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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.
Let us recall some basic tools of Wiener analysis. Let be a one-dimensional Brownian motion starting from 0. It is classically related to the heat equation on : where is the standard Laplacian and 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 is continuous. We construct by this way the Wiener measure on the continuous path space endowed with its Borelian -algebra. Let be the Hilbert space . We consider the symmetric tensor product of this Hilbert space. It is constituted of maps symmetric in such that We consider the symmetric Fock space of set such that We consider the vacuum expectation. With an element of is associated the Wiener chaos The mat realizes a isomorphism between and . 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. , to the book of Obata , and to the book of Meyer  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 . For that, we use the Itô table for the Laplacian which reflect algebraically the Itô formula for the Brownian motion. From this Itô table, we deduce classically that if is an exponential vector, and not .
The law of is absolutely continuous with respect of the law of , and the Radon-Nikodym derivative between these two laws is . 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 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 . 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 , Hochberg , Krylov , and the review paper of Mazzucchi  for other approaches. We refer to the review paper of Albeverio  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 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 as the map Wiener chaos. A breakdown was performed by Getzler  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 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 of Wiener, by using Wiener chaos. But we use formal Wiener chaos. We consider a continuous path . We consider a map with values in . We consider the formal Wiener chaos: We put If , . We use in order to define the Wiener product on formal chaos associated to the Bilaplacian the Itô 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 the set of map from into such that We introduce the bosonic Connes space (a refinement of the traditional bosonic Fock space). To , we associate a formal Wiener chaos . 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 ).
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 has in some sense independent increments. We consider a type of generalization of the exponential martingale of the Brownian motion: We suppose that is continuous. Let be a polynomial on . We put We show the following Cameron-Martin-Maruyama-Girsanov type formula: where
2. Formal Wiener Chaos Associated to a Bilaplacian
We consider the set . is constituted of maps: where is the standard basis of . On , we consider the natural supremum norm . Moreover, there is a natural action of the symmetric group on . Elements which are invariant under this action of the symmetric group are called elements of the symmetric tensor product . is constituted of formal series where belongs to such that
Definition 2.1. The intersection of all is called the bosonic Connes space .
Remark 2.2. In the sequel we could choose an Hida Fock space.
Definition 2.3. The vacuum expectation on is defined by If belongs to , we consider the formal Wiener chaos: We could do the same expression if belongs to .
Definition 2.4. The map defined on is called the map formal Wiener chaos.
Let , . Let be a concatenation (or pairing). It is an increasing injective map from a set with element in into . There is at most pairing of length . We consider where we concatain the time in and in according the pairing, and we shuffle according to the shuffle and the time in and 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 is equal to 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 pairing and shuffle according to the pairing .
Definition 2.5. The Wiener product of and is defined by
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
But On the other hand, by the Stirling formula, We deduce that and therefore the Wiener product is continuous on the bosonic Connes space.Let , and be 3 elements of the bosonic Connes space.Let be a shuffle between the 3 sets , , and .
We perform two concatenations between the times when the shuffle is done:(i)either we concatain 2 contiguous times in and in and two contiguous time in and in ;(ii)either we concatain 2 contiguous times in and in and two contiguous times in and in ;(iii)either we concatain 2 contiguous times in and in and two contiguous times in and in ;(iv)or we concatain 3 contiguous times in , in and in .
When we concatain time, we use the iterated Itô rule: Such a concatenation is called and the final result is called . We deduce the formula 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 and be elements of the bosonic Connes space. They are seen as a function on the involved simplices. Then
Remark 2.8. In the case of the classical Laplacian, this formula justifies the choice of instead of . But in the previous formula, only a prehilbert space appears. So it is not obviously justified to choose instead of to perform our computations. We have chosen because the estimates are simpler with this space.
We say that belongs to if vanishes as soon as one of the . We say that belongs to if vanishes as soon as one of the . We get the next theorem whose proof is obvious.
Theorem 2.9. and are subalgebras of for the Wiener product. Moreover, if and if ,
Remark 2.10. Let us justify heuristically this part. Let be the semi-group generated by . Let us suppose that there is a formal measure on a path space such that (In the case of the standard Laplacian it is the measure of the Brownian motion). We refer to  for a physicist way to construct this measure. We have So the infinitesimal increment of 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 is a polynomial, where We suppose that 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 is a polynomial, where
Proof. Let us consider the case where . We use and the fact that the Wiener product is associative. We get We put such that by the Itô rules on for : We use Theorem 2.9 and the Itô table on . We deduce that Therefore, the result is obtained.
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Copyright © 2012 Rémi Léandre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.