)=∫0<s1<⋯<sn<1hni1,…,in(s1,…,sn)dws1i1⋯dwsnin.
We put
(1.11)dws4=24ds.
If i>4, dwsi=0. We use in order to define the Wiener product on formal chaos associated to the Bilaplacian L the Itô table for the Bilaplacian:(1.12)dwsidwsj=dwsi+j.
In order to simplify the exposition, we use in the sequel Connes space and not a Hida Fock space. We consider L∞ the set of map h from [0,1] into ℝ3 such that
(1.13)sups|h(s)|=∥h∥∞.
We introduce the bosonic Connes space CO∞-(L∞) (a refinement of the traditional bosonic Fock space). To σ∈CO∞-(L∞), we associate a formal Wiener chaos Ψ(σ). We use the Itô table for the Bilaplacian in order to define a Wiener product on the bosonic Connes space:
(1.14)Ψ(σ1·σ2)=Ψ(σ1)Ψ(σ2).
The bosonic Connes space becomes a commutative topological algebra for the Wiener product (For similar consideration for the case of the standard Laplacian, we refer to the book of Meyer [3]).

We consider as classical the vacuum expectation on the bosonic Connes space, and we state a kind of Itô-Segal-Bargmann-Wiener isomorphism, but in this case there is no Hilbert space involved. We show that for the vacuum expectation ws has in some sense independent increments. We consider a type of generalization of the exponential martingale of the Brownian motion:
(1.15)Ψ(σt)=∑∫0<s1<⋯<sn<ths1dws11⋯hsndwsn1.
We suppose that h is continuous. Let f be a polynomial on ℝ. We put
(1.16)Qth[f]=μ[f(wt1)Ψ(σt)].
We show the following Cameron-Martin-Maruyama-Girsanov type formula:
(1.17)∂∂tQth[f]=Qth[Lh,tf],
where
(1.18)Lh,t=L+lowerterm.

2. Formal Wiener Chaos Associated to a Bilaplacian

We consider the set L∞. (L∞)⊗n is constituted of maps:
(2.1)∑i1,…,inhi1,…,in(s1,…,sn)ei1⊗⋯⊗ein=hn(s1,…,sn),
where ei is the standard basis of ℝ3. On (L∞)⊗n, we consider the natural supremum norm ∥hn∥∞. Moreover, there is a natural action of the symmetric group on (L∞)⊗n. Elements which are invariant under this action of the symmetric group are called elements of the symmetric tensor product (L∞)⊗^n. COC,r(L∞)(r>0,C>0) is constituted of formal series σ=∑hn where hn belongs to (L∞)⊗^n such that
(2.2)∥σ∥C=∑Cnn!∥hn∥∞<∞.

Definition 2.1.

The intersection of all COC(L∞) is called the bosonic Connes space CO∞-(L∞).

Remark 2.2.

In the sequel we could choose an Hida Fock space.

Definition 2.3.

The vacuum expectation μ on CO∞-(L∞) is defined by
(2.3)μ(σ)=h0.
If hn belongs to (L∞)⊗^n, we consider the formal Wiener chaos:
(2.4)Ψ(hn)=∑i1,…,in∫0<s1<⋯<sn<1hni1,..,in(s1,…sn)dws1i1⋯dwsnin.
We could do the same expression if hn belongs to (L∞)⊗n.

Definition 2.4.

The map Ψ defined on CO∞-(L∞) is called the map formal Wiener chaos.

Let {1,…,n}, {n+1,…,n+m}. Let {l} be a concatenation (or pairing). It is an increasing injective map from a set with l element in {1,…,n} into {n+1,…,n+m}. There is at most Cn+m pairing of length l. We consider hn1⊗{l},sh{l}hm2 where we concatain the time in hn and in hm according the pairing, and we shuffle according to the shuffle shl and the time in hn1 and hm1 between two continuous times in the pairing. When we concatenate two times, we use the Itô table for the Bilaplacian, and we symmetrized the expression in the time.

The classical product of Ψ(hn1)Ψ(hm2) is equal to ∑{l},sh{l}Ψ(hn1⊗{l},sh{l}hm2) and generalized with this new Itô table the standard formula which gives the product of two Wiener chaos in the Brownian case. There are at most Cn+mCnlCml pairing {l} and shuffle according to the pairing {l}.

Definition 2.5.

The Wiener product of hn1 and hm2 is defined by
(2.5)Ψ(hn1·hm2)=Ψ(hn1)Ψ(hn2).

Theorem 2.6.

The Wiener product endows the symmetric Connes space with a structure of topological commutative algebra.

Proof.

Let us show first of all that the Wiener product is continuous. We have
(2.6)∥hn1⊗{l},sh{l}hm2∥∞≤Cn+m∥hn1∥∞∥hm2∥∞.
Therefore,
(2.7)∥hn1·hm2∥C≤C1n+mCn+m∥hn1∥∞∥hm2∥∞∑{l},sh{l}C-l((n+m-2l)!).

But
(2.8)∑{l},sh{l}C-l≤∑lCnlCmlC3n+mC-l≤C2n+m(1+C-1)n+m≤C4n+m.
On the other hand, by the Stirling formula,
(2.9)(n!)-1(m!)-1(n+m-2l)!≤C3n+m.
We deduce that
(2.10)∥σ1·σ2∥C≤K∥σ1∥C′∥σ2∥C′
and therefore the Wiener product is continuous on the bosonic Connes space.

Let hn1,hn2, and hn3 be 3 elements of the bosonic Connes space.

We perform two concatenations between the times when the shuffle is done:

either we concatain 2 contiguous times in {1,n1} and in {n1+1,n1+n2} and two contiguous time in {1,n1} and in {n1+n2+1,n1+n2+n3};

either we concatain 2 contiguous times in {n1+1,n1+n2} and in {1,n1} and two contiguous times in {n1+1,n1+n2} and in {n1+n2+1,n1+n2+n3};

either we concatain 2 contiguous times in {n1+n2+1,n1+n2+n3} and in {1,n1} and two contiguous times in {n1+n2+1,n1+n2+n3} and in {n1+1,n1+n2};

or we concatain 3 contiguous times in {1,n1}, in {n1+1,n1+n2} and in {n1+n2+1,n1+n2+n3}.

When we concatain time, we use the iterated Itô rule:
(2.11)(dwsi1·dwsi2)·dwsi3=dwsi1+i2+i3.
Such a concatenation is called l1,2,3 and the final result is called hn1⊗sh1,2,3,l1,2,3hn2⊗sh1,2,3l1,2,3hn3. We deduce the formula
(2.12)(hn1·hn2)·hn3=∑l1,2,3,sh1,2,3hn1⊗sh1,2,3,l1,2,3hn2⊗sh1,2,3l1,2,3hn3.
From this formula we deduce the associativity of the Wiener product.

From the product formula, we deduce easily the next theorem.

Theorem 2.7 (Itô-Bargmann-Wiener-Segal).

Let hn1i1,..,in1 and hn2j1,..,jn2 be elements of the bosonic Connes space. They are seen as a function on the involved simplices. Then
(2.13)μ[Ψ(hn1)Ψ(hn2)]=δn1,n2∏δil+jl=424n1×∫0<s1<⋯<sn<1hn1i1,…,in(s1,…,sn)hn1j1,…,jn(s1,…,sn)ds1⋯dsn.

Remark 2.8.

In the case of the classical Laplacian, this formula justifies the choice of ℍ instead of L∞. But in the previous formula, only a prehilbert space appears. So it is not obviously justified to choose ℍ instead of L∞ to perform our computations. We have chosen L∞ because the estimates are simpler with this space.

We say that hn belongs to CO∞-,t](L∞) if hn vanishes as soon as one of the si≥t. We say that hn belongs to CO∞-,[t(L∞) if hn vanishes as soon as one of the si≤t. We get the next theorem whose proof is obvious.

Theorem 2.9.

CO∞-,t](L∞) and CO∞-,[t(L∞) are subalgebras of CO∞-(L∞) for the Wiener product. Moreover, if σ1∈CO∞-,t](L∞) and if σ2∈CO∞-,[t(L∞),
(2.14)μ[Ψ(σ1)Ψ(σ2)]=μ[Ψ(σ1)]μ[Ψ(σ2)].

Remark 2.10.

Let us justify heuristically this part. Let Qt0 be the semi-group generated by L. Let us suppose that there is a formal measure dμ on a path space t→wt such that
(2.15)Qt0[f]=∫f(wt)dμ.
(In the case of the standard Laplacian it is the measure of the Brownian motion). We refer to [19] for a physicist way to construct this measure. We have
(2.16)Qt0[x4]=24t
So the infinitesimal increment (dwt)i of wt should satisfy the Itô table (1.12) and the formal Wiener chaos should be an extension of the classical Wiener chaos in the Brownian case.

3. A Cameron-Martin-Maruyama-Girsanov Formula Associated to a Bilaplacian

We put if f is a polynomial,
(3.1)Qth[f]=μ[f(wt1)Ψ(σt)],
where
(3.2)Ψ(σt)=∑∫0<s1<⋯<sn<ths1dws11⋯hsndwsn1.
We suppose that h is continuous. In this formula, only finite sums appear due to (2.13). We get the following.

Theorem 3.1 (Cameron-Martin-Maruyama-Girsanov).

If f is a polynomial,
(3.3)∂∂tQth[f]=Qth[Lh,tf],
where
(3.4)Lh=-∂4∂x4+αht∂3∂x3.

Proof.

Let us consider the case where f(x)=xn. We use wt1=∫0tdws1 and the fact that the Wiener product is associative. We get
(3.5)(w1+wt+Δt1-wt1)n=∑Cnk(wt1)n-k(wt+Δt1-wt1)k.
We put
(3.6)σΔt=∑𝕀[t,t+Δt]⊗nn!
such that by the Itô rules on [t,t+Δt] for Δt>0:
(3.7)σt+Δt=σt·σΔt.
We use Theorem 2.9 and the Itô table on [t,t+Δt]. We deduce that
(3.8)μ[(wt+Δt1)nΨ(σt+Δt)]=μ[(wt1)nΨ(σt)]+n(n-1)(n-2)(n-3)μ[(wt1)n-4Ψ(σt)]Δt+αhtn(n-1)(n-2)μ[(wt1)n-3Ψ(σt)]Δt+o(Δt).
Therefore, the result is obtained.

HidaT.KuoH.-H.PotthoffJ.StreitL.ObataN.MeyerP.-A.LéandreR.NagyP.RaicsP.Itô-Stratonovitch for a four order operator on a torusLéandreR.ZaslavskyG.BaleanuD.Tenreiro MachadoJ. A.Itô-Stratonovitch for the Schroedinger equation associated to a big order operator on a torusLéandreR.El TawilM. A.Itô-Stratonovitch formula for the wave equation on a torusLéandreR.RuzhanskyM.WirthJ.Itô formula for an integro differential operator without a stochastic processProceedings of the 20th International Symposium on Algorithms and Computation (ISAAC '09)2009World Scientific225231LéandreR.Stochastic analysis without probability: study of some basic toolsLeandreR.Long time behaviour on a path group of the heat semi-group associated to a bilaplacianLéandreR.The Itô transform for an accretive operatorsFunakiT.Probabilistic construction of the solution of some higher order parabolic differential equationHochbergK. J.A signed measure on path space related to Wiener measureKrylovV. Ju.Some properties of the distribution corresponding to the equation ∂u/∂t=(-1)q+1∂2qu/∂x2qMazzucchiS.Probabilistic representations for the solution of higher order differential equationsUniversity Torino, 2011AlbeverioS.MandrekarV.MasaniP. R.Wiener and Feynman—path integrals and their applications52Proceedings of the Norbert Wiener Centenary Congress1996Providence, RI, USAAmerican Mathematical Society163194Proceedings of Symposia in Applied Mathematics1440913ZBL0899.60058GetzlerE.Cyclic homology and the path integral of the Dirac operatorIn pressLéandreR.ZambriniJ. C.Connes-Hida calculus in index theoryLéandreR.NaberG.TsunT. S.Path integrals in noncommutive geometryKlauderJ. R.ShabanovS. V.BernidoC. C.Carpio-BernidoM. V.NakamuraK.WatanabeK.An introduction to coordinate free quantization and its application to constrained systems