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Invariance of Poisson measures under random transformations

We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method applied in [22] on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure.


Introduction
Poisson random measures on metric spaces are known to be quasi-invariant under deterministic transformations satisfying suitable conditions, cf. e.g. [24], [20]. For Poisson processes on the real line this quasi-invariance property also holds under adapted transformations, cf. e.g. [4], [10]. The quasi-invariance of Poisson measures on the real line with respect to anticipative transformations has been studied in [14] and in the general case of metric spaces in [1]. In the Wiener case, random nonadapted transformations of Brownian motion have been considered by several authors using the Malliavin calculus, cf. [23] and references therein.
On the other hand, the invariance property of the law of stochastic processes has important applications, for example to the construction of identically distributed samples of antithetic random variables that can be used for variance reduction in the Monte Carlo method, cf. e.g. § 4.5 of [3]. Invariance results for the Wiener measure under quasi-nilpotent random isometries have been obtained in [22], [21], by means of the Malliavin calculus, based on the duality between gradient and divergence operators on the Wiener space. In comparison with invariance results, quasi-invariance in the anticipative case usually requires more smoothness on the considered transformation.
Somehow surprisingly, the invariance of Poisson measures under non-adapted transformations does not seem to have been the object of many studies to date.
The classical invariance theorem for Poisson measures states that given a deterministic transformation τ : X → Y between measure spaces (X, σ) and (Y, µ) sending σ to µ, the corresponding transformation on point processes maps the Poisson distribution π σ with intensity σ(dx) on X to the Poisson distribution π µ with intensity µ(dy) on Y .
As a simple deterministic example in the case of Poisson jumps times (T k ) k≥1 on the half line X = Y = IR + with σ(dx) = µ(dx) = dx/x, the homothetic transformation τ (x) = rx leaves π σ invariant for all fixed r > 0. However, the random transformation of the Poisson process jump times according to the mapping τ (x) = x/T 1 does not yield a Poisson process since the first jump time of the transformed point process is constantly equal to 1.
In this paper we obtain sufficient conditions for the invariance of random transformations τ : Ω X × X → Y of Poisson random measures on metric spaces X, Y . Here the almost sure isometry condition on IR d assumed in the Gaussian case will be replaced by a pointwise condition on the preservation of intensity measures, and the quasinilpotence hypothesis will be replaced by a cyclic condition on the finite difference gradient of the transformation, cf. Relation (3.7) below. In particular, this condition is satisfied by predictable transformations of Poisson measures, as noted in Example 1 of Section 4.
In the case of the Wiener space W = C 0 (IR + ; IR d ) one considers almost surely defined random isometries given by R(ω)h(t) = U(ω, t)h(t) where U(ω, t) : IR d → IR d , t ∈ IR + , is a random process of isometries of IR d . The Gaussian character of the measure transformation induced by R is then given by checking for the Gaussianity of the (anticipative) Wiener-Skorohod integral δ(Rh) of Rh, for all h ∈ L 2 (IR + ; IR d ). In the Poisson case we consider random isometries given by R(ω)h(x) = h(τ (ω, x)) where τ (ω, ·) : (X, σ) → (Y, µ) is a random transformation that maps σ(dx) to µ(dy) for all ω ∈ Ω X . Here, the Poisson character of the measure transformation induced by R is obtained by showing that the Poisson-Skorohod integral δ σ (Rh) of Rh has same distribution under π σ as the compensated Poisson stochastic integral δ µ (h) of h under π µ , for all h ∈ C c (Y ).
For this we will use the Malliavin calculus under Poisson measures, which relies on a finite difference gradient D and a divergence operator δ that extends the Poisson stochastic integral. Our results and proofs are to some extent inspired by the treatment of the Wiener case in [22], see [16] for a recent simplified proof on the Wiener space. However, the use of finite difference operators instead of derivation operators as in the continuous case makes the proofs and arguments more complex from an algebraic point of view.
As in the Wiener case, we will characterize probability measures via their moments.
Recall that the moment E λ [Z n ] of order n of a Poisson random variable Z with intensity λ can be written as where T n (λ) is the Touchard polynomial of order n, defined by T 0 (λ) = 1 and the recurrence relation also called the exponential polynomials, cf. e.g. §11.7 of [6], Replacing the Touchard polynomial T n (λ) by its centered versionT n (λ) defined byT 0 (λ) = 1 and yields the moments of the centered Poisson random variable with intensity λ > 0 as Poisson stochastic integrals of all orders that are used in this paper, cf. Theorem 5.1.
In Section 6 we prove the main results of Section 3 based on the lemmas on moment identities established in Section 5. In the appendix Section 7 we prove some combinatorial results that are needed in the proofs. Some of the results of this paper have been presented in [15].

Poisson measures and finite difference operators
In this section we recall the construction of Poisson measures, finite difference operators and Poisson-Skorohod integrals, cf. e.g. [12] and [17] Chapter 6 for reviews.
We also introduce some other operators that will be needed in the sequel, cf. Definition 2.5 below.
Let X be a σ-compact metric space with Borel σ-algebra B(X) and a σ-finite diffuse measure σ.
Let Ω X denote the configuration space on X, i.e. the space of at most countable and locally finite subsets of X, defined as Each element ω of Ω X is identified with the Radon point measure where ǫ x denotes the Dirac measure at x ∈ X and ω(X) ∈ IN ∪ {∞} denotes the cardinality of ω. The Poisson random measure N(ω, dx) is defined by The Poisson probability measure π σ on X can be characterized as the only probability measure on Ω X under which for all compact disjoint subsets A 1 , . . . , A n of X, n ≥ 1, the mapping is a vector of independent Poisson distributed random variables on IN with respective intensities σ(A 1 ), . . . , σ(A n ).
The Poisson measure π σ is also characterized by its Fourier transform where E σ denotes expectation under π σ , which satisfies where the compensated Poisson stochastic integral We refer to [8], [9], [13], for the following definition.
for any random variable F : The operator D is continuous on the space ID 2,1 defined by the norm We refer to Corollary 1 of [13] for the following definition.

Definition 2.2
The Skorohod integral operator δ σ is defined on any measurable process u : Ω X × X → IR by the expression Relation (2.5) between δ σ and the Poisson stochastic integral will be used to characterize the distribution of the perturbed configuration points. Note that if D t u t = 0, t ∈ X, and in particular when applying (2.5) to u ∈ L 1 σ (X) a deterministic function, we have i.e. δ σ (u) with the compensated Poisson-Stieltjes integral of u. In addition if X = IR + and σ(dt) = λ t dt we have for all square-integrable predictable processes is a Poisson process with intensity λ t > 0, cf. e.g. the Example page 518 of [13].
The next proposition can be obtained from Corollaries 1 and 5 in [13].

Proposition 2.3
The operators D and δ σ are closable and satisfy the duality relation The operator δ σ is continuous on the space IL 2,1 ⊂ Dom (δ σ ) defined by the norm and for any u ∈ IL 2,1 we have the Skorohod isometry cf. Corollary 4 and pages 517-518 of [13].
In addition, from (2.5) we have the commutation relation The moments identities for Poisson stochastic integrals proved in this paper rely on the decomposition stated in the following lemma.
Proof. We have, applying (2.10) to F = δ σ (u) n , From Relation (2.6) and Lemma 2.4 we find that the moments of the compensated k , n = 0, . . . , N, which is analog to Relation (1.2) for the centered Touchard polynomials and coincides with (2.3) for n = 1.
The Skorohod isometry (2.9) shows that δ σ is continuous on IL 2,1 , and that its moment of order two of δ σ (u) satisfies as in the Wiener case [22]. This condition is satisfied when i.e. u is adapted in the sense of e.g. [18], Definition 4, or predictable when X = IR + .
The computation of moments of higher orders turns out to be more technical, cf.
Theorem 5.1 below, and will be used to characterize the Poisson distribution. From (2.11), in order for δ σ (u) ∈ L n+1 σ (Ω X ) to have the same moments as the compensated Poisson integral of f ∈ n+1 p=2 L p σ (X), it should satisfy the recurrence relation n ≥ 0, which is an extension of Relation (2.11) to the moments of compensated Poisson stochastic integrals, and characterizes their distribution by Carleman's condition [5] when sup In order to simplify the presentation of moment identities for the Skorohod integral δ σ it will be convenient to use the following symbolic notation in the sequel.
Definition 2.5 For any measurable process u : Ω X × X → IR, let Note that the sum in (2.13) includes empty sets. For example we have and ∆ s 0 u s 0 = 0. The use of this notation allows us to rewrite the Skorohod isometry As a consequence of Theorem 5.1 and Relation (6.1) of Proposition 6.1 below, the third moment of δ σ (u) is given by cf. (5.4) and (6.2) below, which reduces to when u satisfies the cyclic conditions of Lemma 7.2 in the appendix, which shows that (2.13) vanishes, see also (6.4) below for moments of higher orders. When X = IR + , (7.2) is satisfied in particular when u is predictable with respect to the standard Poisson process filtration.

Main results
The main results of this paper are stated in this section under the form of Corollary 3.2 and Theorem 3.3.
Let (Y, µ) denote another measure space with associated configuration space Ω Y and σ-finite diffuse intensity measure µ(dy). Given an everywhere defined measurable random mapping In other terms, the random mapping τ * : Ω X → Ω Y shifts each configuration point x ∈ ω according to x → τ (ω, x), and in the sequel we will be interested in finding conditions for τ * : Ω X → Ω Y to map π σ to π µ . This question is well known to have an affirmative answer when the transformation τ : X → Y is deterministic and maps σ to µ, as can be checked from the Lévy-Khintchine representation (2.2) of the characteristic function of π σ . In the random case we will use the moment identity of the next Proposition 3.1, which is a direct application of Proposition 6.2 below with u = Rh. We apply the convention that . . , l a ), and π σ (ω)-a.s. for all l 0 + · · · + l j ≤ N + 1, l 0 ≥ 1, . . . , l j ≥ 1, j = 1, . . . , N, then we have  [22].
and the cyclic condition
We will apply Corollary 3.2 to the random isometry R : where τ : Ω X × X → Y is the random transformation (3.1) of configuration points considered at the beginning of this section. As a consequence we obtain the following invariance result for Poisson measures when (X, σ) = (Y, µ).
Proof. We first show that, under π σ , δ σ (h • τ ) has same distribution as the compen- Let (K r ) r≥1 denote an increasing family of compact subsets of X such that X = r≥1 K r , and let τ r : Ω X × X → Y be defined for r ≥ 1 by hence (3.8) implies that Condition (3.7) holds, and Corollary 3.2 shows that we have for all λ ∈ IR. Next we note that Condition (3.8) implies that i.e. τ r (ω, t) does not depend on the presence or absence of a point in ω at t, and in particular, Hence by (2.6) we have and by (3.9) we get Next, letting r go to infinity we get for all h ∈ C c (Y ), hence the conclusion.

Examples
In this section we consider some examples of transformations satisfying the hypotheses of Section 3, in case X = Y for σ-finite measures σ and µ. Using various binary relations on X we consider successively the adapted case, and transformations that are conditioned by a random set such as the convex hull of a Poisson random measure. Such results are consistent with the fact that given the position of its extremal vertices, a Poisson random measure remains Poisson within its convex hull, cf. the unpublished manuscript [7], see also [25] for a related use of stopping sets.
1. First, we remark that if X is endowed with a total binary relation and if i.e.
then the cyclic Condition (3.8) is satisfied, i.e. we have for all k ≥ 1. Indeed, for all x 1 , . . . , x k ∈ X there exists i ∈ {1, . . . , k} such that Such binary relations on X can be defined via an increasing family (C λ ) λ∈I R of subsets whose reunion is X and such that for all x = y ∈ X there exists λ x , λ y ∈ IR with x ∈ C λx \ C λy and y ∈ C λy , or y ∈ C λy \ C λx and x ∈ C λx , which is equivalent to y x or x y, respectively.
This framework includes the classical adaptedness condition when X has the form where B c (Z) denotes the compact Borel subsets of Z. In this case it is well-known that ω → τ * ω is Poisson distributed with intensity µ under π σ , provided τ (ω, ·) : on Z × IR + by taking C λ = [λ, ∞) × X, λ ∈ IR + , and the cyclic Condition (3.8) is satisfied when τ (ω, ·) : IR + × Z → IR + × Z is predictable in the sense of (4.1).
Next all ω ∈ Ω X , let C(ω) denote the convex hull of ω in IR d with interiorĊ(ω), and let ω e = ω ∩ (C(ω) \Ċ(ω)) denote the extremal vertices of C(ω). Consider a measurable mapping τ : Ω X × X → X such that for all ω ∈ Ω X , τ (ω, ·) is measure preserving, mapsĊ(ω) toĊ(ω), and for all ω ∈ Ω X , x, x ∈ X \Ċ(ω), shows an example of a transformation that modifies only the interior of the convex hull generated by the random measure, in which the number of points is taken to be finite for simplicity of illustration.
Next we prove the invariance of such transformations as a consequence of Theorem 3.3.
This invariance property is related to the intuitive fact that given the positions of the extreme vertices, the distribution of the inside points remains Poisson when they are shifted according to the data of the vertices, cf. e.g. [7].
Here we consider the binary relation ω given by The relation ω is clearly reflexive, and it is transitive since x ω y and y ω z hence x ω z. Note that ω is also total on C(ω) and it is an order relation on X \ C(ω), since it is also antisymmetric on that set, i.e. if x, y / ∈ C(ω) then x ω y and y ω x means x ∈ C(ω ∪ {y}) and y ∈ C(ω ∪ {x}), which implies x = y. We will need the following lemma.
We conclude that D x τ (ω, y) = 0 in both cases.

Moment identities for stochastic integrals
In this section we prove a moment identity for Poisson stochastic integrals of arbitrary orders in Theorem 5.1, whose application will be to prove Proposition 3.1. More precisely, given F : Ω X → IR a random variable and u : Ω X × X → IR a measurable process, we aim at decomposing E σ [δ σ (u) n F ] in terms of the gradient D, while removing all occurrences of δ σ using the integration by parts formula (2.8).
In Theorem 5.1 and in the rest of this section we will use the notation Moreover, by saying that u : Ω X × X → IR has a compact support in X we mean that there exists a compact subset K of X such that u(ω, x) = 0 for all ω ∈ Ω X and x ∈ X \ K.
Theorem 5.1 Let F : Ω X → IR be a bounded random variable and let u : Ω X × X → IR be a bounded process with compact support in X. For all n ≥ 0 we have where dσ b (s b ) = σ(ds 1 ) · · · σ(ds b ), L a = (l 1 , . . . , l a ), and Before turning to the proof of Theorem 5.1 we consider some examples.
1. For n = 2 and F = 1, Theorem 5.1 recovers the Skorohod isometry (2.9) as follows: 2. For n = 3 and F = 1, Theorem 5.1 yields the following third moment identity: The proof of Theorem 5.1 will be done by induction based on the following lemma.

Recursive moment identities
The main results of this section are Propositions 6.1 and 6.2. Their proofs are stated using Lemma 2.4 above and Proposition 6.3 below, and they are used to prove the main results of Section 3. In the next theorem we use the notation ∆ s of Definition 2.5 and let Proposition 6.1 Let N ≥ 0 and let u ∈ IL 2,1 be bounded with u ∈ N +1 p=1 L ∞ (Ω X , L p σ (X)) and E σ X b+1 l 0 + · · · + l a ≤ N + 1, l 0 , . . . , l a ≥ 1, 0 ≤ j ≤ a ≤ b ≤ N. Then for all n = 0, . . . , N we have δ σ (u) ∈ L n+1 (Ω X , π σ ) and where C l 0 ,n La,b = (−1) b−a n l 0 C La,b , and C La,b is defined in (5.2).
Proof. When u : Ω X × X → IR is a bounded process with compact support in X this result is a direct consequence of Lemma 2.4 and Proposition 6.3 below applied with n = k and l 0 + k − b = n − b. We conclude the proof by induction and a limiting argument, as follows. Let (K r ) r≥1 denote an increasing family of compact subsets of X such that X = r≥1 K r . The family of processes u (r) x (ω) := u x 1 Kr (x), r ≥ 1, converges in IL 2,1 to u as r goes to infinity, hence δ σ (u (r) ) converges to δ(u) in L 2 (Ω X , π σ ) as r goes to infinity. Clearly the result holds for N = 0 by applying the formula to the process u (r) which is bounded with compact support by letting r go to infinity. Next, letting N ≥ 0 and assuming that δ σ (u) ∈ L n+1 (Ω X , π σ ) and that (6.1) holds for all n = 0, . . . , N, we note that for all even integer m ∈ {2, . . . , N + 1} we have the bound which, applied to u (r) (ω), allows us to extend (6.1) to the order N + 1 by uniform integrability after taking the limit as r goes to infinity.
Let us consider some particular cases of Proposition 6.1. For n = 1, Relation (6.1) reads which coincides with (5.3). On the other hand for n = 2 Relation (6.1) yields the third moment 2) which recovers (5.4) by the duality relation (2.8). As a consequence of Proposition 6.1 and Lemma 7.2 in the appendix, when the process u satisfies the cyclic condition D t 1 u t 2 (ω) · · · D t k u t 1 (ω) = 0, ω ∈ Ω X , t 1 , . . . , t k ∈ X, (6.3) k ≥ 2, Relation (6.1) becomes i.e. the last two terms of (6.2) vanish when n = 2. In case X = IR + × Z, Condition (6.3) is satisfied when u is predictable, by the same argument as the one leading to (4.3).
The next Proposition follows from Proposition 6.1 and is used to prove Proposition 3.1.
The next proposition is used to prove Proposition 6.1 with the help of Lemma 2.4. Proposition 6.3 Let u : Ω X × X → IR and v : Ω X × X → IR be bounded processes with compact support in X. For all k ≥ 0 we have Proof. This proof is an application of Theorem 5.1 with F = v s . Using Proposition 7.1 below and the expansion we have, up to the symmetrization due to the integral in σ(ds 0 ) · · · σ(ds a ) and the summation on l 1 , . . . , l a , where we identified E σ δ σ (u) k X v s σ(ds) to (6.6) on the last step, by another application of Theorem 5.1 to F = X v s σ(ds).

Appendix
In this appendix we state some combinatorial results that have been used above. for mutually different s n = (s 0 , . . . , s n ) ⊂ X.
Clearly for j = 0 the relation holds since Finally in the next lemma, which is used to prove Corollary 3.2, we show that Relation (6.5) is satisfied provided D s u t (ω) satisfies the cyclic condition (7.2).