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

Abstract

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 𝐵_𝑡 be a one-dimensional Brownian motion starting from 0. It is classically related to the heat equation on ℝ: 𝜕𝐸𝑓𝐵𝜕𝑡^𝑡=12𝐸Δ𝑓𝐵_𝑡, (1.1) 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 ℎ_𝑛(𝑠₁,…,𝑠_𝑛) symmetric in 𝑠_𝑖 such that ℎ_𝑛²=_[]0,1𝑛ℎ²_𝑛𝑠₁,…,𝑠_𝑛𝑑𝑠₁⋯𝑑𝑠_𝑛<∞. (1.2) We consider the symmetric Fock space 𝐹(ℍ) of set 𝜎=∑^∞_𝑛=0ℎ_𝑛 such that ‖𝜎‖²=𝑛!ℎ_𝑛²<∞. (1.3) We consider the vacuum expectation. 𝜇[𝜎]=ℎ₀. (1.4) With an element ℎ_𝑛 of ℍ⊗_𝑛 is associated the Wiener chaos Ψℎ_𝑛=_[]0,1𝑛ℎ_𝑛𝑠₁,…,𝑠_𝑛𝑑𝐵_𝑠1⋯𝑑𝐵_𝑠𝑛. (1.5) 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: ℎ𝜎=^⊗𝑛.𝑛! (1.6) The functional associated to such a coherent vector is a so-called exponential martingale Ψ(𝜎)=exp¹₀ℎ_𝑠𝑑𝐵_𝑠−‖ℎ‖²2. (1.7) 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: Ψ𝜎₁⋅𝜎₂=Ψ𝜎₁Ψ𝜎₂, (1.8) where we consider the ordinary product of the two Ψ(𝜎_𝑖). For that, we use the Itô table for the Laplacian𝑑𝐵_𝑠⋅𝑑𝐵_𝑠=12𝑑𝑠,𝑑𝐵_𝑠⋅𝑑𝑠=𝑑𝑠⋅𝑑𝑠=0 (1.9) 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[∫¹₀ℎ_𝑠𝑑𝐵_𝑠−‖ℎ‖²/2] and not exp[∫¹₀ℎ_𝑠𝑑𝐵_𝑠].

The law of 𝐵_𝑡+∫^𝑡₀ℎ_𝑠𝑑𝑠 is absolutely continuous with respect of the law of 𝐵_𝑡, and the Radon-Nikodym derivative between these two laws is Ψ(𝜎)=exp[−∫¹₀ℎ_𝑠𝑑𝐵_𝑠−‖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:(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 ℎ^{𝑖1,…,𝑖𝑛}_𝑛(𝑠₁,…,𝑠_𝑛)𝑠₁<𝑠₂<⋯<𝑠_𝑛<1 with values in ℝ. We consider the formal Wiener chaos: Ψℎ_𝑛=_{0<𝑠1<⋯<𝑠𝑛<1}ℎ^{𝑖1,…,𝑖𝑛}_𝑛𝑠₁,…,𝑠_𝑛𝑑𝑤^𝑖1_𝑠1⋯𝑑𝑤^𝑖𝑛_𝑠𝑛. (1.10) We put 𝑑𝑤⁴_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 Itô table for the Bilaplacian:𝑑𝑤^𝑖_𝑠𝑑𝑤^𝑗_𝑠=𝑑𝑤_𝑠^𝑖+𝑗. (1.12) 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 sup_𝑠|ℎ(𝑠)|=‖ℎ‖_∞. (1.13) 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: Ψ𝜎₁⋅𝜎₂=Ψ𝜎₁Ψ𝜎₂. (1.14) 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 𝑤_𝑠 has in some sense independent increments. We consider a type of generalization of the exponential martingale of the Brownian motion: Ψ𝜎_𝑡=_{0<𝑠1<⋯<𝑠𝑛<𝑡}ℎ_𝑠1𝑑𝑤¹_𝑠1⋯ℎ_𝑠𝑛𝑑𝑤¹_𝑠𝑛. (1.15) We suppose that ℎ is continuous. Let 𝑓 be a polynomial on ℝ. We put 𝑄^ℎ_𝑡𝑓=𝜇𝑓𝑤¹_𝑡Ψ𝜎_𝑡. (1.16) We show the following Cameron-Martin-Maruyama-Girsanov type formula: 𝜕𝑄𝜕𝑡^ℎ𝑡𝑓=𝑄^ℎ_𝑡𝐿_ℎ,𝑡𝑓, (1.17) where 𝐿_ℎ,𝑡=𝐿+lowerterm. (1.18)

2. Formal Wiener Chaos Associated to a Bilaplacian

We consider the set 𝐿^∞. (𝐿^∞)^⊗𝑛 is constituted of maps: _{𝑖1,…,𝑖𝑛}ℎ^{𝑖1,…,𝑖𝑛}𝑠₁,…,𝑠_𝑛𝑒_𝑖1⊗⋯⊗𝑒_𝑖𝑛=ℎ_𝑛𝑠₁,…,𝑠_𝑛, (2.1) where 𝑒_𝑖 is the standard basis of ℝ³. 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 ‖𝜎‖_𝐶=𝐶^𝑛𝑛!ℎ_𝑛∞<∞. (2.2)

Definition 2.1. The intersection of all 𝐶𝑂_𝐶(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 𝜇(𝜎)=ℎ₀. (2.3) If ℎ_𝑛 belongs to (𝐿^∞)⊗_𝑛, we consider the formal Wiener chaos: Ψℎ_𝑛=_{𝑖1,…,𝑖𝑛0<𝑠1<⋯<𝑠𝑛<1}ℎ^{𝑖1,..,𝑖𝑛}_𝑛𝑠₁,…𝑠_𝑛𝑑𝑤^𝑖1_𝑠1⋯𝑑𝑤^𝑖𝑛_𝑠𝑛. (2.4) 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 ℎ¹_𝑛⊗_{{𝑙},𝑠ℎ{𝑙}}ℎ²_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 ℎ¹_𝑛 and ℎ¹_𝑚 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(ℎ¹_𝑛)Ψ(ℎ²_𝑚) is equal to ∑_{{𝑙},𝑠ℎ{𝑙}}Ψ(ℎ¹_𝑛⊗_{{𝑙},𝑠ℎ{𝑙}}ℎ²_𝑚) 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 𝐶^𝑛+𝑚𝐶^𝑙_𝑛𝐶^𝑙_𝑚 pairing {𝑙} and shuffle according to the pairing {𝑙}.

Definition 2.5. The Wiener product of ℎ¹_𝑛 and ℎ²_𝑚 is defined by Ψℎ¹_𝑛⋅ℎ²_𝑚=Ψℎ¹_𝑛Ψℎ²_𝑛. (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 ℎ¹_𝑛⊗_{{𝑙},𝑠ℎ{𝑙}}ℎ²_𝑚∞≤𝐶^𝑛+𝑚ℎ¹_𝑛∞ℎ²_𝑚∞. (2.6) Therefore, ℎ¹_𝑛⋅ℎ²_𝑚𝐶≤𝐶₁^𝑛+𝑚𝐶^𝑛+𝑚ℎ¹_𝑛∞ℎ²_{𝑚∞{𝑙},𝑠ℎ{𝑙}}𝐶^−𝑙((𝑛+𝑚−2𝑙)!). (2.7)
But _{{𝑙},𝑠ℎ{𝑙}}𝐶^−𝑙≤_𝑙𝐶^𝑙_𝑛𝐶^𝑙_𝑚𝐶₃^𝑛+𝑚𝐶^−𝑙≤𝐶₂^𝑛+𝑚1+𝐶^{−1𝑛+𝑚}≤𝐶₄^𝑛+𝑚. (2.8) On the other hand, by the Stirling formula, (𝑛!)⁻¹(𝑚!)⁻¹(𝑛+𝑚−2𝑙)!≤𝐶₃^𝑛+𝑚. (2.9) We deduce that 𝜎₁⋅𝜎_2𝐶≤𝐾𝜎_1𝐶′𝜎_2𝐶′ (2.10) and therefore the Wiener product is continuous on the bosonic Connes space.Let ℎ_𝑛1,ℎ_𝑛2,  and  ℎ_𝑛3 be 3 elements of the bosonic Connes space.Let 𝑠ℎ_1,2,3 be a shuffle between the 3 sets {1,𝑛₁}, {𝑛₁+1,𝑛₁+𝑛₂}, and {𝑛₁+𝑛₂+1,𝑛₁+𝑛₂+𝑛₃}.
We perform two concatenations between the times when the shuffle is done:(i)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};(ii)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};(iii)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};(iv)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: 𝑑𝑤^𝑖1_𝑠⋅𝑑𝑤^𝑖2_𝑠⋅𝑑𝑤^𝑖3_𝑠=𝑑𝑤^{𝑖1+𝑖2+𝑖3}_𝑠. (2.11) Such a concatenation is called 𝑙_1,2,3 and the final result is called ℎ_𝑛1⊗_{𝑠ℎ1,2,3,𝑙1,2,3}ℎ_𝑛2⊗_{𝑠ℎ1,2,3𝑙1,2,3}ℎ_𝑛3. We deduce the formula ℎ_𝑛1⋅ℎ_𝑛2⋅ℎ_𝑛3=_{𝑙1,2,3,𝑠ℎ1,2,3}ℎ_𝑛1⊗_{𝑠ℎ1,2,3,𝑙1,2,3}ℎ_𝑛2⊗_{𝑠ℎ1,2,3𝑙1,2,3}ℎ_𝑛3. (2.12) 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 ℎ^{𝑖1,..,𝑖𝑛1}_𝑛1 and ℎ^{𝑗1,..,𝑗𝑛2}_𝑛2 be elements of the bosonic Connes space. They are seen as a function on the involved simplices. Then 𝜇Ψℎ_𝑛1Ψℎ_𝑛2=𝛿_𝑛1,𝑛2𝛿_{𝑖𝑙+𝑗𝑙=4}24^𝑛1×_{0<𝑠1<⋯<𝑠𝑛<1}ℎ^{𝑖1,…,𝑖𝑛}_𝑛1𝑠₁,…,𝑠_𝑛ℎ^{𝑗1,…,𝑗𝑛}_𝑛1𝑠₁,…,𝑠_𝑛𝑑𝑠₁⋯𝑑𝑠_𝑛. (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.

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^∞), 𝜇Ψ𝜎₁Ψ𝜎₂=𝜇Ψ𝜎₁𝜇Ψ𝜎₂. (2.14)

Remark 2.10. Let us justify heuristically this part. Let 𝑄⁰_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 𝑄⁰_𝑡𝑓=𝑓𝑤_𝑡𝑑𝜇. (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 𝑄⁰_𝑡𝑥⁴=24𝑡 (2.16) So the infinitesimal increment (𝑑𝑤_𝑡)^𝑖 of 𝑤_𝑡 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, 𝑄^ℎ_𝑡𝑓=𝜇𝑓𝑤¹_𝑡Ψ𝜎_𝑡, (3.1) where Ψ𝜎_𝑡=_{0<𝑠1<⋯<𝑠𝑛<𝑡}ℎ_𝑠1𝑑𝑤¹_𝑠1⋯ℎ_𝑠𝑛𝑑𝑤¹_𝑠𝑛. (3.2) 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, 𝜕𝑄𝜕𝑡^ℎ𝑡𝑓=𝑄^ℎ_𝑡𝐿_ℎ,𝑡𝑓, (3.3) where 𝐿_ℎ𝜕=−⁴𝜕𝑥⁴+𝛼ℎ_𝑡𝜕³𝜕𝑥³. (3.4)

Proof. Let us consider the case where 𝑓(𝑥)=𝑥^𝑛. We use 𝑤¹_𝑡=∫^𝑡₀𝑑𝑤¹_𝑠 and the fact that the Wiener product is associative. We get 𝑤¹+𝑤¹_𝑡+Δ𝑡−𝑤¹_𝑡^𝑛=𝐶^𝑘_𝑛𝑤¹_𝑡^𝑛−𝑘𝑤¹_𝑡+Δ𝑡−𝑤¹_𝑡^𝑘. (3.5) We put 𝜎_Δ𝑡𝕀=^{[]⊗𝑛𝑡,𝑡+Δ𝑡}𝑛! (3.6) such that by the Itô rules on [𝑡,𝑡+Δ𝑡] for Δ𝑡>0: 𝜎_𝑡+Δ𝑡=𝜎_𝑡⋅𝜎_Δ𝑡. (3.7) We use Theorem 2.9 and the Itô table on [𝑡,𝑡+Δ𝑡]. We deduce that 𝜇𝑤¹_𝑡+Δ𝑡^𝑛Ψ𝜎_𝑡+Δ𝑡=𝜇𝑤¹_𝑡^𝑛Ψ𝜎_𝑡+𝑛(𝑛−1)(𝑛−2)(𝑛−3)𝜇𝑤¹_𝑡^𝑛−4Ψ𝜎_𝑡Δ𝑡+𝛼ℎ_𝑡𝑛(𝑛−1)(𝑛−2)𝜇𝑤¹_𝑡^𝑛−3Ψ𝜎_𝑡Δ_𝑡+𝑜(Δ𝑡). (3.8) Therefore, the result is obtained.