A Path-Integral Approach to the Cameron-Martin-Maruyama-Girsanov Formula Associated to a Bilaplacian

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.


Introduction
Let us recall some basic tools of Wiener analysis.Let B t be a one-dimensional Brownian motion starting from 0. It is classically related to the heat equation on R: where Δ ∂ 2 /∂x 2 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 L 2 0, 1 ; R .We consider the symmetric tensor Journal of Function Spaces and Applications product H ⊗n of this Hilbert space.It is constituted of maps h n s 1 , . . ., s n symmetric in s i such that We consider the symmetric Fock space F H of set σ ∞ n 0 h n such that We consider the vacuum expectation.

1.4
With an element h n of H ⊗n is associated the Wiener chaos The mat Ψ realizes a isomorphism between F H and L 2 dP .On the level of the Fock space some important elements are constituted by coherent vectors: The functional associated to such a coherent vector is a so-called exponential martingale 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: where we consider the ordinary product of the two Ψ σ i .For that, we use the It ô 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 /∂x 4 .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: i an algebraic space, generally a kind of Fock space; ii a map Ψ from this algebraic space into a set of functionals on a mapping space; iii 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 w s .We consider a map We consider the formal Wiener chaos: We put dw 4 s 24ds.

1.11
If i > 4, dw i s 0. We use in order to define the Wiener product on formal chaos associated to the Bilaplacian L the It ô table for the Bilaplacian: 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 R 3 such that sup 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: 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 It ô-Segal-Bargmann-Wiener isomorphism, but in this case there is no Hilbert space involved.We show that for the vacuum expectation w s has in some sense independent increments.We consider a type of generalization of the exponential martingale of the Brownian motion: We suppose that h is continuous.Let f be a polynomial on R. We put We show the following Cameron-Martin-Maruyama-Girsanov type formula: where

Formal Wiener Chaos Associated to a Bilaplacian
We consider the set L ∞ .L ∞ ⊗n is constituted of maps: where e i is the standard basis of R 3 .On L ∞ ⊗n , we consider the natural supremum norm h n ∞ .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 .CO C,r L ∞ r > 0, C > 0 is constituted of formal series σ h n where h n belongs to L ∞ ⊗n such that Remark 2.2.In the sequel we could choose an Hida Fock space.

2.3
If h n belongs to L ∞ ⊗n , we consider the formal Wiener chaos: We could do the same expression if h n 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 C n m pairing of length l.We consider h 1 n ⊗ {l},sh {l} h 2 m where we concatain the time in h n and in h m according the pairing, and we shuffle according to the shuffle sh l and the time in h 1  n and h 1 m between two continuous times in the pairing.When we concatenate two times, we use the It ô table for the Bilaplacian, and we symmetrized the expression in the time.
The classical product of Ψ h

2.5
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

2.8
On the other hand, by the Stirling formula,

2.9
We deduce that and therefore the Wiener product is continuous on the bosonic Connes space.
Let h n 1 , h n 2 , and h n 3 be 3 elements of the bosonic Connes space.

2.12
From this formula we deduce the associativity of the Wiener product.
From the product formula, we deduce easily the next theorem.

2.13
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 h n belongs to CO ∞−,t L ∞ if h n vanishes as soon as one of the s i ≥ t.We say that h n belongs to CO ∞−, t L ∞ if h n vanishes as soon as one of the s i ≤ t.We get the next theorem whose proof is obvious.

2.14
Remark 2.10.Let us justify heuristically this part.Let Q 0 t be the semi-group generated by L. Let us suppose that there is a formal measure dμ on a path space t → w t such that Q 0 t f f w t dμ.

2.15
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 So the infinitesimal increment dw t i of w t should satisfy the It ô table 1.12 and the formal Wiener chaos should be an extension of the classical Wiener chaos in the Brownian case.

A Cameron-Martin-Maruyama-Girsanov Formula Associated to a Bilaplacian
We put if f is a polynomial,

3.8
Therefore, the result is obtained.
We suppose that h is continuous.In this formula, only finite sums appear due to 2.13 .We get the following.Cameron-Martin-Maruyama-Girsanov .If f is a polynomial, Proof.Let us consider the case where f xx n .We use w 1 Δt σ t • σ Δ t .3.7We use Theorem 2.9 and the It ô table on t, t Δt .We deduce that such that by the It ô rules on t, t Δt for Δt > 0:σ t Ψ σ t Δ t o Δt .