Invariant measures of Mass Migration Processes

We introduce the Mass Migration Process (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\ge 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence in particular zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We obtain necessary and sufficient conditions for the existence of translation-invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation-invariant, invariant probability measures. We also consider attractiveness properties (through couplings) and condensation phenomena for MMP. We illustrate our results on many examples; in particular, we prove the coexistence of both condensation and attractiveness in one of them.


Introduction
In the study of an interacting particle system, an essential tool is the explicit knowledge of an invariant measure. Conservative systems such as exclusion or zero-range processes possess a oneparameter family of translation invariant and invariant product probability measures, where the parameter represents the average particle density per site [26,19,1]. Under conditions on the rates, this is also the case for misanthropes processes, which include the more recently studied target processes [8,15,21]. All these dynamics (we call them single-jump models) consist in individual jumps of particles between sites, with rates which are the product of two terms: a transition probability giving the direction of the jump, and a function depending on the occupation numbers at the departure and/or arrival sites of the jump. Each dynamics has its particular features, which dictate the precise form of the rates.
However, finding invariant measures is challenging as soon as one departs from these classical models. In this paper we address this question for a class of models generalizing them, that we call mass migration processes (abbreviated as MMP). Those dynamics, of state space N Z d , allow multiple (simultaneous) jumps of particles between sites, according to rates written as above, but where the function depends also on the number k ≥ 1 of particles which jump. We distinguish among them the processes for which the rates do not depend on the occupation number of the arrival site and call them mass migration zero range processes (abbreviated as MM-ZRP), and the processes for which the rates do not depend on the occupation number of the departure site (as long as it is non-empty) and call them mass migration target processes (abbreviated as MM-TP).
Examples of such dynamics have appeared in the literature in various contexts and under various names. Without being exhaustive, we quote some of them that will illustrate our results: The totally asymmetric stick process has a constant jump rate (independent of particles' numbers); it was studied in [25] in the context of Ulam's problem, and a generalization with nearest neighbour jumps was considered in [14]; it possesses a one parameter family of product geometric invariant measures. The MMP is a particular case of the multiple particle jump model studied in [14] in the context of attractiveness properties; under some conditions on the rates, the latter possesses a one parameter family of translation invariant and invariant measures, that are not always explicit. Dynamics with zero range interaction and multiple jumps were studied in a finite volume setup in the context of condensation properties: they were called mass transport models in [10,11,12], and Date: July 8, 2016. generalized zero range processes in [16] (where the number k of particles that may jump together was bounded). In those references, the set of sites is an interval, a general finite graph or a cube; in [11,12], the authors consider a continuous mass instead of particles moving among sites. A generic form of stationary product measures is exhibited for all those dynamics; such an explicit form is required to analyse condensation phenomena, this is why these models are often chosen as examples (see the reviews [9,10,7]). In the context of exactly solvable models and duality, the q-Hahn asymmetric zero range processes are other dynamics with zero range interaction and multiple jumps (see [3] and references therein) with explicit product invariant measures; this knowledge is crucial for exact-solvability. Another possible extension of these models are processes in which multiple births and deaths are superimposed to multiple jumps; their attractiveness properties have been established in [5], and then they have been applied to the study of survival and extinction of species in [6].
In all those cases, a specific derivation is performed for each model to find stationary product measures, under conditions on the rates if necessary. In the present paper, our central result unifies and generalizes those derivations, which in particular opens the door to the study of condensation phenomena and exact solvability for new models, or models not tractable up to now: In Theorem 3.1, we obtain necessary and sufficient conditions on the rates of a MMP (which is a dynamics in infinite volume) to guarantee the existence of translation-invariant, product invariant probability measures for the process. We can state explicitly how these measures look like for single-jump models, that is misanthropes, zero-range and target processes (abbreviated as MP, ZRP and TP), as well as for MM-ZRP and MM-TP. For MMP, we then analyse attractiveness properties, as well as condensation issues; our exact formulas for invariant measures enable us to study the relations between those two properties. Finally, we illustrate our results on various examples, including the ones previously mentioned. In particular, we prove for the first time that attractiveness and condensation can coexist on an example of MM-ZRP.
The paper is organized as follows. In Section 2, we give a formal description and state an existence theorem for MMP; proofs are done in Section 8. Section 3 is devoted to invariant measures of MMP: Theorem 3.1, single-jump models, MM-ZRP and MM-TP. Proofs are done in Section 4. Section 5 is devoted to attractiveness of MMP, and contains the determination of the extremal translation invariant and invariant measures for MM-ZRP and MM-TP. We explain in Section 6 how known results on condensation for ZRP can be applied to MMP, and check whether MMP could also be attractive. In section 7 we bring examples of MMPs and, for each one, we check whether the conditions established in Sections 2, 3, 5, and 6 are satisfied.

The model
2.1. Description, existence results. Let us introduce the mass migration process, abbreviated as MMP. Particles are located on a countable set X of sites, typically Z d . At a given time t ≥ 0, for a configuration η t = (η t (x) : x ∈ X), η t (x) ∈ N is the number of particles on site x ∈ X. Particles move between sites with respect to the mass migration dynamics, that is, k ≥ 1 particles from the total amount α of particles at a departure site x jump simultaneously to a target site y occupied by β particles with a rate p(x, y)g k α,β where (p(x, y), x, y ∈ X) is a transition probability on X, and where, for all α, β ∈ N, g 0 α,β = 0, g k α,β are nonnegative for 0 < k ≤ α, and g k α,β = 0 for k > α or α = 0. (2) This dynamics is conservative: the total number of particles involved in a transition is preserved. When X is finite, rates (1) define straightforwardly a Markov process (η t ) t≥0 on the state space N X . When X is countably infinite, some care is required to define (η t ) t≥0 as a Markov process. For this, we rely on methods by Andjel [1] and Liggett & Spitzer [18] and proceed as follows. Details and proofs are given in Section 8.
To avoid initial configurations that could cause explosions, we first restrict the state space to for a positive function a on X such that, for some positive constant M , y∈X p(x, y)a y + y∈X a y p(y, x) ≤ M a x , for every x ∈ X. (5) and x∈X a x < ∞.
The following proposition and theorem define the infinitesimal generator L and the corresponding semigroup (S(t), t ≥ 0) for the Markov process (η t ) t≥0 . We also introduce the associated Markovian coupled process, and finally characterize invariant measures for (η t ) t≥0 .
We assume from now on that (p(x, y) : x, y ∈ X) is an irreducible transition probability satisfying sup y∈X x∈X p(x, y) = m p < ∞. (7) and that the rates satisfy (2). Let the set of Lipschitz functions on X be (where η − ζ = x∈X |η(x) − ζ(x)|a x ).
Theorem 2.2. Consider the infinitesimal generator L given by (10), and assume that condition (13) is satisfied. Then there exists a Markov semigroup of operators (S(t), t ≥ 0), defined on Lipschitz functions L on X, and a constant c > 0 such that for all f ∈ L, t ≥ 0, η ∈ X, the following items hold: (1) S(t)f ∈ L and L S(t)f ≤ L f e ct ; where (η t ) t≥0 on the right-hand side is a (countable state space) Markov process with rates (1), initial configuration η 0 = η, and E η denotes its expectation.
We construct similarly a (Markovian) coupled process (η t , ζ t ) t≥0 on X × X (where we set (η, ζ) = η + ζ ) whose marginals (η t ) t≥0 , (ζ t ) t≥0 are copies of the original MMP. As detailed in Section 8, such a process in finite volume enables to prove Theorem 2.2, and it can then be extended analogously to infinite volume, with a Markov semigroup (S(t) : t ≥ 0) derived from an infinitesimal generator L. Whereas basic coupling was the natural coupling to use in [1] and [18], it is not valid anymore here, hence we use the one introduced in [14], which will moreover be helpful dealing later on with attractiveness (the purpose for which it was built), see Section 5 for details.
A probability measureμ on N X is called invariant for the process with generator L and Markov semigroup (S(t) : t ≥ 0) ifμ is supported on X, and (14a) Letμ be a probability measure on N X satisfying Let us consider a process with generator L given by (10) where rates satisfy (9). Assume that the Markov semigroup (S(t) : t ≥ 0) is such that statements (i)-(vii) of Lemma 8.11 hold. Then (14b) is equivalent to Lf dμ = 0 for every bounded cylinder function f on N X .
2.2. Alternatives. For some models, Conditions (13) and/or (9) will not be valid, and/or the involved probability measures will not satisfy Condition (15). All these assumptions were sufficient for our construction, so for some examples they could be not necessary, and an alternative construction could work (this includes dynamics in finite volume). Moreover some examples require a state space smaller than X and a different construction (see Section 7).
To deal with some of these cases, we will indicate how to adapt the proofs for invariant measures results in Section 3 for models satisfying the two following assumptions: A probability measureμ on N X is invariant for the process with generator L given by (10) and Markov semigroup (S(t) : t ≥ 0) when (14b) is satisfied, and (14b) is equivalent to (16). (17a) There exists a constant C > 0 such that Note that assumption (17b) implies (9).

2.3.
Examples. All along this paper, we will illustrate our results on the following models, which have only one conservation law. They were all initially defined as single-jump models, that is with rates p(x, y)g k α,β such that g k α,β = 0 for k > 1. We analyse their generalizations to k ≥ 2, which were either already defined or that we introduce in this work.
• In the misanthropes process (MP), introduced by Cocozza in [8], for a nonnegative function g 1 ·,· on N × N, non-decreasing (non-increasing) in its first (second) coordinate. In the works of Godrèche et al. [15,20,21], arbitrary (that is, without monotonicity properties) nonnegative g 1 ·,· on N × N are considered in (18) and the process is then called the dynamic urn model or the migration process. However, as per usual, we keep the denomination MP also for those cases; we denote by mass migration processes (MMP) the dynamics extended to multiple jumps.
• Zero range processes (ZRP) are single-jump dynamics (that is, with rate (18)) introduced by Spitzer in [26], for which the dependence on β is dropped in g 1 α,β . We denote by MM-ZRP their extension to multiple jumps, which was introduced in [16]. To simplify the notation for the rates, g k α denotes a function of k and α (the occupation number on the departure site of the jump) only: • Target processes (TP) are single-jump dynamics (that is, with rate (18)) introduced in [21], for which the dependence on α is (almost) dropped in g 1 α,β : only α > 0 is required. We define in this paper their generalization to multiple jumps (MM-TP). To simplify the notation for the rates, g k * ,β denotes a function of k and β (the occupation number on the arrival site of the jump) only: We will study in detail various examples of MM-ZRP and MM-TP in Section 7.

Product invariant measures
Throughout this section we consider a mass migration process (MMP) (η t ) t≥0 on the set of sites X = Z d , with generator (10), rates satisfying assumption (9), which is translation invariant, that is with (p(x, y), x, y ∈ X) a translation invariant transition probability (hence bistochastic, so that m p = 1 in (7)). Let us denote by S the set of translation-invariant probability measures on N Z d , and by I the set of invariant probability measures for the MMP (η t ) t≥0 . We are interested in product, translation-invariant and invariant probability measures for (η t ) t≥0 . Most proofs are done in Section 4.
3.1. Necessary and sufficient conditions for product invariant measures. In this subsection, we exhibit necessary and sufficient relations between the rates of the MMP (η t ) t≥0 and the single site marginal µ of a product, translation-invariant probability measureμ on N Z d which make the latter invariant for (η t ) t≥0 . Theorem 3.1. Consider a mass migration process with generator L given by (10), whose rates satisfy (9). Letμ ∈ S be a product measure whose single site marginal µ has a finite first moment µ 1 = n∈N nµ(n). Let us denote for all α, β ≥ 0 A necessary condition forμ to be invariant for the process is These two conditions are also sufficient when p(·, ·) is symmetric. When p(·, ·) is asymmetric, (24) has to be combined with the following stronger condition to be necessary and sufficient: there exists a function ψ on N such that Theorem 3.1 includes dynamics with state space {0, · · · , γ} Z d , for some γ > 0, as well as measuresμ such that µ(α) = 0 for various value(s) of α. We will deal with such cases in a forthcoming paper, and we will restrict ourselves in the present paper to measures such that µ(α) > 0 for all α ∈ N. In this case definition (23) reduces to its first line, and condition (24) is absent.
It is also possible to deal with a product measureμ whose single site marginal µ has an infinite first moment, either by taking an alternative norm to the one defined in (4) in the construction of the process, or if assumptions (17a)-(17b) are valid. Remark 3.3. If n≥0 nµ(n) = +∞ but there exists a non-decreasing function f : [0, +∞) → [0, +∞), with f (0) = 0 and increasing to infinity when n → ∞ in such a way that n≥0 f (n)µ(n) < ∞, then it is possible to replace the coefficients a x in the norm · defined in (4) by new coefficients a * x for which a new norm · * defined by η * = x∈X η(x)a * x satisfiesμ( η * < ∞) = 1.
From Theorem 3.1 and Corollary 3.4, we deduce that if the product measureμ is invariant for the MMP, the latter possesses a one-parameter family of invariant product probability measures: Corollary 3.5. Consider a MMP with generator L given by (10). Letμ ∈ S be a product measure invariant for the process. Assume that either the rates of the MMP satisfy assumption (9) and the single site marginal µ ofμ has a finite first moment, or that the rates of the MMP satisfy assumptions (17a)-(17b). Let ϕ c ≥ 1 be the radius of convergence of the series Then for all ϕ < ϕ c , the translation invariant product measureμ ϕ defined by its single site marginal is invariant for the process. Moreover, either if n≥0 nϕ n c µ(n) < ∞, or if Z ϕc < ∞ and assumptions (17a)-(17b) are satisfied, (30) then the measureμ ϕc is also invariant for the process.
Therefore the latter possesses a one-parameter family of invariant product probability measures, and For further use, when Z ϕc < ∞ we define the critical density by 3.2. Applications.
In this subsection, we use Theorem 3.1, Corollaries 3.4 and 3.5 to characterize invariant product probability measures for the various examples introduced in Section 2.3. The necessary and sufficient conditions obtained in Theorem 3.1 will be exploited in two ways: 1) the rates being given, check that a measureμ is invariant; 2) a measureμ being given, define rates for whichμ is invariant. This second point of view will enable us in Section 6 to associate many different dynamics to a measureμ satisfying condensation properties.

Single-jump models.
The single-jump models we consider are misanthropes processes (MP), for which the generic rate is p(x, y)g k α,β with g k α,β given by (18). In this case we derive the following proposition from Theorem 3.1.
Proposition 3.6. Consider a MP with rates p(x, y)g 1 α,β . Assume that the rates (g 1 1,α ) α≥0 and (g 1 α,0 ) α≥1 are positive. Letμ ∈ S be a product probability measure whose single site marginal µ satisfies µ(α) > 0 for all α ∈ N, and such that either µ has a finite first moment or assumptions (17a)-(17b) are verified. Thenμ is invariant for the MP if and only if its single site marginal µ is given by and provided that the following compatibility conditions are fulfilled, respectively in the symmetric case, and (35) together with in the asymmetric case.
In other words, the existence of a product invariant measure for the dynamics implies the existence of a family of such measures. Indeed, from (43), one gets (34).
On the other hand, inserting expression (43) in (42) leads to an expression of detailed balance condition in terms of the jump rates only, that is (35). Condition (35) has to be supplemented by condition (26) in the asymmetric case: the latter, after using Remark 3.2(b) and (38), (41), writes (36).
We now apply Proposition 3.6 to ZRP and TP, that is, we check what become the compatibility conditions (35), (36), and the form of the marginal (34) of a product invariant measure, when it exists. We also explicit Corollary 3.5 in these cases.
• Zero range process (ZRP). Here, both conditions (35) and (36) are always satisfied, so that there are no compatibility conditions on the rates. Equation (43) expresses that for a product invariant measure with marginal µ, there is a constant ϕ µ > 0 such that for all α ≥ 1, Consistently with Remark 3.7, a convenient expression for ϕ µ is thus Let ϕ c ≥ 1 be the radius of convergence of the series (28) which writes here Remark 3.8. Instead of considering equation (36), one can insert Equation (44) in (37), (38) and get the following expression for A(α, β), which is directly the form of condition (26) with ψ(α) = g 1 α . This implies that, for the zero range process, conditions for the existence of product invariant measures are the same for both p(·, ·) symmetric and p(·, ·) asymmetric.
The invariant measures derived here are those studied by [26,1].
• Target process (TP). For a target process, conditions (35) are always satisfied, hence there are no compatibility conditions on the rates in the symmetric case. Condition (36) becomes here Thus, only under condition (50) does an asymmetric target process admit invariant product probability measures.
Equation (43) expresses that for a given product invariant measure with marginal µ, there is a constantφ such that for all β ≥ 0, Let ϕ c ≥ 1 be the radius of convergence of the series Then, by Corollary 3.5, there exists a one-parameter family of product, translation-invariant, invariant probability measures, {μ ϕ , 0 < ϕ < ϕ c } or {μ ϕ , 0 < ϕ ≤ ϕ c } if (30) is satisfied, with single site marginal given by (cf. (34)): Then, only under condition (50), an asymmetric target process admits a family of invariant product probability measures, for which the expression (54) for the single site marginal depends on the two distinct jump rates, and becomes: Remark 3.9. Comparing (54) with the analogous formula (48) for zero range process, we see that there is a family of invariant product measures, common to both a zero range process and a symmetric target process if and only if there exists a constant c 0 > 0 such that holds for all α ≥ 0. In [21], (56) is called a duality relation between ZRP and TP.
The invariant measures derived here were studied in [21].

Mass migration zero range process (MM-ZRP)..
In this subsection and the following one, we will see that the possibility for k to be larger than 1 yields a very different behavior than for single-jump models.
Proposition 3.10. Letμ ∈ S be a product measure whose single site marginal µ satisfies µ(0) > 0. Consider a MM-ZRP with rates p(x, y)g k α for an asymmetric p(·, ·) and g k α . Assume that either the rates satisfy (9) and µ has a finite first moment, or that assumptions (17a)-(17b) are satisfied. Thenμ is invariant for the MM-ZRP if and only if If p(·, ·) is symmetric then condition (57) is only sufficient forμ to be invariant.
The form (57) for the rate is the generic one exhibited in [11,16]. We will study such models in detail along the paper, in Subsection 5.2, examples 3 and 4 in Section 7.
An additional assumption on the rates yields the following corollary.
Remark 3.12. Unlike for ZRP, the rates have here to satisfy a compatibility condition, namely (60).
The result of Proposition 3.10 can be used on one hand to find invariant measures when the rates of a process are given, and on the other hand to set rates of a process such that it has a prescribed invariant measure. Hence we rephrase Proposition 3.10 and Corollary 3.11 as follows.
Proposition 3.13. (a) If µ is a probability measure on N with µ(α) > 0 for all α ∈ N, then formula (57) gives transition rates for a MM-ZRP for which the product measureμ with single site marginal µ is invariant. Here g k k > 0, k ≥ 1 can be arbitrarily chosen such that the rates g k α+k for all α, k ≥ 1 satisfy (9).
Remark 3.14. In (b) above, the measureμ ϕ , which depends only on the rates g k α for k = 1, is the same as in the single-jump case.
The situation is rather different for a symmetric MM-ZRP. In general, the only equivalence result follows from Theorem 3.1 and says that a product, translation-invariant probability measureμ (such thatμ(X) = 1) is invariant for this process with rates g k α if and only if If (57) is satisfied, then all product, translation-invariant probability measuresμ ϕ with marginals given by (48) are invariant for the symmetric MM-ZRP with rates g k α . But there could exist other product translation-invariant probability measures satisfying (61) but not (57).

Mass migration target process (MM-TP)..
Theorem 3.15. Letμ ∈ S be a product measure whose single site marginal µ satisfies µ(α) > 0 for all α ∈ N. Consider a MM-TP with rates p(x, y)g k * ,α for an asymmetric p(·, ·). Assume that either the rates satisfy (9) and µ has a finite first moment, or that assumptions (17a)-(17b) are satisfied. Thenμ is invariant for the MM-TP if and only if where the H α (β, k) depend only on µ(·) and are solution of the following recurrence relations: with for all r > 0 and all s ≥ 0 By iterating (63), we obtain the simplest terms H α (β, k): The general term can be guessed: Like for MM-ZRP, the result of Theorem 3.15 can be used on one hand to find invariant measures of a MM-TP with given rates, and on the other hand to set rates of a MM-TP such that it has a prescribed invariant measure. As in Corollary 3.11 and Proposition 3.13, an additional assumption on the rates enables to write a compatibility condition on the rates.
(a) For a probability measure µ on N with µ(α) > 0 for all α ∈ N, for arbitrarily chosen g α * ,0 > 0 for all α ≥ 1 such that condition (9) is satisfied, equations (62)-(63) give transition rates for a MM-TP for which the product measure with single site marginal µ is invariant, provided that all such rates are non-negative. A sufficient condition for the g α * ,β to be non-negative is that ∆ r (s) ≥ 0 for all r > 0, s ≥ 0, or equivalently that the ratio µ(n + 1)/µ(n) is a nondecreasing function of n.
(b) Consider a MM-TP with the following rates. For k ≥ 1, g k * ,0 and g k * ,1 are given such that For the weights w(α), α ≥ 0, defined as then setting for all r > 0 and all s ≥ 0, and the rates g α * ,β , for all α ≥ 1, β > 1, are given by Then, provided these rates satisfy (9), there is a one-parameter family of product invariant prob- Note that both (a) and (b) hold independently of p(·, ·). If we assume p(·, ·) asymmetric then conditions (66)-(70) on rates are also necessary for the existence of product invariant probability measures.
Note that for a symmetric MM-TP, the set of all product, translation-invariant, invariant measures is in general bigger. In addition to the measuresμ ϕ with single site marginal (71) when (62)-(63) are satisfied, there may exist other product translation-invariant measures satisfying (25) but not (62)-(63). 4. Proofs for Section 3 4.1. Proofs for Subsection 3.1. Hereafter, we prove Theorem 3.1, Remark 3.3, and Corollaries 3.4 and 3.5. We first prove the following lemma.
We have by (9) γ≤N µ(γ) We have also γ≤N k≤γ and finally and we can let N → +∞ to conclude that for any (73) is also satisfied.
where L n is the finite volume approximation of L defined by (180).
Let f be a bounded cylinder function, and denote by V f ∈ Z d its finite support, and by m f an upper bound for |f |. We want to derive an expression for L n f (η) dμ(η), involving A(·, ·) defined in (23). We have For further use, note that since the probability measure µ has a finite first moment µ 1 , we have by (9), for x, y ∈ X n , x = y Step 2. We now prove that condition (24) is necessary. Let α 0 , β 0 such that µ(α 0 )µ(β 0 ) = 0 and consider the function The last line is a sum of non negative terms, thus each term is zero and in particular the one for Step 3. For x, y ∈ X n , x = y, {x, y} ∩ V f = ∅, thanks to (86), the integral in the right hand side of (85) reads where we used Condition (24) in the third equality and introduced in the next one the quantity Step 4. We prove necessity of condition (26) and condition (25), respectively for the asymmetric and symmetric cases. Let us denote Using (87), we have for any n large enough so that In the first equality, we have split the sum over x and y in three terms, depending on whether they belong to the support of f , and used the fact thatμ is a product measure. In the second equality, we introduced notations (89). Consider f (η) = I [η(x0)=α] for some fixed x 0 ∈ Z d and α ∈ S µ . We get in this case from (90), Taking the limit n → +∞ gives by (84) We get from (90), Taking the limit n → +∞ gives by (84) Using (91) and the fact that µ(α)µ(β) = 0, (92) becomes The same relation being valid under exchange of x 0 and y 0 , both the symmetric and antisymmetric parts of equation (93) are separately zero. The symmetric part reads Assuming (25), the antisymmetric part gives If p(·, ·) is symmetric, this equation is identically zero for any choice of A, and we are left with (25) alone. If p(·, ·) is asymmetric, there is a choice of (x 0 , y 0 ) such that p(x 0 , y 0 ) = p(y 0 , x 0 ) and we obtain A(α, β) = ψ(β) − ψ(α).
(95) Since (95) defines ψ up to a constant, we may take ψ(0) = 0 and write ψ(·) in terms of A as Step 5. Let us show now that conditions (24)- (25) and conditions (24)- (26) are sufficient forμ to be invariant, respectively in the symmetric and in the asymmetric case.
Assume first p(x, y) = p(y, x) for all x, y and conditions (24)-(25) satisfied. Equation (87) gives Assume now p(., .) asymmetric and conditions (24)-(26) satisfied. Thus, recalling that f is a cylinder function, and using that by (73), ψ is integrable, we have then For every n such that V f ⊂ X n , In the first equality, we have split the sum on y in two terms and used the fact thatμ is a product measure. In the second equality, we have exchanged the roles of x and y in the second term. The third equality uses that In the fourth equality, we used the fact that p(·, ·) is bistochastic, Thus, using (98), we have which converges to 0 when n → +∞. Thus equation (84) is satisfied.
It follows from Borel-Cantelli Lemma thatμ(B k holds for infinitely many k ∈ N) = 0. Therefore alsoμ(A x holds for at most finitely many Proof. of Corollary 3.4. Note first that in the proof of Lemma 4.1, replacing each bound involving the first moment of µ by the bound (17b) yields the results of the lemma. We now explain where and how to modify the proof of Theorem 3.1 to obtain Corollary 3.4 under assumptions (17a)-(17b). By assumption (17a), the first remark in Step 1 is useless. The first equality in (84) comes from assumption (17a); to derive the second one thanks to assumption (17b), we need to go into the proofs of Subsections 8.  Proof. of Corollary 3.5. For every ϕ < ϕ c , and for ϕ = ϕ c if n≥0 nϕ n c µ(n) < ∞, if the rates of the MMP satisfy assumption (9) and the single site marginal µ ofμ has a finite first moment, then the measureμ ϕ has a finite first moment, and Theorem 3.1 may be applied toμ ϕ .
It follows from the definition (29) ofμ ϕ that the quantities A(., .) defined in (23) coincide for bothμ andμ ϕ for all a, β ∈ N. Since the necessary and sufficient conditions given in Theorem 3.1 are expressed through these quantities, they are thus satisfied forμ ϕ as soon as they are satisfied forμ. Ifμ is invariant for the MMP, then these necessary and sufficient conditions are satisfied.
Thus, by Theorem 3.1 or by Corollary 3.4 for ϕ < ϕ c , and, for ϕ = ϕ c , either by Theorem 3.1 if n≥0 nϕ n c µ(n) < ∞ under assumption (9), or by Corollary 3.4 if n≥0 nϕ n c µ(n) = +∞ with Z ϕc < ∞ under assumptions (17a)-(17b), we have thatμ ϕ is also invariant for the process. Then the invariance condition (26) writes, using (27), To prove that condition (57) implies (100) is straightforward. For the converse, let us denote Taking β = 1 in (101) gives We prove by induction on (n, i), with 0 < i < n, that we have that is, (57) after the change of variables n = α+k, i = α. To initiate the induction for n = 2, i = 1, we choose β = α = 1 in (101). To complete the induction with n > 2, i = n−1, we choose α = n−1 in (102). For the induction step, let us fix n > 2, 0 < i < n and assume that (103) holds for all (n ′ , i ′ ) such that 2 ≤ n ′ < n and 0 < i ′ < n ′ . Use (101) for α = i, β = n − i (that is, α + β = n). We obtain Using Remark 3.2, we set ψ(0) = 0 and equation (27) reads Thus the set of equations (26) is equivalent to We first consider equations (108). Using (105), (106) and notation (64), they can be written as, for all α > 0, β > 0, For β = 1, it reads which is equation (62) for β = 1, using Definition (63). We now prove by induction on β ≥ 1 that equation (62) is satisfied. Suppose that the rates fulfill equations (63) for all values of β up to some valueβ − 1 ≥ 1. We derive (62) for β =β as follows. We have where we used (109) for the first equality and the induction hypothesis for the second one, we exchanged the order of the two sums then the names of indexes in the third and fourth ones, and we used Definition (63) to conclude. Conversely, suppose that equations (62) are valid. We now prove that equations(108) or equivalently equations (109) are verified. We have where we used Definitions (63) for the second and fourth equalities, and we exchanged the order of the two sums then the names of indexes in the third one.
It remains to show that solutions of (62) also verify (107). We do it in Lemma 4.4 below which finishes this proof.
We need two preparatory lemmas to derive Lemma 4.4.
Lemma 4.2. The coefficients H α (β, k) solution of (63) are also solution of the following (dual) recurrence relation Proof. We do an induction on (β, k), with 1 ≤ k ≤ β − 1. Notice first that for all integers β, m, we have, by (63), To initiate the induction with β = 2, k = 1, and to complete it for k = β − 1, β > 2, we compute, for β ≥ 2, using (63), (112): which is the expression of H α (β, β − 1) given by (111). Then we fix 1 ≤ k < β − 1, and we assume (111) satisfied for all (β ′ , k ′ ) such that β ′ < β and 1 ≤ k ′ < β ′ − 1. We have where we applied first (63) to H α (β − k, l) to get the first equality, exchanged the order of the two sums then the names of indexes to get the second one, applied the induction hypothesis with β ′ = β − l < β to get the third one and finally applied (63) in the other direction to conclude.
Proof. We do an induction on (β, k), with 1 ≤ k ≤ β − 1. To initiate the induction with β = 2, k = 1, and to complete it for k = β − 1, β > 2, we compute, for β ≥ 2, using (63), (64): Then we fix 1 ≤ k < β − 1, and we assume (113) satisfied for all (β ′ , k ′ ) such that β ′ < β and 1 ≤ k ′ < β ′ − 1. We have where we applied (111) to get the first equality, we exchanged the order of sums then the names of indexes to get the second one, we used the definition of ψ(., .) to get the third one, we applied the induction hypothesis for the fourth one and (64) for the fifth one. Now we are ready to show that Lemma 4.4. Equation (107) holds with rates g α * ,β as in (62)-(63). Proof. We have, for β ≥ 1, where we used (105) for the first equality, (62) for the second one, we exchanged the order of sums then the names of indexes to get the third one, we applied Lemma 4.3 for the fourth one and (106) for the last one.

Attractiveness and coupling rates
In this section, we take advantage of the work done in [14] on multiple-jump conservative dynamics of misanthrope type, a class of models including MMP, to analyze attractiveness properties of MMP. In Subsection 5.1, we derive the extremal translation invariant and invariant probability measures for MM-ZRP and MM-TP. In Lemma 5.1 and in Subsection 5.2 we give necessary and sufficient conditions for attractiveness of MMP, MM-ZRP and MM-TP based on the conditions established in [14].
Let us first recall the set-up for attractiveness, for which we refer to [19]. We consider the following partial order on the state space X ⊂ N X . For η, ζ ∈ X, η ≤ ζ if and only if η(x) ≤ ζ(x) for every x ∈ X.
A bounded, continuous function f on X is called monotone if f (η) ≤ f (ζ) whenever η ≤ ζ. We call a particle system attractive if for every bounded, monotone continuous function f on X and every time t > 0, the function S(t)f is again a bounded, monotone, continuous function on X.
To check attractiveness of a particle system, the first step is to derive necessary conditions on the rates. The second step is to check that they are sufficient by constructing an increasing, markovian coupling (η t , ζ t ) t≥0 of two copies of the process, that is a coupling which preserves the ordering of its marginals: η 0 ≤ ζ 0 implies η t ≤ ζ t , P (η0,ζ0) -almost surely, for every t > 0. Usually, for dynamics with transitions consisting in the jump, birth or death of a single particle, the so-called basic coupling is used: for jumps, it means that coupled particles try to jump together as much as possible from the same departure site to the same arrival site. But basic coupling is useless when k ≥ 1 particles jump simultaneously, as in the class of conservative dynamics analyzed in [14], which includes the MMP. Let us rewrite the results from [14] we need in the context of models with generator (10).
is defined for (η, ζ) ∈ X × X on the set of functions The rates for coupled jumps are defined in [14,Proposition 2.11]: for all α, β, γ, δ ≥ 0, for all 1 ≤ k ≤ α when 1 ≤ α, for all 1 ≤ l ≤ γ, (123) In other words, for a given time t and a departure site x ∈ X, a target site y is chosen with probability p(x, y), and the simultaneous jump of k particles in the first coordinate together with l particles in the second coordinate from x to y occurs with a rate p(x, y)G k,l α,β,γ,δ where α = η t− (x), β = η t− (y), γ = ζ t− (x), δ = ζ t− (y), 0 ≤ k ≤ α and 0 ≤ l ≤ γ. Other jumps are not allowed.
Even if this coupling was originally built in connection with attractiveness, it is of course always valid, thus we used it in Section 2.
Attractiveness conditions (120a)-(120b) can be reset in the following explicit form for MMP.
Lemma 5.1. Consider (a) the MMP given by generator (10) with rates g k α,β and, respectively in (b) and (c), the MM-ZRP with rates g k α and the MM-TP with rates g k * ,β .

(a) The MMP is attractive if and only if for all
Proof.

Remark 5.2.
For single-jump models, all the inequalities in Lemma 5.1 are restricted to k = 0. First, we recover well-known results: for ZRP, attractiveness means that g 1 α is a nondecreasing function of α (and basic coupling is an increasing coupling process, see [1] for a detailed construction). The misanthrope process was introduced in [8] as an attractive process, that is with the function g 1 α,β nondecreasing in its first coordinate α and nonincreasing in its second one β (thus explaining the name misanthrope).
For TP, attractiveness means that the function g 1 * ,β is nonincreasing in β. When a ZRP and a TP are dual single jump models (see Remark 3.9), attractiveness of the first is equivalent to attractiveness of the second.

5.1.
Characterization of (I ∩ S) e for MM-ZRP and MM-TP. The fact that attractiveness is a very strong property of conservative particle systems is illustrated by the characterization of the set of all translation-invariant and invariant measures of MMP. The latter follows from [14,Theorem 5.13], which requires attractiveness and irreducibility conditions for the coupling transition rates, denoted by (IC). Under Conditions (IC), if an initial coupled configuration (ξ 0 , ζ 0 ) contains a pair of discrepancies of opposite signs (that is, on two sites x, y, (ξ 0 (x) − ζ 0 (x))(ξ 0 (y) − ζ 0 (y)) < 0), there is a positive probability (for the coupled evolution) that it evolves into a locally ordered coupled configuration (that is, for some t ≥ 0, (ξ t (x) − ζ t (x))(ξ t (y) − ζ t (y)) ≥ 0; we refer to [14, Section 2.3] for a detailed analysis of the evolution of discrepancies). We Proposition 5.3. Consider a MM-ZRP for which the g k α 's satisfy (9), (58) and (60). Assume that the attractiveness condition (125) is satisfied. Then where eachμ ϕ is a product probability measure with marginals given by (48).
Proof. By Proposition 3.16, the MM-TP has a one-parameter family of product, translationinvariant, invariant probability measures. Since we assumed that g 1 * ,α > 0 for every α ≥ 0, we can proceed exactly as in the proof of Proposition 5.3 to check conditions (IC) of [14,Theorem 5.13], restricting ourselves to coupled jumps with k, l ∈ {0, 1}.

Attractiveness of MM-ZRP with product invariant measures.
In this paragraph we give equivalent forms of condition (126) for attractiveness of MM-ZRP for the following rate, which is generic for MM-ZRP with product invariant measures (cf. (57)), as will be developed in Examples 3 and 4 of Section 7.
Lemma 5.6. (a) Assume that r is a nonincreasing function. Then condition (135) for attractiveness is equivalent to and the following condition is necessary for the attractiveness of MM-ZRP with g k α given by (134).
(a') If we moreover assume that h(α) = h 0 π(α) ∀α ≥ 1 for some constant h 0 > 0 then condition (138) is sufficient and necessary for the attractiveness of MM-ZRP with g k α of the form (b) Assume that r is a nondecreasing function. Then, to have attractiveness, Condition (137a) is necessary, and Condition is sufficient.
Proof. Preliminaries: Note that the function defined for 1 ≤ m ≤ α, which represents the r.h.s. of (135b) minus its l.h.s., has the same monotonicity as r: if r is nonincreasing (resp. nondecreasing), then φ α is also nonincreasing (resp. nondecreasing).
Therefore for attractiveness properties, it is enough to restrict ourselves to the MM-ZRP with g k α given by (139) with r nonincreasing, for which we arrange condition (138). By (136), we can write for α ≥ 2, i ≥ 2 with i ≤ α,

Condensation
It follows from Section 3 that we are able, for a probability measureμ which is product, translation-invariant on Z d , with single site marginal µ satisfying some conditions, to define rates of a MMP for whichμ is invariant. Moreover this yields a one-parameter family {μ ϕ : ϕ ∈ Rad(Z ′ )} of invariant measures for this process; the single site partition function Z ϕ , defined in (28), has radius of convergence ϕ c , and the critical density ρ c was defined in (33).
The point is that, as we explain below in subsection 6.1, condensation phenomena amount to convergence results concerning only stationary distributions, and not the details of the dynamics. So it is enough to verify convergence properties of the partition function and the average density of particles with respect to the involved probability measure to know whether condensation occurs in the stationary states. Then having a stationary state which produces condensation we can study different MMPs which lead to this stationary state: We give some examples at the end of this section, and others in Section 7.
We end this section checking whether attractiveness and condensation could be compatible for MMPs. We answer 'no' for particular cases. We then turn to this question for the second set-up for condensation, where we show that the answer 'yes' is possible.
6.1. Known results. Consider a finite set Λ ⊂ Z with L sites where N particles evolve according to the restriction on Λ of the infinite volume dynamics, with a periodic boundary condition. To stress dependence on finite volume we denote byμ Λ ϕ the product measure on N Λ with marginals µ ϕ . Such a finite volume process has a unique stationary distribution µ N,L (independent of ϕ) which can be expressed for η ∈ N Λ as The first set-up for condensation [2,9,17] concerns, for a system with finite critical density ρ c < ∞, the thermodynamic limit of the above restricted dynamics when N/L → ρ > ρ c when N, L → ∞. Then there is no appropriate invariant measure and the system can be seen at two levels: a homogeneous background (fluid phase) where occupation numbers per site are distributed according to the product measure at critical densityμ ϕc , and a condensate (condensed phase) in which the excess mass is accumulated on a single site. We say that the process (or dynamics) allows condensation.
In the second set-up for condensation [4,13,23], the number L of sites is fixed and only the total number N of particles goes to infinity. This set-up requires only ϕ c to be finite, but ρ c can be infinite. We talk about condensation on a fixed finite volume.
We mention both approaches since we will present examples for each one. In either case, we say that the process exhibits condensation. Note that both cases require that Z ϕc < +∞.
In the first set-up for condensation, assuming ρ c < ∞, the following equivalence of ensembles result was proved in [17] in the context of zero range processes: For all ρ ≥ 0, n ∈ N, we have (for µ ϕ given by (48)) lim L,N →∞ N/L→ρ for all k 1 , . . . , k n ∈ N.
In the particular case of a ZRP with rate by (48), the product invariant measuresμ ϕ have single site marginals given by The local convergence in (144) was improved in [2] in (strong) convergence in the total variation norm: for ρ > ρ c lim where x L is the last site of linearly ordered Λ, µ Λ\{xL} ϕc * δ N − Λ\{x L } η is a probability measure on N Λ with marginal on N Λ\{xL} given by µ Λ\{xL} ϕc and one site marginal on {x L } given by the Dirac measure at N − x∈Λ\{xL} η(x), and σ x,xL is a permutation of configurations given by If moreover b > 2, (147) implies a stable limit theorem for M L (η) = max x∈Λ η(x) (see [2]).
For the second set-up for condensation, consider the case b ∈ (1, 2] for the measureμ ϕ defined by (146). As mentioned in [17], Z ϕ < ∞ for every ϕ ∈ [0, 1], that is, ϕ c = 1, and there is no finite critical density but Quoting from [17], "a typical configuration for this stationary distribution has a hierarchical structure which can be understood as a precursor for a condensation phenomenon". It is proved in [13] that for a fixed number of sites L ≥ 2, the probability measureŝ on N {1,...,L} , applying an operator on them which orders the coordinates of a configuration and removes the last one (the maximal one). It means that all but finitely many particles accumulate at one site.
6.2. Back to MMP. As indicated at the beginning of this section, let us give some examples of MMPs exhibiting condensation. For instance, consider again the generic model for condensation used firstly in [9] and later in [17] and [2], that is, ZRP with rate (145) and invariant measuresμ ϕ given by their single site marginals (146). Then many possible MMPs have this stationary state and hence they may exhibit condensation.
• If we choose a MM-ZRP, by (57) in Proposition 3.10, it has rates for k ≥ 1, α ≥ 0 where c is an arbitrary positive function on N \ {0} and µ is the special case of (146) for ϕ = 1. We study this case in details later on, as Example 4 in Section 7.
• If we choose a MM-TP, we obtain from Proposition 3.16(a), using that the ratio µ(n + 1)/µ(n) is nondecreasing, a recursive formula for its rates for every α ≥ 1, where c * is an arbitrary positive function on N \ {0} and the function H is given by formulas (64)-(65).
For both these MM-ZRP and MM-TP, the results (144), (147), (148) hold, that is, they may exhibit condensation in both set-ups.
Remark 6.1. We observe a duality between MM-ZRP and MM-TP having rates g k α and g k * ,β which lead to the same family of product, translation-invariant, and invariant measures. But it is no longer a simple formula as it was the case for the duality between ZRP and TP explicited in Remark 3.9. Combining (149) and (64) we obtain that By (150), we then obtain a formula for g α * ,β which is a function of (g k α+k , g k α+k−1 , g l k : 6.3. Condensation vs. attractiveness. An intriguing question is whether an MMP could be at the same time attractive and exhibit condensation. The answer is 'no' in the two following cases. We conclude that an attractive ZRP or an attractive MP has r(n) nondecreasing and so Z ϕc = +∞ by usual criteria for convergence of series. Thus they cannot exhibit condensation. (b) Assume a MM-TP has an invariant product measure with marginals µ(·). If the rates g k * ,β satisfy the conditions (127a)-(127b) for attractiveness then r(n) = µ(n)/µ(n+1) is a nondecreasing function and therefore Z ϕc = +∞. Indeed, from (127a)-(127b) we obtain in particular that g α * ,β − g α * ,0 ≤ 0 for all α, β ≥ 1. Combining this with (110) gives the result.
Let us consider the second set-up for condensation. For the latter it is proven in [23] that: "All spatially homogeneous processes with stationary product measures that exhibit condensation with a finite critical density are necessarily not attractive". However, processes with an infinite critical density could be also attractive. In [23] an example is numerically studied to support this assertion. In Section 7 below, we prove on a particular example, that this coexistence is indeed possible: where π is a positive function on N with π(0) ≥ (1 + b)π(1) is attractive and exhibits condensation in the second set-up for b = 3/2.

Examples
In this section, we present some examples of MMPs on Z d , for which we check existence conditions, whether they have product invariant probability measures, whether they are attractive, and whether they exhibit condensation.
For each dynamics, we thus verify the following conditions: (i) sufficient condition (13) for existence; The following condition is equivalent to (13): there exists C > 0 such that It means that there exists C > 0 such that (i)' sufficient condition (9) (which implies (12)) and condition (17b); (ii) necessary and sufficient conditions for attractiveness, by Lemma 5.1, (124a)-(127b); (iii) necessary and sufficient conditions for the existence of product, translation-invariant, invariant measures, by the appropriate result in Section 3. If there are such product invariant measures, we compute their single site marginals µ ϕ , and ϕ c , ρ c . We also check condensation properties.

Particular cases:
• The process with h(k) = 1/k is attractive; • the process with h(k) = k for k < k 0 , h(k) = 1/k for k ≥ k 0 , is not attractive; and both satisfy the existence condition (13).
• The totally asymmetric q-Hahn ZRP is one of the models studied in [3]; there p(1) = 1, with 1). The totally asymmetric q-Hahn ZRP is attractive, and condition (13) is satisfied.
(i) Condition (13) is satisfied if there exists C > 0 such that for α ≥ 0, (i)' Conditions (9) and (17b) are satisfied if there exists C > 0 such that for α ≥ 0, αR(α) ≤ C. Note that if R is a constant function, both conditions (13), (9) and (17b) fail. It is the case for the "stick process" introduced in [25] and also studied in [14]; for this example, another construction was provided in [25], on a state space smaller than X.
(iii) Unless R is constant (that is for the stick process), Condition (60) is never satisfied (take k = α − 1 ≥ 1 in (60)), so there are no product translation-invariant and invariant probability measures.
Example 3. Generic MM-ZRP with product invariant measures: we consider the rates (134) for which we began to investigate attractiveness in Section 5.2, that is where π is a positive function on N and h is a non-negative function on N \ {0}. For technical reasons we define π(i) = 0 for i ∈ Z − and h(0) = 0.
(i) Condition (13) is satisfied if there exists C > 0 such that for α ≥ 0, (i)' Condition (9) writes: There exists C > 0 such that for every α ≥ 1, Condition (17b) writes: there exists C > 0 such that for every α ≥ 1, (ii) In Section 5.2 we proved that the process is attractive if and only if Condition (135) is satisfied. In Lemma 5.6 we simplified this condition when the function r(n) = π(n)/π(n + 1) (see (136)) is monotone.
(iii) The rate (156) is the generic one to satisfy assumption (57) hence we have a one-parameter family of product and translation-invariant measures and Rad(Z ′ ), Rad(Z) are given by (32) and (31). These measures do not depend on h.
Properties of the ratio r(n) yield the existence of a critical value ϕ c such that Z ϕc < ∞ or moreover ρ c < ∞. If ϕ c ∈ Rad(Z) \Rad(Z ′ ) then the measureμ ϕc is well defined by (160) but its first moment is infinite. If assumption (30) is satisfied, thenμ ϕc is invariant.
(i)' In the case h(k) = π(k), condition (9) is satisfied if and only if for some C > 0 Indeed, (9) writes (158), and if we use that the term π(k)π(α−k) is symmetric we obtain (167), which, by (162), is equivalent to: for some C > 0, which, using Stirling approximation (163), is equivalent to for some C > 0. We have that Lemma 7.3. For every b > 0 there exists C > 0 such that (168) is satisfied.
Let us now check condition (17b), which writes here (159). With the choice h(k) = π(k), it becomes (167). If it is satisfied, it will also be the case for (17b) with the choice h(k) = π(k)/k. is decreasing. In the case h(k) = π(k), Lemma 5.6(a') gives a necessary and sufficient condition for the process to be attractive, namely (138). By (142) and (162), for α ≥ 2 we have, writing from now on S b (α) instead of S(α) to stress the dependence in b, Proposition 7.4. The process is attractive for b = 1 and b = 3/2 and not attractive for all values of b larger or equal to 2.
Proof. First, for all α ≥ 2 and all b > 0, using that Γ(z + 1) = zΓ(z) for all z > 0, we compute the increment • For b = 1, the expression (170) becomes Therefore the inequality in (138) is satisfied for all α ≥ 2, and the process is attractive.
In this example, we chose π such that the process with b > 2 exhibits condensation in the first set-up whereas the process with b > 1 exhibits condensation in the second set-up (that is, on a fixed finite volume), which follows from results in [13,17] (see Section 6). We proved coexistence of attractiveness and condensation in the second set-up for b = 3/2 (cf. Proposition 6.3).
(iii) If conditions (62)-(63) or conditions (66) on rates are satisfied then there exist product invariant probability measures given by formulas (71)-(72). The MM-TP dual of Example 4 has rates (150) (see Remark 6.1). Contrary to duality for singlejump dynamics which conserves attractiveness (see Remark 5.2), we have here a non-attractive dynamics (by Proposition 6.2) dual of a one which can be attractive (by Proposition 7.4).

Appendix: Construction of the process
In this section we shall prove Proposition 2.1, Theorem 2.2, and Proposition 2.3. For our construction, Consider an approximation of the infinite set X by an increasing sequence of finite sets and let us define (as in [18]) a restriction of the transition probabilities (p(x, y), x, y ∈ X) on X n by p(x, x) + z ∈Xn p(x, z) x = y, x ∈ X n 1 x = y, x ∈ X n p(x, y) x = y, x, y ∈ X n 0 otherwise.
The transition probabilities (p n (x, y), x, y ∈ X n ) satisfy (5) with M replaced by M + 1, because p n (x, y) ≤ p(x, y) + I [x=y] . They induce a restriction of the dynamics of MMP to X n , whose generator L n is given by (10) with p(·, ·) replaced by p n (·, ·): for f : Since particles do not move outside X n , this corresponds to a Markov process on the countable state space N Xn . Hence there is a well defined Markov semigroup of operators (S n (t), t ≥ 0) on functions f ∈ L, associated to L n : Similarly, the dynamics of the coupled process restricted to X n has generator L n given by (121) with p(·, ·) replaced by p n (·, ·): for (η, ζ) ∈ X × X, h ∈ L, G k,l η(x),η(y),ζ(x),ζ(y) h S k x,y η, S l x,y ζ − h (η, ζ) .
Our goal is to define, in Subsection 8.3, limits of the generators L n and semigroups S n (t), as n → ∞. For this we will as a first step, in Subsection 8.2, analyze the restriction of (η t ) t≥0 to the finite set of sites X n , keeping by a misuse of language the same notations L n , S n (t), (η t ) t≥0 , L n for the dynamics of state space N Xn . Subsection 8.1 is devoted to preliminary results.
8.1. Preliminaries. In this subsection we explicit some statements done in Section 2, and derive a key inequality about the coupling rates.
• Note that the set L includes all bounded cylinder functions. More precisely let f on N X be such a function. It is bounded by a value m f , and its support is a finite set V f ⊂ X such that f depends only on values (η(x) : x ∈ V f ) and not on the whole η ∈ N X . We can write • Next Lemma provides the analogue for the coupled process of (13) for the MMP, and is crucial for the following results. i (g i γ,δ −g i α,β ) +I [α>γ] α∨γ i=(α∧γ)+1 i (g i α,β −g i γ,δ ).
We have therefore proved thatμ is invariant if and only if Lf dμ = 0 for every f ∈ L. (208) • Now fix a bounded f ∈ L and assume that Lg dμ = 0 for every bounded cylinder function g. Given the sequence (178) of finite sets, one can define a sequence of bounded cylinder functions that satisfy lim n→∞ f n (η) = f (η) for every η. Using the same arguments as before we obtain that Lf (η)dμ(η) = lim n→∞ Lf n (η)dμ(η) = lim n→∞ Lf n (η)dμ(η) = 0.
Hence the proposition is proved.