The Lent Particle Method, Application to Multiple Poisson Integrals
aa r X i v : . [ m a t h . P R ] A p r The Lent Particle Method,Application to Multiple Poisson Integrals
Nicolas BOULEAUEcole des Ponts ParisTech6 Avenue Blaise Pascal, Marne-la-Vallée 77455, France [email protected]
Abstract
We give a extensive account of a recent new way of applying the Dirichlet form theoryto random Poisson measures. The main application is to obtain existence of density for thelaws of random functionals of Lévy processes or solutions of stochastic differential equationswith jumps. As in the Wiener case the Dirichlet form approach weakens significantly theregularity assumptions. The main novelty is an explicit formula for the gradient or for the“carré du champ" on the Poisson space called the lent particle formula because based onadding a new particle to the system, computing the derivative of the functional with respectto this new argument and taking back this particle before applying the Poisson measure.The article is expository in its first part and based on Bouleau-Denis [12] with severalnew examples, applications to multiple Poisson integrals are gathered in the last part whichconcerns the relation with the Fock space and some aspects of the second quantization.
Keywords : Dirichlet form, Poisson random measure, Malliavin calculus, stochastic differen-tial equation, Poisson functional, energy image density, Lévy processes, Lévy measure, gradient,carré du champ.
This lecture is an introduction to Dirichlet forms methods for studying regularity of randomvariables yielded by Lévy processes, solutions of stochastic differential equations driven by Pois-son measures and multiple Poisson integrals. The main part of this study has been done incollaboration with Laurent Denis.A Dirichlet forms is a generalisation of the classical quadratic operator R Ω |∇ f ( x ) | dx earlyintroduced in potential theory. The concept has been developped especially by Beurling and Denyin the 1950’s as an application of Hilbert space methods in potential theory, and by Fukushimain the 1970’s in connection with symmetric Markov processes theory. It received recently astrong development in infinite dimensional spaces where it appears as an alternative approachto Malliavin calculus.The importance of the notion comes from the fact that if P t is a symmetric strongly continuouscontraction semigroup on a space L ( µ ) (for µ σ -finite positive measure) with generator A ,a necessary and sufficient condition that P t be Markov is that “contractions operate" on thequadratic form E [ f ] = − < Af, f > L ( µ ) i.e. E [ ϕ ( f )] E [ f ] for ϕ contraction from R to R (cf[11] Chap.I prop. 3.2.1). Such a quadratic form is called a Dirichlet form.1he case of Malliavin calculus is that of Wiener space taking for P t the Ornstein-Uhlenbecksemi-group. The corresponding Dirichlet form E possesses a “carré du champ" operator, i.e. maybe written E [ f ] = R Γ[ f ] dµ where Γ is a quadratic operator from the domain of E to L ( µ ) .This fact makes it possible the definition of a “gradient" satisfying the chain rule and allowinga differential calculus through stochastic expressions and stochastic differential equations (SDE)and providing integration by parts formulae which yield existence of density results (cf [29]).Using Dirichlet forms in this framework of Wiener space improves several results : contractionarguments show that the Picard iteration method for solving SDE’s holds not only in L butstill for the stronger Dirichlet norm. This gives existence of density for solutions of SDE’s underonly Lipschitz assumptions on the coefficients (cf [10] and [11]). More generally, Dirichlet formsare easy to construct in the infinite dimensional frameworks encountered in probability theoryand this yields a theory of errors propagation through the stochastic calculus (cf Bouleau [7]),also for numerical analysis of PDE and SPDE (cf Scotti [42]).As the Malliavin calculus has been extended to the case of Poisson measures and SDE’s withjumps, either dealing with local operators acting on the size of the jumps (Bichteler-Gravereaux-Jacod [4] Ma-Röckner[30] Léandre [26] [27] etc.) or based on the Fock space representation ofthe Poisson space and finite difference operators (Nualart-Vives [33] Picard [34] Ishikawa-Kunita[20] etc.), it is quite natural to attempt extending the Dirichlet forms arguments to such cases.This has been done first by Coquio [16] when the state space is Euclidean then by Denis [18] bya time perturbation, see also related works of Privault [36], Albeverio-Kondratiev-Röckner [1],Ma-Röckner [30].We shall give a general presentation of Dirichlet forms methods for the Poisson measuresin the spirit of the first approach (Bichteler-Gravereaux-Jacod [4]) which gives rise to a verysimilar situation like in Malliavin Calculus : a symmetric semi-group on the Poisson space anda local gradient satisfying the chain rule. With respect to preceding works in this direction weintroduce a major simplification due to a new tool the lent particle formula [12] which gives thegradient on the Poisson space by a closed formula. Thanks to this representation we obtainedwith Laurent Denis several results of existence of density [12] [13] and the method extends to C ∞ results (forthcoming paper). In this lecture I present the method and the main applicationsobtained up to now and I expose new results about the regularity of multiple Poisson integralsin connection with the Fock space representation that the Poisson space provides. It is organisedas follows : The functional analytic reasoning.
Dirichlet forms and non-Gaussian Malliavin calculus — Poisson random measures — Dirichletform on the Poisson space : the lent particle formula.
Practice of the method.
Other examples — Applications to SDE’s — A useful theorem of Paul Lévy.
Regularity results for multiple Poisson integrals.
Random Poisson measure and Fock space — Decomposition of D in chaos — Density for ( I ( g ) , . . . , I n ( g ⊗ n )) — Density for ( I n ( f ⊗ n ) , . . . , I n p ( f ⊗ n p p )) — Other functionals of Poissonintegrals — Density of I n ( f ) . 2 The functional analytic reasoning.
Let us first introduce the fundamental notions of the theory of local Dirichlet forms.
Let ( X, X , ν, d , γ ) be a local symmetric Dirichlet structure which admits a “carré du champ"operator. This means that ( X, X , ν ) is a measured space, ν is a σ -finite positive measure and thebilinear form e [ f, g ] = R γ [ f, g ] dν is a local Dirichlet form with domain d ⊂ L ( ν ) and carré duchamp γ (cf Fukushima-Oshima-Takeda [19] in the finite dimensional case and Bouleau-Hirsch[11] in a general setting). The form e is closed in L ( ν ) and the bilinear operator γ satisfies thefunctional calculus of class C ∩ Lip : ∀ f, g ∈ d n , ∀ F, G of class C ∩ Lip on R n γ [ F ( f ) , G ( g )] = X ij ∂ i F ( f ) ∂ j G ( g ) γ [ f i , g j ] . We write always γ [ f ] for γ [ f, f ] and e [ f ] for e [ f, f ] .The space d equipped with the norm ( k . k L ( ν ) + e [ ., . ]) is a Hilbert space that we will supposeseparable. It is then possible to generate the quadratic differential computations with γ by anordinary differential calculus thanks to the fact that a gradient exists (see Bouleau-Hirsch [11]ex.5.9 p. 242): there exist a separable Hilbert space H and a continuous linear map D from d into L ( X, ν ; H ) such that • ∀ u ∈ d , k D [ u ] k H = γ [ u ] . • If F : R → R is Lipschitz then ∀ u ∈ d , D [ F ◦ u ] = ( F ′ ◦ u ) Du, where F ′ is the Lebesguealmost everywhere defined derivative of F . • If F is C (continuously differentiable) and Lipschitz from R d into R (with d ∈ N ) then ∀ u = ( u , · · · , u d ) ∈ d d , D [ F ◦ u ] = d X i =1 ( ∂ i F ◦ u ) D [ u i ] . In [11] Chap VII we used for H a copy of the space L ( ν ) , but a wide choice is possible dependingon convenience.This differential calculus gives rise to integration by parts formulae as in classical Malliavincalculus. For all u ∈ d and v ∈ D ( a ) domain of the generator a associated with the Dirichletstructure, we have Z γ [ u, v ] dν = − Z ua [ v ] dν. (1)The space d ∩ L ∞ may be shown to be an algebra, hence if u , u ∈ d ∩ L ∞ Z u γ [ u , v ] dν = − Z u u a [ v ] dν − Z u γ [ u , v ] dν (2)Introducing now the adjoint operator δ of the gradient D , the equality with u ∈ d , U ∈ dom δ Z uδU dν = Z h D [ u ] , U i H dν (3)provides for ϕ Lipschitz Z ϕ ′ ( u ) h D [ u ] , U i H dν = Z ϕ ( u ) δU dν. (4)3ee [7] Chap V to VIII and [9] for applications of such formulae.But the Dirichlet structures do possess pecular features allowing to show existence of densitywithout using integration by parts arguments. This is based on the following important energyimage density property or (EID):For each positive integer d , we denote by B ( R d ) the Borel σ -field on R d and by λ d theLebesgue measure on ( R d , B ( R d )) . For f measurable f ∗ ν denotes the image of the measure ν by f . The Dirichlet structure ( X, X , ν, d , γ ) is said to satisfy (EID) if for any d and for any R d -valuedfunction U whose components are in the domain of the form U ∗ [(det γ [ U, U t ]) · ν ] ≪ λ d where det denotes the determinant. This property is true for any local Dirichlet structure with carré du champ when d = 1 (cfBouleau [5] Thm 5 and Corol 6). It has been conjectured in 1986 (Bouleau-Hirsch [10] p251)that (EID) were true for any local Dirichlet structure with carré du champ. This has been shownfor the Wiener space equipped with the Ornstein-Uhlenbeck form and for some other structuresby Bouleau-Hirsch (cf [11] Chap. II §5 and Chap. V example 2.2.4) but this conjecture being atpresent neither refuted nor proved in full generality, it has to be established in every particularsetting. For the Poisson space it has been proved by A. Coquio [16] when the intensity measureis the Lebesgue measure on an open set and we obtained with Laurent Denis a rather generalcondition ([12] Section 2 Thm 2 and Section 4) based on a criterion of Albeverio and Röckner[2] and an argument of Song [44]. The new regularity results that are presented here are basedon the (EID) property.Let us first explain the framework of Poisson measures and the notation of the configurationspace. We are given ( X, X , ν ) a measured space. We call it the bottom space . We assume that ν is σ -finite, that for all x ∈ X , { x } belongs to X and that ν is continuous or diffuse ( ν ( { x } ) = 0 ∀ x ).We consider a random Poisson measure N on ( X, X ) with intensity measure ν . Such arandom measure is characterized by the fact that for A ∈ X the random variable N ( A ) follows aPoisson law with parameter ν ( A ) and N ( A ) , . . . , N ( A n ) are independent for disjoint A i . Suchan object may be constructed on the space of countable sums of Dirac masses on ( X, X ) (theconfiguration space), by considering first the case where ν is bounded where the constructionis explicit and then proceding by product along a partition of ( X, X ) (see e.g. [6] or [7] ChapVI §3). We denote by (Ω , A , P ) the configuration space where N is defined, A is the σ -fieldgenerated by N and P its law. The space (Ω , A , P ) is called the upper space .The following density lemma (cf [12]) is the key of several proofs. Lemma 1.
For p ∈ [1 , ∞ [ , the set { e − N ( f ) : f > , f ∈ L ( ν ) ∩ L ∞ ( ν ) } is total in L p (Ω , A , P ) and { e iN ( f ) : f ∈ L ( ν ) ∩ L ∞ ( ν ) } is total in L p (Ω , A , P ; C ) . We set ˜ N = N − ν , then the identity E [( ˜ N ( f )) ] = R f dν, for f ∈ L ( ν ) ∩ L ( ν ) can beextended uniquely to f ∈ L ( ν ) and this permits to define ˜ N ( f ) for f ∈ L ( ν ) . The Laplace4haracteristic functional is the basis of all subsequent formulae E [ e i ˜ N ( f ) ] = e − R (1 − e if + if ) dν f ∈ L ( ν ) . (5)The creation and annihilation operators ε + and ε − well-known in quantum mechanics (see Meyer[31], Nualart-Vives [33], Picard [34] etc.) will play a central role for calculus on the configurationspace, they are defined in the following way: ∀ x, w ∈ Ω , ε + x ( w ) = w { x ∈ supp w } + ( w + ε x ) { x/ ∈ supp w } ∀ x, w ∈ Ω , ε − x ( w ) = w { x/ ∈ supp w } + ( w − ε x ) { x ∈ supp w } . (6)One can verify that for all w ∈ Ω , ε + x ( w ) = w and ε − x ( w ) = w − ε x for N w -almost all x (7)and ε + x ( w ) = w + ε x and ε − x ( w ) = w for ν -almost all x (8)We extend these operators to the functionals by setting: ε + H ( w, x ) = H ( ε + x w, x ) and ε − H ( w, x ) = H ( ε − x w, x ) . This extension recommands to be careful with the order of composition since we have for instance ( ε − ε + H )( x, ω ) = H ( x, ε + x ε − x ω ) (= H ( x, ε + x ω ) = ε + H ) (9)It is important to emphasize that since ν is continuous the two measures P × ν and P N = P ( dω ) N ( ω )( dx ) defined on the same sapce (Ω × X, A × X ) are mutually singular. Computationneeds to be careful with respect to negligible sets. The next lemma shows that the image of P × ν by ε + is nothing but P N whose image by ε − is P × ν : Lemma 2.
Let H be A ⊗ X -measurable and non negative, then E Z ε + Hdν = E Z HdN and E Z ε − HdN = E Z Hdν.
We will encounter also another notion, sometimes called a “marked" Poisson measure associ-ated with N , which needs here a rigorous construction.We are still considering N the random Poisson measure on ( X, X , ν ) and we are given anauxiliary probability space ( R, R , ρ ) . We construct a random Poisson measure N ⊙ ρ on ( X × R, X ⊗ R , ν × ρ ) such that if N = P i ε x i then N ⊙ ρ = P i ε ( x i ,r i ) where ( r i ) is a sequence ofi.i.d. random variables independent of N whose common law is ρ .The construction of N ⊙ ρ follows line by line the one of N . Let us recall it. We first study thecase where ν is bounded and we consider the probability space ( N , P ( N ) , P ν ( X ) ) × ( X, X , νν ( X ) ) N ∗ , where P ν ( X ) denotes the Poisson law with parameter ν ( X ) and we put N = Y X i =1 ε x i , ( with the convention X = 0) where Y, x , · · · , x n , · · · denote the coordinates maps. We introduce the probability space ( ˆΩ , ˆ A , ˆ P ) = ( R, R , ρ ) N ∗ , r , · · · , r n , · · · . On the probability space ( N , P ( N ) , P ν ( X ) ) × ( X, X , νν ( X ) ) N ∗ × ( ˆΩ , ˆ A , ˆ P ) , we define the random measure N ⊙ ρ = P Yi =1 ε ( x i ,r i ) . It is a Poissonrandom measure on X × R with intensity measure ν × ρ . For f ∈ L ( ν × ρ )ˆ E [ Z X × R f dN ⊙ ρ ] = Z X ( Z R f ( x, r ) dρ ( r )) N ( dx ) P − a.e. (10)and if f ∈ L ( ν × ρ )ˆ E [( Z X × R f dN ⊙ ρ ) ] = ( Z X Z R f dρdN ) − Z X ( Z R f dρ ) dN + Z X Z R f dρdN, (11)where ˆ E stands for the expectation under the probability ˆ P .If ν is σ -finite, this construction is extended by a standard product argument. Eventually inall cases, we have constructed N on (Ω , A , P ) and N ⊙ ρ on (Ω , A , P ) × ( ˆΩ , ˆ A , ˆ P ) , it is a randomPoisson measure on X × R with intensity measure ν × ρ , and identities (10) and (11) generalizeas follows: Proposition 3.
Let F be an A ⊗ X ⊗ R measurable function such that E R X × R F dνdρ and E R R ( R X | F | dν ) dρ are both finite then the following relation holds ˆ E [( Z X × R F dN ⊙ ρ ) ] = ( Z X Z R F dρdN ) − Z X ( Z R F dρ ) dN + Z X Z R F dρdN, (12) in particular if F is such that R F dρ = 0 P × ν -a.e., then ˆ E [( R X × R F dN ⊙ ρ ) ] = R X R R F dρdN. Proof.
Approximating first F by a sequence of elementary functions and then introducing apartition ( B k ) of subsets of X of finite ν -measure, this identity is seen to be a consequence of(11).Let us take the opportunity to state two formulae that we didn’t mention in our precedingarticles, and which may be useful in some context. Let F be measurable as in Prop 3 and say < F then ˆ E exp Z log F dN ⊙ ρ = exp Z (log Z F dρ ) dN (13) ˆ E Z F dN ⊙ ρ = Z ( Z F dρ ) dN (14)whose proofs follow the same lines as the construction of N ⊙ ρ and Prop 3. Now, after these notions related to the pure probabilistic Poisson space, we shall assume wehave on the bottom space a Dirichlet structure ( X, X , ν, d , γ ) as defined in section 2.1. And weattempt to lift up this structure to the Poisson space in a natural manner. This may be done inseveral ways (see e.g. the introduction of [12]). The method we will follow is not the simplest,we choose it because it enlightens the role of operators ε + and ε − in the upper gradient.First, thanks to (5) we obtain the following relation: for all f ∈ d and all h ∈ D ( a ) , E (cid:20) e i ˜ N ( f ) (cid:18) ˜ N ( a [ h ]) + i N ( γ [ f, h ]) (cid:19)(cid:21) = 0 . (15)6his relation and the explicit construction which may be done when ν is a bounded measure (cf[6]) suggest a candidate for the generator of the upper structure.Let us consider the space of test functions D = L{ e i ˜ N ( f ) with f ∈ D ( a ) ∩ L ( ν ) et γ [ f ] ∈ L ( ν ) } . and for U = P p λ p e i ˜ N ( f p ) in D , let us put A [ U ] = X p λ p e i ˜ N ( f p ) ( i ˜ N ( a [ f p ]) − N ( γ [ f p ])) . (16)The procedure to show that A is uniquely defined and is the generator of a Dirichlet formsatisfying the hoped properties, has two steps : first to construct an explicit gradient, then touse Friedrichs’ property. Gradients.
We will suppose as in section 2.1 that the bottom structure possesses a gradient that we denotefrom now on ( · ) ♭ . For convenience we assume it satisfies the following properties • constants belong to d loc (see Bouleau-Hirsch [11] Chap. I Definition 7.1.3.) ∈ d loc which implies γ [1] = 0 and ♭ = 0 . (17) • ( . ) ♭ is with values in the orthogonal subspace L ( R, R , ρ ) of in the space L ( R, R , ρ ) .This condition is costless since for the gradient only the Hilbert structure of H matters. Fromnow on we denote this gradient ( . ) ♭ .We take for candidate of the upper-gradient for F ∈ D the pre-gradient F ♯ = Z ε − (( ε + F ) ♭ ) dN ⊙ ρ. where N ⊙ ρ is the Poisson measure N “marked” by ρ as defined in section 2.2.Let us remark that thanks to Prop 3 and (17) we have ˆ E [( Z X × R ε − (( ε + F ) ♭ ) dN ⊙ ρ ) ] = Z ε − ( γ ( ε + F )) dN P - a.e. (18)For f ∈ D ( a ) ∩ L ( m ) , γ [ f ] ∈ L , we have e i ˜ N ( f ) ∈ D and ( e i ˜ N ( f ) ) ♯ = Z e i ˜ N ( f ) ( if ) ♭ dN ⊙ ρ what yields on D : ˆ E [ F ♯ G ♯ ] = X p,q λ p µ q e i ˜ N ( f p − g q ) N ( γ ( f p , g q )) (19)7 riedrichs’ argument. This enables us to show that the representation (16) does not depend on the expression of U and that A is indeed a symmetric negative operator on the dense subspace D of L ( P ) so thatFriedrichs’ argument applies (see [11] p.4 or [7] Lemma III.28 p.48) : it can be extended to a selfadjoint operator which may be proved to generate a Dirichlet form with domain D admitting acarré du champ Γ with a gradient extending ( . ) ♯ .It remains only a technical point to verify: the fact that D be dense in L ( P ) . This is notobvious because of the condition γ [ f ] ∈ L ( ν ) that we need in D in order A take its valuesin L ( P ) . In [12] we called it bottom core hypothesis (BC), it is not a real constraint in theapplications. We can state (cf [12]) : Theorem 4.
The formula ∀ F ∈ D , F ♯ = Z E × R ε − (( ε + F ) ♭ ) dN ⊙ ρ, extends from D to D , it is justified by the following decomposition : F ∈ D ε + − I ε + F − F ∈ D ε − (( . ) ♭ ) ε − (( ε + F ) ♭ ) ∈ L ( P N × ρ ) d ( N ⊙ ρ ) F ♯ ∈ L ( P × ˆ P ) where each operator is continuous on the range of the preceding one and where L ( P N × ρ ) isthe closed set of elements G in L ( P N × ρ ) such that R R Gdρ = 0 P N -a.s. Furthermore for all F ∈ D Γ[ F ] = ˆ E ( F ♯ ) = Z E ε − γ [ ε + F ] dN. This main result — that we call the lent particle formula — implies the validity of a functionalcalculus for the obtained Dirichlet structure (Ω , A , P , D , Γ) on the Poisson space that may besketched as follows:Let be H = Φ( F , . . . , F n ) with Φ ∈ C ∩ Lip ( R n ) and F = ( F , . . . , F n ) with F i ∈ D , wehave : a ) γ [ ε + H ] = P ij Φ ′ i ( ε + F )Φ ′ j ( ε + F ) γ [ ε + F i , ε + F j ] P × ν -a.e. b ) ε − γ [ ε + H ] = P ij Φ ′ i ( F )Φ ′ j ( F ) ε − γ [ ε + F i , ε + F j ] P N -a.e. c ) Γ[ H ] = R ε − γ [ ε + H ] dN = P ij Φ ′ i ( F )Φ ′ j ( F ) R ε − γ [ ε + F i , ε + F j ] dN P -a.e. Remark 5.
Let F ∈ D , by the theorem applying formula (13) to F ♯ gives ˆ E exp F ♯ = ˆ E exp Z ε − ( ε + F ) ♭ N ⊙ ρ = exp Z (cid:18) log Z exp ε − ( ε + F ) ♭ dρ (cid:19) dN = exp Z (cid:18) ε − log Z exp ( ε + F ) ♭ dρ (cid:19) dN (20)what may yield the characteristic function of the law of F ♯ under P × ˆ P : if we put R exp( iuε + F ) ♭ dρ =exp Ψ( u ) we obtain E ˆ E e iuF ♯ = E exp Z ε − Ψ( u ) dN . xample 1. Let Y t be a centered Lévy process with Lévy measure σ integrating x and such that a localDirichlet structure may be constructed on R \{ } with carré du champ γ [ f ] = x f ′ ( x ) . Withour notation ( X, X , ν ) = ( R + × R \{ } , Borelian sets , dt × σ ) .We define the gradient ♭ associated with γ by choosing ξ on the auxiliary space ( ˆΩ , ˆ A , ˆ P ) such that R ξ ( r ) dr = 0 and R ξ ( r ) dr = 1 and putting f ♭ = xf ′ ( x ) ξ ( r ) . The operator ♭ acts as a derivation with the chain rule ( ϕ ( f )) ♭ = ϕ ′ ( f ) .f ♭ (for ϕ ∈ C ∩ Lip or even only Lipschitz). N is the Poisson random measure associated with Y with intensity dt × σ such that R t h ( s ) dY s = R [0 ,t ] ( s ) h ( s ) x ˜ N ( dsdx ) for h ∈ L loc ( R + ) . (These hypotheses imply Y s = 0 a.s.)Let us study the existence of density for the pair ( Y t , E xp ( Y ) t ) where E xp ( Y ) is the Doléansexponential of Y . E xp ( Y ) t = e Y t Y s t (1 + ∆ Y s ) e − ∆ Y s . / We add a particle ( α, y ) i.e. a jump to Y at time α t with size y : ε +( α,y ) ( E xp ( Y ) t ) = e Y t + y Y s t (1 + ∆ Y s ) e − ∆ Y s (1 + y ) e − y = E xp ( Y ) t (1 + y ) . / We compute γ [ ε + E xp ( Y ) t ]( y ) = ( E xp ( Y ) t ) y . / We take back the particle : ε − γ [ ε + E xp ( Y ) t ] = (cid:0) E xp ( Y ) t (1 + y ) − (cid:1) y we integrate in N and that gives the upper carré du champ operator (lent particle formula): Γ[ E xp ( Y ) t ] = R [0 ,t ] × R (cid:0) E xp ( Y ) t (1 + y ) − (cid:1) y N ( dαdy )= P α t (cid:0) E xp ( Y ) t (1 + ∆ Y α ) − (cid:1) ∆ Y α . By a similar computation the matrix Γ of the pair ( Y t , E xp ( Y t )) is given by Γ = X α t (cid:18) E xp ( Y ) t (1 + ∆ Y α ) − E xp ( Y ) t (1 + ∆ Y α ) − (cid:0) E xp ( Y ) t (1 + ∆ Y α ) − (cid:1) (cid:19) ∆ Y α . Hence under hypotheses implying (EID) the density of the pair ( Y t , E xp ( Y t )) is yielded by thecondition dim L (cid:18)(cid:18) E xp ( Y ) t (1 + ∆ Y α ) − (cid:19) α ∈ J T (cid:19) = 2 where
J T denotes the jump times of Y between 0 and t .Making this in details we obtain Let Y be a real Lévy process with infinite Lévy measure with density dominating a positivecontinuous function = 0 near , then the pair ( Y t , E xp ( Y ) t ) possesses a density on R . xample 2. Let Y be a real Lévy process as in the preceding example.Let us consider a real càdlàg process K independent of Y and put H s = Y s + K s . Putting M = sup s t H s and computing successively ( ε + M ) , γ [ ε + M ] and applying the lent particleformula gives Proposition 6. If σ ( R \{ } ) = + ∞ and if P [sup s t H s = H ] = 0 , the random variable sup s t H s possesses a density. It follows that any real Lévy process X starting at zero and immediately entering R ∗ + , whoseLévy measure dominates a measure σ satisfying Hamza’s condition ([19] p105) and infinite, issuch that sup s t X s has a density. Example 3. Lévy’s stochastic area.
This example will show that the method can detect densities even when both the Malliavinmatrix is non invertible and the Lévy measure is singular.Let X ( t ) = ( X ( t ) , X ( t )) be a Lévy process with values in R with Lévy measure σ . Wesuppose that the hypotheses of the method are fulfilled, we shall explicit this later on.Let us consider first a general gradient on the bottom space : f ♭ = f ′ ξ + f ′ ξ where f ′ i = ∂f∂x i , and ξ , ξ are functions defined on R × R which satisfy: R R ξ ( · , r ) ρ ( dr ) = R R ξ ( · , r ) ρ ( dr ) = 0 , R R ξ ( x , x , r ) ρ ( dr ) = α ( x , x ) , R R ξ ( x , x , r ) ξ ( x , x , r ) ρ ( dr ) = α ( x , x ) , R R ξ ( x , x ) ρ ( dr ) = α ( x , x ) , so that γ [ f ] = α f ′ + 2 α f ′ f ′ + α f ′ . Let us consider the following vector involving Lévy’s stochastic area V = ( X ( t ) , X ( t ) , Z t X ( s − ) dX ( s ) − Z t X ( s − ) dX ( s )) . We have for < α < t and x = ( x , x ) ∈ R ,ε +( α,x ) V = V + ( x , x , X ( α − ) x + x ( X ( t ) − X ( α )) − X ( α − ) x − x ( X ( t ) − X ( α ))= V + ( x , x , x ( X ( t ) − X ( α )) − x ( X ( t ) − X ( α ))) because ε + V is defined P × ν × dα -a.e. and ν × dα is diffuse, so ( ε + V ) ♭ = ( ξ , ξ , ξ ( X ( t ) − X ( α )) − ξ ( X ( t ) − X ( α ))) and γ [ ε + V ] = α α Aα − Bα α α Aα − Bα Aα − Bα Aα − Bα A α − ABα + B α denoting A = ( X ( t ) − X ( α )) and B = ( X ( t ) − X ( α )) .10his yields ε − A = X ( t ) − ∆ X ( α ) − X ( α − ) let us denote it ˜ Aε − B = X ( t ) − ∆ X ( α ) − X ( α − ) let us denote it ˜ B and eventually Γ[ V ] = X α t α (∆ X α ) α (∆ X α ) ˜ Aα (∆ X α ) − ˜ Bα (∆ X α ) ∼ α (∆ X α ) ˜ Aα (∆ X α ) − ˜ Bα (∆ X α ) ∼ ∼ ˜ A α (∆ X α ) − A ˜ Bα (∆ X α ) + ˜ B α (∆ X α ) the symbol ∼ denoting the symmetry of the matrix.Considering the case α = 0 let us take the Lévy measure of ( X , X ) expressed in polarcoordinates as ν ( dρ, dθ ) = g ( θ ) dθ. ]0 , ( ρ ) dρρ with g locally bounded and such that it dominates a continuous and positive function near .Then V = ( X ( t ) , X ( t ) , R t X ( s − ) dX ( s ) − R t X ( s − ) dX ( s )) has a density (and condition (0.4)of [14] or of [34] prop1.1 are not fulfilled).Considering now the case ξ = λ ( x , x ) ξ which applies to V = ( X ( t ) , [ X ] t , R t X ( s − ) d [ X ]( s ) − R t [ X ]( s − ) dX ( s )) .The Lévy measure of ( X , [ X ]) is carried by the curve x = x . We have λ ( x , x ) = 2 x .We arrive to the sufficient condition : V has a density as soon as the Lévy measure of X isinfinite and satisfies hypotheses for (BC) and (EID). (cf [12] and [13]). Computation with the lent particle formula.
The presence of operators ε + and ε − in the lent particle formula (Thm 4) which exchange themutually singular measures P N and P × ν , requires to be more careful than in the usual stochasticcalculus where all is defined P -a.s. We make some remarks and give some examples to help thereader to become familiar with this tool. The lent particle formula extends to D loc . The space D loc is a remarkable specific feature of local Dirichlet forms with carré du champ :the carré du champ operator extends to functions locally – in a measurable sense – in D (cf [11]Chap I §7.1). We denote D loc the set of applications F : Ω R such that there exists a sequence Ω n ∈ A suchthat ∪ n Ω n = Ω and ∃ F n ∈ D with F = F n on Ω n . The fact that (EID) is always true for d = 1 (cf [5]) shows that, for F ∈ D loc , Γ[ F ] is uniquelydefined and may be evaluated by Γ[ F n ] on Ω n . The operator ♯ extends to D loc by putting F ♯ = F ♯n on Ω n . For F in D loc , the formulae F ♯ = Z ε − (( ε + F ) ♭ ) dN ⊙ ρ Γ[ F ] = Z ε − ( γ [ ε + F ]) dN resume a computation done on each Ω n . 11 egligible sets. As it was recalled above at the beginning of section 3, it is recommended to write down thenegligible sets at each equality e.g. ε + ( ˜ N f ) = ˜
N f + f P × ν -a.e. ε − ( ˜ N f ) = ˜
N f − f P N -a.e. ε + ( e i ˜ Nf g ) = e i ˜ Nf e if g P × ν -a.e. ε − ( e i ˜ Nf g ) = e i ˜ Nf e − if g P N -a.e. Remark 7.
Let us observe that if H ( ω, x ) = G ( ω ) g ( x ) where G is defined P -a.s. and g ν -a.e.then H belongs necessarily to a single class P N -a.e. So that we may apply to H both operators ε + and ε − without ambiguity. This will be used further about multiple Poisson integrals. A simplified sufficient condition.
Theorem 4 gives a method for obtaining Γ[ F ] for F ∈ D or F ∈ D n , then with the hypothesesgiving (EID) it suffices to prove det Γ[ F ] > P -a.s. to assert that F has a density on R n . Letus mention a stronger condition which may be also useful in some applications. By the followinglemma that we leave to the reader Lemma 8.
Let M α be random symmetric positive matrices and µ ( dα ) a random positive mea-sure. Then { det R M α µ ( dα ) = 0 } ⊂ { R det M α µ ( dα ) = 0 } , it is enough to have R det ε − ( γ [ ε + F ]) dN > P -a.s. hence enough that det ε − ( γ [ ε + F ]) be > P N -a.e. We obtain, by lemma 2, that a sufficient condition for the density of F is det γ [ ε + F ] > P × ν × dt -a.e. (or equivalently that the components of the vector ( ε + F ) ♭ be P × ν × dt -a.e.linearly independent in L ( ρ ) ). The energy image density property (EID). We gave in Bouleau-Denis [12] general conditions on the bottom structure ( X, X , ν, d , γ ) to satisfy (EID) and for this property to be lifted up to the upper space (Ω , A , P , D , Γ) . Here arethese conditions in a simplified form: Proposition 9.
Suppose ( X, X , ν ) = ( R d , B ( R d ) , k ( x ) dx ) with k continuous on an open set offull Lebesgue measure and suppose the carré du champ operator is defined on the test functions C ∞ K infinitely differentiable with compact support by the formula X ij ξ ij ( x ) ∂ i f ( x ) ∂ j f ( x ) (21)where ξ is locally bounded and locally elliptic i.e. for every compact K there are constants C K < ∞ and c K > such that ∀ x ∈ K, ∀ c ∈ R d C K | c | > P di,j =1 ξ ij ( x ) c i c j > c K | c | , then thebilinear form e [ u, v ] = 12 Z R r X i,j ξ ij ( x ) ∂ i u ( x ) ∂ j v ( x ) k ( x ) dx. (22)defined on C ∞ K is closable and its closure defines a Dirichlet form ( e, d ) with carré du champgiven by (21), and this structure satisfies (EID) and (BC).12t is useful for many examples to remark that the preceding case allows to extend (EID) and(BC) to situations where ν is singular w.r. to Lebesgue measure.Let ( R p \{ } , B ( R p \{ } ) , ν, d , γ ) be a Dirichlet structure on R p \{ } satisfying (EID). Let U : R p \{ } 7→ R q \{ } be an injective map ( p < q ) such that U ∈ d q . Then U ∗ ν is σ -finite. Ifwe put d U = { ϕ ∈ L ( U ∗ ν ) : ϕ ◦ U ∈ d } e U [ ϕ ] = e [ ϕ ◦ U ] γ U [ ϕ ] = d U ∗ ( γ [ ϕ ◦ U ] .ν ) d U ∗ ν then the term ( R q \{ } , B ( R q \{ } ) , U ∗ ν, d U , γ U ) is a Dirichlet structure satisfying (EID). Addi-tional regularity assumptions make U transport also property (BC).Now it is possible to lift up (EID) from the bottom to the upper space if two conditions arefulfilled. First to be able to share the bottom space on a partition of sets of finite ν -measure.Second that the obtained Dirichlet structures are such that any finite product satisfies (EID). Theprecise formulation is given in Bouleau-Denis [12] Section 4. This covers all cases encounteredin practice. Example 4. Nearest point of the origin.
This example shows the quickness of the method which has, in some sense, to be paid by thecare to put on negligible sets.Let us take for the bottom space ( R d , B ( R d ) , ν, d , γ ) satisfying (BC), assuming the identitymap j on R d belong to d d and γ [ | j | ] > , the measure ν being infinite, possibly carried by asurface or a curve. Let us consider the functional H defined on (Ω , A , P ) H ( ω ) = inf x ∈ supp ( ω ) | x | . The inf is reached because the measure ν is σ -finite. We have ε + x H = | x | ∧ H P × ν -a.e.We will suppose that the measure ν does not charge the level surfaces of | x | i.e. the spherescentered at O. Then for fixed ω , x ε + x H belongs to d and we have ( ε + x H ) ♭ = ( | j | ) ♭ | j | H = ( | j | ) ♭ | j |
1] : | x + B t ( w ) | = K ( w ) } .It follows that Γ[ K ] = ˆ E [( K ♯ ) ] = T > a.s. if x = 0 . B) Let us come back to our usual notation. For the bottom space we take ( X, X , ν ) = ( R × W, B ( R ) × W , λ × m ) where λ is the 3-dimensional Lebesgue measure, that we equip with theproduct Dirichlet structure of the zero form on R and the O-U-form on the Wiener space. Thestructure ( X, X , ν, d , γ ) is thus naturally endowed with a gradient induced by the gradient usedin part A) and that we denote now ♭ as usual, it is with values in L ( ˆ m ) . The hypothesis (BC)is fulfilled.We construct the upper structure (Ω , A , P , D , Γ) which describes a gas of Brownian particles.We denote ( x, w ) the current point of X and we consider the functional H ( ω ) = inf t ∈ [0 , x, w ) ∈ supp ω | x + B t ( w ) | . We apply the lent particle method : ε +( x,w ) H = ( inf t ∈ [0 , | x + B t ( w ) | ) ∧ H Here the measure λ × m does not charge the level sets of (inf t ∈ [0 , | x + B t ( w ) | ) and we have ( ε + H ) ♭ = (inf t ∈ [0 , | x + B t ( w ) | ) ♭ { (inf t ∈ [0 , | x + B t ( w ) | ) H } = (inf t ∈ [0 , | x + B t ( w ) | ) ♭ { (inf t ∈ [0 , | x + B t ( w ) | ) Let ( ξ, η ) be a 2-dimensional Lévy process starting from (0,0). The process X t = e ξ t ( x + Z t e − ξ s dη s ) t > x ∈ R is a homogeneous Markov process called generalized O-U process driven by ( ξ, η ) (cf [15]). It ispossible to see by the classical Malliavin calculus that if ( ξ, η ) possesses a Brownian part then X t has a density. We exclude this case now and suppose that the Lévy measure carries a Dirichlet15orm satisfying (BC) in order to apply the method (without care of (EID) because X t is onedimensional).Let us begin by computing Γ[ X t ] by the lent particle method.Let ( α, ξ, η ) denote the current point of X = R + × R × R , ε +( α,ξ,η ) ξ t = ξ t + ξ α t ε +( α,ξ,η ) ξ t − = ξ t − + ξ α Several forms of interaction potential are encountered in physics for an infinite system of inter-acting particles: exp {− β P ij Ψ( X i − X j ) } , αβ n Q ij g ( | X i − X j | ) or exp { P ij a ( X i ) a ( X j ) b ( | X i − X j | ) } etc.Let us consider the functional Φ = R ϕ ( x ) ϕ ( y ) ψ ( | x − y | ) N ( dx ) N ( dy ) where the functions ϕ and ψ are regular, ψ (0) = 0 , N being a random Poisson measure on R .After computing as usual ε + x Φ , ( ε + Φ) ♭ and ε − ( ε + Φ) ♭ , the lent particle theorem gives Γ[Φ] = Z V ( x ) t γ [ j, j t ] V ( x ) N ( dx ) where j is the identity on R and V ( x ) is the column vector V ( x ) = Z (cid:0) ϕ ( α ) ψ ( | x − α ] ) ∇ ϕ ( x ) + 4 ϕ ( x ) ϕ ( α ) ψ ′ ( | x − α | )( x − α ) (cid:1) N ( dx ) . If the bottom structure is such that γ [ j, j t ] may be chosen to be the identity matrix, we have Γ[Φ] = Z (cid:12)(cid:12)(cid:12)(cid:12)Z F ( x, y ) N ( dy ) (cid:12)(cid:12)(cid:12)(cid:12) N ( dx ) (23)with F = [2 ψ ( | x − y | ) ∇ ϕ ( x ) + 4 ϕ ( x ) ψ ′ ( | x − y | )( x − y )] ϕ ( y ) .In order to study the positivity of Γ[Φ] , we will use the following lemma (due to Paul Lévy1931) on which we will come back in the next section.17 emma 10. Let f be measurable on the bottom space such that R | f | ∧ dν < + ∞ .If ν { f = 0 } = + ∞ then the law of N ( f ) is continuous.That gives us the following result Proposition 11. If F is such that (i) ∃ G ∈ L ( ν ) : | F ( x, y ) | G ( y ) , (ii) ∀ y x F ( x, y ) iscontinuous, (iii) ∀ x ν { F ( x, . ) } = + ∞ , then ( R F ( x, y ) N ( dy ) = 0) P -a.s. Proof. For ω outside a negligible set F ( x, . ) is bounded in modulus by an integrable functionfor N ( ω, dy ) , hence x R F ( x, y ) N ( dy ) is continuous by dominated convergence, hence theset { x : R F ( x, y ) N ( dy ) = 0 } is open; by the property (iii) and the lemma this set contains acountable dense set, hence all the space. (cid:3) It follows that if the bottom structure satisfies (BC) Φ has a density. Let d ∈ N ∗ , we consider the following SDE : X t = x + Z t Z X c ( s, X s − , u ) ˜ N ( ds, du ) + Z t σ ( s, X s − ) dZ s (24)where x ∈ R d , c : R + × R d × X → R d and σ : R + × R d → R d × n , Z is a semi-martingale and ˜ N a compensated Poisson measure.The lent particle method allows to apply the machinery of Malliavin calculus faster thanusual and under a set of hypotheses that express the Lipschitz character of the coefficient andsome other regularity assumptions for the details of which we refer to [13].Let us emphasize that applying the method to SDE’s uses reasoning in complete functionalspaces in which may be computed and solved the stochastic differential equations giving the ♯ of the solution. This takes full advantage of the fact that the lent particle formula is proved notonly on a set of test functions but on the space D itself.In [13] applications are given to McKean-Vlasov type equation driven by a Lévy process andto stable like processes. Example 9. A regular case violating Hörmander conditions. The following SDE driven by a two dimensional Brownian motion X t = z + R t dB s X t = z + R t X s dB s + R t dB s X t = z + R t X s dB s + 2 R t dB s . (25)is degenerate and the Hörmander conditions are not fulfilled. The generator is A = ( U + U )+ V and its adjoint A ∗ = ( U + U ) − V with U = ∂∂x + 2 x ∂∂x + x ∂∂x , U = ∂∂x + 2 ∂∂x and V = − ∂∂z − ∂∂z . The Lie brackets of these vectors vanish and the Lie algebra is of dimension2: the diffusion remains on the quadric of equation x − x + x − t = C. Let us now consider the same equation driven by a Lévy process :18 Z t = z + R t dY s Z t = z + R t Z s − dY s + R t dY s Z t = z + R t Z s − dY s + 2 R t dY s (26)under hypotheses on the Lévy measure such that the bottom space may be equipped with thecarré du champ operator γ [ f ] = y f ′ + y f ′ satisfying (BC) and (EID). Applying the lentparticle method is as usual and shows easily that if the Lévy measures of Y and Y are infinite Z t has a density on R . See [12] for details. The regularizing property is related to the fact thatequation (26) is not under the canonical form in the sense of Kunita [23] [24]. The next example,on the contrary shows a Lévy process in R living on a hyperbolic paraboloid. Example 10. For α ∈ R , let us consider the diffusion solution of X t = α + Z t U ( X s ) ◦ dB s + Z t U ( X s ) ◦ dB s where B = ( B , B ) is a standard Brownian motion with values in R , integrals being in theStratonovich sense, and vectors U and U being given by U ( x ) = x x − a x x x x + a x x x ( a + x ) U ( x ) = x x + a x x x x − a x x x ( a + x ) with a = α + α − α .Then the diffusion ( Z t ) remains on the quadric of equation x + x − x = a . (27)Now let us consider two independent Lévy processes ( Y t ) , ( Y t ) and the equation Z t = α + Z t U ( Z s − ) dY s + Z t U ( Z s − ) dY s (28)the Markov process with jumps Z remains on the hyperbolic paraboloid (27) as seen by applyingIto formula. This is due to the fact that the HP is a ruled manifold and at each point of itthe jumps of Z are in the direction of either generatrix crossing at this point. Equation (28) iscanonical in Kunita’s sense. Using a map from the HP to R the method allows to show thedensity of the law of Z t w.r. to the area measure on the HP. It is the occasion to rectify a historical injustice about the remarkable article of Paul Lévy “Surles séries dont les termes sont des variables éventuelles indépendantes" which appeared in StudiaMathematica in 1931 [28]. This article is almost never cited up to now (today the search enginsdo not mention any citation of this article) and the textbooks of K.I. Sato [41] and of J. Bertoin[3] do not quote it. One of his theorems, that we recall below, is generally attributed to Hartman19nd Wintner “On the infinitesimal generator of integral convolutions" Amer. J. Math. 64, (1942)273-298, which was published ten years later.Paul Lévy’s results may be stated as follows: Theorem 12. Let X n be a sequence of independent real random variables such that the series P X n converges almost surely.a) If for any sequence of constants ( a n ) , P P { X n = a n } diverges, P X n has a continuouslaw.b) If there is a sequence ( a n ) s.t. P P { X n = a n } converges and if the lower bound of the totalmass of the discrete part of the laws of the X n ’s is zero, then the law of P X n is continuous. It follows from this theorem that any process with independent increments whose Lévy mea-sure in infinite has a continuous law. In the framework of random Poisson measures it giveseasily Lemma 10 above. Remark 13. If f ∈ L ( ν ) and ν { f = 0 } = + ∞ then the law of N ( f ) is continuous butits characteristic function does not necessarily tend to zero at infinity, in other words is notnecessarily a Rajchman measure (cf [37] [38] or [8]) . This gives an easy way to constructcontinuous measures which are not Rajchman. Let m = f ∗ ν , m is σ -finite and integrates x 7→ | x | .Since E e iuNf = e R ( e iuf − iuf ) dν the law of N f is Rajchman iff lim | u |→ + ∞ R (1 − cos ux ) m ( dx ) =+ ∞ . If we choose a step function for f so that m = P ε n , we have R (1 − cos 2 k πx ) m ( dx ) = P ∞ j =0 (1 − cos π j ) < + ∞ so that the law of N f is continuous and not Rajchman. Let us first recall some links of our study with the Fock space. We recall that ν is continuous (i.e. diffuse). Let us call simple the measurable functions f definedon ( X m , X ⊗ m ) which are symmetric, finite sums of weighted indicator functions of sets of theform A × · · · × A m with disjoint A i ’s.On simple functions if we define I m ( f ) = Z X m f ( x , . . . , x m ) ˜ N ( dx ) · · · ˜ N ( dx m ) it is easily seen that E [ I m ( f ) I n ( g )] = δ m,n n ! h f, g i L ( X m , X ⊗ m ,ν × m ) . Thanks to this equality I m ( f ) may be extended to f ∈ L ( X m , X ⊗ m , ν × m ) so that denoting ˜ f the symmetrized f , I m ( f ) = I m ( ˜ f ) and E [ I m ( f ) I n ( g )] = δ m,n n ! h ˜ f , ˜ g i L ( X m , X ⊗ m ,ν × m ) . Let us observe that for f ∈ L ( X m , X ⊗ m , ν × m ) the formula I m ( f ) = Z X m f ( x , · · · , x m ) {∀ i = j,x i = x j } ˜ N ( dx ) · · · ˜ N ( dx m ) . There is an obvious misprint in this paper p128 line 20 where = has to be change into = . 20s a symbolic notation, because on the right hand side, the quantities to be substracted to theintegral on X m are generally not defined for non regular functions f .It has a sense if f is well defined on diagonals by continuity, X being supposed topological.A sense may also be yielded by Hilbertian methods, supposing f allows to define trace operators.The sub-vector space of L (Ω , A , P ) generated by the variables I n ( f ) , f ∈ L ( X n , X ⊗ n , ν × n ) is the Poisson chaos of order n denoted C n . The equality L (Ω , A , P ) = R ⊕ + ∞ n =1 C n . (29)has been proved by K. Ito (see [21]) in 1956. This proof is based on the fact that the set { N ( E ) · · · N ( E k ) , ( E i ) disjoint sets in X } is total in L (Ω , A , P ) .There are now several proofs of this result. A combinatorial proof is possible by counting therole of successive diagonals (cf [39] and [12] §4.1.) By transportation of structure, the density ofthe chaos has a short proof using stochastic calculus for the Poisson process on R + (cf Dellacherie-Maisonneuve-Meyer [17] p207).Thanks to the density of the chaos the following expansion is easily obtained (cf [45]) for u ∈ L ∩ L ∞ ( ν ) with small k u k ∞ , e N (log(1+ u )) − ν ( u ) = 1 + + ∞ X n =1 n ! I n ( u ⊗ n ) . (30)Let us mention the relationship between the strongly continuous semigroup of the bottom struc-ture p t in L ( ν ) and the one of the upper structure P t in L ( P ) (see [12] for a proof). For all u measurable function with − u , ∀ t > , P t [ e N (log(1+ u )) ] = e N (log(1+ p t u )) . (31)By (30) and (31) the vector spaces C n are preserved by P t and P t ( I n ( u ⊗ n )) = I n (( p t u ) ⊗ n )) . (32)It is generally spoken of second quantization for the transform ( p t ) ( P t ) . More precisely thesecond quantization maps the generator a of p t to an operator on the Fock space which maybe then lifted up either on the Wiener space or on the Poisson space and in this later casecorresponds to the generator A of P t . Remark 14. Let us suppose that the bottom semigroup p t be generated by a transition kernel ˜ p t ( x, dy ) from ( X, X ) into itself, which be simulatable in the sense that there exists a probabilityspace – that we choose here for the sake of simplicity of notation to be ( R, R , ρ ) – and a familyof random variables η t ( x, r ) such that the law of η t ( x, r ) under ρ ( dr ) be ˜ p t ( x, dy ) .Then, using our notation in which we have ω = R ε x N ( dx ) , the fact that the upper semigrouprepresents the evolution of independent particles each governed by p t and with initial law N (seethe introduction of [12]) may be expressed, for F A -measurable and bounded, by the formula P t F = ˆ E F ( Z ε η t ( x,r ) N ⊙ ρ ( dxdr )) (33)in analogy with the Mehler formula for the Ornstein-Uhlenbeck semigroup on the Wiener spaceor extensions of it (see [7] p116). Applying (33) to F = exp N log(1 + g ) for − g gives P t F = ˆ E exp Z log(1 + g ( η t ( x, r )) N ⊙ ρ ( dxdr ) P t F = exp N log( R (1 + g ( η t ( x, r )) ρ ( dr )) = exp N log(1 + p t g ) . (cid:3) Remark 15. Surgailis [45] has shown that in the correspondence between p t and P t given by(32) a necessary and sufficient condition P t be Markov is that p t and its adjoint be Markovoperators (i.e. positivity preserving and s.t. p t ).In our framework p t is selfadjoint and so is P t . (cid:3) D in chaos. Let us precise some notation. On the upper space (Ω , A , P , D , Γ) the Dirichlet form is denoted E . The product structure ( X, X , ν, d , γ ) n will be denoted ( X n , X ⊗ n , ν × n , d n , γ n ) (cf [11] ChapV). It is endowed with the Dirichlet form e n [ f ] = R γ n [ f ] dν . The functions in d n which aresymmetric define a sub-structure of ( X n , X ⊗ n , ν × n , d n , γ n ) denoted ( X n , X ⊗ nsym , ν × n , d n,sym , γ n ) .The semigroup associated with e n is denoted p ⊗ nt . Our choice of gradient for the bottom space(see §2.3 above) induces a gradient for ( X n , X ⊗ n , ν × n , d n , γ n ) that we denote ( · ) ♭ n with valuesin ( L ( R, R , ρ )) ⊗ n : ( f ♭ n )( x , r , x , r , · · · , x n , r n ) = ( f ( · , x , · · · , x n )) ♭ ( x , r ) + ( f ( x , · , x , · · · , x n )) ♭ ( x , r ) + · · · let us note that if f is symmetric, then f ♭ n is symmetric of the pairs ( x i , r i ) .Let be f ( x , . . . , x m ) = f ( x ) · · · f m ( x m ) ∈ d m and g ( x , . . . , x n ) = g ( x ) · · · g n ( x n ) ∈ d n . Bypolarization of (32) P t I m f = I m p ⊗ mt f gives E t [ I m f, I n g ] = t h I m f − P t I m f, I n g i L ( P ) = t h I m ( f − p ⊗ mt f ) , I n g i = δ mn m ! h f − p ⊗ mt ft , g i L ( ν × m ) . By the theory of symmetric strongly continuous contraction semigroups, we have F ∈ D if andonly if lim t ↓ ↑ E t [ F ] < + ∞ and E [ F ] = lim t ↓ E t [ F ] . Taking f = g , we obtain that I m f ∈ D and E [ I m f ] = m ! e m [ f ] . Then by density we obtain Proposition 16. For f ∈ d m the random variable I m f (= I m ( ˜ f )) belongs to D . The vectorspaces D m generated by I m f for f ∈ d m , are closed and orthogonal in D . The sum D = R M n > D n is direct in the sense of the Hilbert structure of D ( k · k D = k · k L + E [ · ]) .Every function F in D decomposes uniquely F = E [ F ] + X n > I n ( F n ) with F n ∈ d n . roof. It remains only to prove the density of the Dirichlet chaos D n . Let be F ∈ D and let F = P I n ( F n ) be its L -chaos expansion. Then t h F − P t F, F i L ( P ) = t P n > h I n ( F n − p ⊗ nt F n ) , I n F n i L ( P ) = P n > n ! h F n − p ⊗ nt F n t , F n i L ( ν × n ) . Since on the left-hand side t h F − P t F, F i ↑ E [ F ] < + ∞ it follows that all terms on the right-handside, which are increasing, possess limits what yields F n ∈ d n and the proposition follows.Let us emphasize that this proof is only based on the relation of second quantization (32) andwould be still valid on the Wiener space for instance equipped with a generalized Mehler typestructure (cf e.g. [7] p113 et seq.) or on the Poisson space equipped with a non local Dirichletform on the bottom space.Let u ∈ L ∞ ∩ d , applying the gradient operator ♯ to the two sides of (30) gives e N log(1+ tu ) − tν ( u ) Z tu ♭ tu dN ⊙ ρ = X n > t n n ! ( I n ( u ⊗ n )) ♯ what yields, taking terms in t n on both sides ( I n ( u ⊗ n )) ♯ = n − X q =0 ( − q n !( n − − q )! I n − − q ( u ⊗ ( n − − q ) ) Z u q u ♭ dN ⊙ ρ. (34)and i ! 1 j ! Γ[ I i u ⊗ i , I j v ⊗ j ] = Z i X k =1 I i − k u ⊗ ( i − k ) ( i − k )! ( − k u k − ! j X ℓ =1 I j − ℓ v ⊗ ( j − ℓ ) ( j − ℓ )! ( − ℓ v ℓ − ! γ [ u, v ] dN. (35)If f is the symmetrized of f ( x ) · · · f m ( x m ) then (34) writes I m ( f ) ♯ = Z (cid:16) mI m − f ♭ − m ( m − I m − f ♭ + m ( m − m − I m − f ♭ − · · · (cid:17) dN ⊙ ρ (36)where I m − p acts on the m − p first arguments of f and ♭ acts on the last one, all free argumentsbeing taken on the same point x .Extending formulae (34)-(36) from tensor products to general functions f ∈ d m supposes apriori that f does possess traces on diagonals. Indeed let us suppose f and g be regular so thatvalues on diagonals make sense, then defining for regular symmetric functions f ( x , . . . , x m ) and g ( y , . . . , y n ) the ( k, ℓ ) - γ -contraction, for k m and ℓ n , denoted f γ ≍ k, ℓ g as follows f γ ≍ k, ℓ g ( x , · · · , x m − k , y , · · · , y n − ℓ , x ) = γ [ f ( x , · · · , x m − k , x, · · · , x, · ) , g ( y , · · · , y n − ℓ , x, · · · , x, · )]( x ) , f γ ≍ k, ℓ g is symmetric in ( x , · · · , x m − k ) and in ( y , · · · , y n − ℓ ) . Then formulae (34)-(36) extend to symmetric functions f and g as Γ[ I m ( f ) , I n ( g )] = m X k =1 n X ℓ =1 ( − k + ℓ m ! n !( m − k )! n − ℓ )! I m − k I n − ℓ Z ( f γ ≍ k, ℓ g ) dN. (37)where I m − k operates on the x i ’s, I n − ℓ operates on the y j ’s and N on x .But this formula is unsatisfactory because we know that I m ( f ) is defined and in D for generalfunctions f ∈ d m which dont have defined values on diagonals in general. Actually the values ondiagonals cancel in formula (37). To see this we have to consider the Fock space for the gradientand to come back to the lent particle formula.The random Poisson measure N ⊙ ρ (cf §2.2) is defined on (Ω × ˆΩ , A ⊗ ˆ A , P × ˆ P ) with intensity ν × ρ on ( X × R, X ⊗ R ) . It possesses an expansion in chaos : ∀ F ∈ L ( P × ˆ P ) F = E ˆ E F + X n > J n ( F n ) where J n denotes the multiple integral for ^ N ⊙ ρ and where F n ∈ L sym (( ν × ρ ) × n ) .Let us remark that the random Poisson measure N may be seen as a function of N ⊙ ρ andthat the multiple integrals I n are nothing else but J n applied to a function G ( x , r , · · · , x n , r n ) not depending on the r i ’s. We can now state Proposition 17. Let be f ∈ d m,sym , by Prop 16 the multiple integral I m ( f ) belongs to D .a) Its gradient is given by ( I m ( f )) ♯ = Z ε − ( I m − ( f )) ♭ dN ⊙ ρ = m Z I m − ( ϕ ) N ⊙ ρ ( dxdr ) (38) where we note ψ ( x , . . . , x m − , x, r ) = ( f ( x , . . . , x m − , · )) ♭ ( x, r ) and ϕ is defined as ϕ ( x , . . . , x m − , x, r ) = ψ ( x , . . . , x m − , x, r )1 { x i = x ∀ i =1 ,...,m − } so that ϕ ( · , · · · , · , x, r ) ∈ L sym ( ν × ( m − ) and I m − ( ϕ ) is defined.b) This gradient may also be written ( I m f ) ♯ = J m ( f ♭ m ) (39) so that Γ[ I m ( f ) , I n ( g )] = ˆ E [ J m ( f ♭ m ) J n ( g ♭ n )] . (40) Proof. Let be f ∈ d m,sym . Let us apply the lent particle formula to I m ( f ) . We have ε + I m ( f ) = I m ( f ) + mI m − f P × ν -a.e.and since I m ( f ) does not depend on x ( ε + I m ( f )) ♭ = m ( I m − f ) ♭ P × ν × ρ -a.e.24ow, applying the operator ε − amounts to take the preceding relation with ω changed into ε − ω and to work under the measure P N instead of P × ν . That means that a functional F ( ˜ N ( u ) , x ) is changed into ε − x ( F ( ˜ N ( u ) , x )) = F ( Z u ( y )1 { y = x } ˜ N ( dy ) , x ) P N -a.e.Taking m = 2 for instance, we see that ε − ( I f ) must be written P N -a.e. R f ( y, x )1 { y = x } ˜ N ( dy ) instead of ( I f )( x ) − f ( x, x ) . Thus the part a) of the statement is a direct application of thelent particle formula.b) Since ♭ takes its values in L ( R, R , ρ ) , it is equivalent to use the compensated randommeasure ^ N ⊙ ρ instead of N ⊙ ρ in (38).Now m R I m − ( ϕ ) d ^ N ⊙ ρ = J m ( f ♭ m ) as seen by beginning with f = u ⊗ m , then polarizing to f symmetrized of u ⊗ · · · ⊗ u m and then to general f ∈ d n,sym by density.Let us remark that formula (38) allows a new simple proof of the orthogonality of the chaosin D . Let f be as in the proposition. We have E [ I m f ] = E Γ[ I m f ] = E m R γ [ I m − ( f { x i = x ∀ i } )] N ( dx )= m R ε − γ [ I m − f ] dN d P = m R γ [ I m − f ] d P dν (by Lemma 2) = m ( m − R γ [ f ] dν × ( m − dν = m !2 e m [ f ] . and similarly with the scalar products. Now (39) yields an even shorter proof using the orthog-onality of the chaos generated by J n under P × ˆ P , since h f ♭ m , g ♭ m i L ( ν × ρ ) m = 2 e m [ f, g ] .Contrarily to the Wiener case the random variables I m ( f ) are not regular in general. Theirdistributions may contain Dirac masses. Even in the first chaos the ♯ or the Γ applied to I u = ˜ N u yields a non deterministic result, and the sharp operator does not diminish the orderof the chaos. Studying regularity of multiple integrals needs therefore additional hypotheses. ( I ( g ) , . . . , I n ( g ⊗ n )) . Relation (34) yields immediately i ! 1 j ! Γ[ I i , I j ] = Z i X k =1 I i − k ( i − k )! ( − k g k − ! j X ℓ =1 I j − ℓ ( j − ℓ )! ( − ℓ g ℓ − ! γ [ g ] dN. (41)Let us denote I the column vector of ( I , . . . , I n ) , we have Γ[ I , I t ] = Z V V t γ [ g ] dN with V the column vector of ( − , − I + g, . . . , n ! P nk =1 I n − ( n − k )! ( − k g k − ) .Let us precise now some hypotheses. We suppose ν { γ [ g ] > } = + ∞ and that assumptions arefulfilled such that we have (BC) on the bottom space and (EID) on the upper space, as usual.If for some ω ∈ Ω the matrix Γ[ I , I t ] is singular, this means that all the vectors V ( ω, X i ( ω )) for X i ∈ supp ( ω ) ∩ { γ [ g ] > } R n , in other words, this implies that there exist λ ( ω ) , . . . , λ n − ( ω ) not all null such that: − λ ( ω ) + λ ( ω )( − I + g ) + · · · + λ n − ( ω ) n ! n X k =1 I n − ( n − k )! ( − k g k − = 0 on all the points of supp ( ω ) ∩ { γ [ g ] > } . Since g ∈ d , by (EID) on the bottom space — which is always true for scalar functions — themeasure g ∗ [1 { γ [ g ] > } .ν ] is absolutely continuous hence continuous (diffuse). As ν { γ [ g ] > } = + ∞ the random Poisson measure image by g of the points of N which are in { γ [ g ] > } do possessinfinitely many distinct points. Hence the g ( X i ( ω )) cannot annul a polynomial except if it isidentically sero.The question is therefore to know whether − λ ( ω ) + λ ( ω )( − I + x ) + · · · + λ n − ( ω ) n ! n X k =1 I n − ( n − k )! ( − k x k − ≡ implies λ ( ω ) = · · · = λ n − ( ω ) = 0 .But this is due to the fact that the annulation of the coefficients of this polynomial builds atriangular linear system whose diagonal terms are − λ ( ω ) , . . . , n !( − n λ n − ( ω ) . We have proved Proposition 18. If the upper structure satisfies (EID) , for g ∈ L ∞ ∩ d such that ν { γ [ g ] > } = + ∞ the vector ( I ( g ) , . . . , I n ( g ⊗ n )) has a density on R n . Remark 19. This result is quite different from what happens on the Wiener space since therethe law of ( I ( f ) , . . . , I n ( f ⊗ n )) is carried by the algebraic curve of equation x = 2! H ( k f k , x ) ... x n = n ! H n ( k f k , x ) where H n ( λ, x ) is the Hermite polynomial given by exp( tx − t λ ∞ X n =0 t n H n ( λ, x ) . ( I n ( f ⊗ n ) , . . . , I n p ( f ⊗ n p p )) . Let f = ( f , . . . , f p ) ∈ ( L ∩ L ∞ ∩ d ) p and let be J the column vector ( I n ( f ⊗ n ) , . . . , I n p ( f ⊗ n p p )) where we suppose n i > ∀ i. Defining the polynomials P i by P i ( x ) = i ! (cid:16)P ik =1 I i − k ( i − k )! ( − k x k − (cid:17) we have by (41) theequality between p × p -matrices Γ[ J , J t ] = Z (cid:0) P n i ( f i ) P n j ( f j ) γ [ f i , f j ] (cid:1) ij dN. By Lemma 8 { det Γ[ J , J t ] = 0 } ⊂ { Z det γ [ f, f t ] P n ( f ) · · · P n p ( f p ) dN = 0 } . ν { det γ [ f, f t ] > } = + ∞ , and that we have (EID) below and above. The imageby f of { det[ f,f t ] > } · ν is absolutely continuous w.r. to Lebesgue measure and the Poisson randommeasure image of N | { det[ f,f t ] > } has an absolutely continuous and infinite intensity measure, itpossesses necessarily points outside the finite union (less than P pi =1 ( n i − ) of hyperplans definedby P n i ( x i ) = 0 whose term of highest degree is ( n i )!( − n i x n i − i . We obtain Proposition 20. If (EID) holds below and above, and if γ [ f, f t ] is invertible ν -a.e. ( I n ( f ⊗ n ) , . . . , I n p ( f ⊗ n p p )) has a density as soon as n i > ∀ i. Remark 21. Let us compare with the situation on the Wiener space. We dispose only ofsufficient conditions of regularity, but we can nevertheless compare the thread of the arguments.We have DI n ( g ⊗ n ) = nI n − ( g ⊗ ( n − ) g and Γ[ I n i ( f ⊗ n i i ) , I n j ( f ⊗ n j n j )] = n i n j I n i − ( f ⊗ ( n i − i ) I n j − ( f ⊗ ( n j − n j − ) Z f i f j dt. Since (EID) holds on the Wiener space a sufficient condition of density of J is that almost surelythe vector (cid:16) n I n − ( f ⊗ ( n − ) f ( t ) , . . . , n p I n p − ( f ⊗ ( n p − p ) f p ( t ) (cid:17) generates a p -dimensional space when t varies. It is easily seen by induction on n that ∀ f ∈ L ( dt ) , k f k 6 = 0 : P { I n ( f ⊗ n ) = 0 } = 0 . It follows that on the Wiener space, J has adensity as soon as n i > ∀ i and ( f , . . . , f p ) are linearily independent in L ( dt ) . Density of ( N ( f ( g ) , . . . , N ( f n ( g ))) . Let g ∈ L ∞ ∩ d and let f i be regular real functions on R . Let us denote K = ( N ( f ( g ) , . . . , N ( f n ( g ))) t and suppose ν { γ [ g ] > } = + ∞ . From Γ[ N ( f i ( g )) , N ( f j ( g ))] = R f ′ i ( g ) f ′ j ( g ) γ [ g ] dN we obtainthat the matrix Γ[ K , K t ] is singular if the vectors ( f ′ ( g ) , . . . , f ′ n ( g )) taken on the points of ω arein a same hyperplan. Now the points g ( x ) , x ∈ supp ( ω ) , have an accumulation point at zero.We obtain Proposition 22. Suppose (EID) holds above, g ∈ L ∞ ∩ d , ν { γ [ g ] > } = + ∞ , and the functions f i be analytic at the neighborhood of O such that (1 , f , . . . , f n ) be linearily independent, then ( N ( f ( g )) , . . . , N ( f n ( g ))) has a density. Since there are infinitely many distinct points g ( x ) , x ∈ supp ( ω ) , we see also that withoutanalyticity hypothesis it suffices that any hyperplan cuts the curve ( f ′ ( t ) , . . . , f ′ m ( t )) t ∈ R at afinite number of points, the f i being supposed C ∩ Lip . Density of ( P j N ( f j ) , . . . , P j ( N ( f j )) n ) . Let us consider Φ the column vector of the polynomials Φ k ( x , . . . , x n ) = P nj =1 x kj , f = ( f , . . . , f n ) ∈ d n and let us pose V the column vector ofthe Φ k ( N ( f ) , . . . , N ( f n )) . We obtain Γ[ V, V t ] = ∇ Φ( N f , . . . , N f n )Γ[ N f, N f t ]( ∇ Φ) t ( N f , . . . , N f n )= ∇ Φ( N f , . . . , N f n ) R γ [ f, f t ] dN ( ∇ Φ) t ( N f , . . . , N f n )det Γ[ V, V t ] = (det ∇ Φ( N f , . . . , N f n )) det R γ [ f, f t ] dN ∇ Φ is the Jacobian matrix of Φ . det ∇ Φ is a Vandermonde determinant, if ν { f i = f j } = + ∞ , det ∇ Φ( N f , . . . , N f n ) cannotvanish by Paul Lévy’s theorem. R γ [ f, f t ] dN is an infinite sum of non negative symmetric matrices, as before we can state Proposition 23. Supposing (EID) above, ν { f i = f j } = + ∞ ∀ i = j , and ν { det γ [ f, f t ] > } =+ ∞ , then V has a density. I n ( f ) for f ∈ d n,sym . If f ( x , · · · , x n ) is a symmetric element of d n , the function ( f ( x , · · · , x n − , · )) ♭ ( x, r ) may beseen as a symmetric Hilbert valued function in d n − ( H ) with H = L ( ν × ρ ) . So that we caniterate the operator ♭ going down on the arguments ( f ( x , · · · , x n − , · , · )) ♭♭ ∈ d n − ( H ⊗ H ) .Let us apply this with Prop 17: Γ[ I n ( f )] = n Z ε − γ [ I n − ( f )] dN. By Lemma 2 for Γ[ I n ( f )] to be > it suffices γ [ I n − ( f )] > P × ν -a.e.i.e. that I n − f ♭ be = 0 P × ν × ρ -a.e. hence it suffices that ν × ρ -a.e. I n − f ♭ have a continuouslaw.Now getting down the induction and using Paul Lévy’s theorem yields that it suffices that ( ν × ρ ) n − -a.e. ν { x : f ( n − ♭ ( x , x , r , . . . , x n , r n ) = 0 } = + ∞ . Applying this to the classical case where the bottom space is R + equipped with the Lebesguemeasure and the form e [ f ] = R f ′ ( t ) dt , where we can choose f ♭ = f ′ · ξ with ξ reducedGaussian, we obtain Proposition 24. For n > , I n ( f ) has a density if the Lebesgue measure of the set { x : ∂ n − ˜ f∂x ··· ∂x n = 0 } is infinite dx · · · dx n -a.e. This extends to the classical case on R d taking f ♭ = ∂f∂x ξ + · · · + ∂f∂x n ξ n with the ξ i i.i.d.reduced Gaussian. Remark 25. There is a major difference with the case of the Brownian motion about the sumof the series ∞ X n =0 t n n ! I n ( f ⊗ n ) . In the case of Wiener space this sum is a function of R f dB = I ( f ) since it is equal to e t R fdB − t k f k . On the Poisson space it is not a function of I ( f ) = N ( f ) but of N (log(1 + tf )) and for f ∈ L ∞ ∩ d and small t by our usual argument using Paul Lévy’s theorem the pair ( N f, N log(1 + tf )) do have a density if ν { γ [ f ] > } = + ∞ .It is natural to ask about the density of the vector ( N log(1 + t f ) , . . . , N log(1 + t n f )) . For f ∈ L ∞ ∩ d , supposing < t , . . . , t n < k f k ∞ , by the method it suffices to have (BC) down,(EID) above, ν { γ [ f ] > } = + ∞ and the t i to be distinct.28 emark 26. In the Wiener case multiple integrals obey a product formula (cf. Shigekawa [43]p276) allowing to express explicitely I m [ f ] I n [ g ] as linear combination of multiple integrals oforder less or equal to m + n .A similar formula exists on the Poisson space slightly more complicated. It may be obtainedin the following way. Let u, v ∈ L ∩ L ∞ ( ν ) with small uniform norm. By the relation e N (log(1+ su )) − sν ( u ) e N (log(1+ tv )) − tν ( v ) = e N (log(1+ su + tv + stuv )) − ν (( su + tv + stuv ) e stν ( uv ) thanks to (30) we have (1 + ∞ X m =1 s m m ! I m ( u ⊗ m )(1 + ∞ X n =1 t n n ! I n ( v ⊗ n ) = (1 + ∞ X p =1 p ! I p (( su + tv + stuv ) ⊗ p ) e stν ( uv ) and the product formula is obtained by identification of the term in s m t n of the two sides. Thenit may be extended by polarization to ˜ f and ˜ g for f = f ⊗ · · · ⊗ f m and g = g ⊗ · · · ⊗ g n andthen for general f ∈ L ( ν × m ) , g ∈ L ( ν × n ) by density. See [22], [35], [46] for different forms ofsuch a formula, also [40], [39], and [17] p261 for a general expression and proof.If we apply this product formula to J m ( f ♭ m ) J n ( g ♭ n ) using ˆ E J k ( h ) = I k ( R hρ ( dr ) · · · ρ ( dr k )) for h ( x , r , · · · , x k , r k ) ∈ L sym ( ν × ρ ) × k ) we could obtain another expression of Γ[ I m f, I n g ] =ˆ E J m ( f ♭ m ) J n ( g ♭ n ) to be compared with (40). References [1] Albeverio S., Kondratiev Y. and Röckner M. "Differential geometry of Poissonspace" C. R. Acad. Sci. Paris , t. 323, sI, 1129-1134, (1996); and "Analysis and geome-try on configuration spaces" J. Funct. Analysis Albeverio , M. Röckner , “Classical Dirichlet forms on topological vectors spaces – Clos-ability and a Cameron -Martin formula" J. Funct. Analysis 88 (1990) 395-436.[3] Bertoin J., Lindner A., Maller R. “On continuity properties of the law of integralsof Lévy processes" Sém. Probabilités XLI, Lect. 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