aa r X i v : . [ m a t h . D S ] O c t The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2013
ERGODICITY OF POISSON PRODUCTS AND APPLICATIONS
By Tom Meyerovitch
Ben-Gurion University of the Negev
In this paper we study the Poisson process over a σ -finite measure-space equipped with a measure preserving transformation or a groupof measure preserving transformations. For a measure-preserving trans-formation T acting on a σ -finite measure-space X , the Poisson sus-pension of T is the associated probability preserving transformation T ∗ which acts on realization of the Poisson process over X . We proveergodicity of the Poisson-product T × T ∗ under the assumption that T is ergodic and conservative. We then show, assuming ergodicityof T × T ∗ , that it is impossible to deterministically perform natu-ral equivariant operations: thinning , allocation or matching . In con-trast, there are well-known results in the literature demonstrating theexistence of isometry equivariant thinning, matching and allocationof homogenous Poisson processes on R d . We also prove ergodicityof the “first return of left-most transformation” associated with ameasure preserving transformation on R + , and discuss ergodicity ofthe Poisson-product of measure preserving group actions, and relatedspectral properties.
1. Introduction.
It is straightforward that the distribution of a homoge-nous Poisson point process on R d is preserved by isometries. In the liter-ature, various translation-equivariant and isometry-equivariant operationson Poisson process have been considered: • Poisson thinning : A (deterministic)
Poisson-thinning is a rule for selectinga subset of the points in the Poisson process which are equal in distributionto a lower intensity homogenous Poisson process. Ball [4] demonstrated adeterministic Poisson-thinning on R which was translation equivariant —that is, if a translation is applied to the original process, the new pointsselected are translations of the original ones by the same vector. This wasextended and refined by Holroyd, Lyons and Soo [11] to show that for any d ≥
1, there is an isometry-equivariant
Poisson-thinning on R d . Received July 2011; revised November 2012.
AMS 2000 subject classifications.
Key words and phrases.
Poisson suspension, equivariant thinning, equivariant alloca-tion, infinite measure preserving transformations, conservative transformations.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2013, Vol. 41, No. 5, 3181–3200. This reprint differs from the original inpagination and typographic detail. 1
T. MEYEROVITCH • Poisson allocation : Given a realization ω of a Poisson process on R d ,a Poisson allocation partitions R d up to measure 0 by assigning to eachpoint in ω a cell which is a finite-measure subset of R d . Hoffman, Hol-royd and Peres [9] constructed an isometry-equivariant allocation schemefor any stationary point process of finite intensity. The above allocationscheme had the characteristic property of being “stable.” Subsequentwork demonstrated isometry-equivariant Poisson allocations with othernice properties such as connectedness of the allocated cells [15] or goodstochastic bounds on the diameter of the cells [5]. • Poisson matching : A
Poisson matching is a deterministic scheme whichfinds a perfect matching of two identically distributed independent Poissonprocesses. Different isometry-equivariant Poisson matching schemes havebeen constructed [10, 12].Consider a transformation of R d which preserves Lebesgue measure. Doesthere exist a Poisson thinning which is equivariant with respect to the giventransformation? What about an equivariant Poisson allocation or matching?To have a couple of examples in mind, consider the following transforma-tions T RW , T Boole : R → R of the real line given by T RW ( x ) = ⌊ x ⌋ + (2 x mod 1) − · (0 , / ( x mod 1)(1)and T Boole ( x ) = x − x (2) T Boole is known as Boole’s transformation. It is a is a classical example of anergodic transformation preserving Lebesgue measure. See [3] for a proof ofergodicity and discussions of this transformation. You may notice that T RW is isomorphic to the shift map on the space of forward trajectories of thesimple random walk on Z .From our perspective, it is natural (although mathematically equivalent)to consider an abstract standard σ -finite measure space ( X, B , µ ), insteadof R d with Lebesgue measure. We consider a Poisson point process on thisspace, which denoted by ( X ∗ , B ∗ , µ ∗ ). Any measure preserving transforma-tion T : X → X naturally induces a map T ∗ : X ∗ → X ∗ on the Poisson pro-cess. This transformation T ∗ is the Poisson suspension of T [17].We prove the following theorem: Theorem 1.1.
Let T : X → X be any conservative and ergodic measurepreserving transformation of ( X, B , µ ) with µ ( X ) = ∞ . There does not exista T -equivariant Poisson thinning, allocation or matching. We prove Theorem 1.1 by studying ergodic properties of the map T × T ∗ ,which acts on the product space ( X × X ∗ , B × B ∗ , µ ∗ × µ ). We refer to thissystem as the Poisson-product associated with T . The space X × X ∗ can RGODICITY OF POISSON PRODUCTS be considered as a countable set of “indistinguishable” points in X , with aunique “distinguished” point. The Poisson-product T × T ∗ acts on this byapplying the same map T to each point, including the distinguished point.Our main result about Poisson-products is the following theorem: Theorem 1.2.
Let ( X, B , µ, T ) be a conservative, measure-preservingtransformation with µ ( X ) = ∞ . Then the Poisson-product T × T ∗ is ergodicif and only if T is ergodic. Before concluding the introduction and proceeding with the details, werecall a couple of results regarding nonexistence of certain equivariant op-erations on Poisson processes. Evans proved in [6] that with respect to anynoncompact group of linear transformations there is no invariant Poisson-thinning on R d . Gurel-Gurevich and Peled proved the nonexistence of trans-lation equivariant Poisson thickening on the real line [7], which means thatthere is no measurable function on realizations of the a homogenous Pois-son process that sends a Poisson process to a higher intensity homogenousPoisson process.This paper is organized as follows: In Section 2 we briefly provide someterminology and necessary background. Section 3 contains a short proof ofTheorem 1.2 stated above, based on previous work in ergodic theory. InSection 4 we prove any T -equivariant thinning is trivial, assuming T × T ∗ is ergodic. In Section 5 we show that under the same assumptions thereare no T -equivariant Poisson allocations or Poisson matchings, using anintermediate result about nonexistence of positive equivariant maps into L .Section 6 discusses the “leftmost position transformation” and contains aproof of ergodicity, yet another application of Theorem 1.2. Section 7 is adiscussion of ergodicity of Poisson products for measure preserving groupactions.
2. Preliminaries.
In this section we briefly recall some definitions andbackground from ergodic theory required for the rest of the paper. We alsorecall some properties of the Poisson point process on a σ -finite measurespace.2.1. Ergodicity, conservative transformations and induced transformations.
Throughout this paper ( X, B , µ ) is a standard σ -finite measure space. Wewill mostly be interested in the case where µ ( X ) = ∞ . Also throughout thepaper, T : X → X is a measure preserving transformation, unless explicitlystated otherwise, where T denotes an action of a group by measure preserv-ing transformations of ( X, B , µ ). The collection of measurable sets of positivemeasure by will be denoted by B + := { B ∈ B : µ ( B ) > } .Recall that T is ergodic if any set A ∈ B which is T -invariant has either µ ( A ) = 0 or µ ( A c ) = 0. Equivalently, T is ergodic if any measurable function T. MEYEROVITCH f : X → R satisfying f ◦ T = f µ -almost everywhere is constant on a set offull measure.A set W ∈ B is called a wandering set if µ ( T − n W ∩ W ) = 0 for all n > T is called conservative if there are no wandering setsin B + . The Poincar´e recurrence theorem asserts that any T which preservesa finite measure is conservative.For a conservative T and A ∈ B + , the first return time function is definedfor x ∈ A by ϕ A ( x ) = min { n ≥ T n ( x ) ∈ A } . ϕ A is finite µ -a.e; this is adirect consequence of T being conservative.The induced transformation on A is defined by T A ( x ) := T ϕ A ( x ) ( x ). If T isconservative and ergodic and A ∈ B + , T A : A → A is a conservative, ergodictransformation of ( A, B ∩
A, µ | A ).See [1] for a comprehensive introduction to ergodic theory of infinite mea-sure preserving transformations.2.2. Cartesian product transformations.
Suppose T is conservative, and S : Y → Y is a probability preserving transformation of ( Y, C , ν ), namely ν ( Y ) = 1. It follows (as in Proposition 1.2.4 in [1]) that the Cartesian producttransformation T × S : X × Y → X × Y is a conservative, measure-preservingtransforation of the Cartesian product measure-space ( X × Y, B ⊗ C , µ × ν ).2.3. L ∞ -eigenvalues of measure preserving transformations. A function f ∈ L ∞ ( X, B , µ ) is an L ∞ -eigenfunction of T if f = 0 and T f = λf for some λ ∈ C . The corresponding λ is called an L ∞ - eigenvalue of T . We brieflyrecall some well-known results:If T is ergodic and f is an L ∞ -eigenfunction, it follows that | f | is constantalmost-everywhere. The L ∞ -eigenvalues of T are e ( T ) := { λ ∈ C : ∃ f ∈ L ∞ ( X, B , µ ) f = 0 and T f = λf } . If T is conservative, then | λ | = 1 for any eigenvalue λ , for otherwise theset { x ∈ X : | f ( x ) | ∈ ( | λ | k , | λ | k +1 ] } would be a nontrivial wandering set for some k ∈ Z if | λ | >
1. Thus, for anyconservative transformation T , e ( T ) is a subset if the unit sphere S = { x ∈ C : | x | = 1 } .e ( T ) is a group with respect to multiplication, and carries a natural Polishtopology, with respect to which the natural embedding in S is continuous.When T preserves a finite measure, e ( T ) is at most countable. For ageneral infinite-measure preserving T , however, e ( T ) can be uncountable,and quite “large,” for instance, the arbitrary Hausdorff dimension α ∈ (0 , e ( T ) can RGODICITY OF POISSON PRODUCTS be. For instance, e ( T ) is a weak Dirichlet set. This means thatlim inf n →∞ Z | − χ n ( s ) | dp ( s ) = 0whenever p is a probability measure on S with p ( e ( T )) = 1, and χ n ( s ) :=exp(2 πins ). In particular the set e ( T ) has measure zero with respect to Haarmeasure on S .We refer the reader to existing literature for further details [1, 2, 16, 19].2.4. The L -spectrum. Let U T : L ( µ ) → L ( µ ) denote the unitary oper-ator defined by U T ( f ) := f ◦ T .The spectral type of a unitary operator U on a Hilbert space H , denoted σ U , is a positive measure on S satisfying(a) h U n f, g i = Z S χ n ( s ) h ( f, g )( s ) dσ U ( s ) , where h : H × H → L ( σ U ) is a sesquilinear map;(b) σ U is minimal with that property, in the sense that it satisfies σ U ≪ σ for any measure σ on S satisfying (a).In (b) above and throughout the paper, we write µ ≪ µ to indicate thatthe measure µ is absolutely continuous with respect to µ . If µ ≪ µ and µ ≪ µ , we say they are in the same measure class.The spectral type σ U is defined only up to measure class. Existence of σ U is a formulation of the scalar spectral theorem .For a measure-preserving transformation T , The spectral type of T σ T is the spectral type of the associated unitary operator U T on L ( µ ). For aprobability preserving transformation S , the restricted spectral type is thespectral type the unitary operator U S restricted to L -functions with integralzero.Our brief exposition here follows Section 2.5 of [1].2.5. Poisson processes and the Poisson suspension.
For a standard σ -finite measure space ( X, B , µ ), ( X ∗ , B ∗ , µ ∗ ) denotes the associated Poissonpoint process , which we now describe. X ∗ is the space of countable subsetsof X . We will typically denote an element of X ∗ by ω , ω , ω and so on.The σ -algebra B ∗ is generated by sets of the form[ | ω ∩ B | = n ] := { ω ∈ X ∗ : | ω ∩ B | = n } (3)for n ≥ B ∈ B .The probability measure µ ∗ is is uniquely defined by requiring that forany pairwise disjoint A , A , . . . , A n ∈ B , if ω ∈ X ∗ is sampled accordingto µ ∗ , then | ω ∩ A i | are jointly independent random variables individually T. MEYEROVITCH distributed Poisson with expectation µ ( A i ) µ ∗ ( | ω ∩ A | = k ) = e − µ ( A ) µ ( A ) k k ! . (4)The underlaying measure µ ∗ is called the intensity of the Poisson process.We will assume that the measure µ has no atoms, namely µ ( { x } ) = 0 forany x ∈ X . This is a necessary and sufficient condition to avoid multiplicityof points almost surely with respect to µ ∗ .A Poisson point process can be defined on very general measure spaces,under milder assumptions than “standard.” Details of the construction andgeneral properties of Poisson processes can be found, for instance, in [13, 14].To make various measurability statements in the following sections moretransparent, we assume the following technical condition: There is a fixedsequence { β n } ∞ n =1 of countable partitions of X into B -measurable sets, suchthat β n +1 refines β n , with the additional property that the mesh of thesepartitions goes to 0, namely, λ ( β n ) := sup { µ ( B ) : B ∈ β n } → n → ∞ . We assume that B = W ∞ n =1 σ ( β n ) is the σ -algebra generated by the unionof these partitions. For instance, if ( X, B , µ ) is the real line with Lebesguemeasure on the Borel sets, we can take β n to be the partition into half-openintervals with endpoints on the lattice n Z .The σ -algebra B ∗ can now be defined by B ∗ = ∞ _ n =1 β ∗ n , where β ∗ n is the σ -algebra generated by sets of the form (3) with B ∈ β n and n ∈ { , , , . . . } . Different sequences β n with the above properties will notchange the completion with respect to µ ∗ of the resulting σ -algebra B ∗ .The Poisson suspension of a measure preserving map T : X → X , is thenatural map obtained by applying T on X ∗ . As in [17], we denote it by T ∗ : X ∗ → X ∗ . This transformation is formally defined by T ∗ ( ω ) = { T ( x ) : x ∈ ω } .T ∗ is a probability-preserving transformation of ( X ∗ , B ∗ , µ ∗ ).The following proposition relates the spectral measures of T and T ∗ [17]: Proposition 2.1. If σ is the spectral-type of T , the restricted spectraltype of T ∗ is given by σ T ∗ = X n ≥ n ! σ ⊗ n . It is a classical result that a probability-preserving transformation is er-godic if and only if its restricted spectral type has no atom at λ = 1, and is RGODICITY OF POISSON PRODUCTS weakly mixing if and only if its restricted spectral type has no atoms in S (this property is also equivalent to ergodicity of T × T ). It follows that T ∗ is ergodic if and only if T ∗ is weakly mixing if and only if there are no T -invariant sets of finite measure in B + [17].In the following sections we will use the map π : X × X ∗ → X ∗ given by π ( x, ω ) = { x } ∪ ω. (5)The map π defined by (5) is a measurable map from between the measurespaces ( X × X ∗ , B ⊗ B ∗ ) and ( X ∗ , B ∗ ). This is can be verified directly usingthe following equalities of sets: π − [ | ω ∩ A | = 0] = ( X \ A ) × [ | ω ∩ A | = 0]and π − [ | ω ∩ A | = n ] = (( X \ A ) × [ | ω ∩ A | = n ]) ∪ ( A × [ | ω ∩ A | ∈ { n − , n } ])for A ∈ B and n ∈ N .In fact, π is a ∞ -factor map between the measure preserving maps T × T ∗ and T ∗ , in the sense of Chapter 3 of [1]: This means that π ◦ T ∗ = ( T × T ∗ ) ◦ π and for A ∈ B ∗ ( µ × µ ∗ ) ◦ π − ( A ) = (cid:26) , if µ ∗ ( A ) = 0, ∞ , otherwise.
3. Ergodicity of Poisson product for conservative transformations.
Wenow provide a proof of Theorem 1.2. The argument we use is an adaptationof [2]. To prove our result, we invoke the following condition for ergodicityof Cartesian products, due to M. Keane:
Theorem (The ergodic multiplier theorem).
Let S be a probability pre-serving transformation and T a conservative, ergodic, nonsingular transfor-mation. S × T is ergodic if and only if σ S ( e ( T )) = 0 , where: • σ S is the restricted spectral type of S ; • e ( T ) is the group of L ∞ -eigenvalues of T . A proof of this result is provided, for instance, in Section 2 . T ∗ is a linear combination of convolution powers of the spectral type of T .We make use of the following basic lemma about convolution of measuresand equivalence of measure classes. A short proof is provided here for thesake of completeness: Lemma 3.1.
Let µ and µ be Borel probability measures on S withthe same null-sets. For any Borel probability measure ν on S , the measures µ ∗ ν and µ ∗ ν have the same null-sets. T. MEYEROVITCH
Proof.
We will prove that µ ≪ µ implies that µ ∗ ν ≪ µ ∗ ν whichsuffices by symmetry.We assume µ ≪ µ , and show that for any ε >
0, there exists δ > A ∈ P ( S ) with ( µ ∗ ν )( A ) ≥ ε has ( µ ∗ ν )( A ) ≥ δ .Fix ε > A ∈ B ( S ) with ( µ ∗ ν )( A ) ≥ ε . It follows that ν (cid:18)(cid:26) x ∈ S : µ ( A · x ) ≥ ε (cid:27)(cid:19) ≥ ε . Since µ ≪ µ , there exists δ ′ > µ ( B ) ≥ ε implies µ ( B ) ≥ δ ′ .Thus, ν ( { x ∈ S : µ ( A · x ) ≥ δ ′ } ) ≥ ε . It follows that ( µ ∗ ν )( A ) ≥ δ ′ · ε , which establishes the claim with δ = δ ′ · ε . (cid:3) From this we deduce the following lemma.
Lemma 3.2.
Let T be a conservative, measure-preserving transforma-tion. For any n ≥ , the group e ( T ) acts nonsingularly on σ ⊗ nT , the n thconvolution power of the restricted spectral type of T . Proof.
Our claim is that ∀ t ∈ e ( T ) σ ⊗ nT ∼ δ t ∗ σ ⊗ nT , (6)where δ t denotes dirac measure at t , and ∼ denotes equivalence of measureclasses. For n = 1, a proof can be found in [2, 8].Equation (6) follows for n > t ∈ e ( T ), σ T and δ t ∗ σ T substituting for µ and µ , respectively, and σ ⊗ ( n − T substituting for ν . (cid:3) Completing the proof of Theorem 1.2 .By the ergodic multiplier theorem above, proving ergodicity of the Poisson-product amounts to proving σ T ∗ ( e ( T )) = 0. Since σ T ∗ = P n ≥ n ! σ ⊗ nT , it issufficient to prove that for all n ≥ σ ⊗ nT ( e ( T )) = 0 . (7)A proof that σ T ( e ( T )) = 0 is provided in [8]; see also [2]. This is the case n = 1 of equation (7). We also refer to the discussion in Chapter 9 of [16].For convenience of the reader and in preparation for the discussion inSection 7, we briefly recall the arguments leading to this result: Supposethe contrary, σ T ( e ( T )) >
0. Since e ( T ) acts nonsingularly on σ T , it fol-low that σ T | e ( T ) is a quasi-invariant measure on e ( T ). Thus, e ( T ) can befurnished with a locally-compact second-countable topology, respecting the RGODICITY OF POISSON PRODUCTS Borel structure inherited from S . Haar measure on e ( T ) must be is equiv-alent to σ T | e ( T ) . With respect to this topology, we have that e ( T ) is alocally compact group, continuously embedded in S , where the topologicalembedding is also a group embedding. In this situation, it follows as in [2]that e ( T ) is either discrete or e ( T ) = S . The possibility that e ( T ) is dis-crete is ruled out since this would imply σ T has atoms, which means T has L ( µ ) eigenfunctions. This is impossible since T is an ergodic transformationpreserving an infinite measure. The alternative is that e ( T ) = S . This is im-possible since e ( T ) is weak Dirichlet, thus must be a null set with respectto Haar measure on S [19].To prove the equality in (7) for n >
1, note that the convolution powerof an atom-free measure is itself atom-free and that by Lemma 3.2 above e ( T ) also acts nonsingularly on σ ⊗ nT . The result now follows using the samearguments outlined above for the case n = 1.This completes the proof of Theorem 1.2.
4. Nonexistence of equivariant thinning.
Here is a formalization of thenotion of a (deterministic) thinning . This is a B ∗ -measurable map Ψ : X ∗ → X ∗ , satisfying µ ∗ ([ | Ψ( ω ) ∩ B | ≤ | ω ∩ B | ]) = 1 ∀ B ∈ B . This essentially means that Ψ is a measurable map on the space X ∗ ofcountable sets of X , for which almost-surely Ψ( ω ) ⊂ ω .A Poisson thinning satisfies the extra condition that µ ∗ ◦ Ψ − = ( θµ ) ∗ forsome θ ∈ (0 , θµ ) ∗ we mean the measure on ( X ∗ , B ∗ ) which corre-sponds to a Poisson process with intensity given by θ · µ . In other words, thelaw of the countable set Ψ( ω ) is that of a lower-intensity Poisson process.Given a measure preserving transformation T : X → X , a thinning Ψ iscalled T -equivariant if Ψ ◦ T ∗ = T ∗ ◦ Ψ. A thinning Ψ is trivial if µ ∗ ([Ψ( ω ) = ∅ ]) = 1 or µ ∗ ([Ψ( ω ) = ω ]) = 1 . Proposition 4.1.
Let T be a group-action by measure preserving trans-formations. If T × T ∗ is ergodic, there does not exist a nontrivial T -equivariantthinning. Proof.
Suppose by contradiction that Ψ is a nontrivial T -equivariantthinning. Consider the set A = { ( x, ω ) ∈ X × X ∗ : x ∈ Ψ( ω ∪ { x } ) } . (8)Measurability of the set A is verified by the following: A = ∞ \ n =1 [ B ∈ β n ( B × X ∗ ) ∩ ((Ψ ◦ π ) − [ | ω ∩ B | > µ × µ ∗ , where { β n } ∞ n =1 is a “decreasing net” of countable partitions, as in Section 2. T. MEYEROVITCH
Since Ψ is T -equivariant, the set A is a T × T ∗ invariant set. By ergodicityof T × T ∗ , either ( µ × µ ∗ )( A ) = 0 or ( µ × µ ∗ )( A c ) = 0.Intuitively, A is the subset of X × X ∗ where applying the thinning Ψ onthe union of the “indistinguishable points” with the “distinguished point”does not delete the distinguished point. We will complete the proof by show-ing that this implies that the thinning Ψ is trivial.For j ∈ N , define π ( j ) : j z }| { X × · · · × X × X ∗ → X ∗ by π ( x , . . . , x j , ω ) = j [ k =1 { x k } ∪ ω.π ( j ) is B ⊗ j ⊗ B ∗ -measurable. This follows from measurability of the map π given by (5), which coincides with π (1) .For any B ∈ B with 0 < µ ( B ) < ∞ , and j ∈ N , we consider the followingprobability measures:(i) µ ∗ B,j ( · ) := µ ∗ ( · | [( ω ∩ B ) = j ]) . This is a probability measure on ( X ∗ , B ∗ ) corresponding to a Poisson processwith intensity µ , conditioned to have exactly j points in the set B ,(ii) ˆ µ B,j ( · ) := ( µ × µ ∗ ) | B × [( ω ∩ B )= j ] µ ( B ) · µ ∗ ([ ω ∩ B ] = j ) ( · ) . ˆ µ B,j is a probability measure on X × X ∗ given by the product of a randompoint in B , distributed according to µ | B and an independent Poisson processwith intensity µ , conditioned to have exactly j points inside the set B ,(iii) ˜ µ B,j ( · ) := j z }| { µ | B × · · · × µ | B × ( µ | B c ) ∗ µ ( B ) j ( · ) . This is the probability on ( X j × X ∗ , B ⊗ j ⊗ B ∗ ) which corresponds to j in-dependent random points identically distributed according to µ | B and anindependent Poisson process of intensity µ | B c .From the properties of the Poisson process, it directly follows that theprobability measures defined above are related as follows:ˆ µ B,j ◦ π − = ˜ µ B,j +1 ◦ π − j ) = µ ∗ B,j +1 (9)and ˆ µ B,j = ˜ µ B,j +1 ◦ π − ,j ] , (10) RGODICITY OF POISSON PRODUCTS where π [2 ,j ] : j z }| { X × · · · × X × X ∗ → X × X ∗ is given by π [2 ,j ] ( x , . . . , x j , ω ) = x , j [ k =2 { x k } ∪ ω ! . In particular, it follows that π ( j ) is a nonsingular map for all j ≥
1, in thesense that the inverse image of a µ ∗ -null set is always j z }| { µ × · · · µ × µ ∗ -null.Assuming Ψ is not a trivial thinning implies that there exist B ∈ B with0 < µ ( B ) < ∞ so that µ ∗ (0 < | Ψ( ω ) ∩ B | < | ω ∩ B | ) > . It follows that for some j > µ ∗ B,j (cid:18) < | Ψ( ω ) ∩ B || ω ∩ B | < (cid:19) > . (11)Now by (9) and (10), using symmetry of ˜ µ B,j with respect to the variables( x , . . . , x j ), it follows that the probability ˆ µ B,j ( x ∈ Ψ( π ( x, ω ))) is equal tothe expectation of | Ψ( ω ) ∩ B || ω ∩ B | under µ ∗ B,j . By (11) this expectation must bestrictly positive and smaller than one. This contradicts triviality of theset A : Either ( µ × µ ∗ )( A ) = 0 in which case ˆ µ B,j ( x ∈ Ψ( π ( x, ω ))) = 0 or( µ × µ ∗ )( A c ) = 0 in which case ˆ µ B,j ( x ∈ Ψ( π ( x, ω ))) = 1. (cid:3)
5. Nonexistence of equivariant allocation and matching.
The aim of thissection is to establish the nonexistence of T -equivariant Poisson allocationand Poisson matching, under an ergodicity assumption of a certain exten-sion of T . Combined with Theorem 1.2, this will establish the last part ofTheorem 1.1.We begin with an intermediate result about measure-preserving systems.Consider a measurable function Φ : X → L ( µ ), sending x ∈ X to Φ x ∈ L ( µ ),which is T -equivariant in the sense that Φ T x ◦ T = Φ x . Such a function Φ canbe interpreted as a T -equivariant “mass allocation” scheme. For instance, on X = R d with Lebesgue measure, Φ x ( y ) = 1 B ( x ) ( y ) and Φ x ( y ) = exp( −k x − y k ) both define isometry-equivariant “mass allocations.” The later can beconsidered a “fractional allocation,” in the sense that it obtains values in theinterval (0 , T -equivariant Poisson allocation and Poissonmatching will be a consequence of the following: Proposition 5.1.
Let T be a measure-preserving group action on ( X, B , µ ) .If T × T ∗ is ergodic, and µ ( X ) = ∞ , any T -equivariant measurable function Φ : X → L ( µ ) must be equal to µ -a.e. Proof.
Suppose Φ : X → L ( µ ) satisfies Φ T x ◦ T = Φ x . Note that ergod-icity of T implies that k Φ x k L ( µ ) is constant µ -a.e, as this is a T -invariant T. MEYEROVITCH function. Consider the function F : X × X ∗ → R given by F ( x, ω ) = X y ∈ ω | Φ x ( y ) | . We verify that F indeed coincides with a B ⊗ B ∗ -measurable function on aset of full µ × µ ∗ -measure.Indeed, Φ x = X B ∈ β X y ∈ ω ∩ B | Φ x ( y ) | , by Martingale convergence, X y ∈ ω ∩ B | Φ x ( y ) | = lim n →∞ E µ ∗ (cid:18) X y ∈ ω ∩ B | Φ x ( y ) | | β ∗ n (cid:19) for µ × µ ∗ -almost-every ( x, ω ). For B ∈ β and n ≥ E µ ∗ (cid:18) X y ∈ ω ∩ B | Φ x ( y ) | | β ∗ n (cid:19) = X D ∈ β n ∩ B E µ ∗ (cid:18) X y ∈ ( ω ∩ D ) | Φ x ( y ) | (cid:19) , and the right-hand side is clearly B × β ∗ n -measurable.Let ˜ F ( x ) := Z | F ( x, ω ) | dµ ∗ ( ω ) = Z X y ∈ ω | Φ x ( y ) | dµ ∗ ( ω ) , and it follows from the definition of µ ∗ that ˜ F = k Φ x k L ( µ ) . Thus, by ergod-icity of T , ˜ F is equal to a nonzero (finite) constant µ -almost everywhere. Inparticular, F is finite µ × µ ∗ -almost everywhere.Observe that F is T × T ∗ -invariant, so by ergodicity of T × T ∗ must beconstant µ × µ ∗ -a.e. On the other hand, for any ε > M >
0, we have F ( x, ω ) > M whenever ( x, ω ) ∈ X × X ∗ satisfy | ω ∩ { y ∈ X : | Φ x ( y ) | > ε }| > Mε . From the definition of the Poisson process, it thus follows that( µ × µ ∗ )([ F ≥ M ]) ≥ µ ( { x ∈ X : k Φ x k L ( µ ) ≥ ε } ) · ε M/ε M ! exp (cid:18) − Mε (cid:19) . Because the right-hand side is strictly positive for any
M >
0, whenever ε > F is not essentially bounded, whichcontradicts F being almost-everywhere constant. (cid:3) Together with Theorem 1.2, Proposition 5.1, immediately gives the fol-lowing corollary, which does not seem to involve Poisson processes at all:
Corollary 5.2.
Let T : X → X be a conservative and ergodic measurepreserving transformation of ( X, B , µ ) with µ ( X ) = ∞ . Any measurable func-tion Φ : X → L ( µ ) satisfying Φ T x ◦ T = Φ x must be equal to µ -a.e. RGODICITY OF POISSON PRODUCTS We now turn to define and establish a nonexistence result for equivariantPoisson allocations:By a
Poisson allocation rule we mean a B ∗ ⊗ B -measurable map Υ : X × X ∗ → L ( µ ) satisfying the following properties:(A1) nonnegativity: Υ ( x,ω ) ( y ) ≥ partition of unity: P x ∈ ω ( y )Υ ( x,ω ) = 1 µ ∗ -a.e.;(A3) Υ ( x,ω ) ≡ x / ∈ ω .If x ∈ ω , we think of Υ ( x,ω ) as the “the cell allocated to x .” Properties(A1) and (A2) above guarantee that Υ essentially takes values in the interval[0 , ( · ,ω ) corresponds to a partition of X up to a null set between the points in ω ,which assigns each x ∈ ω finite mass. For a “proper” allocation, we wouldrequire that Φ ( x,ω ) only takes values in { , } , but this extra requirement isnot necessary in order to prove our result.For it is often useful to consider a wider class of Poisson allocation rules,where Υ ( x,ω ) is undefined for a null set of ( x, ω )’s, and Υ is only measurablewith respect to the µ × µ ∗ -completion of the σ -algebra B ∗ ⊗ B . However,conditions (A2) and (A3) above apply to µ × µ ∗ -null sets, so we need to becareful and restate them as follows:(A1) nonnegativity: Υ ( x,ω ) ( y ) ≥ ′ ) partition of unity: R X Υ ( x,ω ) dµ ( x ) = 1 µ ∗ -a.e.;(A3 ′ ) R A Υ ( x,ω ) dµ ( x ) ≡ µ ∗ -a.e on { ω ∈ X ∗ : ω ∩ A = ∅ } whenever A ∈ B .A poisson allocation Υ is T -equivariant if Υ ( T x,T ∗ ω ) ◦ T = Υ ( x,ω ) . Proposition 5.3.
Let T be a group-action by measure preserving trans-formations, and denote S := T × T ∗ . If S × S ∗ is ergodic, there does not exista T -equivariant Poisson-allocation. Proof.
Given a Poisson allocation Υ : X × X ∗ → L ( µ ), we will definea T × T ∗ -equivariant function Φ : X × X ∗ → L ( µ × µ ∗ ) , which by ergodicityof S = T × T ∗ will contradict Proposition 5.1. This is given byΦ ( x,ω ) ( y, ω ) = Υ ( x,ω ∪{ x } ) ( y ) . It follows directly that k Φ ( x,ω ) k L ( µ × µ ∗ ) = k Υ ( x,ω ∪{ x } ) k L ( µ ) , which is positive and finite µ × µ ∗ -a.e.Measurability of Φ follows from the measurability assumptions on Υ andfrom measurability of the map ( x, ω ) → { x } ∪ ω . (cid:3) We now consider the existence of equivariant Poisson matching schemes: T. MEYEROVITCH
Given a pair of independent Poisson processes realizations a (determin-istic)
Poisson matching assigns a perfect matching (or bijection) betweenthe points of the two realizations, almost surely. To formalize this we de-fine a Poisson matching as a measurable-function Ψ : X ∗ × X ∗ → ( X × X ) ∗ ,satisfying the following:(M1) µ ∗ ( { ω ∈ X ∗ : | Ψ( ω , ω ) ∩ ( B × B ) | ≤ min {| ω ∩ B | , | ω ∩ B |}} ) = 1for µ ∗ -a.e ω and all B , B ∈ B ;(M2) µ ∗ ( { ω ∈ X ∗ : | Ψ( ω , ω ) ∩ ( B × X ) | = | ω ∩ B |} ) = 1for µ ∗ -a.e ω and all B ∈ B ;(M3) µ ∗ ( { ω ∈ X ∗ : | Ψ( ω , ω ) ∩ ( X × B ) | = | ω ∩ B |} ) = 1for µ ∗ -a.e ω and all B ∈ B . Proposition 5.4.
Under the assumptions of Proposition 5.3, there doesnot exist a nontrivial T -equivariant Poisson matching. Proof.
Suppose Ψ is a T -equivariant Poisson matching. We will definea “fractional” T -equivariant Poisson allocation Υ : X × X ∗ → L ( µ ), contra-dicting Proposition 5.3.The (implicit) definition of Υ is given by Z A Υ ( x,ω ) ( y ) dµ ( y ) = µ ∗ ( { ω : | Ψ( ω , ω ) ∩ ( { x } × A ) | > } )(12)for all A ∈ B , ω ∈ X ∗ and x ∈ X .In other words, if x ∈ ω , Υ ( x,ω ) is the density with respect to Lebesguemeasure of the conditional distribution of the partner of x under the match-ing Ψ, given ω . This defines Υ up to a null set.It follows from the properties of Ψ that Υ satisfies the conditions (A1),(A2 ′ ) and (A3 ′ ) above.Thus, Υ is indeed a Poisson allocation. Because Ψ is a T -equivariantmatching, it follows directly that Υ is a T -equivariant allocation. (cid:3) To complete the proof of the last part of Theorem 1.1, we note that if T isa conservative and ergodic measure-preserving transformation, S = T × T ∗ isalso conservative and ergodic by Theorem 1.2, and so S × S ∗ is also ergodic,again by Theorem 1.2.
6. The leftmost position transformation.
In this section X = R + is theset of positive real numbers, B is the Borel σ -algebra on X and µ is Lebesguemeasure on the positive real numbers. T : X → X is an arbitrary conserva-tive, ergodic, Lebesgue-measure-preserving map of the positive real numbers. RGODICITY OF POISSON PRODUCTS In order to have a concrete example for such transformation T in hand,the reader can consider the unsigned version of Boole’s transformation, givenby T ( x ) = | x − x | . We define the following function: t : X ∗ → X by t ( ω ) = inf ω. (13)The map t is well defined on a set of full µ ∗ -measure, namely whenever ω = ∅ . Note that t ( ω ) is the leftmost point of ω whenever ω is a discretecountable subset of R + . The map t is B ∗ -measurable since t − ( a, b ) = { ω ∈ X ∗ : ω ∩ (0 , a ] = ∅ and ω ∩ ( a, b ) = ∅ } . From this, it also follows directly that µ ∗ ◦ t − ( a, b ) = e − µ (0 ,a ) (1 − e − µ ( a,b ) ) = e − a − e − b . In particular it follows that µ ∗ ◦ t − ≪ µ .Define the leftmost return time κ : X ∗ → N ∪ { + ∞} by κ ( ω ) = inf { k ≥ t ( T k ∗ ( ω )) = T k ( t ( ω )) } . (14) µ ∗ -almost surely, κ ( ω ) is the smallest positive number of iterations of T ∗ which must be applied to ω in order for the leftmost point to return tothe leftmost location. A priori, κ T is could be infinite. Nevertheless, we willsoon show that when T is conservative and measure preserving, κ is finite µ ∗ -almost surely. Finally, the leftmost position transformation associatedwith T , T κ ∗ : ω → ω , is defined by T κ ∗ ( ω ) : = T κ ( ω ) ∗ ( ω ) . This is the map of X ∗ obtained by reapplying T ∗ till once again there areno points to the left of the point which was originally leftmost.The reminder of this section relates the leftmost transformation associatedwith T with the Poisson-product T × T ∗ .Let X = { ( x, ω ) ∈ X × X ∗ : ω ∩ (0 , x ] = ∅ } . (15)The set X is simply the subset of X × X ∗ in which the “distinguishedpoint” is strictly to the left of any “undistinguished point.” The formulabelow verifies measurability of X : X = \ n ∈ N [ q ∈ Q (cid:18)(cid:18) q − n , q + 1 n (cid:19) × (cid:26) ω ∈ X ∗ : ω ∩ (cid:18) , q + 2 n (cid:19) = ∅ (cid:27)(cid:19) mod µ × µ ∗ . Proposition 6.1.
Let T : R + → R + be conservative and Lebesgue-meas-ure-preserving. Then the leftmost position transformation associated with T is well defined and is isomorphic to the induced map of the Poisson product T. MEYEROVITCH on the set X defined by equation (15), ( X ∗ , B ∗ , µ ∗ , T κ ∗ ) ∼ = ( X , B , µ , ( T × T ∗ ) X ) , where µ = ( µ × µ ∗ ) | X is the restriction of the measure product µ × µ ∗ tothe set X , and B = ( B ⊗ B ∗ ) ∩ X is the restriction of the σ -algebra on theproduct space to subset of X .In particular, µ ( X ) = 1 , so ( X , B , µ ) is a probability space. Proof.
Consider the map π : X → X ∗ which is the restriction to X of the map π ( x, ω ) = { x } ∪ ω described in Section 2.5 above.For a nonempty, discrete ω ∈ X ∗ we have π − ( ω ) = ( t ( ω ) , ω \ t ( ω )) . Thus π is invertible on a set of full µ ∗ -measure in X ∗ .As T is conservative and T ∗ is a probability preserving transformation,the Poisson product T × T ∗ is also conservative. We will show below that µ × µ ∗ ( X ) >
0. Therefore, the return time ϕ X is finite almost everywhereon X .Since κ ◦ π = π ◦ ϕ X , it follows that κ is finite µ ∗ -a.e.We also have π ( T n x, T n ∗ ω ) = T n ∗ ( π ( x, ω ))whenever ( x, ω ) and ( T n x, T n ∗ ω ) are in X . Thus, π ◦ ( T × T ∗ ) X = T κ ∗ ◦ π . It remains to check that π − µ ∗ = µ . It is sufficient to verify that µ ∗ ( A ) = µ ( π − ( A )) for sets A ∈ B ∗ of the form A = N \ k =1 [ | ω ∩ A k | = n k ] , where A i = ( a i − , a i ], 0 = a < a < a < · · · < a N and n k ≥ k = 1 , . . . N .Given the definition of µ ∗ , this amounts to an exercise in elementarycalculus. By definition of µ ∗ , µ ∗ ( A ) = N Y k =1 µ ( A k ) n k n k ! exp( − µ ( A k )) , which simplifies to µ ∗ ( A ) = exp( − a N ) N Y k =1 ( a k − a k − ) n k n k ! . (16) RGODICITY OF POISSON PRODUCTS Assuming the n k ’s are not all zero, let k the smallest index for which n k >
0. We have π − ( A ) = \ j = k ( X × [ | ω ∩ A j | = n j ]) ∩ [ x ∈ A k { x } × ([ | ω ∩ [ a k − , x ) | = 0] ∩ [ | ω ∩ [ x, a k ) | = n k − . Thus µ (Φ − ( A )) = T Z A k exp( − ( x − a k − )) exp( − ( a k − x )) ( a k − x ) n k − ( n k − dx, where T = Y j = k ( a j − a j − ) n j n j ! exp( a j − a j − ) . Integrating this rational function of a single variable, we see that the lastexpression is equal to the expression on right-hand side of (16).In particular, it follows that µ ( X ) = 1.It remains to check the case that n k = 0 for all k = 1 , . . . , N : In this casethen A = [ ω ∩ (0 , a N ] = 0] and π − ( A ) = { ( x, ω ) ∈ X : x > a n } . Thus µ ( π − ( A )) = Z [ a N , ∞ ) e − µ [ x, ∞ ) dµ ( x ) = exp( − a N ) , which is equal to µ ∗ ( A ). (cid:3) Corollary 6.2.
Let T : R + → R + be a conservative and ergodic Lebesgue-measure-preserving transformation. Then the leftmost position transforma-tion T κ ∗ : ( R + ) ∗ → ( R + ) is an ergodic probability preserving transformation. Proof.
Let T be as above. By Proposition 6.1, T κ ∗ is isomorphic to themap obtained by inducing the Poisson product T × T ∗ onto the set X . It iswell known that inducing a conservative and ergodic transformation on a setof positive measure results in an ergodic transformation. By Theorem 1.2, T × T ∗ is indeed ergodic. (cid:3) It would be interesting to establish other ergodic properties of T κ . Forexample, what conditions on T are required for T κ ∗ to be weakly mixing?
7. Poisson-products and measure-preserving group actions.
The pur-pose of this section is to discuss counterparts of our pervious results onergodicity of Poisson products, and various equivariant operations in thecontext of a group of measure preserving transformations. Some motivating T. MEYEROVITCH examples for this are groups of R n -isometries, which naturally act on R n preserving Lebesgue measure.Briefly recall the basic setup: We fix a topological group G and a σ -finitemeasure space ( X, B , µ ). A measure-preserving G -action T on the σ -finitemeasure space ( X, B , µ ) is a representation g T g ∈ Aut( X, B , µ ) of G intothe measure preserving automorphisms of ( X, B , µ ).A G -action T is ergodic if for some A ∈ B , µ ( T g A \ A ) = 0 for all g ∈ G then either µ ( A ) = 0 or µ ( X \ A ) = 0.Any measure preserving G -action T induces an action T ∗ on the Poissonprocess by probability preserving transformations [18]. The Poisson-product G -action T × T ∗ is thus defined the same way as in the case of a singletransformation.The proofs of Propositions 4.1, 5.1, 5.3 and 5.4 above are still valid in thisgenerality.Let us recall the definition of a conservative G -action: Say W ∈ B is a wandering set with resect to the action T of a locally-compact group G if µ ( T ( g, W ) ∩ W ) = 0 for all g in the complement of some compact K ⊂ G .Call a G -action conservative if there are no nontrivial wandering sets.If in the statement of Theorem 1.2 we let T be a conservative ergodic G -action for a group other than Z , ergodicity of T × T ∗ may fail. This canhappen even for conservative and ergodic Z -actions, as we demonstrate inthe example below:Let a, b ∈ R \ { } with ab / ∈ Q .Define a Z -action T on R by T ( m,n ) ( x ) = x + am + bn for ( m, n ) ∈ Z . It is a simple exercise to show that the Z -action above is both conserva-tive and ergodic. Nevertheless, it is easy to see that T × T ∗ is not ergodic,for instance, by noting that { ( x, ω ) ∈ R × R ∗ : ( x + 1 , x − ∩ ω = ∅ } is a nontrivial T × T ∗ -invariant set. Since this action T consists of trans-lations, as noted in the Introduction, there do exist T -equivariant Poissonallocations, Poisson matchings and Poisson thinning.Although the example above demonstrates Theorem 1.2 does not gen-eralize, for abelian group actions most components of the proof given inSection 3 remain intact. Our next goal is to explain this, and point outwhere the proof of Theorem 1.2 breaks down for the example above:Let G be a locally compact abelian group, and let b G denote its dual.Generalizing the discussion in Section 2, the L ∞ - spectra of a G -action T ,denoted Sp( T ), is the set of homomorphisms χ : G → C ∗ such that f ( T g x ) = χ ( g ) f ( x ) for some nonzero f ∈ L ∞ ( X, µ ). In case G = Z , the spectra issimply the group L ∞ -eigenvalues. As in the case G = Z discussed earlier,the L ∞ -spectra is a weak-Dirichlet set in b G [19]. RGODICITY OF POISSON PRODUCTS The L - spectral type of T is an equivalence class of Borel measures σ T on b G for any nonzero f ∈ L ( µ ) σ f ≪ σ T , where the measure σ f is given byˆ σ f ( g ) = Z f ( T g ( x )) f ( x ) dµ ( x ) . The spectral type of σ T is the minimal equivalence class of measures on b G with respect to which all the σ f ’s are absolutely continuous.With these definitions, Keane’s ergodic multiplier theorem above gener-alizes as follows: The product of an ergodic measure preserving G -action T and a probability preserving G -action S is ergodic if and only if Sp( T ) isnull with respect to the restricted spectral type of σ T . The discussion in theend of Section 3 following [2, 19] still shows that in this case Sp( T ) mustbe a locally compact group continuously which embeds continuously in b G .However, when G = Z , this does not imply that Sp( T ) is either discrete orequal to b G .Getting back to the example of the Z -action T above, we note that forany τ ∈ R , the function f τ ∈ L ∞ ( R ) defined by f τ ( x ) = exp( iτ x ) , is an L ∞ eigenfunction of T , since it satisfies f τ ( T ( m,n ) ( x )) = exp( iτ ( x + am + bn )) = χ ( ta,tb ) ( m, n ) exp( iτ x ) , where χ ( a,b ) ( m, n ) = exp( iam + bn ). The map t → χ ( ta,tb ) is a continuousgroup embedding of R in Sp( T ) ( c Z . Acknowledgments.
Thanks to Emmanual Roy for inspiring conversationsand in particular for suggesting the “leftmost position transformation” andasking about its ergodicity. This work is indebted to Jon Aaronson for nu-merous contributions, in particular for recalling the paper [2], which con-tains key points of the main result. To Omer Angel and Ori Gurel-Gurevich,thanks for helpful discussions about equivariant operations on Poisson pro-cesses. REFERENCES [1]
Aaronson, J. (1997).
An Introduction to Infinite Ergodic Theory . Mathematical Sur-veys and Monographs . Amer. Math. Soc., Providence, RI. MR1450400[2] Aaronson, J. and
Nadkarni, M. (1987). L ∞ eigenvalues and L spectra of non-singular transformations. Proc. Lond. Math. Soc. (3) Adler, R. L. and
Weiss, B. (1973). The ergodic infinite measure preserving trans-formation of Boole.
Israel J. Math. Ball, K. (2005). Poisson thinning by monotone factors.
Electron. Commun. Probab. T. MEYEROVITCH[5]
Chatterjee, S. , Peled, R. , Peres, Y. and
Romik, D. (2010). Gravitational allo-cation to Poisson points.
Ann. of Math. (2)
Evans, S. N. (2010). A zero-one law for linear transformations of L´evy noise. In
Algebraic Methods in Statistics and Probability II . Contemp. Math.
Gurel-Gurevich, O. and
Peled, R. (2013). Poisson thickening.
Israel J. Math.
Toappear. Available at arXiv:0911.5377.[8]
Hahn, P. (1979). Reconstruction of a factor from measures on Takesaki’s unitaryequivalence relation.
J. Funct. Anal. Hoffman, C. , Holroyd, A. E. and
Peres, Y. (2006). A stable marriage of Poissonand Lebesgue.
Ann. Probab. Holroyd, A. E. (2011). Geometric properties of Poisson matchings.
Probab. TheoryRelated Fields
Holroyd, A. E. , Lyons, R. and
Soo, T. (2011). Poisson splitting by factors.
Ann.Probab. Holroyd, A. E. , Pemantle, R. , Peres, Y. and
Schramm, O. (2009). Poissonmatching.
Ann. Inst. Henri Poincar´e Probab. Stat. Kingman, J. F. C. (1993).
Poisson Processes . Oxford Studies in Probability . TheClarendon Press Oxford Univ. Press, New York. MR1207584[14] Kingman, J. F. C. (2006). Poisson processes revisited.
Probab. Math. Statist. Krikun, M. (2007). Connected allocation to Poisson points in R . Electron. Com-mun. Probab. Nadkarni, M. G. (2011).
Spectral Theory of Dynamical Systems . Texts and Readingsin Mathematics . Hindustan Book Agency, New Delhi. MR2847984[17] Roy, E. (2009). Poisson suspensions and infinite ergodic theory.
Ergodic TheoryDynam. Systems Roy, E. (2010). Poisson–Pinsker factor and infinite measure preserving group actions.
Proc. Amer. Math. Soc.
Schmidt, K. (1982). Spectra of ergodic group actions.