Invariance of Poisson measures under random transformations

www.imstat.org/aihp 2012, Vol. 48, No. 4, 947–972

DOI:10.1214/11-AIHP422

© Association des Publications de l’Institut Henri Poincaré, 2012

Invariance of Poisson measures under random transformations

Nicolas Privault

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, SPMS-MAS, 21 Nanyang Link, Singapore 637371. E-mail:nprivault@ntu.edu.sg

Received 28 September 2010; revised 10 February 2011; accepted 23 February 2011

Abstract. We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite differ- ence 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 of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) 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.

Résumé. Nous montrons que les mesures de Poisson sont invariantes par les transformations aléatoires qui préservent les mesures d’intensité, et dont le gradient aux différences finies satisfait une condition d’annulation cyclique. La preuve de ce résultat repose sur des identités de moments d’intérêt indépendant pour les intégrales stochastiques de Poisson adaptées et anticipantes, et est inspirée par la méthode de Üstünel et Zakai (Probab. Theory Related Fields103(1995) 409–429) sur l’espace de Wiener, bien que l’algèbre correspondante soit plus compliquée que dans le cas Wiener. Les exemples d’application incluent des transformations conditionnées par des ensembles aléatoires tels que l’enveloppe convexe d’une mesure aléatoire de Poisson.

MSC:60G57; 60G30; 60G55; 60H07; 28D05; 28C20; 37A05

Keywords:Poisson measures; Random transformations; Invariance; Skorohod integral; Moment identities

1. Introduction

Poisson random measures on metric spaces are known to be quasi-invariant under deterministic transformations sat- isfying suitable conditions, cf. e.g. [20,24]. For Poisson processes on the real line this quasi-invariance property also holds under adapted transformations, cf. e.g. [4,11]. The quasi-invariance of Poisson measures on the real line with respect to anticipative transformations has been studied in [13] and in the general case of metric spaces in [1]. In the Wiener case, random non-adapted transformations of Brownian motion have been considered by several authors using the Malliavin calculus, cf. [23] and references therein.

On the other hand, theinvarianceproperty of the law of stochastic processes has important applications, for exam- ple 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. Section 4.5 of [3]. Invariance results for the Wiener measure under quasi-nilpotent random isometries have been obtained in [21,22], 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.

1Supported by the Grant GRF 102309 from the Research Grants Council of the Hong Kong Special Administrative Region of the People’s Republic of China.

The classical invariance theorem for Poisson measures states that given adeterministictransformationτ:X→Y between measure spaces(X, σ )and(Y, μ)sendingσtoμ, the corresponding transformation on point processes maps the Poisson distribution π_σ with intensityσ (dx)on X to the Poisson distribution π_μ with intensityμ(dy)on Y. As a simple deterministic example in the case of Poisson jumps times(T_k)_k≥1 on the half lineX=Y =R₊ with σ (dx)=μ(dx)=dx/x, the homothetic transformationτ (x)=rx leavesπ_σ invariant for all fixedr >0. However, the random transformation of the Poisson process jump times according to the mappingτ (x)=x/T₁does 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 ofrandomtransformations τ:Ω^X×X→Y of Poisson random measures on metric spacesX,Y. Here the almost sure isometry condition onR^d assumed in the Gaussian case will be replaced by a pointwise condition on the preservation of intensity measures, and the quasi- nilpotence 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 Example1of Section4.

In the case of the Wiener spaceW=C0(R+;R^d)one considers almost surely defined random isometries R(ω):L²

R₊;R^d

→L²

R₊;R^d

, ω∈W,

given byR(ω)h(t )=U (ω, t )h(t )whereU (ω, t):R^d→R^d,t∈R+, is a random process of isometries ofR^d. The Gaussian character of the measure transformation induced byR is then given by checking for the Gaussianity of the (anticipative) Wiener–Skorohod integralδ(Rh)ofRh, for allh∈L²(R+;R^d). In the Poisson case we consider random isometries

R(ω):L²_μ(Y )→L²_σ(X)

given by R(ω)h(x)=h(τ (ω, x))whereτ (ω,·):(X, σ )→(Y, μ) is a random transformation that mapsσ (dx) to μ(dy)for allω∈Ω^X. Here, the Poisson character of the measure transformation induced byRis obtained by show- ing that the Poisson–Skorohod integralδ_σ(Rh)ofRh has same distribution underπ_σ as the compensated Poisson stochastic integralδ_μ(h)ofhunderπ_μ, for allh∈Cc(Y ).

For this we will use the Malliavin calculus under Poisson measures, which relies on a finite difference gradient Dand a divergence operatorδ 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 [15] for a recent simplified proof on the Wiener space.

However, the use offinite differenceoperators instead ofderivationoperators 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 momentE_λ[Zⁿ] of ordernof a Poisson random variableZwith intensityλcan be written as

E_λ Zⁿ

=T_n(λ),

whereTn(λ)is the Touchard polynomial of ordern, defined byT0(λ)=1 and the recurrence relation Tn+1(λ)=λ

n

k=0

n k

Tk(λ), n≥0, (1.1)

also called the exponential polynomials, cf. e.g. Section 11.7 of [6], Replacing the Touchard polynomialTn(λ)by its centered versionT˜n(λ)defined byT˜0(λ)=1 and

T˜_n+1(λ)=λ

n−1

k=0

n k

T˜_k(λ), n≥0, (1.2)

yields the moments of thecenteredPoisson random variable with intensityλ >0 as T˜n(λ)=Eλ

(Z−λ)ⁿ

, n≥0.

Our characterization of Poisson measures will use recurrence relations similar to (1.2), cf. (2.12) below, and iden- tities for the moments of compensated Poisson stochastic integrals which are another motivation for this paper, cf.

Theorem5.1below.

The paper is organized as follows. The main results (Corollary3.2and Theorem3.3) on the invariance of Poisson measures are stated in Section 3after recalling the definition of the finite difference gradientD and the Skorohod integral operatorδ under Poisson measures in Section2. Section4 contains examples of transformations satisfying the required conditions which include the classical adapted case and transformations acting inside the convex hull generated by Poisson random measures, given the positions of the extremal vertices. Section5contains the moment identities for Poisson stochastic integrals of all orders that are used in this paper, cf. Theorem5.1. In Section6we prove the main results of Section3based on the lemmas on moment identities established in Section5. In theAppendixwe prove some combinatorial results that are needed in the proofs. Some of the results of this paper have been presented in [14].

2. 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. [9] and [16], Chapter 6, for reviews. We also introduce some other operators that will be needed in the sequel, cf. Definition2.5below.

LetXbe aσ-compact metric space with Borelσ-algebraB(X)and aσ-finite diffuse measureσ. LetΩ^Xdenote the configuration space onX, i.e. the space of at most countable and locally finite subsets ofX, defined as

Ω^X= ω=(x_i)^N_i=1⊂X, x_i=x_j ∀i=j, N∈N∪ {∞}

.

Each elementωofΩ^Xis identified with the Radon point measure ω=

ω(X)

i=1 xi,

where xdenotes the Dirac measure atx∈Xandω(X)∈N∪ {∞}denotes the cardinality ofω. The Poisson random measureN (ω,dx)is defined by

N (ω,dx)=ω(dx)=

ω(X)

k=1

x_k(dx), ω∈Ω^X. (2.1)

The Poisson probability measureπ_σonXcan be characterized as the only probability measure onΩ^Xunder which for all compact disjoint subsetsA₁, . . . , A_nofX,n≥1, the mapping

ω→

ω(A1), . . . , ω(An)

is a vector of independent Poisson distributed random variables onNwith respective intensitiesσ (A₁), . . . , σ (A_n).

The Poisson measureπ_σ is also characterized by its Fourier transform ψ_σ(f )=E_σ

exp

i

X

f (x)

ω(dx)−σ (dx)

, f∈L²_σ(X), whereE_σ denotes expectation underπ_σ, which satisfies

ψ_σ(f )=exp

X

e^{if (x)}−if (x)−1 σ (dx)

, f∈L²_σ(X), (2.2)

where the compensated Poisson stochastic integral

Xf (x)(ω(dx)−σ (dx))is defined by the isometry E_σ

X

f (x)

ω(dx)−σ (dx)²

=

X

f (x)²σ (dx), f∈L²_σ(X). (2.3)

We refer to [8,10,12], for the following definition.

Definition 2.1. LetDdenote the finite difference gradient defined as

D_xF (ω)=ε_x⁺F (ω)−F (ω), ω∈Ω^X, x∈X, (2.4)

for any random variableF:Ω^X→R,where ε_x⁺F (ω)=F

ω∪ {x}

, ω∈Ω^X, x∈X.

The operatorDis continuous on the spaceD2,1defined by the norm F²2,1= F²_L2(Ω^X,πσ)+ DF²_L2(Ω^X×X,πσ⊗σ ), F ∈D2,1. We refer to Corollary 1 of [12] for the following definition.

Definition 2.2. The Skorohod integral operatorδσ is defined on any measurable process u:Ω^X×X→Rby the expression

δ_σ(u)=

X

u_t

ω\ {t}

ω(dt )−σ (dt )

, (2.5)

providedE_σ[

X|u(ω, t )|σ (dt )]<∞.

Relation (2.5) betweenδ_σ and the Poisson stochastic integral will be used to characterize the distribution of the perturbed configuration points. Note that ifDtut=0,t∈X, and in particular when applying (2.5) tou∈L¹_σ(X)a deterministic function, we have

δσ(u)=

X

u(t )

ω(dt )−σ (dt )

(2.6) i.e.δ_σ(u)with the compensated Poisson–Stieltjes integral ofu. In addition ifX=R₊andσ (dt )=λ_tdt we have

δ_σ(u)= _∞

0

u_t(dNt−λ_tdt ) (2.7)

for all square-integrable predictable processes(ut)t∈R+, where Nt =ω([0, t]),t ∈R+, is a Poisson process with intensityλ_t>0, cf. e.g. the Example, p. 518, of [12].

The next proposition can be obtained from Corollaries 1 and 5 in [12].

Proposition 2.3. The operatorsDandδσ are closable and satisfy the duality relation Eσ

DF, uL²_σ(X)

=Eσ

F δσ(u)

(2.8) on theirL²domainsDom(δσ)⊂L²(Ω^X×X, π_σ⊗σ )andDom(D)=D2,1⊂L²(Ω^X, π_σ)under the Poisson mea- sureπ_σ with intensityσ.

The operatorδ_σ is continuous on the spaceL2,1⊂Dom(δσ)defined by the norm u²2,1=Eσ

X|ut|²σ (dt )

+Eσ

X|Dsut|²σ (ds)σ (dt )

,

and for anyu∈L2,1we have the Skorohod isometry E_σ

δ_σ(u)²

=E_σ u²_L2

σ(X)

+E_σ

X

X

D_su_tD_tu_sσ (ds)σ (dt )

, (2.9)

cf. Corollary 4 and pp. 517–518 of [12].

In addition, from (2.5) we have the commutation relation ε⁺_t δ_σ(u)=δ_σ

ε⁺_t u

+u_t, t∈X, (2.10)

providedD_tu∈L2,1,t∈X.

The moments identities for Poisson stochastic integrals proved in this paper rely on the decomposition stated in the following lemma.

Lemma 2.4. Letu∈L2,1be such thatδ_σ(u)ⁿ∈D2,1,D_tu∈L2,1,σ (dt )-a.e.,and Eσ

X

|ut|^n−k+1δσ

ε_t⁺u^kσ (dt )

<∞, Eσ

δσ(u)^k

X

|ut|^n−k+1σ (dt )

<∞, 0≤k≤n.Then we have

E_σ

δ_σ(u)ⁿ⁺¹

=

n−1

k=0

n k

E_σ

δ_σ(u)^k

X

uⁿ_t^−k+1σ (dt )

+ n

k=1

n k

Eσ

X

uⁿ_t^−k+1 δσ

ε_t⁺uk

−δσ(u)^k σ (dt )

for alln≥1.

Proof. We have, applying (2.10) toF=δσ(u)ⁿ, E_σ

δ_σ(u)ⁿ⁺¹

=E_σ

X

u_tD_tδ_σ(u)ⁿσ (dt )

=Eσ

X

ut

ε_t⁺δσ(u)n

−δσ(u)ⁿ σ (dt )

=E_σ

X

u_t

u_t+δ_σ ε⁺_t un

−δ_σ(u)ⁿ σ (dt )

=

n−1

k=0

n k

E_σ

X

uⁿ_t^−k+1δ_σ ε⁺_t uk

σ (dt )

=

n−1

k=0

n k

E_σ

X

uⁿ_t^−k+1δ_σ(u)^kσ (dt )

+ n

k=1

n k

Eσ

X

uⁿ_t^−k+1 δσ

ε_t⁺uk

−δσ(u)^k σ (dt )

.

From Relation (2.6) and Lemma 2.4we find that the moments of the compensated Poisson stochastic integral

Xf (t)(ω(dt )−σ (dt ))off ∈_N+1

p=1L^p_σ(X)satisfy the recurrence identity Eσ

X

f (t )

ω(dt )−σ (dt )ⁿ⁺¹

=

n−1

k=0

n

k _Xf^n−k+1(t)σ (dt )Eσ

X

f (t )

ω(dt )−σ (dt )^k

, (2.11)

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 onL2,1, and that its moment of order two ofδ_σ(u) satisfies

E_σ δ_σ(u)²

=E_σ u²_L2

σ(X)

, provided

X

X

D_su_tD_tu_sσ (ds)σ (dt )=0

as in the Wiener case [22]. This condition is satisfied when D_tu_sD_su_t=0, s, t∈X,

i.e.uis adapted in the sense of e.g. [18], Definition 4, or predictable whenX=R₊.

The computation of moments of higher orders turns out to be more technical, cf. Theorem5.1below, and will be used to characterize the Poisson distribution. From (2.11), in order forδ_σ(u)∈Lⁿ_σ⁺¹(Ω^X)to have the same moments as the compensated Poisson integral off ∈_n+1

p=2L^p_σ(X), it should satisfy the recurrence relation

E_σ

δ_σ(u)ⁿ⁺¹

=

n−1

k=0

n

k _Xf^n−k+1(t)σ (dt )Eσ

δ_σ(u)^k

, (2.12)

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_p≥1f_L^p_{σ(Y )}<∞.

In order to simplify the presentation of moment identities for the Skorohod integralδ_σ it will be convenient to use the following symbolic notation in the sequel.

Definition 2.5. For any measurable processu:Ω^X×X→R,let

_s0· · ·_sj n

p=0

u_sp=

Θ₀∪···∪Θ_n={s₀,s₁,...,s_j} s0∈/Θ0,...,sj∈/Θj

D_Θ0u_s0· · ·D_Θnu_sn, (2.13)

s0, . . . , sn∈X, 0≤j≤n,whereDΘ:=

s_j∈ΘDs_j whenΘ⊂ {s0, s1, . . . , sj}. Note that the sum in (2.13) includes empty sets. For example we have

_s0 n

p=0

u_sp=u_s0

Θ₁∪···∪Θ_n={s₀} s₀∈/Θ₀,...,s_j∈/Θ_j

D_Θ1u_s1· · ·D_Θnu_sn=u_s0D_s0 n

p=1

u_sp,

and_s0u_s0=0. The use of this notation allows us to rewrite the Skorohod isometry (2.9) as E_σ

δ_σ(u)²

=E_σ

X

u²_sσ (ds)

+E_σ

X

X

_st(u_su_t)σ (ds)σ (dt )

, since by definition we have

st(usut)=DsutDtus, s, t∈X.

As a consequence of Theorem5.1and Relation (6.1) of Proposition6.1below, the third moment ofδ_σ(u)is given by E_σ

δ_σ(u)³

=E_σ

X

u³_sσ (ds)

+3Eσ

δ(u)

X

u²_sσ (ds)

+3Eσ

X³

s₁s₂

us₁u²_s

2

σ (ds1)σ (ds2)

+E_σ

X³

_s1s2s3(u_s1u_s2u_s3)σ (ds1)σ (ds2)σ (ds3)

, (2.14)

cf. (5.4) and (6.2) below, which reduces to E_σ

δ_σ(u)³

=E_σ

X

u³_sσ (ds)

+3Eσ

δ(u)

X

u²_sσ (ds)

whenusatisfies the cyclic conditions

D_t1u_t2D_t2u_t1=0 and D_t1u_t2D_t2u_t3D_t3u_t1=0, t₁, . . . , t₃∈X,

of LemmaA.2in theAppendix, which shows that (2.13) vanishes, see also (6.4) below for moments of higher orders.

WhenX=R₊, (A.2) is satisfied in particular when uis predictable with respect to the standard Poisson process filtration.

3. Main results

The main results of this paper are stated in this section under the form of Corollary3.2and Theorem3.3.

Let (Y, μ)denote another measure space with associated configuration spaceΩ^Y andσ-finite diffuse intensity measureμ(dy). Given an everywhere defined measurable random mapping

τ:Ω^X×X→Y, (3.1)

indexed byX, letτ_∗(ω),ω∈Ω^X, denote the image measure ofωbyτ, i.e.

τ_∗:Ω^X→Ω^Y (3.2)

maps ω=

ω(X)

i=1

xi∈Ω^X to τ_∗(ω)=

ω(X)

i=1

τ (ω,x_i)∈Ω^Y.

In other terms, the random mappingτ_∗:Ω^X→Ω^Y shifts each configuration pointx∈ωaccording tox→τ (ω, x), and in the sequel we will be interested in finding conditions forτ_∗:Ω^X→Ω^Y to mapπ_σtoπ_μ. This question is well known to have an affirmative answer when the transformationτ:X→Y is deterministic and mapsσ toμ, as can be checked from the Lévy–Khintchine representation (2.2) of the characteristic function ofπ_σ. In the random case we will use the moment identity of the next Proposition3.1, which is a direct application of Proposition6.2below with u=Rh. We apply the convention that0

i=1l_i=0, so that{l0, l1≥0: 0

i=1l_i=0}is an arbitrary singleton.

Proposition 3.1. LetN≥0and letR(ω):L^pμ(Y )→L^pσ(X),ω∈Ω^X,be a random isometry for allp=1, . . . , N+1.

Then for allh∈_N+1

p=1L^p_μ(Y )such thatRh∈L2,1is bounded and E_σ

X^a+1

s₀· · ·s_a

_a

p=0

Rh(sp)lp

σ (ds0)· · ·σ (dsa)

<∞, (3.3)

l₀+ · · · +l_a≤N+1,l₀, . . . , l_a≥1,a≥0,we haveδ_σ(Rh)∈Lⁿ⁺¹(Ω^X, π_σ)and

E_σ

δ_σ(Rh)ⁿ⁺¹

=

n−1

k=0

n

k _Yh^n−k+1(y)μ(dy)Eσ

δ_σ(Rh)^k

+ n

a=0

a

j=0

n

b=a

l₀+···+l_a=n−b l0,...,la≥0 la+1,...,lb=0

a j

C_L^l0,n

a,b

× _b

q=j+1

Y

h^1+lq(y)μ(dy)

E_σ

X^j+1

_s0· · ·_{sj j}

p=0

Rh(s_p)1+lp

dσ^j+1(s_j)

,

n=0, . . . , N,wheredσ^j+1(sj)=σ (ds0)· · ·σ (dsj),La=(l1, . . . , la),and

C_L^l0,n

a,a+c=(−1)^c n

l0

0=rc+1<···<r0=a+c+1

c

q=0

r_q−1−(c−q)

p=rq+1+1−(c−q)

l₁+ · · · +l_p+p+q−1 l1+ · · · +l_p−1+p+q−1

. (3.4)

As a consequence of Proposition3.1, if in additionR(ω):L^p_μ(Y )→L^p_σ(X)satisfies the condition

X^j+1

_t0· · ·_{tj j}

p=0

Rh(ω, t_p)lp

σ (dt0)· · ·σ (dtj)=0, (3.5)

π_σ(ω)-a.s. for alll₀+ · · · +l_j≤N+1,l₀≥1, . . . , lj≥1,j=1, . . . , N, then we have

E_σ

δ_σ(Rh)ⁿ⁺¹

=

n−1

k=0

n

k _Yh^n−k+1(y)μ(dy)Eσ

δ_σ(Rh)^k

, (3.6)

n=0, . . . , N, i.e. the moments ofδ_σ(Rh)satisfy the extended recurrence relation (2.11) of the Touchard type.

Hence Proposition3.1and LemmaA.2yield the next corollary in which the sufficient condition (3.7) is a strength- ened version of the Wiener space condition trace(DRh)ⁿ=0 of Theorem 2.1 in [22].

Corollary 3.2. LetR:L^p_μ(Y )→L^p_σ(X)be a random isometry for allp∈ [1,∞].Assume thath∈_∞

p=1L^p_μ(Y )is such thatsup_p≥1h_L^p_{μ(Y )}<∞,and thatRhsatisfies(3.3)and the cyclic condition

Dt₁Rh(t2)· · ·Dt_kRh(t1)=0, t1, . . . , tk∈X, (3.7) πσ⊗σ^⊗k-a.e.for allk≥2.Then,underπσ,δσ(Rh)has same distribution as the compensated Poisson integralδμ(h) ofhunderπ_μ.

Proof. LemmaA.2below shows that Condition (3.5) holds under (3.7) since

D_s Rh(t )l

=ε⁺_s Rh(t )l

− Rh(t )l

= l

k=1

l

k Rh(t )l−k D_s

Rh(t )k

=0,

s, t∈X,l≥1, hence by Proposition3.1, Relation (3.6) holds for alln≥1, and this shows by induction from (2.12) that underπ_σ,δ_σ(Rh)has same moments asδ_μ(h)underπ_μ. In addition, since sup_p≥1h_L^p_{μ(Y )}<∞, Relation (3.6)

also shows by induction that the moments ofδ_σ(Rh)satisfy the boundE_σ[|δ_σ(Rh)|ⁿ] ≤(Cn)ⁿ for someC >0 and alln≥1, hence they characterize its distribution by the Carleman condition

∞ k=1

E_σ

δ_σ(Rh)²ⁿ₋1/(2n)

= +∞,

cf. [5] and p. 59 of [19].

We will apply Corollary3.2to the random isometryR:L^p_μ(Y )→L^p_σ(X)is given as Rh=h◦τ, h∈L^p_μ(Y ),

whereτ: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).

Theorem 3.3. Let τ:Ω^X×X→Y be a random transformation such thatτ (ω,·):X→Y mapsσ toμ for all ω∈Ω^X,i.e.

τ_∗(ω,·)σ=μ, ω∈Ω^X, and satisfying the cyclic condition

D_t1τ (ω, t₂)· · ·D_tkτ (ω, t₁)=0 ∀ω∈Ω^X,∀t₁, . . . , t_k∈X, (3.8) for allk≥1.Thenτ_∗:Ω^X→Ω^Y mapsπσ toπμ,i.e.

τ_∗π_σ =π_μ

is the Poisson measure with intensityμ(dy)onY.

Proof. We first show that, underπ_σ,δ_σ(h◦τ )has same distribution as the compensated Poisson integralδ_μ(h)ofh underπ_μ, for allh∈Cc(Y ).

Let(Kr)r≥1denote an increasing family of compact subsets ofXsuch thatX=

r≥1Kr, and letτr:Ω^X×X→Y be defined forr≥1 by

τ_r(ω, x)=τ (ω∩K_r, x), x∈X, ω∈Ω^X.

LettingR_rh=h◦τ_r defines a random isometryR_r:L^p_μ(Y )→L^p_σ(X)for allp≥1, which satisfies the assumptions of Corollary3.2. Indeed we have

D_sR_rh(t)=D_sh

τ_r(ω, t)

=1Kr(s) h

τ_r(ω, t)+D_sτ_r(ω, t)

−h

τ_r(ω, t)

=1K_r(s) h

τr

ω∪ {s}, t

−h

τr(ω, t)

, s, t∈X,

hence (3.8) implies that Condition (3.7) holds, and Corollary3.2shows that we have E_σ

e^{iλδμ(h◦τr)}

=E_μ e^iλδμ(h)

(3.9) for allλ∈R. Next we note that Condition (3.8) implies that

D_tτ_r(ω, t)=0 ∀ω∈Ω^X,∀t∈X, (3.10)

i.e.τ_r(ω, t)does not depend on the presence or absence of a point inωatt, and in particular, τr(ω, t)=τr

ω∪ {t}, t

, t /∈ω,

and

τr(ω, t)=τr

ω\ {t}, t

, t∈ω.

Hence by (2.6) we have δ_μ(h)◦τ_r∗=

Y

h(y)

τ_r∗ω(dy)−μ(dy)

=

X

h

τ_r(ω, x)

ω(dx)−σ (dx)

=

X

h τr

ω\ {x}, x

ω(dx)−σ (dx)

=δ_σ(h◦τ_r), and by (3.9) we get

Eσ

exp

i

Y

h(y)

τr∗ω(dy)−μ(dy)

=E_σ

exp

i

Y

h(y)

ω(dy)−μ(dy)

◦τ_r∗

=Eσ

e^{iδμ(h)◦τr∗}

=E_μ e^iδμ(h)

=Eμ

exp

i

Y

h(y)

ω(dy)−μ(dy) .

Next, lettingrgo to infinity we get Eσ

exp

i

Y

h(y)

τr∗ω(dy)−μ(dy)

=Eμ

exp

i

Y

h(y)

ω(dy)−μ(dy)

for allh∈Cc(Y ), hence the conclusion.

In Theorem3.3above the Identity (3.8) is interpreted fork≥2 by stating thatω∈Ω^X, and t₁, . . . , t_k∈X, the k-tuples

τ

ω∪ {t1}, t2

, τ

ω∪ {t2}, t3

, . . . , τ

ω∪ {tk−1}, tk

, τ

ω∪ {tk}, t1

and

τ (ω, t₂), τ (ω, t₃), . . . , τ (ω, t_k), τ (ω, t₁)

coincide on at least one component n^oi∈ {1, . . . , k}inY^k, i.e.Dt_iτ (ω, ti+1modk)=0.

4. Examples

In this section we consider some examples of transformations satisfying the hypotheses of Section3, in caseX=Y forσ-finite measuresσ andμ. Using various binary relations onXwe 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.

Example 1. First,we remark that ifXis endowed with a total binary relationand ifτ:Ω^X×X→Y is(backward) predictable in the sense that

xy ⇒ D_xτ (ω, y)=0, (4.1)

i.e.

τ

ω∪ {x}, y

=τ (ω, y), xy, (4.2)

then the cyclic condition(3.8)is satisfied,i.e.we have

Dx₁τ (ω, x2)· · ·Dx_kτ (ω, x1)=0, x1, . . . , xk∈X, ω∈Ω^X, (4.3) for all k≥1.Indeed,for allx1, . . . , xk ∈X there exists i∈ {1, . . . , k}such thatxixj,for all 1≤j ≤k,hence Dx_iτ (ω, xj)=0, 1≤j ≤k,by the predictability condition(4.1),hence(4.3)holds.Consequenly,τ_∗:Ω^X→Ω^Y mapsπσ toπμby Theorem3.3,providedτ (ω,·):X→Y mapsσ toμfor allω∈Ω^X.

Such binary relations onX can be defined via an increasing family(Cλ)λ∈R of subsets whose reunion isXand such that for allx=y∈Xthere existsλ_x, λ_y∈Rwithx∈C_λx \C_λy andy∈C_λy,ory∈C_λy \C_λx andx∈C_λx, which is equivalent toyxorxy,respectively.

This framework includes the classical adaptedness condition whenXhas the formX=R+×Y.For example,if XandY are of the formX=Y=R₊×Z,consider the filtration(Ft)_t∈R+,whereFt is generated by

σ

[0, s] ×A

: 0≤s < t, A∈Bb(Z) ,

whereBc(Z)denotes the compact Borel subsets ofZ.In this case it is well-known thatω→τ_∗ωis Poisson distributed with intensityμunderπ_σ,providedτ (ω,·):R₊×Z→R₊×Zis predictable in the sense thatω→τ (s, z)isFt- measurable for all0≤s≤t,z∈Z,cf.e.g.Theorem3.10.21of[2].Here,Condition(4.1)holds for the partial order

(s, x)(t, y) ⇐⇒ s≥t (4.4)

onZ×R₊by takingC_λ= [λ,∞)×X,λ∈R₊,and the cyclic condition(3.8)is satisfied whenτ (ω,·):R₊×Z→ R₊×Zis predictable in the sense of(4.1).

Next, we consider other examples in which the binary relation is configuration dependent. This includes in particular transformations of Poisson measures within their convex hull, given the positions of extremal vertices.

Example 2. LetX= ¯B(0,1)\ {0}denote the closed unit ball inR^d.For allω∈Ω^X,letC(ω)denote the convex hull ofωinR^d with interiorC˙(ω),and letωe=ω∩(C(ω)\ ˙C(ω))denote the extremal vertices ofC(ω).Consider a measurable mappingτ:Ω^X×X→Xsuch that for allω∈Ω^X,τ (ω,·)is measure preserving,mapsC˙(ω)toC˙(ω), and for allω∈Ω^X,

τ (ω, x)=

τ (ω_e, x), x∈ ˙C(ω),

x, x∈X\ ˙C(ω), (4.5)

i.e.the points ofC˙(ω)are shifted byτ (ω,·)depending on the positions of the extremal vertices of the convex hull of ω,which are left invariant byτ (ω,·).Figure1shows 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 Theorem3.3.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 relationωgiven by xωy ⇐⇒ x∈C

ω∪ {y}

, ω∈Ω^X, x, y∈X.

Fig. 1. Example of random transformation.

The relationωis clearly reflexive,and it is transitive sincexωyandyωzimplies x∈C

ω∪ {y}

⊂C ω∪ {z}

,

hencexωz.Note thatωis also total onC(ω)and it is an order relation onX\C(ω),since it is also antisymmetric on that set,i.e.ifx, y /∈C(ω)then

xωy and yωx

meansx∈C(ω∪ {y})andy∈C(ω∪ {x}),which impliesx=y.We will need the following lemma.

Lemma 4.1. For allx, y∈Xandω∈Ω^Xwe have

xωy ⇒ D_xτ (ω, y)=0 (4.6)

and

xωy ⇒ D_yτ (ω, x)=0. (4.7)

Proof. Letx, y∈Xandω∈Ω^X. First, ifxωythen we havex /∈C(ω∪ {y})henceτ (ω∪ {y}, x)=τ (ω, x)=xby (4.5). Next, ifxωy, i.e.x∈C(ω∪ {y}), we can distinguish two cases:

(a) x∈C(ω). In this case we haveC(ω∪ {x})=C(ω), henceτ (ω∪ {x}, y)=τ (ω, y)for ally∈X.

(b) x∈C(ω∪ {y})\C(ω). Ify∈C(ω∪ {x})thenx=y /∈ ˙C(ω∪ {x}), henceτ (ω∪ {x}, y)=τ (ω, y). On the other hand ify /∈C(ω∪ {x})thenyωx andτ (ω∪ {x}, y)=τ (ω, y)=y as above.

We conclude thatD_xτ (ω, y)=0 in both cases.

Let us now show thatτ:Ω^X×X→Ω^Xsatisfies the cyclic condition(3.8).Lett₁, . . . , t_k∈X.First,ift_i∈C(ω) for somei∈ {1, . . . , k},then for allj =1, . . . , kwe havet_iωt_j and by Lemma4.1we get

D_tiτ (ω, t_j)=0,

thus(3.8) holds, and we may assume that t_i ∈/C(ω) for all i=1, . . . , k.In this case,if t_i+1modkωt_i for some i=1, . . . , k,then by Lemma4.1we have

D_tiτ (ω, t_i+1modk)=0,

which shows that(3.8)holds.Finally,ift1ωtkω· · · ωt2ωt1,then by transitivity ofωwe havet1ωtkωt1, which impliest₁=t_k∈/C(ω)by antisymmetry on X\C(ω),henceD_tkτ (ω, t₁)=0,and τ:Ω^X×X→X satisfies the cyclic condition(3.8)for all k≥2.Hence τ satisfies the hypotheses of Theorem3.3,and τ_∗π_σ =π_μ provided τ (ω,·):X→Y mapsσ toμfor allω∈Ω^X.