Stable transports between stationary random measures

We give an algorithm to construct a translation-invariant transport kernel between ergodic stationary random measures $\Phi$ and $\Psi$ on $\mathbb R^d$, given that they have equal intensities. As a result, this yields a construction of a shift-coupling of an ergodic stationary random measure and its Palm version. This algorithm constructs the transport kernel in a deterministic manner given realizations $\varphi$ and $\psi$ of the measures. The (non-constructive) existence of such a transport kernel was proved in [8]. Our algorithm is a generalization of the work of [3], in which a construction is provided for the Lebesgue measure and an ergodic simple point process. In the general case, we limit ourselves to what we call constrained densities and transport kernels. We give a definition of stability of constrained densities and introduce our construction algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable constrained densities, we study existence, uniqueness, monotonicity w.r.t. the measures and boundedness.


Introduction
For a random measure Ψ on R d , there are a number of equivalent definitions for the Palm distribution of Ψ (see Section 2). Heuristically speaking, in the case that Ψ is stationary and ergodic, the Palm version of Ψ is obtained by viewing Ψ from a typical point of Ψ. The result of [13] shows that there exists a random point Y such that by viewing Ψ from Y ; i.e. by translating Ψ by vector −Y , we get exactly the Palm version of Ψ. In other words, there exists a coupling of Ψ and its Palm version such that almost surely each one is a translated version of the other one. Such a coupling is called a shift-coupling.
To obtain a shift-coupling, one can use a (random) balancing transport kernel T that transports a multiple of the Lebesgue measure to Ψ, where by a (random) transport kernel we mean a (random) function that assigns to each point s ∈ R d and ω in the probability space, a probability measure T ω (s, ·) on R d . This measure can be interpreted as how the infinitesimal mass at s is distributed in the space. Given that T depends on Ψ in a translation-invariant manner (which is called flow-adapted here), then choosing Y with distribution T ω (0, ·) gives a shift-coupling of Ψ and its Palm version (see [9] and [5] as explained after Theorem 4.9). A case of special interest is when T is a balancing allocation in which the measure T ω (s, ·) is a Dirac measure a.e. In this case, given Ψ, the vector Y defined above is deterministic and the shift-coupling does not need extra randomness.
Let Φ and Ψ be jointly stationary and ergodic random measures on R d . Considering flow-adapted transport kernels balancing Φ and Ψ has been of great interest recently.
The abstract existence of such transports was proved in [9] (which is merely based on [13]) provided that the intensities are equal, finite and positive. Nevertheless, there has been tremendous interest in the construction of such transports in special cases in the recent years. One of the reasons for this interest is that such transport kernels lead to explicit shift-coupling of the Palm distributions of Φ and Ψ, as explained above. The constructions were motivated by Liggett [10], who constructed a balancing allocation for an ergodic simple point process in dimension one. The landmark in this topic is [3] which generalizes [10] to arbitrary dimensions. There are several other constructions in the literature for the case of simple point processes, such as gravitational allocation [1], optimal transport [6], one-sided stable allocation on the line [8], etc.
In this paper we give an algorithm that works in the general case and enables us to • construct a flow-adapted transport kernel balancing two arbitrary jointly stationary and ergodic random measures on R d with equal intensities (Theorem 4.8) • and constructs a shift-coupling for an arbitrary stationary ergodic random measure on R d and its Palm version (Theorem 4.9).
To do this, we generalize the notion of stable allocations introduced in [3] to what we call stable constrained transport densities, where the notion of constrained transport densities is a special case of capacity constrained transport kernels introduced in [7]. The first, and in fact deterministic, result (Theorem 4.13) is that stable constrained transport densities exist and one can be given by our algorithm (Algorithm 4.4) which is inspired by the continuum version of the Gale-Shapley algorithm in [3]. Another important result is considering the algorithm in the random case described above. Other results deal with monotonicity (Theorem 4.21) and optimality (Corollary 4.22) properties of stable constrained transport densities, uniqueness (Theorem 4.24) and boundedness of the support of the mass transported from and to a region (Theorem 4.25). These results are in the spirit of the seminal papers [3] and [5] and generalize some of their results. We also introduce the notion of Voronoi transport kernel with respect to a measure, which generalizes the notion of Voronoi diagram for a discrete set. It helps us in proving some statements, but it can be interesting in its own. The construction and results in this paper can be generalized to random measures on a locally compact Abelian group and also to non-ergodic cases (only equality of sample intensities is important) in the setting of [9]. We don't go through these general cases to stay focused on the main ideas.
We show that using transport kernels is inevitable in the general case by providing examples where no flow-adapted balancing allocation exists. However, some results and open problems on the existence and construction of flow-adapted balancing allocations are addressed in Subsection 4.6.
In this paper, by a transport kernel we mean a non-weighted transport kernel. Being non-weighted is equivalent to the condition that the total mass transported from a set B, which is B T (s, R d )ϕ(ds), is equal to ϕ(B). Being (ϕ, ψ)-balancing means that the mass transported into B is equal to ψ(B). If T is (ϕ, ψ) − balancing transport kernel, then T is a Markovian kernel which transports ϕ to ψ.
In this work, we fix a measurable space (Ω, F) equipped with a measurable flow θ s : Ω → Ω for s ∈ R d ; i.e. (ω, s) → θ s ω is measurable, θ 0 is the identity on Ω and With an abuse of notation, we use θ s also for natural flows on the space of functions on R d or R d × R d , on M , and on the space of weighted transport kernels; i.e.  Using these conventions, a measurable function F from Ω to a flow-equipped space is called flow-adapted if F (θ s ω) = θ s F (ω). We denote F (ω) by F ω too.
A random measure is a pair (P, Φ), where P is a probability measure on Ω and Φ : Ω → M is a measurable function. The distribution of (P, Φ) is the push-forward measure P(Φ −1 (·)) on M .
A probability measure P on Ω is stationary if it is invariant under θ s for all s in R d . A stationary random measure is a random measure (P, Φ) such that P is a stationary probability measure and Φ is flow-adapted. This implies stationarity in the usual sense, which is translation-invariance of its distribution; i.e. P [Φ ∈ A] = P [θ s Φ ∈ A] for any A ∈ M and s ∈ R d . Moreover, if (P, Ψ) is another stationary random measure (with the same P), then Φ and Ψ are jointly stationary in the usual sense; i.e. the distribution of the pair (Φ, Ψ) is invariant by translations. When there is no ambiguity about P, we may say that Φ is a stationary random measure in the sense given above. Therefore, when we mention two stationary random measures without definig the probability measure(s), they are assumed to be jointly stationary.
A stationary probability measure P on Ω is ergodic if for any event A ∈ F that is invariant under all θ s , we have P(A) ∈ {0, 1}. The stationary random measure (P, Φ) is ergodic if P is ergodic. It is clear that if Ψ is a measure derived from Φ in a translationinvariant manner, then stationarity (resp. ergodicity) of (P, Φ) implies stationarity (resp. ergodicity) of (P, Ψ). For simplicity, we use the term ergodic random measure instead of ergodic stationary random measure.
Since we deal with random measures in this work, we have two different notions of 'almost everywhere'; one corresponding to probability and the other corresponding to the measures on R d . To avoid confusion, we preserve the phrase almost sure(ly) (or a.s.) for probability measures on Ω and we use almost everywhere and almost all (denoted by a.e. and a.a.) for measures on R d . Lemma 2.1. Let V ⊆ R d be a subset that contains arbitrarily large balls. If Φ is a stationary random measure such that a.s. Φ(R d ) > 0, then a.s. Φ(V ) = ∞.
The intensity of a stationary random measure (P, Φ) is the unique constant λ ∈ [0, ∞] such that E [Φ(B)] = λL d (B) for an arbitrary Borel set B ∈ G. If λ is positive and finite, then the Palm version of (P, Φ) is the random measure (P Φ , Φ) in which Stable transports between stationary random measures where ds is a short form of L d (ds). If (P, Ψ) is also a stationary random measure on the same space, we have Neveu's exchange formula where λ Φ and λ Ψ are the intensities of Φ and Ψ respectively. Interested readers may refer to [12], sections 3.3 and 3.4, for more details on the properties of Palm distributions.
A shift-coupling of random measures (P 1 , Φ 1 ) and (P 2 , Φ 2 ) is a random vector Y (possibly on an extension of Ω) such that (P 1 , θ Y (Φ 1 )) has the same distribution as (P 2 , Φ 2 ). Note that this provides a coupling of the distributions of the two random measures such that they are a translated version of each other in every realization. The necessary and sufficient condition for the existence of a shift-coupling is proved in [13] in a more general setting. However, the proof in [13] is non-constructive (See Theorem 4.9).
Shift-couplings are of special interest in the case that (P 1 , Φ 1 ) is a stationary random measure and (P 2 , Φ 2 ) is its Palm version. They can be used to construct the Palm version of a random measure or to reconstruct the random measure from its Palm version by a random translation (see Theorem 4.9). In particular, when Φ 1 is a simple point process, Y is a called an extra head scheme for Φ 1 in [5]. The name comes from the fact that if Φ 1 is the point process on Z d formed by the heads in i.i.d. coin tosses (resp. the Poisson point process), then by shifting Y to the origin we get the same thing (in distribution) as placing a head (resp. point) on the origin.

Motivation: extra head scheme for a point process
To understand the idea behind the definitions in this paper, it is helpful to recall the novel algorithm of [3] (Algorithm 3.1) which we will generalize. This algorithm is inspired by the stable marriage algorithm of Gale and Shapley in bipartite graphs ( [2]) and appears to be the first generalization to a continuum setting. The main goal in the paper is to construct a translation-invariant balancing allocation between the Lebesgue measure and a realization of an ergodic stationary simple point process in R d .
Let Ξ be a discrete subset of R d . By sites and centers we mean the points of R d and Ξ respectively. The idea of the following algorithm is that each site and each center prefer to be allocated as close as possible. Forget the sites that are equidistant from two or more centers. Algorithm 3.1. For each natural number n, stage n consists of the following two parts: (a) Each site x applies to the closest center to x which has not rejected x at any earlier stage.
(b) For each center ξ, Let A (which depends on n and ξ) be the set of sites which applied to ξ in the previous step. Let B be the smallest ball centered at ξ such that L(A ∩ B) ≥ 1. Then ξ rejects the sites in A\B.
Easily seen, each site x either is rejected by all centers or for some center ξ, x applies to ξ and is never rejected for sufficiently large n. In the first case let τ (x) = ∞ (where ∞ is treated as a single point added to the space whose distance to every other point is ∞) and in the second case let τ (x) := ξ.
It is proved in [3] that the function τ defined in this algorithm is stable in a sense that is a generalization of stable matchings, which is, roughly speaking, there are no sites Definition 4.32 for the exact definition). An interesting theorem in [3] is that if Ψ is an ergodic simple EJP 21 (2016), paper 51. point process in R d with intensity 1, then almost surely the (random) function τ defined in Algorithm 3.1 for the discrete set Ψ ω defines a balancing allocation between the Lebesgue measure and Ψ; i.e. almost surely τ is defined for almost every site and L(τ −1 (ξ)) = 1 for all centers ξ. One can try to apply Algorithm 3.1 to find an allocation for two arbitrary measures ϕ and ψ (instead of the Lebesgue measure and a counting measure). For example, let them be the Lebesgue measure restricted on (−∞, 0] and [0, ∞) respectively. The sites and the centers are the points in the supports of ϕ and ψ respectively. In the first step, all sites apply to center 0, but the next steps are vague. Same issue may happen in the stationary case. If we consider the allocation as a transport kernel T (x, ·) := δ τ (x) , our idea is to force the infinitesimal mass at every point to be spreaded instead of being transported to a single point. To do so, We will force an upper bound on the measure T (x, ·). Since x wants to apply to the support of ψ, a natural upper bound is T (x, B) ≤ ψ(B). This justifies the choice of the name 'constrained' (see definitions 4.1 and 4.2). The above example will be addressed in Example 6.8.

Constrained transport kernels and constrained transport densities
Let ϕ and ψ be given locally finite non-negative Borel measures on R d .
As we will see in Subsection 4.6, a (ϕ, ψ)-balancing ψ-constrained transport kernel cannot be an allocation except maybe when ψ is discrete.
Since T (x, ·) is absolutely continuous w.r.t. ψ for any x, we will work with its Radon-Nykodim derivative. The following definition and Remark 4.3 establish the setup in a measurable way.

Definition 4.2.
A non-negative measurable function f (x, ξ) on R d × R d is called a subbalancing transport density (given ϕ and ψ) if We call f balancing if equality happens for ϕ-a.e. x and ψ-a.e. ξ. Furthermore, f is called constrained if it is sub-balancing and f (x, ξ) ≤ 1 for every (x, ξ) ∈ R d × R d .  f (x, ξ) can be thought as the infinitesimal mass going from x to ξ. If f is balancing, then T is a (ϕ, ψ)-balancing transport kernel. Conversely, If T is a (ϕ, ψ)-balancing ψconstrained transport kernel, then (a suitable version of) the Radon-Nykodim derivative of the measure on R d × R d defined as (A × B) → A T (x, B)ϕ(dx), with respect to ϕ ⊗ ψ, is a balancing constrained transport density. Hence, we might think of f as the density of T w.r.t. ψ. Similar correspondence holds in the random case.
Note that the definition of constrained transport densities depends on both ϕ and ψ, but we don't use a prefix for simplicity. The notion of balancing constrained transport densities is a special case of capacity constrained transport kernels defined in [7]. It can EJP 21 (2016), paper 51. be interpreted as there is a transportation capacity between any pair of points. More explicitly, the total mass going from a set A to a set B is at most ϕ(A)ψ(B).

Construction of a shift-coupling
Let ϕ, ψ ∈ M be given. Since we have different procedures regarding ϕ and ψ, it is helpful to use the names sites and centers for points in the supports of ϕ and ψ respectively, following the terminology of [3]. Imagine we have two copies of R d , in one copy, we have sites and measure ϕ and in the other copy we have centers and measure ψ. Nevertheless, we will measure the distance between a site and a center as if they are on the same space. We use roman letters for naming sites and Greek letters for naming centers.
Here is an overview of Algorithm 4.4. The algorithm consists of infinitely many stages and each stage has two steps. At stage n, each site x 0 applies to the closest possible centers with weight A n (x 0 , ·) : R d → [0, 1] (tries to construct f (x 0 , ·)). Then each center ξ 0 rejects some of the weights applied to ξ 0 if it has reached its capacity. The amount of rejection is denoted by R n (·, ξ 0 ). Note that even if R n (x 0 , ξ 0 ) > 0, x 0 will still apply to ξ 0 at all later stages. The functions A n and R n will be non-decreasing with respect to n. A heuristic for choosing A n and R n in the algorithm is that the sites prefer to apply to centers which are as close as possible, in a greedy manner. Similarly, the centers prefer to reject no portion of the applications of the sites which are as close as possible.
For an illustration of Algorithm 4.4 see Example 6.2. Example 6.1 shows that this algorithm generalizes the algorithm in [3]. The name site-optimal is justified in Corollary 4.22.

Algorithm 4.4 (site-optimal Gale-Shapley algorithm).
Given measures ϕ and ψ on R d , let the rejection function be zero at the beginning; i.e. R 0 (x, ξ) For each natural number n, stage n consists of the following steps: (i) For each site x 0 , define its application radius at stage n as a n (x 0 ) := sup a : Define the n-th application function as and we let c = 1 (or any arbitrary constant) if a n (x 0 ) = ∞ or ψ(∂B an(x0) (x 0 )) = 0.
We say x 0 applies to the centers with weight A n (x 0 , ·) at stage n.
(ii) For each center ξ 0 , define its rejection radius at stage n as r n (ξ 0 ) := sup r : Define the n-th rejection function as We say ξ 0 rejects the application weights according to R n (·, ξ 0 ) at stage n.
It is shown in the proof of Lemma 4.5 that c n (x 0 ) and c n (ξ 0 ), which are defined in the algorithm, exist and are well defined.
Lemma 4.5. In Algorithm 4.4, A n , R n and a n are non-decreasing with respect to n, r n is non-increasing. Moreover, they depend on ϕ and ω in a measurable and flow-adapted manner.
This lemma allows us to provide the following definitions.
Definition 4.6. In Algorithm 4.4, the (final) application radius, rejection radius, application function, rejection function and the site-optimal transport density are defined as follows a(x) := lim n→∞ a n (x), The center-optimal Gale-Shapley algorithm for ϕ and ψ is just Algorithm 4.4 for ψ and ϕ; i.e. we swap the roles of sites and centers. The center-optimal transport density f c is defined similar to Definition 4.6 using the center-optimal Gale-Shapley algorithm. intensities. If the intensities are equal, then the site-optimal transport density which is constructed by Algorithm 4.4 is almost surely a balancing constrained transport density.
According to Theorem 4.8, Algorithm 4.4 answers a question in [9] asking for a constructive algorithm to find a flow-adapted balancing transport kernel, although the existence of such a transport kernel is proved to be equivalent to the equality of the sample intensities in [9], which is granted here by ergodicity.
As a result of Theorem 4.8, one can construct a shift-coupling between an ergodic random measure and its Palm version as in the following theorem. The key tool for this construction is the following well-known theorem in the literature. Theorem 4.9 (Shift-coupling). Let Ψ be an ergodic random measure on R d with positive and finite intensity λ Ψ and let Φ = λ Ψ L d . Let F = F ω (x, ξ) be a flow-adapted function which is almost surely a balancing transport density for Φ ω and Ψ ω ; e.g. the site-optimal transport density for Φ ω and Ψ ω .
(i) If Y is a random vector such that its conditional distribution given Ψ is F (0, ·)Ψ, then Y gives a shift-coupling of Ψ and its Palm version; i.e. θ Y Ψ has the same distribution as the Palm version of Ψ.
(ii) On the probability space (Ω, F, P Ψ ), if Y is a random vector such that its conditional distribution given Ψ is F (·, 0)Φ, then the random measure (P Ψ , θ Y Ψ) has the same distribution as (P, Ψ). In words, θ Y Ψ is a reconstruction of Ψ from its Palm version. Theorem 4.9 is a direct implication of Theorem 4.1 in [9] (see also Theorem 16 in [5] for point processes). Nevertheless, we state it for the purpose of this paper. In part (i), we call the random vector Y an extra head scheme for Ψ with the terminology of [5]. One can also replace F (0, ·)Ψ by T (0, ·), where T is a flow-adapted transport kernel which is almost surely (Φ, Ψ)-balancing. If moreover T is a balancing allocation that is a function of Ψ, provided that it exists, then Y will be a deterministic vector conditional on Ψ. This is called a non-randomized extra head scheme (note that to construct Y in the general case, one may need extra randomness; i.e. to extend the probability space) and its existence is proved in the case when Ψ is an ergodic simple point process in [3] by providing a construction. Algorithm 4.4 is a generalization of that construction. Although it does not provide a non-randomized extra head scheme in Theorem 4.9, it has the property that the distribution of Y conditional on Ψ depends only on the realization of Ψ. We don't know whether non-randomized extra head schemes always exist for general random measures or not, as stated in Open Problem 4.40. However, we show that a flow-adapted balancing allocation may not exist for two general random measures, as shown in examples 6.6 and 6.7. Some results and open problems about this problem are provided in Subsection 4.6.
The claim of Theorem 4.9 is similar to the inversion formula ( [11], Satz 2.4. See also [9], (2.6)). This formula recovers the distribution of Ψ from its Palm distribution. Here, the difference is that the balancing property of F ensures that the recovery can be done using a shift-coupling as defined in the theorem.
If this is done such that Lemma 4.5 and Remark 5.1 hold, then all of our results remain valid.

Remark 4.11.
Being stationary is crucial in Theorem 4.14. As an example, Example 6.3 shows cases in which the site-optimal transport density is not balancing although the (non-random) measures in the example may have equal spatial intensities; i.e.
ϕ([−r, r]) ∼ ψ([−r, r]) as r → ∞ (see also [3]). As mentioned in [4], it seems difficult to give a sufficient condition for deterministic discrete sets (and therefore, for measures) to ensure that the site-optimal transport density is balancing.

Stability of constrained transport densities
Definition 4.12. Let ϕ, ψ ∈ M be given and f be a given constrained transport density. We say that a site We drop the prefix 'f -' when there is no confusion. A constrained transport density f is called stable if there is no (x 0 , ξ 0 ) ∈ R d × R d such that x 0 desires ξ 0 and ξ 0 desires x 0 . Theorem 4.13 (Stability). The site-optimal transport density is a stable constrained transport density and depends on ϕ and ω in a measurable and flow-adapted manner.
Theorem 4.14. Let Φ and Ψ be ergodic random measures on R d with positive and finite intensities λ Φ and λ Ψ . Let F = F ω (x, ξ) be a flow-adapted function which is almost surely a stable constrained transport density for Φ ω and Ψ ω . Then almost surely

the set of unexhausted sites has zero
Φ ω -measure and the set of unsated centers has zero Ψ ω -measure.
(ii) if λ Φ < λ Ψ , then there is no unexhausted site but the set of unsated centers has an infinite Ψ ω -measure.
(iii) if λ Φ > λ Ψ , then there is no unsated center but the set of unexhausted sites has an infinite Φ ω -measure.
Proposition 4.20 quantifies how far the centers (resp. sites) are from being sated (resp. exhausted) in average in the second (resp. third) case of Theorem 4.14.
Remark 4.15. Theorems 4.8, 4.9 and 4.14 can be extended to stationary non-ergodic cases. The necessary and sufficient condition on Φ and Ψ is the equality of the sample intensities lim r→∞ Φ(B r (0))/L(B r (0)) and lim r→∞ Ψ(B r (0))/L(B r (0)). This condition is proved to be necessary and sufficient for the existence of a flow-adapted balancing transport kernel in Theorem 5.1 in [9]. To do so, we can slightly change the proofs and use conditional expectation with respect to the invariant sigma-filed (using the ideas in [9]) or obtain the general versions simply by applying the ergodic decomposition theorem.

Other properties of stable constrained transport densities
In this subsection we study monotonicity, optimality, uniqueness and boundedness of territories of stable constrained transport densities. In general, uniqueness is not granted in the deterministic case (see [3] for a counter example). The manner of choosing between equidistant points in Algorithm 4.4 (see Remark 4.10) is another obstacle for uniqueness; e.g. when ϕ and ψ are measures on Z d . Assumption 4.16 gives a sufficient condition for uniqueness of the choice in Remark 4.10 for almost all points. However, we will prove uniqueness of stable constrained transport densities only in the stationary case in Theorem 4.24. It can be seen that the first condition in Assumption 4.16 means that, for ϕ-a.e. site x, the boundary of no ball centered at x can be partitioned in two disjoint sets with positive ψ-measure. This assumption is not difficult to satisfy as shown by Proposition 4.17. Assumption 4.16. For ϕ-a.e. site x and ψ-a.e. center ξ, we have ∀r > 0, ∃s ∈ ∂B r (x) : ψ(∂B r (x) \{s}) = 0, (4.5) ∀r > 0, ∃s ∈ ∂B r (ξ) : ϕ(∂B r (ξ) \{s}) = 0. The following auxiliary functions measure how far the mass is transported from sites and to centers, given a sub-balancing transport density f .
Note that the left hand sides in the above equations are the spatial averages of Ψ x (F, ∞) and Φ ξ (F, ∞). This is a measure of how far the sites and the centers are from being satisfied.
(iii) For µ-a.e. site x we have (iv) For ψ-a.e. center ξ we have Intuitively, it means that when there are less sites and more centers, the situation is better for sites and worse for centers. The following corollary is immediately obtained from Theorem 4.21. Note that the inequalities that contain f c are equivalent to the ones that contain f s as seen by swapping the roles of the sites and the centers.  (ii) For ψ-a.e. center ξ we have In words, among all stable constrained transport densities, the site-optimal transport density is the best for sites and the worst for centers. This justifies the names site-optimal and center-optimal for the Gale-Shapley algorithm.
As mentioned in [4], it seems difficult to express a simple condition in terms of ϕ and ψ that ensures uniqueness of stable constrained transport densities. But Assumption 4.16 is enough in the stationary case, as shown in Theorem 4.24. The key for proving the theorem is the following proposition.
then there is a (ϕ ⊗ ψ)-a.e. unique stable constrained transport density for ϕ and ψ.
Here, by (ϕ ⊗ ψ)-a.e. unique, we mean that any two stable constrained transport densities are identical except on a set of zero (ϕ ⊗ ψ)-measure. Here, we mean that there is an event with probability one such that uniqueness holds on that event.
Given a stable constrained transport density f , the territory of a site x is the set {ξ ∈ R d : f (x, ξ) > 0}. Similarly, the territory of a center ξ is the set {x ∈ R d : f (x, ξ) > 0}. (ii) The union of the territories of the sites (resp. centers) in a bounded set, has finite Ψ-measure (resp. finite Φ-measure).

Voronoi transport kernel corresponding to a measure
In this subsection, we generalize the notion of Voronoi tessellation and define it for a measure. We will use it only for proving Theorem 4.25 here, but it can be interesting in its own.
where c is the constant in [0, 1] such that v(x 0 , ξ)ψ(dξ) = 1. We let c = 1 if s(x 0 ) = ∞ or ψ(∂B s(x0) (x 0 )) = 0. The Voronoi transport density and the Voronoi transport kernel with respect to ψ are the function v and the transport kernel V (x, ·) := v(x, ·)ψ respectively. The Voronoi territory of center ξ with respect to ψ is the set {x ∈ R d : v(x, ξ) > 0}. Note that the Voronoi transport density is the same as the function A 1 in Algorithm 4.4.
Remark 4.27. The condition ψ(R d ) ≥ 1 ensures that the Voronoi transport kernel is a non-weighted transport kernel. If ψ is the counting measure on a discrete set Ξ, then the Voronoi territories with respect to ψ give the usual Voronoi tessellation of Ξ. Therefore, the notion of Voronoi transport kernel generalizes the notion of Voronoi tessellation.
Remark 4.28. It is easy to see that x is in the Voronoi territory of ξ if and only if ψ(B • |x−ξ| (x)) ≤ 1 and if equality happens, then ψ(∂B |x−ξ| (x)) = 0. Lemma 4.29. The Voronoi territory of ξ is star-shaped with center ξ but not necessarily convex or closed. Moreover, it is a closed polyhedral region in R d provided that ψ has a discrete support. Proposition 4.30. If every half-space has infinite ψ-measure, then all Voronoi territories with respect to ψ are bounded. In particular, if Ψ is a stationary random measure which is almost surely non-zero, then almost surely all Voronoi territories with respect to Ψ are bounded.

On allocations
As explained in the introduction, a problem of special interest is constructing flowadapted balancing allocations. But since a ψ-constrained weighted transport kernel T satisfies T ≤ ψ (as in Definition 4.1), a balancing ψ-constrained transport kernel cannot be an allocation except maybe when ψ is a discrete measure. The reason is, if B is a set of centers such that 0 < ψ(B) < 1 (which exists in the non-discrete case), then on a set of sites with positive measure (under ϕ) we have T (x, B) > 0. On the other hand, T (x, B) < 1 and so T (x, ·) is not a Dirac measure. However, we will see that under some conditions in the case when ψ is a counting measure, the site-optimal transport density is guarantied to give an allocation.
A similar notion of one-sided stable allocations in dimension one is also defined in [8]. The authors construct a balancing stable allocation (in that sense) between two ergodic stationary and mutually singular diffuse (i.e. without atom) random measures on R.

Remark 4.34.
In the case that each one of ϕ and ψ is either a diffuse (i.e. without atom) measure or a counting measure (not necessarily satisfying Assumption 4.16), Algorithm 4.4 can be slightly changed to find a {0, 1}-valued stable constrained transport density. It is enough to re-define the functions A n (x 0 , ·) and R n (·, ξ 0 ) on ∂B an(x0) (x 0 ) and ∂B rn(ξ0) (ξ 0 ) for example using the lexicographic order on the boundary of the balls (see Remark 4.10).

Proposition 4.35.
A constrained transport density which is {0, 1}-valued on (ϕ ⊗ ψ)a.e. points gives an allocation provided that ψ is a counting measure. Therefore, by Remark 4.34, for the existence (and construction) of stable allocations it is enough that (i) ϕ is diffuse and ψ is a counting measure, (ii) or both ϕ and ψ are counting measures.
Here, by 'gives an allocation' we mean that the weighted transport kernel given by Remark 4.3 coincides with an allocation on almost all sites.

Remark 4.36.
When ψ is a measure with discrete support ψ = ∞ i=1 w i δ ξi and ϕ is diffuse, we can slightly change Algorithm 4.4 to obtain a stable allocation. We limit ourselves toψ-constrained transport kernels, whereψ = ∞ i=1 δ ξi is the counting measure with the same support as ψ. Equivalently, f (x, ξ i ) ≤ 1 wi for defining constrained transport densities. We then change the definition of A n and a n in Algorithm 4.4 in a similar manner. We similarly get the following proposition. Proposition 4.37. Suppose Φ is a diffuse random measure and Ψ is a random measure whose support is a.s. discrete in the setting of Theorem 4.8. One can construct an allocation which is almost surely stable and (Φ, Ψ)-balancing.

We finish this section with two open problems.
Open Problem 4.38. Does there exist a stable allocation for any two measures ϕ and ψ on R d provided that ϕ is absolutely continuous with respect to the Lebesgue measure?
If a stable allocation exists and can be chosen as a measurable flow-adapted function of (ϕ, ψ), then it leads to a solution of the following questions in a way similar to theorems 4.9 and 4.14. letB r ⊆ V \B R (0) be a ball with radius r, which exists by the assumption on V . By stationarity we get

Remark 5.1. By the definition of
and if a n (x 0 ) < ∞, equality holds. Moreover, if ψ has infinite total mass, then one can prove by induction that a n (·) < ∞ for all n. Similarly, by the definition of R n we have and if r n (ξ 0 ) < ∞, equality holds. Moreover, if equality holds at some stage n, then it holds at all later stages; i.e. ξ 0 is sated at stage n afterwards.
Proof of Lemma 4.5. It is clear that R 1 ≥ R 0 . For n ≥ 2, assume R n−1 ≥ R n−2 . We will conclude a n ≥ a n−1 , A n ≥ A n−1 , r n ≤ r n−1 and R n ≥ R n−1 , which proves the claim by induction. By the same induction, it is easily proved that the functions depend on ϕ and ψ in a measurable and flow-adapted manner.

Lemma 5.2.
In the site-optimal transport density, if a site x 0 desires a center ξ 0 , then A n (x 0 , ξ 0 ) = 1 for sufficiently large n. Similarly, if ξ 0 desires x 0 then R n (x 0 , ξ 0 ) = 0 for all n.
Proof. Suppose x 0 desire ξ 0 . If there is a center ξ 1 such that |ξ 1 − x 0 | > |ξ 0 − x 0 | and f (x 0 , ξ 1 ) > 0, then x 0 has applied to ξ 1 and so it has applied to all points closer than ξ 1 with weight 1. Hence x 0 has applied to ξ 0 with weight 1. In the other case, x 0 is unexhausted. It is enough to prove that a n (x 0 ) > |x 0 − ξ 0 | for sufficiently large n. If this is not true, a n (x 0 ) < ∞ for all n and Remark 5.1 gives , which is integrable with respect to ψ due to locally finiteness of ψ. So, Lebesgue's dominated convergence theorem gives R d f s (x 0 , ξ)ψ(dξ) = 1; i.e. x 0 is exhausted, a contradiction. Now, suppose that ξ 0 desires x 0 . If ξ 0 is unsated, then it has not rejected any weights. So suppose there is a site x 1 such that |x 1 − ξ 0 | > |x 0 − ξ 0 | and f (x 1 , ξ 0 ) > 0. Therefore r n (ξ 0 ) ≥ |x 1 − ξ 0 | for all n, or else ξ 0 would fully reject x 1 at all stages after stage n since the rejection radius is non-increasing. Thus r n (ξ 0 ) > |x 0 − ξ 0 | for all n and so ξ 0 has not rejected any application weight of x 0 . EJP 21 (2016), paper 51.
Proof of Theorem 4.13. Let f s be the site-optimal transport density. By Lemma 4.5 we get that f s depends on ϕ and ψ in a measurable and flow-adapted manner. We have Moreover, A n ≥ R n−1 and A n ≥ R n . By Remark 5.1 and Fato's lemma, it follows that f s is a sub-balancing transport density. Also, f s is constrained since f s ≤ 1. Now, suppose that f s is unstable. So we can find a site x 0 and a center ξ 0 that desire each other. By Lemma 5.2 we get that A n (x 0 , ξ 0 ) = 1 and R n (x 0 , ξ 0 ) = 0 for sufficiently large n. So f s (x 0 , ξ 0 ) = 1 − 0 = 1, a contradiction. Lemma 5.3. Let f be any stable constrained transport density for ϕ and ψ as in Definition 4.12. Let X be the set of unexhausted sites and Ξ be the set of unsated centers. If X = ∅, then ψ(Ξ ) < 1 and if Ξ = ∅, then ϕ(X ) < 1. In particular, we have either ϕ(X ) < 1 or ψ(Ξ ) < 1.
Note that it is possible that both X and Ξ in Lemma 5.3 have positive measure, as shown in Example 6.2.
Lemma 5.4. Let Φ and Ψ be stationary random measures on R d satisfying the assumptions of Theorem 4.14. For any flow-adapted sub-balancing transport density F we Intuitively, this means that the average mass that is transported to a typical center from the sites of distance at most t is equal to the mass that is transported from a typical site to the centers of distance at most t. This is a version of mass transport principles. A more general equation can be found in [9].
where the last equality is due to the fact that Φ 1 = 1 {Us(F )>0} Φ. Since U s (F ) is bounded and Φ 1 is ergodic, we get that Similarly, Since F is stable almost surely, Lemma 5.3 gives that, almost surely,  Proof of Theorem 4.8. By Lemma 4.5, the site-optimal transport density for Φ and Ψ gives a flow-adapted transport kernel. Moreover, it is a stable constrained transport density by Theorem 4.13. Now the theorem is a direct consequence of part (i) of Theorem 4.14.
Proof of Theorem 4.9. Let H be any measurable function on M . Since P Φ = P, it is a direct consequence of Corollary 4.7 in . This finishes the proof. In fact, case (i) is just Example 4.8 in [9].
Proof of Proposition 4.17. Suppose ϕ assigns zero to all spheres and all affine hyperplanes. So ϕ(∂B) = 0 for every ball B. Thus, (4.6) holds for all ξ. For 0 ≤ k < d, by a k-dimensional sphere, we mean the intersection of the boundary of a ball with a non-tangent affine subspace of dimension k + 1. We call a k-dimensional sphere S bad if ψ(S\{s}) > 0 for every point s. Let T be the set of atoms of ψ, which is countable due to locally finiteness of ψ.
zero-dimensional bad spheres are just pairs of atoms and so they are countable. Also, a EJP 21 (2016), paper 51.
positive-dimensional minimal bad sphere S contains at most one atom and so ψ(S\T ) > 0. Suppose there are an uncountable number of minimal bad spheres. So, there exist R > 0 and > 0 such that there are infinitely many minimal bad spheres S i , i ∈ N such that S i ⊆ B R (0) and ψ(S i \T ) > . For i = j, since S i ∩ S j is not a bad sphere, we have ψ ((S i \T ) ∩ (S j \T )) = 0. Therefore ψ(B R (0)) ≥ i ψ(S i \T ) = ∞, a contradiction.
Proof of Proposition 4.20. The claim is a direct consequence of (5.5) and Birkhoff's theorem.
By the definition of T , we have R n (x, ξ 0 ) > 1 − f (x, ξ 0 ) ≥ 0 for x ∈ T . So ξ 0 has rejected some weight from all sites in T at stage n and so ξ 0 is sated at that stage. Thus, if we let and moreover, T is disjoint from the interior of B. we have x is closer than some point in T to ξ 0 .
To prove the lemma, we consider three cases. Note that if µ is absolutely continuous w.r.t. the Lebesgue measure, then only the first case happens.
Therefore, the inequality in (5.8) is strict. Thus, the integrand is positive on a set with non-zero ϕ-measure, which is the desired set. On the other hand, (5.6) gives ϕ(∂B\T ) = 0. Now (5.8) gives So the integrand is positive on a set with positive ϕ-measure, which is the desired set.
This completes the proof of Lemma 5.5.
For x ∈ B ξ0 , part (a) of Lemma 5.5 implies f (x, ξ 0 ) < 1. By the definition of T we get f (·, ξ 0 ) > 0 on T . So ξ 0 , f -desires x because of part (b) of Lemma 5.5; i.e. it desires all sites in B ξ0 .
By Fubini's theorem for the set {(x, ξ) : ξ ∈ Ξ 1 , x ∈ B ξ } and Assumption 4.16 for µ and ψ we get that there is a site x 0 such that the set has positive ψ-measure and ∀r > 0, ∃s ∈ ∂B r (x 0 ) : ψ(∂B r (x 0 ) \{s}) = 0. (5.9) Note that our construction of B ξ0 is given in terms of some inequalities. So the above set is measurable and Fubini's theorem is valid. Let B 1 be the smallest closed ball centered at x 0 (with possibly infinite radius) that contains Ξ 2 . Part (a) of Lemma 5.5 implies As a result, x 0 has applied to all centers in Ξ 2 at stage n. Therefore, A n (x 0 , ·) ≡ 1 on B • 1 . Furthermore, by the choice of n we could choose x 0 such that ψ-a.e. we have f (x 0 , ·) + R n−1 (x 0 , ·) ≤ 1. Thus (5.10) and (5.11) give where the second inequality is due to the definition of A n . Therefore, by Remark 4.19 we get that x 0 , f -desires the centers in Ξ 2 ∩ B • 1 . This gives an unstable pair for f , a contradiction. So suppose ψ(Ξ 2 ∩ B • 1 ) = 0. We should have ψ(Ξ 2 ∩ ∂B 1 ) > 0. But (5.9) gives ψ(∂B 1 \Ξ 2 ) = 0 and as before we get So x 0 , f -desires the centers in Ξ 2 ∩ ∂B 1 , a contradiction.
(ii) If x fully applies to ξ, i.e. A(x, ξ) = 1, then we have f s (x, ξ) = 1 − R(x, ξ) by the definition of f s . Hence, the claim is a direct consequence of (i).
(iii) Let X be the set of sites x such that for ψ-a.e. ξ the claim of (ii) holds for (x, ξ). By (ii) we have µ(X c ) = 0. We prove that all sites in X satisfy the claim of (iii). Suppose x 0 ∈ X and ν x0 (f, t) > ψ x0 (f s , t). Therefore, ψ x0 (f s , t) < 1. It follows that either x 0 is f s -unexhausted or f s (x 0 , ·) is positive somewhere outside B t (x 0 ). Since f s is obtained by the site-optimal Gale-Shapley algorithm, in both cases x 0 has applied to all centers in B t (x 0 ) with weight 1 (for the first case use Lemma 5.2 and for the second case EJP 21 (2016), paper 51.
(iv) By the right-continuity of µ ξ (f, ·) and ϕ ξ (f s , ·), it is enough to prove the claim for rational t. If this doesn't hold, we can find t ∈ [0, ∞] and S ⊆ R d such that ψ(S) > 0 and µ ξ (f, t) < ϕ ξ (f s , t) for all ξ ∈ S. For arbitrary ξ 0 ∈ S, since µ ≥ ϕ, the set of sites has positive µ-measure. Moreover, we have µ ξ0 (f, t) < 1 and f (·, ξ 0 ) < 1 on T ξ0 . Therefore, Remark 4.19 gives that ξ 0 , f -desires all points of T ξ0 . We will show that ξ 0 can be chosen such that some point of T ξ0 also f -desires ξ 0 and contradiction follows. Assumption 4.16 for µ and ψ, (ii) and Fubini's theorem on {(x, ξ) : ξ ∈ S, x ∈ T ξ } (which is measurable) imply that there exists a site x 0 such that (a) (5.9) holds, (b) statement (ii) is valid for x = x 0 and ψ-a.e. ξ, In fact, these conditions are satisfied by µ-a.e. x 0 . Consider the smallest ball B centered at x 0 (possibly with infinite radius) that contains C. Since f s (x 0 , ·) > f (x 0 , ·) ≥ 0 on C, we get that x 0 has applied to the interior of B with full weight. Therefore, (b) implies Therefore B • f (x 0 , ξ)ν(dξ) < 1. So x 0 f -desires all centers in C ∩ B • by Remark 4.19, which gives an unstable pair for f , a contradiction. Case 2. ψ(C ∩ B • ) = 0. Therefore ψ(C ∩ ∂B) > 0. By (a) we get ψ(∂B\C) = 0. Since f s (x 0 , ·) > f (x 0 , ·) on C we get as before Therefore ν x0 (f, s) < 1, where s is the radius of B. So x 0 f -desires the centers in C ∩ ∂B by Remark 4.19, a contradiction again.
Proof of Proposition 4.23. Let f be an arbitrary stable constrained transport density for ϕ and ψ. Corollary 4.22 implies that for ϕ-a.e. site x we have We can use this equation for t < a s (x) (where a s (x) is the application radius of x in the site-optimal Gale-Shapley algorithm) and part (ii) of Theorem 4.21 to obtain that for (ϕ ⊗ ψ)-a.e. (x, ξ), if |x − ξ| < a s (x), then f s (x, ξ) = f (x, ξ). Assumption 4.16 and (5.12) for t = a s (x) imply that this is also valid for |x − ξ| = a s (x); i.e. for (ϕ ⊗ ψ)-a.e. (x, ξ), if |x − ξ| ≤ a s (x), then f s (x, ξ) = f (x, ξ). Since f s (x, ·) ≡ 0 outside B as(x) (x), we use (5.12) for t = ∞ to obtain that (ϕ ⊗ ψ)-a.e. we have f s = f . This proves the claim.
Proof of Theorem 4.24. We should prove that almost surely any two stable constrained transport densities are equal except on a set with zero (Φ ω ⊗ Ψ ω )-measure. Let f s and f c be the site-optimal and the center-optimal transport densities respectively. We take expectations in Corollary 4.22 and apply Lemma 5.4 to get Therefore, all inequalities are indeed equality. Hence By an argument similar to (5.3) we get Since the integrand is non-negative by Corollary 4.22, for a given t, we almost surely have for Φ ω -almost every Considering this for rational t and using right-continuity of Ψ x (f s , ·) and Ψ x (f c , ·) we get that almost surely for Φ ω -a.e. x we have Now, for a sample ω ∈ Ω such that the above equation holds for Φ ω -a.e. x and Assumption 4.16 holds for Φ ω and Ψ ω , the claim is a direct consequence of Proposition 4.23.
Proof of Lemma 4.29. Remark 4.28 easily implies that the Voronoi territory of ξ is starshaped with center ξ. As an example for non-convex territories, let ψ be half of the counting measure on the vertices of an equilateral triangle in the plane. Also, if ψ = L+δ 0 , we see that the Voronoi territory of center 0 is not closed. Now, suppose the support of ψ is a discrete set {ξ 1 , ξ 2 , . . .}. Let D i be the Voronoi territory of a center ξ i . In this case, Remark 4.28 gives that x ∈ D i if and only if ψ(B • |x−ξi| (x)) < 1. This easily implies that D i is closed. With the notations of Definition 4.26, let A be the set of points x ∈ R d such that ∂B s(x) (x) contains more than one atom of ψ. In each component of A c , the set of atoms in B s(x) (x) is fixed. Therefore, all Voronoi territories are bound by A.
j is contained in a hyperplane. Therefore, it suffices to prove that for each compact set K, only finitely many of the A i,j 's hit K. Let S ⊃ K be a compact set such that ψ(S) > 1 (in the case ψ(R d ) = 1 the claim is trivial). Each open ball B • s(x) (x) for x ∈ K doesn't contain S and therefore it is contained in a compact set T which depends only on K and S. Now, if A i,j hits K, then ξ i and ξ j lie in T . It follows that such pairs (i, j) are finite and we are done.
Proof of Proposition 4.30. Suppose the Voronoi territory C of center ξ 0 is unbounded. According to Lemma 4.29, C is star-shaped and hence there is a half-line l starting at ξ 0 which completely lies in C. For x ∈ l, Remark 4.28 gives ψ(B • |x−ξ0| (x)) ≤ 1. By taking union over x ∈ l, we find ψ(H) ≤ 1, where H is an open half-space orthogonal to l, a contradiction. Now, suppose Ψ is a stationary non-zero random measure. One can obtain from Lemma 2.1 that every half-space has infinite Ψ-measure a.s. and the claim follows.
EJP 21 (2016), paper 51. Lemma 5.6. Let A ⊆ R d , a ∈ R d and r > 0. Let C be the set of points x such that |x − a| < |x − a | + r for all a ∈ A. C is bounded if and only if B r (a) is contained in the interior of Conv (A), where Conv (A) stands for the convex hull of A.
Proof. One has C is star-shaped with center a. Hence, it is unbounded if and only if it contains a half-line {a + tv : t ∈ [0, ∞)} for some unit vector v ∈ R d . This is equivalent to Proof of Theorem 4.25. Using the ergodic decomposition theorem, we can assume that P is an ergodic measure for the family (θ s ) s∈R d .
(i) We call a center bad if its territory is unbounded. Let Ξ be the set of bad centers. Suppose Ψ(Ξ ) = 0 with positive probability. By ergodicity, this happens almost surely. Since F is flow-adapted, the restriction Ψ of Ψ to Ξ is an ergodic random measure with positive intensity. Let ξ 0 ∈ Ξ and let C be the Voronoi territory of ξ 0 with respect to 1 2 Ψ . We claim that the territory of ξ 0 is a subset of C and contradiction follows directly by applying Proposition 4.30 for Ψ . If the claim is not true, let x 0 ∈ C such that F (x 0 , ξ 0 ) > 0. By Remark 4.28 we get Ψ (B • |x0−ξ0| (x 0 )) ≥ 2 > 1. Therefore, there is a bad center ξ 1 ∈ B • |x0−ξ0| (x 0 ) such that F (x 0 , ξ 1 ) < 1. Now x 0 desires ξ 1 since it prefers ξ 1 over ξ 0 . Also ξ 1 desires x 0 since its territory is unbounded. So (x 0 , ξ 1 ) is an unstable pair, a contradiction.
(ii) By stationarity, it is enough to show that for a deterministic point z, the union of the territories of the sites in B 1 (z) has finite Ψ-measure almost surely. Call a lattice point z ∈ Z d bad if this property doesn't hold for z. Suppose by contradiction that 0 is bad with positive probability. Then the set of bad lattice points form an ergodic simple point process on Z d with positive intensity. Let Z be the set of bad lattice points, which contains z 0 := 0. Since every half-space contains some element of Z, we have Conv(Z) = R d . Therefore, we can find z 1 , . . . , z n ∈ Z\{0} such that B 2 (z 0 ) ⊆ Conv (z 1 , . . . , z n ). Let C 0 be the region defined in Lemma 5.6 for A = {z 1 , . . . , z n }, a = z 0 and r = 2, which is bounded by the claim of the lemma. For i = 1, . . . , n let C i be the set of points x ∈ C 0 that are closer to z i than other points of A. Let T be the union of the territories of the sites in B 1 (z 0 ). Since z 0 is a bad lattice point, we have Ψ(T ) = ∞. Since Ψ(C 0 ) < ∞, there is i > 0 and a bounded Borel set D ⊆ T ∩ C i such that Ψ(D) > 1. Since z i is a bad lattice point, there is a site x i ∈ B 1 (z i ) such that its territory contains some centers further away than all of the centers in D. Since Ψ(D) > 1, we can find a center ξ i ∈ D such that F (x i , ξ i ) < 1. We claim that (x i , ξ i ) is an unstable pair.
Since ξ i ∈ T , there is a site x 0 ∈ B 1 (z 0 ) such that F (x 0 , ξ i ) > 0. The fact that ξ i ∈ C 0 gives |ξ i − z 0 | ≥ |ξ i − z i | + 2. Therefore |ξ i − x 0 | > |ξ i − x i | and thus ξ i desires x i . On the other hand, x i desires ξ i since its territory contains centers further away than ξ i by the definition of x i . So (x i , ξ i ) is an unstable pair, a contradiction.

Examples
In the following examples we let Z d be the counting measure on Z d ; i.e.  presented in [3]. In this setting, definitions 4.2 and 4.12 also generalize the definitions of allocations and stability in [3].  Figure 1(a) and 1(b) illustrate the first stage of Algorithm 4.4. It can be seen that the site-optimal and the center-optimal transport densities agree a.e. with the function depicted in Figure 1(c). Therefore, we can use Proposition 4.23 to see that this is the unique stable constrained transport density for ϕ and ψ. Note that the sites (resp. centers) that have distance less than 1 2 (1 − 1 φ ) to the boundary are unexhausted (resp. unsated), where φ is the golden number; i.e. it satisfies φ − 1 φ = 1. Example 6.3. Let ϕ = L 1 and ψ be the counting measure on a subset A ⊆ Z. For x ∈ R and ξ ∈ A one has f s (x, ξ) = 1, |x − ξ| ≤ 1 2 0, otherwise It is easy to verify that f s is a balancing stable constrained transport density only when A = Z. f (x, ξ) = 1, max{|x 1 − ξ 1 | , |x 2 − ξ 2 |} ≤ 1 2 , 0, otherwise. This function is a balancing constrained transport density, but is not stable since the center ξ 0 = (0, 0) and the site x 0 = (a, 0) desire each other for 1 2 < a < 5 8 . However, Algorithm 4.4 gives f s in one step, which is a balancing stable constrained transport density. For x = (x 1 , x 2 ) and ξ = (ξ 1 , ξ 2 ), the site-optimal transport density is f s (x, ξ) = 1, |x 2 − ξ 2 | ≤ min{ 1 2 , 5 4 − 2 |x 1 − ξ 1 |}, 0, otherwise.
The territory of each center is a hexagon as illustrated in Figure 2.  A similar equation holds for other values of x. By applying a random translation, Theorem 4.8 holds and f s is a balancing stable constrained transport density. Example 6.6. Let Φ and Ψ be jointly stationary ergodic counting measures (i.e. simple point processes) in R, with positive and finite intensities λ Φ and λ Ψ . The random measures Φ := 1 λΦ Φ and Ψ := 1 λΨ Ψ have unit intensity, but there is no (Φ , Ψ )-balancing allocation provided that should be the number of pre-images of a center). However, Theorem 4.8 shows that there is a flow-adapted balancing transport kernel between them. Example 6.7. In the setting of Example 6.6, let Φ := Φ × L d−1 and Ψ := Ψ × L d−1 , which are stationary ergodic random measures on R d with unit intensity. Theorem 4.8 shows the existence of a flow-adapted (Φ , Ψ )-balancing transport kernel, but we claim that there is no such allocation.
Let τ be a flow-adapted (Φ , Ψ )-balancing allocation. By translation invariance, we get that for a site x, the vector τ ω (x) − x depends only on π 1 (x) and ω, where π 1 is projection on the first coordinate. It follows easily that the allocation on R defined by x → π 1 (τ ((x, 0))) is (Φ , Ψ )-balancing, which is a contradiction by Example 6.6. Example 6.8. Let ϕ be the Lebesgue measure on (0, ∞) and ψ be the Lebesgue measure on (−∞, 0). It is easy to see that f s (x, ξ) ∈ {0, 1} and f s (x, ξ) = 1 if and only if x = −ξ , where a is the smallest integer not less than a.