aa r X i v : . [ m a t h . P R ] N ov ERGODIC POISSON SPLITTINGS ´ELISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE
Abstract.
In this paper we study splittings of a Poisson point process whichare equivariant under a conservative transformation. We show that, if theCartesian powers of this transformation are all ergodic, the only ergodic split-ting is the obvious one, that is, a collection of independent Poisson processes.We apply this result to the case of a marked Poisson process: under the samehypothesis, the marks are necessarily independent of the point process andi.i.d. Under additional assumptions on the transformation, a further applica-tion is derived, giving a full description of the structure of a random measureinvariant under the action of the transformation. Introduction
Thinning and splitting are classical operations when studying point processes.Thinning consists in removing points according to some rule, whereas the relatednotion of splitting means decomposing the point process as the sum of severalother point processes. It is well known that thinning a Poisson point process bychoosing to remove points according to independent coin tosses yields a new Poissonprocess of lower (but proportional) intensity. Moreover, this procedure gives riseto a splitting of the original Poisson process into a sum of two independent Poissonprocesses.In recent years, some new results on thinnings of Poisson process have emerged.In particular, it is shown in [2] that it is possible to deterministically choose pointsfrom a homogeneous Poisson point process on R to get another homogeneous Pois-son process of lower intensity. Moreover, it is possible to proceed in a translationequivariant way. This result has been further refined in [8] by extending it to R d and replacing translation equivariance by isometry equivariance. Moreover, theremaining points were also shown to form a homogeneous Poisson point process.The isometry or translation equivariance plays a key role here as Meyerovitchshowed in [14] that it is not possible to deterministically thin a Poisson point processin an equivariant way with respect to a transformation which is conservative ergodicon the base space (the translations of R d yields dissipativity).In the present paper, we are also interested in thinnings and splittings of Poissonprocesses which are equivariant under some dynamics. The difference with [2, 8] isthat we consider thinnings/splittings which are equivariant with respect to a con-servative transformation. Moreover, contrary to the above-mentioned result of [14],we allow additional randomness (yet keeping ergodicity) in the thinning/splittingprocedure. We get the following result, phrased in our terminology where equivari-ance and homogeneity are expressed through the concepts of T -point processes and Key words and phrases.
Poisson point process, Random measure, Splitting, Thinning, Poissonsuspension, Joinings.Research partially supported by French research group GeoSto (CNRS-GDR3477). T -splittings : If T is an infinite measure preserving map with infinite ergodic index ( i.e. all its Cartesian products are ergodic), then any ergodic T -splitting of a Pois-son T -point process yields independent Poisson T -point processes (Theorem 2.6).In the second part of the paper we derive some applications. In Section 3, weshow that, for T with infinite ergodic index, the only way to get an ergodic marked T -point process out of a Poisson T -point process is to take i.i.d. marks, independentof the underlying process (Theorem 3.1). Then we come back to the problem weraised in [10]. In that paper we gave conditions on T under which an ergodic T -point process with moments of all orders is necessarily a cluster-Poisson process(see [3, p. 175]), best described as an independent superposition of shifted Poissonprocesses (a so-called SuShi ). We extend this result in Section 4 to general randommeasures (Proposition 4.2 and Theorem 4.3). This more general framework allowsto simplify and improve some disjointness results from [10] in the last section.1.1. T -random measures and T -point processes. Let X be a complete sep-arable metric space and A be its Borel σ -algebra. Define e X to be the space of boundedly finite measures (also called locally finite measures ) on ( X, A ), that is tosay measures giving finite mass to any bounded Borel subset of X . We refer to [12]for the topological properties of e X . In particular, e X can be turned into a completeseparable metric space, that we equip with its Borel σ -algebra e A .We denote by X ∗ ⊂ e X the subspace of simple counting measures, i.e. whoseelements are of the form ξ = X i ∈ I δ x i , where I is at most countable, x i = x j whenever i = j , and any bounded subset A ⊂ X contains finitely many points of the family { x i } i . We define A ∗ as therestriction of e A to X ∗ .Throughout the paper, we fix a boundedly finite and continuous measure µ on X with µ ( X ) = ∞ , and an invertible transformation T on X preserving µ . We set A f := { A ∈ A , < µ ( A ) < + ∞} . For any measure ξ on X , we define T ∗ ( ξ ) as the pushforward measure of ξ by T :for any A ∈ A , T ∗ ( ξ )( A ) := ξ ( T − A ) . In particular, if ξ = P i ∈ I δ x i , then T ∗ ( ξ ) = P i ∈ I δ T ( x i ) .As we already noticed in [10], the property of bounded finiteness may be lost bythe action of T . Nevertheless, if m is a σ -finite measure on e X which is concentratedon T n ∈ Z T − n ∗ e X , it makes sense to consider the T ∗ -invariance of m . In this case, T ∗ ( ξ ) ∈ e X for m -almost all ξ ∈ e X , and ( e X, e A , m, T ∗ ) is an invertible measurepreserving dynamical system.The following definition generalizes the notion of T -point process introducedin [10]. Definition 1.1. A T -random measure is a random variable N defined on someprobability space (Ω , F , P ) with values in (cid:16) e X, e A (cid:17) such that • for any set A ∈ A , N ( A ) = 0 P -a-s. whenever µ ( A ) = 0 ; • there exists a measure preserving invertible transformation S on (Ω , F , P ) ,such that for any set A ∈ A , N ( A ) ◦ S = N (cid:0) T − A (cid:1) . RGODIC POISSON SPLITTINGS 3
Thus, a T -random measure N implements a factor relationship between thedynamical systems (Ω , F , P , S ) and (cid:16) e X, e A , m, T ∗ (cid:17) , where m is the pushforwardmeasure of P by N . We say that N is ergodic whenever (cid:16) e X, e A , m, T ∗ (cid:17) is ergodic.In particular N is ergodic as soon as (Ω , F , P , S ) is itself ergodic.The intensity of a T -random measure N is the T -invariant measure on X definedby the formula A ∈ A 7→ E [ N ( A )]. It is absolutely continuous with respect to µ and if it is σ -finite, it is a multiple of µ , by ergodicity of ( X, A , µ, T ). In this case,we say that N is integrable . More generally, the higher order moment measures canbe defined as follows. Definition 1.2.
Let n ≥ . A T -random measure N on (Ω , F , P , S ) is said to have moments of order n if, for all bounded A ∈ A , E [( N ( A )) n ] < + ∞ . In this case,the formula M Nn ( A × · · · × A n ) := E [ N ( A ) × · · · × N ( A n )] ( A , . . . , A n ∈ A ) defines a boundedly finite T × n -invariant measure M Nn on ( X n , A ⊗ n ) called the n -order moment measure .A T -random measure with moments of order is said to be square integrable . A T -point process is a T -random measure taking values in X ∗ . In this case,for ω ∈ Ω, we identify N ( ω ) with the corresponding set of points in X . Themost important T -point processes are Poisson point processes, let us recall theirdefinition. Definition 1.3.
A random variable N with values in ( X ∗ , A ∗ ) is a Poisson pointprocess of intensity µ if for any k ≥ , for any mutually disjoint sets A , . . . , A k ∈A f , the random variables N ( A ) , . . . , N ( A k ) are independent and Poisson dis-tributed with respective parameters µ ( A ) , . . . , µ ( A k ) . Such a process always exists, and its distribution µ ∗ on X ∗ is uniquely determinedby the preceding conditions.Since T preserves µ , one easily checks that T ∗ preserves µ ∗ . And defining N onthe probability space ( X ∗ , A ∗ , µ ∗ ) as the identity map provides an example of a T -point process, the underlying measure-preserving transformation being S = T ∗ in this case. Definition 1.4.
The probability-preserving dynamical system ( X ∗ , A ∗ , µ ∗ , T ∗ ) iscalled the Poisson suspension over the base ( X, A , µ, T ) . The basic result (see e.g. [15]) about Poisson suspensions states that ( X ∗ , A ∗ , µ ∗ , T ∗ )is ergodic (and then weakly mixing) if and only if there is no T -invariant set in A f .In particular this is the case if ( X, A , µ, T ) is ergodic and µ infinite.We also recall the classical isometry formula that will be useful several times inthis paper: for f, g ∈ L ( µ ) ∩ L ( µ ),(1) E µ ∗ (cid:20)(cid:18)Z X f ( x ) N ( dx ) − Z X f ( x ) µ ( dx ) (cid:19) (cid:18)Z X g ( x ) N ( dx ) − Z X g ( x ) µ ( dx ) (cid:19)(cid:21) = Z X f ( x ) g ( x ) µ ( dx ) . Remark 1.5.
The notion of T -random measure with intensity µ can be interpretedin terms of quasifactors as introduced by Glasner and Meyerovitch. Glasner defined ´ELISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE in [7] a quasifactor of a probability measure preserving system ( X, A , µ, T ) as aprobability measure preserving system (cid:16) e X, e A , m, T ∗ (cid:17) where E m [ N ( A )] = µ ( A ) .(Here, N is the random variable defined by the identity on e X , and in the case where µ is a probability measure, m is in fact concentrated on the subset of probabilitymeasures on X .) Meyerovitch in [13] extended this definition to the case where µ is infinite (but m is still a probability measure). Thus • Poisson suspensions appear as natural example of ergodic quasifactors. • any T -random measure N with intensity µ on (Ω , F , P , S ) gives rise to thequasifactor defined by m := N ∗ ( P ) ; • any quasifactor (cid:16) e X, e A , m, T ∗ (cid:17) is associated to the T -random measure N :=Id on the probability space (cid:16) e X, e A , m, T ∗ (cid:17) .In Section 4.2.3, we will consider yet another case, namely when m is an infinitemeasure, and use the terminology ∞ -quasifactor in this case. Splittings.Definition 1.6.
Let N be a T -point process defined on the dynamical system (Ω , F , P , S ) . For ≤ k ≤ ∞ , a T -splitting of order k of N is a finite or count-able family of T -point processes { N i } ≤ i 1, there is a transformation with first k Cartesian productsergodic whereas the k + 1 Cartesian product is not. The same authors also give RGODIC POISSON SPLITTINGS 5 examples of infinite measure preserving transformations with infinite ergodic index.All these examples fall into the Markov chain category.Since then, as the zoo of infinite measure preserving transformations developed,various examples of transformations having infinite ergodic index or not were built(see [1] where the so-called infinite Chacon Transformation — an infinite measurepreserving version of the classical Chacon transformation — is shown to have infiniteergodic index).In the last part of the paper, we will assume a much stronger property of T ,which can be viewed as a strong version of the Radon minimal self-joinings propertyintroduced by Danilenko in [4], and roughly saying that the Cartesian powers ofthe transformation admit as few invariant measures as possible (see Definition 4.1).An example of a transformation enjoying this property, the nearly finite Chacontransformation , is described in [9].2. Splitting of Poisson T -point processes For each n ≥ 1, we denote by P n the set of all partitions of { , . . . , n } . Given π ∈ P n , we define a measure on X n by m π ( A × · · · × A n ) := Y P ∈ π µ ( ∩ i ∈ P A i ) . For a given n , these measures are T × n -invariant and mutually singular. The mea-sure corresponding to the trivial partition with a single atom is called the n -diagonalmeasure, and is concentrated on D n := { ( x , . . . , x n ) ∈ X n : x = · · · = x n } .It is well known that the n -order moment measure of the Poisson process ofintensity µ takes the form X π ∈ P n c π m π . for positive coefficients c π , π ∈ P n . Moreover, if N is a Poisson process of intensity αµ , then the n -order moment measure equals X π ∈ P n c π α π m π . In the context of a T -point process, it turns out that the existence for eachmoment measure of a decomposition as a linear combination of the measures m π characterizes Poisson processes. This is the object of the following theorem, whoseproof is hidden in Theorem 3.2 in [10]. Although the argument is almost word-for-word the same, we repeat the proof here since the assumptions are far moregeneral, and it would be cumbersome to explain the differences without giving allthe details. Theorem 2.1. Let ( X, A , µ, T ) be an infinite measure preserving dynamical systemwith no invariant set of positive finite measure. Let N be a T -point process withmoments of all orders defined on (Ω , F , P , S ) .Then N is a Poisson process if and only if N is ergodic and for all n ≥ , thereexist nonnegative numbers α π such that M Nn = X π ∈ P n α π m π . ´ELISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE Proof. Only one direction needs to be detailed.We can assume that (Ω , F , P , S ) is ergodic. For n = 1, we obtain the intensityof N as a multiple of µ , say αµ for some α ≥ n ≥ 1, the weight of the n -diagonal measure is α (this is valid for any point process of intensity αµ ). Indeed, using a set A ∈ A f ,and (cid:16)(cid:0) A ℓi (cid:1) ≤ i ≤ p ℓ (cid:17) ℓ ≥ a generating sequence of partitions of A , we get p ℓ X i =1 M Nn (cid:0) A ℓi × · · · × A ℓi (cid:1) = E " p ℓ X i =1 N (cid:0) A ℓi (cid:1) · · · N (cid:0) A ℓi (cid:1) −−−→ ℓ →∞ E [ N ( A )] = αµ ( A ) = αµ ( A ∩ · · · ∩ A ) . On the other hand, p ℓ X i =1 M Nn (cid:0) A ℓi × · · · × A ℓi (cid:1) = M Nn p ℓ G i =1 A ℓi × · · · × A ℓi ! −−−→ ℓ →∞ M Nn ( D n ∩ A × · · · × A ) . Therefore M Nn ( D n ∩ A × · · · × A ) = αµ ( A ∩ · · · ∩ A ). Since the n -diagonal mea-sure is the only measure m π charging D n , this implies as claimed that the weightof the n -diagonal measure is α .We now want to prove by induction that, for all n ≥ M Nn is the n -order momentmeasure of a Poisson process of intensity αµ . The property is of course satisfiedfor n = 1. Let us assume it is satisfied up to some n ≥ 1, and let A , . . . , A n +1 besets in A f . Pick a nonempty subset K ( { , . . . , n + 1 } . By the ergodic theorem,we get(2) 1 ℓ ℓ X k =1 E " Y i ∈ K N ( A i ) Y i ∈ K c N ( A i ) ◦ S k ! −−−→ ℓ →∞ E " Y i ∈ K N ( A i ) E " Y i ∈ K c N ( A i ) = M N K × i ∈ K A i ! M N ( n +1 − K ) × i ∈ K c A i ! . On the other hand,1 ℓ ℓ X k =1 E " Y i ∈ K N ( A i ) Y i ∈ K c N ( A i ) ◦ S k ! (3) = 1 ℓ ℓ X k =1 M Nn +1 (cid:16) T − ǫ k (1) A × · · · × T − ǫ k ( n +1) A n +1 (cid:17) = X π ∈ P n +1 α π ℓ ℓ X k =1 m π (cid:16) T − ǫ k (1) A × · · · × T − ǫ k ( n +1) A n +1 (cid:17) RGODIC POISSON SPLITTINGS 7 where ǫ k ( i ) := 0 if i ∈ K , and ǫ k ( i ) := k otherwise. Coming back to the definitionof m π , we write m π (cid:16) T − ǫ k (1) A × · · · × T − ǫ k ( n +1) A n +1 (cid:17) = Y P ∈ π µ \ i ∈ P T − ǫ k ( i ) A i ! . Observe that, if K is a union of atoms of π , we have for any 1 ≤ k ≤ ℓ (4) m π (cid:16) T − ǫ k (1) A × · · · × T − ǫ k ( n +1) A n +1 (cid:17) = m π ( A × · · · × A n +1 ) . Otherwise, there exists an atom P ∈ π containing indices i ∈ K and j / ∈ K ,hence with ǫ k ( i ) = 0 and ǫ k ( j ) = k . We get that for some constant C , m π (cid:16) T − ǫ k (1) A × · · · × T − ǫ k ( n +1) A n +1 (cid:17) ≤ Cµ ( T − k A j ∩ A i ) . But, since there is no T -invariant set of positive finite measure,1 ℓ ℓ X k =1 µ (cid:0) T − k A j ∩ A i (cid:1) −−−→ ℓ →∞ . Let P Kn +1 be the set of partitions π ∈ P n +1 where K is a union of atoms of π . Theabove proves that, in the limit as ℓ → ∞ , the contribution in (3) of all partitions π ∈ P n +1 \ P Kn +1 vanishes. Thus we get, using (2), (3) and (4), M N K × i ∈ K A i ! M N ( n +1 − K ) × i ∈ K c A i ! = X π ∈ P Kn +1 α π m π ( A × · · · × A n +1 ) . Since ∅ 6 = K ( { , . . . , n + 1 } , the decompositions of M N K and M N ( n +1 − K ) onlyinvolve the coefficients α π , π ∈ P ∪ · · · ∪ P n . Using the mutual singularity ofthe measures on both sides of the above equality, we see that all the coefficients α π , π ∈ P Kn +1 , are completely determined by the coefficients corresponding topartitions in P ∪ · · · ∪ P n . Moreover, the above argument is valid in particularwhen N is the Poisson process of intensity αµ . By letting K run through all strictsubsets of { , . . . , n + 1 } , and using the induction hypothesis, we identify all butone coefficients of the decomposition of M Nn +1 as those of the Poisson point processof intensity αµ . The only coefficient that cannot be determined by this method isthe one associated to the trivial partition of { , . . . , n + 1 } into a single atom. Butthis corresponds to the ( n + 1)-diagonal measure, and we already know that thiscoefficient is α .We have proved that the moment measures of any order of N are those of aPoisson point process of intensity αµ . Lemma 3.1 in [10] ensures then that N is aPoisson point process of intensity αµ . (cid:3) Observe that the conclusion of the proof fails if one does not assume the ergod-icity of the point process N : think of a mixture of two Poisson point processes withdifferent intensities.The action of T × n on ( X n , A ⊗ n , m π ) is isomorphic to (cid:0) X π , A ⊗ π , µ ⊗ π , T × π (cid:1) .It is therefore ergodic if we assume that T has infinite ergodic index. With thisassumption, we get the following easy consequence for a thinning of Poisson T -pointprocess. ´ELISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE Proposition 2.2. Assume T has infinite ergodic index. Let N and N ′ be T -pointprocesses defined on the system (Ω , F , P , S ) such that N is Poisson of intensity µ and N ′ ≤ N . If N ′ is ergodic, then it is Poisson of intensity αµ for some ≤ α ≤ . Proof. If T has infinite ergodic index, all the measures m π , π ∈ P n , n ≥ 1, areergodic with respect to T × n , therefore, the formula P π ∈ P n c π m π is precisely theergodic decomposition of the moment measure M Nn of the Poisson process N , withrespect to T × n .Now, as N ′ ≤ N , we get for each n ≥ N ′ ⊗ · · · ⊗ N ′ ≤ N ⊗ · · · ⊗ N , and thus M N ′ n = E [ N ′ ⊗ · · · ⊗ N ′ ( · )] ≤ E [ N ⊗ · · · ⊗ N ( · )] = M Nn . We therefore obtain theergodic decomposition of M N ′ n in the form M N ′ n = X π ∈ P n α π m π for some non negative numbers α π . We can now apply the preceding theorem tothe ergodic T -point process N ′ to get the result. (cid:3) Example 2.3. Let us describe an example where the conclusion of the Propositionfails. Consider the so-called “homogeneous Poisson process”, that is, the classicalPoisson process on the real line with intensity equal to the Lebesgue measure. Thebase transformation is the translation T : x x + 1 (in particular, the Poisson T -point process is ergodic). Now we can form a thinning N ′ by keeping the pointsof N that are separated by at least κ > from every other points. It is easy to seethat N ′ is an ergodic T -point process but obviously not a Poisson process. Herethe proposition does not apply, not really because T is not ergodic (in fact T canbe embedded in an ergodic action of R ), but because the Cartesian powers fail tobe ergodic (even if we consider the R -action). For the proof of the next theorem, we shall need some definitions and the fol-lowing lemma which is a particular case of Lemma 2.6 in [10]. Definition 2.4. We say that a T -point process N defined on (Ω , F , P , S ) is free iffor P -almost all ω , for all k ∈ Z ∗ , N ( ω ) ∩ N (cid:0) S k ω (cid:1) = ∅ .Two T -point processes N and N defined on (Ω , F , P , S ) are said to be disso-ciated if, for P -almost all ω , for all k ∈ Z , N ( ω ) ∩ N (cid:0) S k ω (cid:1) = ∅ . Note that if N is a Poisson T -point process, then N is free (Proposition 2.7in [10]). Lemma 2.5. Let N , . . . , N n be n T -point processes defined on the ergodic system (Ω , F , P , S ) , having moments of all orders. Assume there exist a real number c > and some π ∈ P n such that for any sets A , . . . , A n in A f , E [ N ( A ) · · · N n ( A n )] ≥ cm π ( A × · · · × A n ) . Then, for any atom P ∈ π , any A ∈ A f , P A ∩ \ i ∈ P N i = ∅ ! > . In particular, for i, j ∈ P , the processes N i and N j are not dissociated. Theorem 2.6. If T has infinite ergodic index, then any ergodic splitting of a Pois-son T -point process is Poisson. RGODIC POISSON SPLITTINGS 9 Proof. We start with a splitting of order k < ∞ of a Poisson T -point process N defined on the ergodic dynamical system (Ω , F , P , S ), that is, we have k T -pointprocess N , . . . , N k so that N = N + · · · + N k .From Proposition 2.2, the N j ’s are Poisson processes with respective intensities α µ, . . . , α k µ such that α + · · · + α k = 1Let n , . . . , n k be positive numbers, n := n + · · · + n k , and let { Q , . . . , Q k } be the partition of { , . . . , n } in subsets of consecutive integers of respective size n , . . . , n k . For any { A i } ≤ i ≤ n in A f , set(5) σ ( A × · · · × A n ) := E k Y j =1 Y i ∈ Q j N j ( A i ) . This defines a T × n -invariant measure σ on ( X n , A ⊗ n ), for which, as above, sincefor all 1 ≤ j ≤ k we have N j ≤ N , σ ≤ M Nn . Hence σ has at most countably manyergodic components, of the form m π for some π ∈ P n . Observe that the processes N , . . . , N k are mutually dissociated as, for all 1 ≤ j ≤ k , N j ≤ N and N is free.Therefore, by Lemma 2.5, if the measure m π appears in the ergodic decompositionof σ , then π refines the partition { Q , . . . , Q k } . Hence, any ergodic component m π of σ has the form m π ( A × · · · × A n ) = k Y j =1 Y P ∈ π,P ⊂ Q j µ \ i ∈ P A i ! = k Y j =1 ν j × i ∈ Q j A i ! , where each ν j is a T × n j -invariant measure. In particular, any ergodic componentof σ is invariant by the transformation ( x , . . . , x n ) ( y , . . . , y n ), where y i := T x i if i ∈ Q k , and y i := x i otherwise. It follows that σ itself is invariant bythis transformation, hence the expression defining σ ( A × · · · × A n ) on the right-hand side of (5) is unchanged if we replace N k ( A i ) by N k ( T − A i ) for all i ∈ Q k simultaneously. Therefore, we can write for any { A i } ≤ i ≤ n in A f and any L ≥ E k Y j =1 Y i ∈ Q j N j ( A i ) = 1 L X ≤ ℓ ≤ L E k − Y j =1 Y i ∈ Q j N j ( A i ) Y i ∈ Q k N k ( T − ℓ A i ) = E k − Y j =1 Y i ∈ Q j N j ( A i ) L X ≤ ℓ ≤ L Y i ∈ Q k N k ◦ S ℓ ( A i ) . By the ergodic theorem, this converges as L → ∞ to E k − Y j =1 Y i ∈ Q j N j ( A i ) E Y i ∈ Q k N k ( A i ) . A straightforward induction on k then yields the equality E k Y j =1 Y i ∈ Q j N j ( A i ) = k Y j =1 E Y i ∈ Q j N j ( A i ) , and this is sufficient to obtain the independence between the Poisson processes.The case k = ∞ is easily deduced from the finite order case. (cid:3) Remark 2.7. For the conclusion of Theorem 2.6 to hold, it is in fact enough to as-sume only that (with the notations of the proof) all but one of the T -point processes N j are ergodic. Indeed, we can then apply the proof in any ergodic component ofthe joining ( N , . . . , N k ) , and see that in each of these ergodic components we havethe same structure of independent Poisson processes. A posteriori we see thatthere is only one ergodic component. In particular, with the assumptions of Propo-sition 2.2, we get that N ′ and ( N − N ′ ) are independent Poisson processes andthus form a Poisson splitting. Moreover, if we remove the assumption of ergodicityof N ′ in Proposition 2.2, we get that the ergodic components of ( N ′ , N − N ′ ) arenecessarily Poisson splittings. Application to marked Poisson point processes Here, we deal with so-called marked point processes. Roughly speaking a markedpoint process on X with marks in some measurable space ( K, K ) is a point processon X whose points carry some information, a mark, taking values in K . For a T -point process, we require the mark to have the shadowing property , meaning thatit follows the point when the dynamics on the point process is applied. We thusconsider a marked T -point process as a ( T × Id)-point process on the bigger space( X × K, A ⊗ K ) with intensity measure e µ that projects on µ . Theorem 3.1. Let ( X, A , µ, T ) be an infinite measure preserving dynamical systemwith infinite ergodic index. Let N be an ergodic ( T × Id) -point process on X × K .Assume that N := N ( · × K ) is a Poisson point process with intensity µ . Then N is a Poisson point process with intensity µ ⊗ ρ where ρ is some probability measureon K . Remark 3.2. It is well known that when a Poisson point process is independentlyendowed with i.i.d. marks, then the resulting point process on the product spaceis a Poisson process whose intensity is the product measure of the original inten-sity and the distribution of the marks. The above result means that, for T withinfinite ergodic index, the only way to get an ergodic marked T -point process outof a Poisson T -point process is precisely to take i.i.d. marks, independent of theunderlying process. Proof. Let us denote by e µ the intensity of N on X × K . Let E = F i ∈ I A i × B i ⊂ X × K be a finite union of pairwise disjoint product sets of finite e µ -measure. Let( ˜ B j ) j ∈ J be the finite partition of K into nonempty subsets generated by the sets B i . For each j ∈ J , denote by I j the subset of i ∈ I such that ˜ B j ⊂ B i , thenobserve that the sets A i , i ∈ I j are pairwise disjoint. Therefore, E can be alsowritten as the disjoint union E = G j ∈ J G i ∈ I j A i × ˜ B j , and this refines the original partition of E into product sets A i × B i . Since theergodic T -point processes N ˜ B j := N (cid:16) · × ˜ B j (cid:17) , j ∈ J , form an ergodic splitting of N , we get by Theorem 2.6 that they are independent Poisson T -point processes.Since for each j ∈ J the sets A i , i ∈ I j , are pairwise disjoint, the random variables N ( A i × ˜ B j ), j ∈ J , i ∈ I j are independent Poisson random variables. Finally, therandom variables N ( A i × B i ), i ∈ I are also independent Poisson random variables.This is enough to prove that N is a Poisson point process. RGODIC POISSON SPLITTINGS 11 To get the intensity, observe that, for each B ∈ K , N ( · × B ) is a thinning of N whose intensity is µ . By ergodicity of ( X, A , µ, T ), there exists 0 ≤ ρ ( B ) ≤ ρ ( B ) µ is the intensity of N ( · × B ). It is now clear that the map B ρ ( B )defines a probability measure on ( K, K ). Finally e µ = µ ⊗ ρ . (cid:3) Extended SuShis: From simple point processes to general randommeasures In this section, we aim to extend the “rigidity” result obtained in [10] for simplepoint processes to general random measures. The hypothesis on ( X, A , µ, T ) willbe the same as in [10] but rephrased with the notation introduced earlier. Definition 4.1. We say that the infinite measure preserving system ( X, A , µ, T ) hasthe (P) property if, for each n ≥ the following is true: whenever σ is a boundedlyfinite, T × n -invariant measure on X n , with marginals absolutely continuous withrespect to µ , then σ is conservative, and its ergodic components are all of the form (cid:0) T k × · · · × T k n (cid:1) ∗ m π for some π ∈ P n and integers k , . . . , k n . Note that the (P) property implies in particular that T has infinite ergodic index(otherwise the ergodic components of some product measure µ ⊗ n would not satisfythe required assumption).Let e X c ⊂ e X and e X d ⊂ e X be respectively the spaces of continuous and discreteboundedly finite measures on ( X, A ).If N is an integrable T -random measure, we can write N as N c + N d where N c ∈ e X c and N d ∈ e X d (both N c and N d can be obtained deterministically from N ). Of course N c and N d are still integrable T -random measures whose respectiveintensities are multiples of µ (by ergodicity of T ). Thanks to this decomposition, wecan study separately the case of a.s. continuous T -random measures, and the caseof a.s. discrete T -random measures, which will be the objects of the two followingsections.4.1. The continuous case.Proposition 4.2. Assume that T has the (P) property.If N is a square integrable T -random measure defined on some ergodic system (Ω , F , P , S ) , whose realizations are a.s. continuous, then there exists α ≥ suchthat N is constant and a.s. equal to αµ .If the realizations are a.s. absolutely continuous with respect to µ , no hypothesison moments are required to get the same conclusion, although we might have α =+ ∞ . Proof. First assume N is square integrable. We know that its intensity is of theform αµ for some α ≥ 0, and the moment measure of order two, defined on A ⊗ A by M N ( A × B ) := E [ N ( A ) N ( B )] = E [ N ⊗ N ( A × B )]is boundedly finite. This measure is T × T -invariant, and its marginals are absolutelycontinuous with respect to µ . Moreover, as N is a.s. continuous, N ⊗ N gives zeromeasure to the graphs of T k , for all k ∈ Z . Thanks to property (P), M N = βµ ⊗ µ for some β ≥ 0. Applying the ergodic theorem, we get for all A ∈ A f n n X k =1 E (cid:2) N ( A ) N ( A ) ◦ S k (cid:3) −−−−→ n →∞ E [ N ( A )] = α µ ( A ) . On the other hand,1 n n X k =1 E (cid:2) N ( A ) N ( A ) ◦ S k (cid:3) = 1 n n X k =1 M N ( A × T − k A ) = 1 n n X k =1 βµ ( A ) µ (cid:0) T − k A (cid:1) , which by invariance of µ is equal to βµ ( A ) . Therefore β = α and E (cid:20)(cid:16) N ( A ) − αµ ( A ) (cid:17) (cid:21) = M N ( A × A ) − α µ ( A ) = 0 , which implies the result.If we assume that the realizations are a.s. absolutely continuous, then we canwrite for all set A ∈ A N ( ω ) ( A ) = Z A f ( ω, x ) µ ( dx ) . We can therefore define N n by N n ( ω ) ( A ) = Z A ( f ∧ n ) ( ω, x ) µ ( dx ) .N n is still a continuous ergodic T -random measure but is now square integrable.From the first part of the proof, N n = α n µ a.s. Since N n increases to N , we get N = αµ a.s. where α is the increasing limit of α n . (cid:3) The discrete case. We consider the set of sequences ℓ +1 ( Z ) := ( ( a k ) k ∈ Z : ∀ k ∈ Z , a k ≥ X k ∈ Z a k < ∞ ) . The ℓ norm turns ℓ +1 ( Z ) into a complete separable metric space. The goal of thissection is to obtain the following result: Theorem 4.3. Assume T has the (P) property. Let N be a nonzero T -randommeasure with moments of all orders defined on some ergodic system (Ω , F , P , S ) and whose realizations are almost-surely discrete. Then there exists a probabilitydistribution κ on ℓ +1 ( Z ) and a positive number c such that N is distributed as A Z X × ℓ +1 ( Z ) X k ∈ Z a k A (cid:0) T k x (cid:1) N (cid:0) dx, d { a k } k ∈ Z (cid:1) where N is a Poisson point process on X × ℓ +1 ( Z ) with intensity cµ ⊗ κ . This result says that N has a cluster form which can be obtained in the followingway: start from a Poisson point process of intensity cµ , then replace independentlyeach point x output by this Poisson point process with a random measure (thecluster) of the form X k ∈ Z a k δ T k x where { a k } k ∈ Z is chosen according to κ .In the following, replacing if necessary µ by the intensity of N which is of theform αµ for some α > T , we assume that the intensity of N is µ . RGODIC POISSON SPLITTINGS 13 Removing points with small weights. Consider N as in the statement of The-orem 4.3. For ǫ > 0, we define N ǫ from N by removing points with weights lessthan ǫ . We also define N ǫ, as the simple point process obtained from N ǫ with allweights set to 1. We have for all A ∈ A f N ǫ, ( A ) ≤ ǫ N ǫ ( A ) ≤ ǫ N ( A ) , therefore N ǫ and N ǫ, are both T -random measures with moments of all orders.In particular, thanks to Proposition 2.1 in [10], N ǫ, almost surely belongs to thesubset e X d,f ⊂ e X d of measures ν satisfying the following property: ∀ x ∈ X, { n ∈ Z : ν ( T n x ) > } < ∞ . Of course, N ǫ ∈ e X d,f almost surely as well.We construct an injective map Φ from e X d,f to (cid:0) X × ℓ +1 ( Z ) (cid:1) ∗ . For ν ∈ e X d,f , Φ( ν )is the simple counting measures supported on the following collection of points in X × ℓ +1 ( Z ): we select in each orbit seen by ν the first element x in the orbit withmaximal weight, and consider the point (cid:0) x, ( ν ( T n x )) n ∈ Z (cid:1) . In other words, a point (cid:0) x, ( β n ) n ∈ Z (cid:1) belongs to Φ ( ν ) if and only if • β > β n for each n < • β ≥ β n for each n ≥ • ν ( { T n x } ) = β n for each n ∈ Z .Observe that we can recover ν from Φ ( ν ) by the formula(6) ν ( A ) = Z X × ℓ +1 ( Z ) X k ∈ N β k A (cid:0) T k x (cid:1) Φ ( ν ) (cid:0) dx, d { β k } k ∈ N (cid:1) . Now, Φ ( N ǫ ) is P -a.s. well defined as a ( T × Id)-point process on X × ℓ +1 ( Z ).Its projection on the first coordinate (Φ ( N ǫ )) := Φ ( N ǫ ) (cid:0) · × ℓ +1 ( Z ) (cid:1) is a T -pointprocess on X with moments of all orders (since Φ ( N ǫ ) (cid:0) · × ℓ +1 ( Z ) (cid:1) ≤ N ǫ, ( · )). Byconstruction, this T -point process is free since we have selected one point in eachorbit seen by N ǫ .Applying Theorem 3.2 in [10] (and this is where we need the assumptions ofmoments of all orders), we get that (Φ ( N ǫ )) is a Poisson process. Then by Theo-rem 3.1, we obtain that Φ ( N ǫ ) is a Poisson process on (cid:0) X × ℓ +1 ( Z ) (cid:1) . Applying (6),we almost surely have N ǫ ( A ) = Z X × R Z + X k ∈ N α k A (cid:0) T k x (cid:1) Φ ( N ǫ ) (cid:0) dx, d { α k } k ∈ N (cid:1) , and thus we get the theorem for N ǫ , for any ǫ > N , we need to take advantage of the infinite divisibilitycharacter of N ǫ , that N will inherit in the limit. We therefore have to recall somegeneral results about infinite divisibility.4.2.2. Infinitely divisible random measures. The notion of infinite divisibility canbe defined on any measurable semi-group although we only give it in the contextwe are interested in. Definition 4.4. let ( Z, Z ) be complete separable metric space. A probability dis-tribution σ on (cid:16) e Z, e Z (cid:17) is infinitely divisible (ID) if , for any k ∈ N , there exists aprobability distribution σ k such that σ = σ k ⋆ · · · ⋆ σ k | {z } k times , where ⋆ is the convolution of measures induced by the addition on e Z . By extension,we say that a random measure on Z is ID if its distribution is. The first examples of ID random measures are Poisson measures, their ID char-acter comes directly from the well known fact that the sum of two independentPoisson processes on the same space with intensities µ and µ is again a Poissonprocess, but with intensity µ + µ .We recall the fundamental representation result that can be found in [12, Theo-rem 3.20]. Theorem 4.5. A probability measure m on e X is ID if and only if there exist ameasure γ ∈ e X and a σ -finite measure ρ on e X \ { } satisfying, for all bounded B ⊂ X Z e X ( ξ ( B ) ∧ dρ ( ξ ) < ∞ , such that m is the distribution of the following random measure: γ + Z e X ξ ( · ) dω ( ξ ) , where ω is a random element of (cid:16) e X (cid:17) ∗ chosen according to the Poisson measure ρ ∗ on e X with intensity ρ .The measures γ and ρ are uniquely determined by m , and ρ is called the L´evymeasure of m . Here we summarize useful properties about the case we are interested in. Proposition 4.6. Let ( X, A , µ, T ) be a conservative ergodic infinite measure pre-serving system, and let m be the distribution of an ergodic square integrable IDrandom measure with intensity µ , and whose realizations are almost surely discrete.Assume that m corresponds to ( γ, ρ ) as in Theorem 4.5, then: • γ = 0 , • ρ is T ∗ -invariant and supported by e X d , • R e X ξ ( A ) ρ ( dξ ) = µ ( A ) for all A ∈ A f , • R e X ξ ( A ) ρ ( dξ ) = R e X ( ξ ( A ) − µ ( A )) m ( dξ ) for all A ∈ A f , • (cid:16) e X, e A , ρ, T ∗ (cid:17) has no T ∗ -invariant sets of finite, non zero measure. Proof. If m corresponds to ( γ, ρ ), then ( T ∗ ) ∗ ( m ) is ID too and corresponds to (cid:0) T ∗ ( γ ) , ( T ∗ ) ∗ ( ρ ) (cid:1) . Therefore if m is T ∗ -invariant then γ is T -invariant and ρ is RGODIC POISSON SPLITTINGS 15 T ∗ -invariant. For all A ∈ A f , µ ( A ) = Z e X ξ ( A ) m ( dξ )= γ ( A ) + Z e X ∗ Z e X ξ ( A ) ω ( dξ ) ρ ∗ ( dω ) , = γ ( A ) + Z e X ξ ( A ) ρ ( dξ ) . In particular, γ ≪ µ , and by ergodicity of µ , γ is some multiple of µ . But then m could not be the distribution of a point process as it would possess a continuous part,unless γ vanishes. From the Poisson construction, we also get that ρ is supportedon e X d .Next, the second formula is an application of the isometry formula (1).The last fact should come as no surprise for anyone familiar with ergodic prop-erties of ID systems, however there is no available proof for this particular case.We give one here:Assume (cid:16) e X, e A , m, T ∗ (cid:17) is ergodic and (cid:16) e X, e A , ρ, T ∗ (cid:17) possesses a T ∗ -invariant set K with ρ ( K ) < ∞ . Let (cid:16)(cid:16) e X (cid:17) ∗ , (cid:16) e A (cid:17) ∗ , ρ ∗ , ( T ∗ ) ∗ (cid:17) be the Poisson suspension over (cid:16) e X, e A , ρ, T ∗ (cid:17) . The map ω Z e X ξ ( · ) ω ( dξ )is a factor map between the suspension and the ergodic system (cid:16) e X, e A , m, T ∗ (cid:17) .Moreover we have another factor in the suspension, generated by the stationaryprocess ω n ( T ∗ ) k ∗ ( ω ) ( K ) o k ∈ Z . But as K is T ∗ -invariant, ( T ∗ ) k ∗ ( ω ) ( K ) = ω ( K ) for all k ∈ Z , therefore ( T ∗ ) ∗ actsas the identity on this factor. By disjointness between the identity map and anyergodic map [5], we obtain that these two factors are independent inside the Poissonsuspension. In particular, for any A ∈ A f , ω R e X ξ ( A ) ω ( dξ ) and ω ω ( K ) areindependent. It follows that Z ( e X ) ∗ (cid:18)Z e X ξ ( A ) ω ( dξ ) − µ ( A ) (cid:19) (cid:16) ω ( K ) − ρ ( K ) (cid:17) ρ ∗ ( dω ) = 0 . By the isometry formula (1), we get: Z e X ξ ( A ) K ( ξ ) ρ ( dξ ) = 0By monotone convergence, we can replace A by X in the preceding integral and,as ξ ( X ) > ρ -a.e., we get that K = 0 ρ -a.e., i.e. ρ ( K ) = 0. (cid:3) Now, let us explain how N inherits the ID character of N ǫ in the limit. We have N ǫ ≪ N , thus we can introduce the random density g ǫ := dN ǫ dN . Almost surely, g ǫ increases to 1 as ǫ → 0. Let f be a bounded continuous function on X withbounded support. Then by the dominated convergence theorem, Z X f dN ǫ = Z X f g ǫ dN −−−→ ǫ → Z X f dN a.s. Let also h be a bounded continuous function on R . Again by the dominated con-vergence theorem, we get E (cid:20) h (cid:18)Z X f dN ǫ (cid:19)(cid:21) −−−→ ǫ → E (cid:20) h (cid:18)Z X f dN (cid:19)(cid:21) . By [12, Theorem 4.11], this characterizes the weak convergence of the distributionof N ǫ to the distribution of N . Using [12, Lemma 4.24], we can conclude that N isID as a limit of ID random measures. N is thus square integrable and ID, hence admits a L´evy measure ρ whichis σ -finite measure on e X d , and is T ∗ -invariant. By ergodicity of N , the system (cid:16) e X, e A , ρ, T ∗ (cid:17) enjoys all the properties stated in Proposition 4.6.When T has the (P) property, this system has a very simple structure that wefully describe below.4.2.3. Infinite quasifactors. We start by extending the notion of quasifactor of Glas-ner and Meyerovitch to the case of an infinite measure on e X . Definition 4.7. An ∞ -quasifactor of ( X, A , µ, T ) is a dynamical system (cid:16) e X, e A , ρ, T ∗ (cid:17) where ρ is an infinite, σ -finite, T ∗ -invariant measure, and ∀ A ∈ A f , Z e X ξ ( A ) ρ ( dξ ) = µ ( A ) . The ∞ -quasifactor (cid:16) e X, e A , ρ, T ∗ (cid:17) is said to be square integrable if ∀ A ∈ A f , Z e X ( ξ ( A )) ρ ( dξ ) < ∞ . As a simple example of ∞ -quasifactor, we can consider the infinite measurepreserving system ( X ∗ , ρ, T ∗ ), where ρ is the pushforward of µ by x δ x . Itis actually the L´evy measure of the Poisson point process of intensity µ . Moregenerally, from Proposition 4.6, any ergodic infinitely divisible and square integrable T -random measure with intensity µ gives rise to a square integrable ∞ -quasifactorthrough its L´evy measure.Let us first consider the case of a simple ∞ -quasifactor , i.e. a measure ρ con-centrated on the set X ∗ of simple counting measures on X . Proposition 4.8. If T has the (P) property and ( X ∗ , A ∗ , ρ, T ∗ ) is a simple squareintegrable ∞ -quasifactor without T ∗ -invariant set of non-zero finite ρ -measure, then ρ -a.e. ξ ∈ X ∗ is concentrated on a finite subset of a single T -orbit. Proof. As ( X ∗ , A ∗ , ρ, T ∗ ) is square integrable, the formula m ( A × B ) := Z X ∗ ξ ( A ) ξ ( B ) ρ ( dξ )defines a boundedly finite T × T -invariant measure on X × X . Thanks to por-perty (P), it can be written as m = α ∞ µ ⊗ µ + X k ∈ Z α k ∆ k , RGODIC POISSON SPLITTINGS 17 where ∆ k is the measure supported on the graph of T k defined by ∆ k ( A × B ) := µ ( A ∩ T − k B ). Observe that, as there is no T ∗ -invariant set of non-zero finite ρ -measure, 1 n n X ℓ =1 m (cid:0) A × T − ℓ B (cid:1) = 1 n n X ℓ =1 Z X ∗ ξ ( A ) T ℓ ∗ ξ ( B ) ρ ( dξ ) → . However, 1 n n X ℓ =1 α ∞ µ ⊗ µ (cid:0) A × T − ℓ B (cid:1) = α ∞ µ ⊗ µ ( A × B )therefore α ∞ = 0.This means that m is concentrated on the graphs of the maps T k , k ∈ Z , thereforefor ρ -a.e. ξ ∈ X ∗ the product ξ ⊗ ξ is concentrated on these graphs. It follows that ρ -a.e. ξ ∈ X ∗ is concentrated on a single T -orbit. It remains to verify that ρ isalmost surely concentrated on a finite number of points in this orbit. For this, weobserve that Proposition 2.1 in [10] can be generalized to the case of an infinitemeasure ρ on X ∗ . Indeed, the Palm measures of ρ are probability measures since R X ∗ ξ ( A ) ρ ( dξ ) = µ ( A ) < + ∞ . Since ρ has moments of order 2, the propositionyields that ρ -a.e. ξ ∈ X ∗ is concentrated on a finite subset of a single T -orbit. (cid:3) Proposition 4.9. If T has the (P) property and if (cid:16) e X d , e A , ρ, T ∗ (cid:17) is a square in-tegrable ∞ -quasifactor without T ∗ -invariant set of non-zero finite ρ -measure, thenthere exists c > and a factor map ϕ from (cid:16) e X d , e A , ρ, T ∗ (cid:17) to ( X, A , cµ, T ) and some T ∗ -invariant maps ξ a k ( ξ ) ≥ , k ∈ Z , such that ξ = X k ∈ Z a k ( ξ ) δ T k ϕ ( ξ ) . Proof. First observe that { } is a T ∗ -invariant set. It cannot have infinite ρ measurebecause ρ is σ -finite, hence ρ ( { } ) = 0 from the hypotheses.Let ξ ∈ e X d and set ξ | ε , ε > ξ where we have forgotten points with weightsless than ε and set the other weights to be 1. Then ξ ξ | ε is a factor map and ξ | ε turns out to induce on e X d \ (cid:8) ξ | ε = { } (cid:9) a simple square integrable ∞ -quasifactorwithout T ∗ -invariant set of non-zero finite measure. Therefore by Proposition 4.8, ξ | ε has a finite number of points on its support, which all lie on a single T -orbit.It follows that all the points of the support of ξ are ρ -a.s. on a single T -orbit,and only a finite number of them have a weight greater than any fixed positiveconstant. We can therefore see that the map ϕ : ξ ϕ ( ξ ) where ϕ ( ξ ) is the pointof the support with the highest weight and the lowest place in the orbit is welldefined. It satisfies ϕ ( T ∗ ξ ) = T ϕ ( ξ ) . Now with this “origin” ϕ ( ξ ), we can define maps ξ a k ( ξ ) ≥ ξ := X k ∈ Z a k ( ξ ) δ T k ϕ ( ξ ) . We have T ∗ ξ = X k ∈ Z a k ( ξ ) δ T k +1 ϕ ( ξ ) = X k ∈ Z a k ( ξ ) δ T k ϕ ( T ∗ ξ )8 ´ELISE JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE in the one hand, and in the other hand T ∗ ξ := X k ∈ Z a k ( T ∗ ξ ) δ T k ϕ ( T ∗ ξ ) . Therefore the maps a k are T ∗ -invariant.We have for all Aµ ( A ) = Z e X d ξ ( A ) ρ ( dξ )= X k ∈ Z Z e X d a k ( ξ ) δ T k ϕ ( ξ ) ( A ) ρ ( dξ )= X k ∈ Z Z e X d a k (cid:0) T k ∗ ξ (cid:1) δ ϕ ( T k ∗ ( ξ )) ( A ) ρ ( dξ ) by T ∗ -invariance of a k = X k ∈ Z Z e X d δ ϕ ( ξ ) ( A ) a k ( ξ ) ρ ( dξ ) by T ∗ -invariance of ρ .Let us define for each k ∈ Z the measure ρ k by dρ k dρ := a k . Then we get µ ( A ) = X k ∈ Z ϕ ∗ ρ k ( A ) , and in particular ϕ ∗ ρ ≪ µ . But, as a > ρ -a.e., ρ ∼ ρ and we also have ϕ ∗ ρ ≪ µ . By ergodicity ϕ ∗ ρ = cµ for some c > ϕ induces a factor map between (cid:16) e X d , e A , ρ, T ∗ (cid:17) and ( X, A , cµ, T ). (cid:3) Corollary 4.10. Assume that T has the (P) property. Let (cid:16) e X d , e A , ρ, T ∗ (cid:17) be aninfinite measure-preserving square integrable ∞ -quasifactor without T ∗ -invariantset of non-zero finite ρ -measure. Then there exists a probability measure κ on R Z + such that (cid:16) e X d , e A , ρ, T ∗ (cid:17) is isomorphic to (cid:0) X × R Z + , A ⊗ B ⊗ Z , µ ⊗ ( cκ ) , T × Id (cid:1) (where c is given in Proposition 4.9).Moreover, we have c = Z R Z + X k ∈ Z a k ! κ ( d { a k } k ∈ Z ) . In particular, { a k } k ∈ Z ∈ ℓ +1 ( Z ) κ -a.s. Proof. Define Φ from e X d to X × R Z + byΦ ( ξ ) := (cid:0) ϕ ( ξ ) , ( a k ( ξ )) k ∈ Z (cid:1) . Then T × Id preserves m := Φ ∗ ρ and Φ is an isomorphism between (cid:16) e X d , e A , ρ, T ∗ (cid:17) and (cid:0) X × R Z + , A ⊗ B ⊗ Z , m, T × Id (cid:1) . Since the σ -algebra generated by ϕ ( ξ ) is σ -finite by the preceding proposition, we can disintegrate m with respect to the firstcoordinate: we get a family ( κ x ) x ∈ X of probability measures on R Z + , such that m ( A × B ) = Z A κ x ( B ) c dµ ( x ) , RGODIC POISSON SPLITTINGS 19 where c is given in the preceding proposition. By invariance of m under T × Id, weget κ x = κ T x for µ -almost every x , and by ergodicity of T we conclude that thereexists κ such that κ x = κ µ -almost everywhere. This yields m = µ ⊗ ( cκ ).Now, for each A ∈ A f , we have µ ( A ) = Z e X d ξ ( A ) ρ ( dξ )= c Z R Z + Z X X k ∈ Z a k A ( T k x ) µ ( dx ) ! κ ( d { a k } k ∈ Z )= c µ ( A ) Z R Z + X k ∈ Z a k ! κ ( d { a k } k ∈ Z ) . (cid:3) End of the proof of Theorem 4.3. We come back to the end of the proof of themain theorem of this section. Recall that under the assumptions of this theorem,we were left with the following situation: N is a T -random measure with momentsof all orders, we showed it is infinitely divisible. Hence the conclusion follows fromthe next proposition. Proposition 4.11. Assume that T has the (P) property. Let N be a square in-tegrable ID T -random measure defined on some ergodic system (Ω , F , P , S ) whoserealizations are almost surely discrete. Then there exists a probability distribution κ on (cid:0) ℓ +1 ( Z ) , B ⊗ Z (cid:1) and c > such that N is distributed as A Z X × ℓ +1 ( Z ) X k ∈ Z a k A (cid:0) T k x (cid:1) N (cid:0) dx, d { a k } k ∈ Z (cid:1) where N is a Poisson point process on X × ℓ +1 ( Z ) with intensity cµ ⊗ κ . Proof. By Theorem 4.5 and Proposition 4.6, the ID square integrable T-randommeasure N can be described by its L´evy measure ρ . The latter is nothing elsethan a square integrable ∞ -quasifactor, whose structure is completely given inCorollary 4.10: the measure space (cid:16) e X d , ρ (cid:17) is isomorphic to ( X × ℓ +1 ( Z ) , cµ ⊗ κ ).In this context, the representation of N as an integral with respect to a Poissonrandom measure takes the more concrete form explicited in the statement of theproposition. (cid:3) Remark 4.12. Note that the assumption that N has moments of all orders hasonly been used to obtain the ID character of the random measure. Once this isestablished, square integrability of N is sufficient to conclude. We do not know ifsquare integrability alone implies the conclusion of Theorem 4.3. Improved disjointness results This last, short, section deals with joinings and disjointness. The notion ofjoinings in Ergodic Theory is the dynamical counterpart of couplings in ProbabilityTheory. It is particularly relevant for the classification of dynamical systems as wedo below. For the reader unfamiliar with this notion, we refer to the seminalpaper [5] and the book [6], that present modern ergodic theory through joiningsand disjointness. In [10], we obtained a series of disjointness results for Poisson suspensions overtransformations satisfying the (P) property with the additional assumption thatthe base transformation should have a measurable law of large numbers, which is avery particular property. We were already convinced that this assumption was notnecessary. The results proved in the present paper allow to get rid of it.The following proposition is an example of how the simplification occurs. Proposition 5.1. Assume T has the (P) property. If an ergodic probability pre-serving system (Ω , F , P , S ) is not disjoint from ( X ∗ , A ∗ , µ ∗ , T ∗ ) then it possesses (cid:0) X ∗ , A ∗ , ( αµ ) ∗ , T ∗ (cid:1) as a factor for some α > . Proof. Consider a non-trivial joining λ of (Ω , F , P , S ) with the Poisson suspension( X ∗ , A ∗ , µ ∗ , T ∗ ), and denote by Ψ : L ( µ ∗ ) → L ( P ) the associated Markov opera-tor. Denote by N the canonical Poisson T -point process defined on ( X ∗ , A ∗ , µ ∗ , T ∗ ).By positivity of Ψ, the map A ∈ A f Ψ( N ( A )) extends to a T -random measureon (Ω , F , P , S ). Indeed, for A ∈ A f we haveΨ( N ( T − ( A ))) = Ψ( N ( A ) ◦ T ∗ )= Ψ( U T ∗ N ( A ))= U S Ψ( N ( A ))= Ψ( N ( A )) ◦ S, and E P [Ψ( N ( A ))] = E µ ∗ [ N ( A )] = µ ( A ).Moreover, this T -random measure has moments of all orders, as for any A ∈ A f and any n ≥ 1, since Ψ can be interpreted as a conditional expectation, we have E P [(Ψ( N ( A )) n ] ≤ E P [Ψ ( N ( A ) n )] = E µ ∗ [ N ( A ) n ] < ∞ . From Proposition 4.2, there exists 0 ≤ c ≤ T -random measure M ofintensity µ defined on (Ω , F , P ), supported on discrete measures, such thatΨ ( N ( · )) = cµ + (1 − c ) M If c = 1, then for all A ∈ A f , Ψ ( N ( A ) − µ ( A )) = 0, which means that Ψ vanisheson the first chaos. Let Ψ ∗ : L ( P ) → L ( µ ∗ ) be the adjoint Markov operator, weget that Ψ ∗ Ψ is a Markov operator on L ( µ ∗ ) that vanishes on the first chaos. Itcan be written as an integral of indecomposable operatorsΨ ∗ Ψ = Z W Ψ w ρ ( dw ) , where ( W, W , ρ ) is an auxilliary probability space. Now the proof follows the samelines as Proposition 4.11 in [10]. We get that Ψ ∗ Ψ is the projection on constantsand this implies in turn that the initial joining is trivial, hence a contradiction.Therefore c < 1. We deduce that M is a factor of (Ω , F , P , S ), and much asin Section 4.2.1, we obtain a further factor which is a Poisson point process ofintensity αµ for some α > 0. (In Section 4.2.1, we got such a factor by considering(Φ ( N ǫ )) .) (cid:3) In particular, following the same proof as in Theorem 5.14 in [10], we obtain: Theorem 5.2. If T has the (P) property then ( X ∗ , A ∗ , µ ∗ , T ∗ ) is disjoint from anyrank one transformation and any Gaussian dynamical system. RGODIC POISSON SPLITTINGS 21 References 1. Terrence Adams, Nathaniel Friedman, and Cesar E. Silva, Rank-one weak mixing for nonsin-gular transformations , Israel J. Math. (1997), 269–281.2. Karen Ball, Poisson thinning by monotone factors , Electron. Comm. Probab. (2005),60–69.3. D.J. Daley and D. 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Parry, Infinite measure preserving transformations with “mixing” , Bull.Amer. Math. Soc. (1963), 752–756.12. Olav Kallenberg, Random measures, theory and applications , Probability Theory and Sto-chastic Modelling, vol. 77, Springer, Cham, 2017.13. Tom Meyerovitch, Quasi-factors and relative entropy for infinite-measure-preserving trans-formations , Israel J. Math. (2011), 43–60.14. , Ergodicity of Poisson products and applications , Ann. Probab. (2013), no. 5,3181–3200.15. Emmanuel Roy, Ergodic properties of Poissonian ID processes , Ann. Probab. (2007), no. 2,551–576. ´Elise Janvresse: Laboratoire Ami´enois de Math´ematique Fondamentale et Appliqu´ee,CNRS-UMR 7352, Universit´e de Picardie Jules Verne, 33 rue Saint Leu, F80039 Amienscedex 1, France. E-mail address : [email protected] Emmanuel Roy: Laboratoire Analyse, G´eom´etrie et Applications, Universit´e Paris13 Institut Galil´ee, 99 avenue Jean-Baptiste Cl´ement F93430 Villetaneuse, France. E-mail address : [email protected] Thierry de la Rue: Laboratoire de Math´ematiques Rapha¨el Salem, Universit´e deRouen, CNRS, Avenue de l’Universit´e, F76801 Saint ´Etienne du Rouvray, France. E-mail address ::