Consistent particle systems and duality statistical

We consider consistent particle systems, which include independent random walkers, the symmetric exclusion and inclusion processes, as well as the dual of the Kipnis-Marchioro-Presutti model. Consistent systems are such that the distribution obtained by ﬁrst evolving n particles and then removing a particle at random is the same as the one given by a random removal of a particle at the initial time followed by evolution of the remaining n − 1 particles. In this paper we discuss two main results. Firstly, we show that, for reversible systems, the property of consistency is equivalent to self-duality, thus obtaining a novel probabilistic interpretation of the self-duality property. Secondly, we show that consistent particle systems satisfy a set of recursive equations. This recursions implies that factorial moments of a system with n particles are linked to those of a system with n − 1 particles, thus providing substantial information to study the dynamics. In particular, for a consistent system with absorption, the particle absorption probabilities satisfy universal recurrence relations. Since particle systems with absorption are often dual to boundary-driven non-equilibrium systems, the consistency property implies recurrence relations for expectations of correlations in non-equilibrium steady states. We illustrate these relations with several examples.


Introduction
Non-equilibrium systems, i.e. systems carrying a current, such as systems coupled to reservoirs or bulk-driven systems, are of great interest in non-equilibrium statistical Consistent particle systems and duality [8].
A second line of research of this paper is concerned with exploiting the consistency property to obtain detailed information about the dynamics, especially in the context of non-equilibrium systems, i.e., in the presence of boundary reservoirs where particles can enter and leave the system. Indeed, such boundary driven systems are dual to absorbing systems, where the reservoirs are replaced by absorbing boundaries. We show that the addition of absorbing sites does not change the consistency property, and as a consequence we obtain a set of recursive equations from the consistency property of the absorbing dual. These recursions imply that factorial moments of a system with n particles are linked to those of a system with n−1 particles. Although the set of recursive equations is not enough to fully solve the dynamics, it yields substantial simplifications. We apply these recursion relations to show universal properties of non-equilibrium steady states of boundary-driven systems, such as (generalized) exclusion and inclusion processes coupled to boundary reservoirs.
To conclude this introduction, we summarize the main results of the paper. a) We characterize particle systems which have the consistency property. We show the relation between consistency and duality, thus providing a probabilistic interpretation of duality.
b) We show that consistency is conserved upon adding absorbing sites, which provides consistency for duals of systems with boundary reservoirs. c) We show that consistency of absorbing systems leads to universal recurrence relations for the factorial moments of the particle occupation numbers. Via duality this translates into universal recurrence relations for the factorial moments in the non-equilibrium steady state.
The rest of our paper is organized as follows. In section 2 we give the basic definitions yielding the distinction between the coordinate process, which specifies the positions of all the particles, and the configuration process, which instead provides the number of particles at each site. We also recall the definition of the annihilation operator and its interpretation as a particle removal operator. In section 3 we introduce the notion of consistent permutation-invariant particle systems and show that this property is equivalent to a commutation property between the generator of the configuration process and the annihilation operator. We also write the set of recursion relation implied by consistency in the general setting. In section 4, we show the relation between consistency and (self-)duality. In section 5 we show that consistency is preserved by adding sites where particles can be absorbed (independently for different particles) and specify the recursion relations in terms of the absorption probabilities. Sections 6 and 7 are dedicated to a special class of consistent particle systems, including the case of an integrable systems (the open symmetric exclusion process that is solved by matrix product ansatz) and a non-integrable case (the symmetric inclusion process). We show in both cases the form taken by consistency equations.

Preliminary definitions 2.1 Coordinate process and configuration process
We consider a system of n particles moving on a countable set of vertices V , with cardinality |V |. Their positions are denoted by (x 1 , . . . , x n ) ∈ V n . The set of functions g : V n → R is denoted by C n . A configuration of n particles is the n-tuple of their positions modulo permutations of labels. In the following configurations will be denoted by η = (η x ) x∈V , where η x will be an element of Λ ⊆ N to be interpreted as the number of EJP 26 (2021), paper 125. Consistent particle systems and duality particles at vertex x. For x = (x 1 , . . . , x n ) ∈ V n the associated configuration is denoted by ϕ(x) := n i=1 δ xi (2.1) where δ z is the configuration having only one particle located at z ∈ V , i.e. for y ∈ V , (δ z ) y = 1 if y = z, 0 otherwise.
We view ϕ as a map from n-tuples with arbitrary n to configurations, i.e., ϕ : ∪ ∞ n=1 V n → N V . We then define Ω n as the set of configurations of n particles, i.e. Ω n = η ∈ Λ V : |η| = n , with |η| := x∈V η x where Λ ⊆ N is the single-site state space, namely the set of possible occupation numbers for each site. We denote by E n the set of functions f : Ω n → R and define Ω = {η ∈ Λ V : |η| < ∞} the set of finite particle configurations, namely Ω = ∪ n∈N Ω n .
In this paper, we shall consider both examples with a finite state space, such as the partial exclusion processes [11,14] (with a restriction on the number of particles per site, i.e. Λ = {0, . . . , α} where α ∈ N denotes the maximal number of particles per site), as well as examples with Λ = N, such as the inclusion process [8] or the independent random walk process (see Section 6 for a treatment of these processes).
In some cases not all elements of V n give rise to allowed configurations of Ω. For instance, for the partial exclusion process, it is not guaranteed that x ∈ V n does not contain more than α particles at any site, i.e. ϕ(x) ∈ {0, . . . , α} V . For this reason we define the set V n of n-tuples x ∈ V n such that the associated configuration ϕ(x) is an element of Ω n . For the examples of independent random walkers and inclusion process V n = V n , whereas, in the case of the partial exclusion process, V n = V n .
With these preliminaries we next specify the distinction between a coordinate process and a configuration process. Definition 2.1 (Coordinate process). We shall call a coordinate process with n particles, denoted by {X (n) (t), t ≥ 0}, the stochastic process taking values in V n that describes the positions of particles in the course of time. Namely, for i = 1, . . . , n, the random variable X (n) i (t) denotes the position of the i th particle at time t ≥ 0. We denote by {X(t), t ≥ 0} a family of coordinate-processes {X (n) (t), t ≥ 0}, n ∈ N , labeled by the number of particles n ∈ N.
Throughout this paper we shall restrict to coordinate processes that are Markov processes.

Definition 2.2 (Configuration process).
We shall call a configuration process, denoted by {η(t), t ≥ 0}, the stochastic process, taking values in Ω that describes the joint occupancy numbers of all the sites in the course of time. More precisely, for i ∈ V , the random variable η i (t) denotes the number of particles at site i at time t ≥ 0.
Throughout this paper we shall restrict to configuration processes that conserve the number of particles, i.e. if the process {η(t), t ≥ 0} is started from η ∈ Ω n then η(t) ∈ Ω n for all later times t > 0. A configuration process is naturally induced by a coordinate process using the map ϕ defined in (2.1). There can be several coordinate processes whose image under the map ϕ yields the same configuration process. This leads us to the following definition.
Of course it is not guaranteed that, starting from a Markov coordinate process, the mapping ϕ defined in (2.1) induces a compatible configuration process that is also Markov. To further discuss this point we need to introduce the notion of permutation invariance. We denote by Σ n the set of permutations of n elements. Moreover we define the operator U ϕ mapping functions f :

Definition 2.4 (Permutation invariance).
a) A family of coordinate Markov processes {X(t), t ≥ 0} is said to be permutationinvariant if, for every n ∈ N and permutation σ ∈ Σ n , the processes: Equivalently, a function g ∈ C n is permutation-invariant if there exists a function f : Ω n → R such that g = U ϕ f . c) A probability measure µ n on V n is called permutation-invariant if, for all n ∈ N and σ ∈ Σ n , under µ n , the random vectors (X 1 , . . . , X n ) and (X σ(1) , . . . , X σ(n) ) have the same distribution.
We denote by L n the infinitesimal generator of the n-particle coordinate process {X (n) (t), t ≥ 0}, and by S n (t) the related semigroup, i.e., for g ∈ C n S n (t)g(x) : where E x denotes expectation when the coordinate process is started from x ∈ V n .
The following lemma shows that to permutation-invariant coordinate processes one can naturally associate a compatible configuration process enjoying the Markov property. Lemma 2.5. Let {X(t), t ≥ 0} be a family of permutation-invariant coordinate Markov processes with generators L n , n ∈ N. Define the operator L acting on functions f : Ω → R as for all η ∈ Ω n and x ∈ V n : ϕ(x) = η (2.2) or, equivalently, Then L is the infinitesimal generator of a Markov process {η(t), t ≥ 0} that is a configuration process compatible with {X(t), t ≥ 0}.
This means that L n U ϕ f is permutation-invariant, hence there exists a functionf : Ω → R such that Then it is possible to define the operator L acting on functions f : Ω → R such that L f =f , and then (2.3) is satisfied. From (2.6) we have that and then also S n (t)U ϕ f is a permutation-invariant function at all times. Now, if we denote by S (t) the semigroup associated to L , it follows that From this it follows that L is the generator of a Markov process that is a configuration process compatible with {X(t), t ≥ 0}.

Annihilation operator
We continue by introducing the operators that remove particles either in the coordinate process (see (2.10) below) or in the configuration process (see (2.13)). Definition 2.6 (Particle removal operators). For n ∈ N, 1 ≤ i ≤ n we denote by π (n) i : C n−1 → C n the removal operator of the i th labeled particle, acting on functions g ∈ C n−1 as follows: and we denote by Π (n) : C n−1 → C n the operator acting on g ∈ C n−1 via We define the "single-site annihilation operator" a acting on functions f : and the "annihilation operator" working on functions f : Ω → R as i.e. a x denotes the operator a working on the variable η x .
We remind the reader here that we are restricting to configurations η ∈ Ω, i.e. with a finite number of particles and therefore the sum in (2.13) is a finite sum. The annihilation operator is crucially important in the explanation of self-dualities for several particle systems [6,7]. In particular, as it will be shown later, the fact that the generator L of the configuration process and the annihilation operator A commute, i.e. [L , A ] = 0, is enough to obtain a self-duality when the process has a reversible measure. Such a commutation relation for the configuration process is equivalent to an intertwining relation between the coordinate process with n particles and the coordinate process with n − 1 particles. This equivalence is the object of the next theorem. Theorem 2.7. Let {X(t), t ≥ 0} be a family of coordinate Markov processes with generators L n , n ∈ N and {η(t), t ≥ 0} a compatible Markov configuration process with generator L defined in (2.2). Then the following statements are equivalent: a) The generators of the coordinate process with n and n − 1 particles restricted to permutation invariant functions are intertwined via Π (n) , i.e., for every n ∈ N, and for all g ∈ C n−1 permutation-invariant (L n Π (n) )(g) = (Π (n) L n−1 )(g) (2.14) b) The generator of the configuration process commutes with the total annihilation operator, i.e.
[L , A ] = 0 This means that for all η ∈ Ω n and x ∈ V n : ϕ(x) = η As a consequence: where the last step follows because every x ∈ V is counted exactly η x times in the sum n l=1 f (η − δ x l ). This proves (2.16). Suppose now that [L , A ] = 0, then on E n we have where the equalities follow from (2.16), (2.3) and the commutation relation. Then (2.14) follows since, for all g ∈ C n permutation-invariant, g = U ϕ f for some f ∈ E n . The reverse implication is proved analogously.
Remark 2.8. The probabilistic interpretation of (2.14) is as follows: if we remove a randomly chosen particle, evolve the process, evaluate a permutation-invariant function at time t > 0 and finally take expectation, then we can as well first evolve the process, remove a randomly chosen particle at time t > 0, evaluate the same permutationinvariant function and take expectation. In other words, the operations "removing a randomly chosen particle" and "time evolution in the process followed by expectation" commute as long as we restrict to permutation-invariant functions. See also Proposition 3.6.

Consistency
In this section we first define the consistency property and give some characterization of consistent particle systems. We then discuss the implication of consistency in the form a set of recursive equations.

Definition of consistency and relation to annihilation operator
Based on Theorem 2.7 we define consistent particle systems as follows.  c) A family of probability measures µ n on V n , indexed by n ∈ N, is called consistent if for all n ≥ 2, under µ n , the distribution of (x 1 , . . . , x n−1 ) equals µ n−1 .
Let {µ n , n ∈ N} be a consistent collection of probability measures on V n , n ∈ N. If additionally {µ n , n ∈ N} is permutation-invariant, then every m-dimensional marginal of µ n , m ≤ n, coincides with µ m . A simple example of such a consistent permutationinvariant family is for some x ∈ V n Analogously, if {X(t), t ≥ 0} is a family of permutation-invariant coordinate processes, consistency implies that, for all n ∈ N, any m- In the following theorem we show that, if a configuration process is consistent then a compatible coordinate process is also consistent provided that its initial distribution at time t = 0 is consistent and permutation invariant.
Theorem 3.2. Let {X(t), t ≥ 0} be a family of coordinate Markov processes with generators L n , n ∈ N, and let {η(t), t ≥ 0} be a compatible configuration process.
Assume that: ii) the probability measures {µ n , n ∈ N} on V n , n ∈ N, form a consistent family which is also permutation-invariant.
Then we have consistency of the family of coordinate processes starting from {µ n , n ∈ N}, i.e., for all n ∈ N, g ∈ C n−1 permutation-invariant, where E n µn denotes expectation w.r.t. the Markov process {X (n) (t), t ≥ 0}, started initially with distribution µ n .

PROOF.
Let g ∈ C n−1 be a permutation-invariant function, then we have π (n) l g = π (n) k g for k, l ∈ {1, . . . , n}. Then, by consistency and permutation invariance of µ n , we have for all n ∈ N and all l ∈ {1, . . . , n}. Therefore, using (2.14), we have This concludes the proof.

Characterization of consistent configuration processes
In this section we provide a class of configuration processes exhibiting the consistency property. We restrict ourselves to models where only one particle jumps at a time. We further assume the jump rate to depend only on the number of particles hosted by the departure and arrival sites. We consider here generators of the form where the summation is over the set E of (non-oriented) edges {i, j} of the complete graph with vertices V and with η i,j := η − δ i + δ j and c i,j : Λ × Λ → R + being the hopping rate for a particle to jump from site i to site j.
We then call L i,j the single-edge generator corresponding to the (non-oriented) edge {i, j}. We will characterize the processes such that the annihiliation operator commutes with each single edge generator. These processes are then automatically consistent.
for some θ : E → R and α : V 2 → R + such that θ({i, j}) · m + α(i, j) ≥ 0 for all possible values of (κ, m) ∈ Λ × Λ. As a consequence the corresponding configuration process with rates (3.6) is consistent. Remark 3.4. By requiring that the r.h.s. of (3.6) are rates we have implicitly assumed they are non-negative. This implies both restrictions on the functions θ and α and on the set Λ, i.e the allowed values of the occupations (η i ) i∈V .

PROOF.
We have have to characterize the generators L such that for all i, j ∈ V Recalling that A = x∈V a x , using the assumption that the rates only depend on the number of particles in the departure and arrival sites (3.5), we have that Therefore we have that (3.7) is satified if and only if for all i, j We thus impose for all functions f : Λ × Λ → R of two variables. One one hand we have that on the other hand Imposing (3.10) for all f is equivalent to imposing the following conditions for all m, κ ∈ Λ: and EJP 26 (2021), paper 125.
Since the conditions (3.18) must be satisfied for all values of κ and m, the only possibility is that there exists a constant θ = θ({i, j}) such that Hence, iterating both for b i,j and for b j,i we deduce that there exist two constants, α i,j and α j,i such that that concludes the proof.
Remark 3.5. The class of models with rate of the form (3.6) contains some classical models such as symmetric partial exclusion process, independent random walkers and symmetric inclusion process, see section 6 and 7 below. Compared to the class of models with factorized self-duality, described in [13], the class of consistent processes introduced in Theorem 3.3 is larger and in particular contains models which do not have product invariant measures (namely when α(i, j) = α(j, i)).

Recursion relations
In this section we analyze the consequences of consistency. More precisely we prove recursive relations that give the transition probabilities of a system with m particles in terms of the transition probabilities for the system with m − 1 particles. As a consequence we obtain recursive relations for time dependent factorial moments of consistent configuration processes. Specifically, in Theorem 3.8 below, we show that time dependent factorial moments of order m in a system with n > m particles can be expressed in terms of m-particle transition probabilities. These recursion relations will be particularly useful when we deal with systems having absorbing sites (cf. sections 5.1 and 5.2). In the reversible setting they are equivalent with self-duality (cf. section 4).
To prove Theorem 3.8, we state two preparatory propositions.
Proposition 3.6. Let {η(t), t ≥ 0} a consistent configuration process on a finite lattice V , then, for all f :

PROOF.
From the commutation of the generator of the configuration process L with the annihilation operator A , we obtain the commutation of the semigroup of the configuration process S (t) = e tL with A . We then remark that (3.23) can be written this concludes the proof.
We define now the function and prove a recursive relation for its expectation.
Proposition 3.7. Let {η(t), t ≥ 0} a consistent configuration process on a finite lattice V then, for all η, ξ ∈ Ω such that 1 ≤ |ξ| < |η|, we have The right-hand side of (3.26) is equal to The function F (ξ, η) defined in (3.24) has a precise combinatorial meaning that is useful to understand with the help of the coordinate notation. Let us first give some preliminary notations. We denote by [n] the set of the first n natural numbers: [n] := {1, . . . , n}, for n ∈ N. Then, for m ≤ n we define the set C m,n of combinations of m elements chosen in [n]: i.e. the vector of the positions of the particles with labels in I. We define, moreover the following ordering between elements of the coordinates state spaces. For x, y ∈ ∪ n∈N V n , we say that y ≤ x if and only if |y| ≤ |x| and ∃ I ∈ C |y|,|x| s.t. y = x I . for any fixed particles labelling of the configuration η, i.e. for any x ∈ V |η| such that ϕ(x) = η, we can write In other words, for any fixed labelling x of particles in the configuration η, it is the number of ways to select |ξ| particles out of |η| in such a way that the corresponding configuration in Ω is ξ. In view of the ordering (3.30), we can also rewrite (3.31) as follows: This suggests that the term F disappears when switching from the configurations to coordinate variables in the summations of the following type: We denote by J = (j 1 , . . . , j κ ) an element of the set C κ,n defined in (3.28), hence, iterating the argument used in (3.36) κ times we get: The theorem follows from the fact that now, for I ∈ C m,n , |ϕ(x I )| = m = |ξ| and from the observation that . This concludes the proof. give the full distribution of the system with n particles. This is due to the fact that the factorial-moments of order n are still missing.
We close this section with a specialization of Theorem 3.8 that will be useful later.
Remark 3.11. Notice that the random walk X rw is only associated to the configuration process started with a single particle at time 0. Later on, in section 5 below, we will then consider independent copies of these walks and a configuration process {η irw (t); t ≥ 0} associated to them via the map ϕ.
where P u , is the path space measure of the random walk {X rw (t), t ≥ 0} on V starting from u ∈ V associated to the configuration process {η(t), t ≥ 0} as defined above in Definition 3.10.
PROOF. The result immediately follows by applying Theorem 3.8 to the case ξ = δ v .

Consistency and self-duality
In this section we show that consistency implies a form of self-duality whenever a process admits a strictly-positive reversible measure. We start by recalling the definition of duality (we refer to [11] for more background on duality, see also [1,10]).
be two Markov processes with state spaces Ω and Ω and D : for all y ∈ Ω, y ∈ Ω and t > 0. In (4.1) E y is the expectation with respect to the law of the process {Y t } t≥0 started at y, while E y denotes expectation with respect to the law of the process { Y t } t≥0 initialized at y.
We say that a process is self-dual when the dual process coincides with the original process.
It is useful to express the duality property in terms of generators of the two processes.
If L denotes the generator of {Y t } t≥0 and L denotes the generator of { Y t } t≥0 , then (assuming that the duality functions are in the domain of the generators), the above definition is equivalent to LD(·, y)(y) = LD(y, ·)( y) where L acts on the first variable and L acts on the second variable. The conditions to have the equality between semigroup duality and generator duality are further discussed in [10].
In order to prove our theorem on the relation between consistency and self-duality we recall two general results on self-duality from [7]. a) Trivial duality function from a reversible measure.
If a Markov process {Y t : t ≥ 0} with countable state-space Ω has a strictly-positive reversible measure ν, then the function D : Ω × Ω → R given by D(y, y) = δ y, y ν(y)   We then obtain the following result.
is a self-duality function. If we now act with e A on D rev in the η-variable we produce (4.5) via Lemma 4.2. This produces a new self-duality function because, by assumption, A is a symmetry of L . Now we prove that reversibility and self-duality with duality function (4.5) implies consistency. Fix η ∈ Ω n and let ξ = η − δ x for some x ∈ V , then ξ ∈ Ω n−1 . Then, Let L n be the operators implicitly defined by (2.3). Because D is a self-duality function and ν is a strictly-positive function on Ω, satisfying detailed balance w.r.t. L , the operator K : intertwines between L n and L n−1 , i.e., K(L n−1 f ) = L n Kf for f ∈ E n−1 (see [9] for the connection between duality functions and corresponding intertwining kernel operators). Now, via (4.6) we obtain and conclude that A (L n−1 f ) = L n A f for f ∈ E n−1 , which is exactly the commutation property [L , A ] = 0. The process is thus consistent.

Consistency for systems with absorbing sites
In this section we study consistency for systems of interacting particles with absorbing sites. These systems are important as they emerge as dual processes of boundary-driven non-equilibrium systems connected to reservoirs. In Section 7 we will analyze the consequences of consistency in the context of non-equilibrium systems.
Let {η(t), t ≥ 0} denote a configuration process with generator L . Let V * ⊃ V be a countable set of sites containing V , and call V abs := V * \V . We will define a configuration process on N V * as follows: inside V particles move according to the generator L , and additionally every particle at site i moves with rate r(i, j) to a site j ∈ V abs , independently from each other. Particles arriving at sites j ∈ V abs are absorbed and, after absorption, do not move anymore. We thus obtain what we call the process with absorbing sites V abs and absorption rates r(i, j). The generator of this process is given by where L only works on the variables {η i , i ∈ V } and where H denotes the absorption part of the generator, i.e., We can rewrite H as follows H = i∈V,j∈V abs where a i is the annihilation operator at site i defined in (2.13) and a † i is the creation operator defined via a † i f (η) = f (η + δ i ). A process with generator (5.1) is called an absorbing extension of the generator L .
EJP 26 (2021), paper 125. Definition 5.1. Let {X(t), t ≥ 0} be a family of coordinate Markov processes on the lattice V . Then we define its absorbing extension {X abs (t), t ≥ 0} to the lattice V * as the family of coordinate Markov processes {X abs,(n) (t), t ≥ 0}, n ∈ N, on (V * ) n defined by adding to the jumps of {X (n) (t), t ≥ 0}, n ∈ N, the additional jumps from i ∈ V to j ∈ V abs at rate r(i, j), that particles perform independently from each other.
Define the set Ω abs of configurations on absorbing sites: We have the following lemma.
Lemma 5.2. Let {X(t), t ≥ 0} be a family of coordinate Markov processes on the lattice V compatible with the configuration process on Ω having generator L . Then its absorbing extension {X abs (t), t ≥ 0} to the lattice V * is compatible with the configuration process on Ω * := Ω × Ω abs with generator L abs in (5.1).

PROOF.
It follows immediately from the definition of absorbing extensions. We then have the following results. Then its absorbing extension {X abs (t), t ≥ 0} to V * ⊃ V is consistent if started from a consistent family {µ n , n ∈ N} of probability measures on (V * ) n , n ∈ N, which is also permutation-invariant. Namely for all n ∈ N, g : (V * ) n → R permutation-invariant, where E n µn denotes expectation w.r.t. the Markov process {X abs,(n) (t), t ≥ 0}, started initially with distribution µ n .
PROOF. It follows from Lemma 5.2, Lemma 5.3 and Theorem 3.2.

Recursion relations for absorption probabilities
In what follows we denote by P η (η(∞) = ζ) the probability that eventually η(t) settles in the absorbing configuration ζ ∈ Ω abs starting from the initial configuration η ∈ Ω * , i.e.,   The relations (5.9) express combinations of absorption probabilities from the initial configuration ϕ(x) in terms of combinations of absorption probabilities from an initial configuration η with less particles. Although these equations are not sufficient to determine the absorption probabilities in closed form, they are still considerably simplifying the problem of computing them as they imply severe restrictions.
The following Corollary immediately follows by specializing (5.9) to the case ζ = δ v for some v ∈ V abs . Corollary 5.6. Let {η(t), t ≥ 0} be a consistent configuration process on a finite lattice V * with generator L abs = L + H . Let x = (x 1 , . . . , x n ) ∈ (V * ) n , then, for all v ∈ V abs we have where P u is the path-space measure of the random walk {X rw (t), t ≥ 0} on V * starting from u ∈ V * associated to the configuration process {η(t), t ≥ 0} as in Definition 3.10.

Systems with two absorbing states
We now consider the situation in which the system contains only two absorbing states, |V abs | = 2, say V = {1, . . . , N } and V abs = {0, N + 1}. In this case, due to the conservation of particle number, it is sufficient to know the absorption probability in one of the two states, say state 0. The following proposition is an immediate consequence of Theorem 5.5.
Remark 5.8. Notice that (5.11) can be viewed as a linear system of n − 1 independent equations in the n + 1 variables {P η (η 0 (∞) = m) , m = 0, . . . , n}. Indeed we consider inductively the r.h.s. of (5.11) as a "known quantity" because it concerns an absorption probability of less than n particles. For instance if n = 2 and m = 1 the r.h.s. of (5.11) contains only the absorption probability of one particle. Complementing these with the normalization condition n m=0 P η (η 0 (∞) = m) = 1 we obtain n independent equations. This is still not sufficient to get a closed form expression for the absorption probability (which represent n + 1 unknowns), since one independent equation is still missing.

Generating function method
Let {η(t), t ≥ 0} be a consistent process on the lattice V * = V ∪ V abs , V = {1, . . . , N }, V abs = {0, N + 1}, with generator L abs = L + H . We define the function The function G(η, ·) is the probability generating function of the number of absorbed particles at 0 starting from the configuration η. Here we define as usual G(η, 0) by continuous extension, i.e., which is equal to the probability that all the particles in η are eventually absorbed at N + 1. We then have the following recursion relation.
The recursion (5.14) can be iterated until we are left with one particle configurations for which, in ideal situations, the generating function can be computed, as we will see in the last section. It is crucial then to obtain a formula for the function G (η, z) := G(η, z) − G irw (η, z) (5.15) that is the difference between the generating function of our process {η(t), t ≥ 0} and the generating function G irw (η, z) of the auxiliary process {η irw (t), t ≥ 0} of independent random walkers defined as follows. Let {X rw (t), t ≥ 0} be the random walker on V * associated to the configuration process {η(t), t ≥ 0} as in Definition 3.10. Now let {X irw (t), t ≥ 0} be the family of coordinate processes {X irw,(n) (t), t ≥ 0}, n ≥ 1 on (V * ) n whose coordinates are n independent copies of X rw (t): X irw,(n) (t) = (X rw 1 (t), . . . , X rw n (t)), t ≥ 0.

PROOF.
We proceed by induction on n. We first consider the case n = 2. In this case because G(ζ, z) and G irw (ζ, z) coincide on configuration ζ with one particle. Therefore, and we obtain from Proposition 5.9, which is (5.16) for n = 2. Now we assume that (5.16) is true for n − 1 and prove the induction step. First of all we notice that we can rewrite where we used that for η = ϕ(x) we have η i = n j=1 δ xj ,i and we then exchanged the sums. Using (5.14) and the induction hypothesis we have that that concludes the proof.

A class of consistent processes
In this section we consider a natural class of consistent configuration processes {η(t), t ≥ 0}. These can be obtained, with a certain choice of the parameters, as particular cases of the more general class of processes produced in the characterization Theorem 3.3. These processes do not constitute the entire class of processes exibithing the consistency property. Nevertheless they are paradigmatic examples within this class, as they are well known in the literature. The processes we consider are of three types: partial exclusion processes, inclusion processes and independent random walkers. For the sake of synthesis we formally define a unique generator and let this be parametrized by a constant θ ∈ R tuning the attractive or repulsive nature of particle-interaction. The generator is given by acting on functions f : Ω θ → R, with Ω θ to be defined. Here V is a finite set, p : V ×V → R is a symmetric function: p(i, j) = p(j, i) and η i,j := η − δ i + δ j .
Remark that if the configuration contains only one particle then this particle moves according to a continuous-time random walk jumping from i to j at rate p(i, j), which is not depending on the interaction parameter θ. The parameter θ ∈ R can also be negative, under the condition that 1 |θ| is integer: θ ∈ R + ∪ {α < 0 : −1/α ∈ N} and, according to its sign we recover one of three cases: the partial symmetric exclusion-process SEP(1/|θ|), EJP 26 (2021), paper 125. the symmetric inclusion process SIP(1/θ), and the independent-random-walkers process IRW: Also the state space Ω θ where configurations η live changes according to the choice of θ, we have: These processes have been introduced in [7] and broadly studied due to their algebraic properties. In [7] it has been proved that [L , A ] = 0 where A is the annihilation operator defined in (2.13). Notice that it is possible to rewrite the generator as where now the summation is over the edges {i, j} ∈ E of the complete graph with vertices V and It is possible to verify that the commutation relation with the annihilation operator holds true at the level of each bond, namely [L θ,i,j , A ] = 0 for all i, j ∈ V .

Adding absorbing sites
Let now V * = V ∪ V abs ⊆ Z d with V abs a finite set of absorbing sites. Define L abs θ := L θ + H with L θ as in (6.1) and H as in (5.2): From Lemma 5.3 we know that [L abs θ , A abs ] = 0, and as a consequence for these processes we can apply Theorem 5.5 when studying their absorption probabilities.

Adding reservoirs
We now add to the bulk generator (6.1) additional terms describing the action of external reservoirs, each acting on one of the sites of a subset V ext ⊆ V . The generator has the following form: for some ρ, c : V ext → [0, +∞). Particles are injected and removed from the system through the sites i ∈ V ext (which we call the sites coupled to reservoirs) via additional birth-and-death processes. Here c(i) is the global rate at which the external reservoirs acts on the site i and ρ i is the density imposed by the reservoir on that site. This means that the birth-and-death process at site i ∈ V ext has ν ρi,θ as stationary measure, where we recall that the single-site measure ν ρ,θ was defined in (6.6). In particular if the densities of all reservoirs are equal, i.e. ρ i = ρ for all i ∈ V ext , then the process with generator (6.10) is reversible with respect to the product measure ⊗ i∈V ν ρ,θ , which is called the "equilibrium state" in statistical physics. In all other cases, we have that the system has a stationary state which is called the "non-equilibrium steady state", with non-trivial correlations as soon as θ = 0. In order to avoid uninteresting degenerate cases, we will always choose p(i, j) and c(i) in such a way that there exists a unique stationary measure, which we denote by µ st θ . The rates of the birth-and-death processes modeling the reservoirs preserve the duality property. Indeed, in [1] we proved that the process with generator (6.10) is dual to a system with absorbing sites, i.e. generated by (6.9). Here the set of absorbing site V abs is a "copy" of V ext , i.e., such that |V abs | = |V ext | and the rates r(·, ·) of the absorbing part of the generator (6.9) are given by: where * is a bijection * : V ext → V abs that assigns to each site i ∈ V ext a corresponding absorbing site i * ∈ V abs . From now on we will denote by {η(t), t ≥ 0} the process with reservoirs with generator (6.10) and state space Ω θ = Λ V θ (as in (6.3)) and by {ξ(t), t ≥ 0} the dual process with absorbing sites and with generator: (6.11) and state space Ω * θ = Λ V * θ , V * = V ∪ V abs . The duality function D θ : Ω * θ × Ω θ → R between the two processes is the following: with d as in (6.8) and where we extend the definition of density function ρ to the absorbing sites, ρ : V ext ∪ V abs , by identifying: ρ i = ρ i * . See [1] for a proof of the duality relation.
An important consequence of the duality property is the information that it gives about the non-equilibrium stationary measure µ st θ of the process generated by (6.10).
Indeed, the expectations of duality functions under µ st θ can be expressed in terms of the absorption probabilities of the dual process. The key relation is which is obtained from the duality relation by taking the limit as time goes to infinity, and using that asymptotically the dual process voids all the sites i ∈ V implying that D θ (ξ(∞), η) = 1. Notice that now, thanks to the consistency property of the dual process {ξ(t), t ≥ 0}, we can apply Theorem 5.5 to get information about the moments in (6.13).
In the next section we specialize to the case of two reservoirs (and then two dual absorbing sites) for which we can use the generating-function method developed in Section 5.2.
We remark that independently of the value of the parameter θ, if the dual configuration ξ = n i=1 δ xi where all x i are mutually different elements of V , then η xi (6.14) In particular, when the dual process is initialized with only one particle, then this particle will move as a symmetric random walk. As a consequence we have for all x ∈ V y∈V abs ρ y q(x, y) (6.15) where q(x, y) is the probability for the walker starting at x to be absorbed at y eventually: q(x, y) := P x (X rw (∞) = y) (6.16) In the case θ = 0 the process with generator (6.10) is made of independent walkers with reservoirs, and the dual process is made of independent walkers absorbed at the absorbing sites. In such a case one can show by duality that the non-equilibrium steady state is an inhomogeneous product of Poisson measures with parameters given by the local density (6.15). See [1] for a proof in the case with two reservoirs.

Instantaneous thermalization models
Another class of models sharing the consistency property is the class of processes obtained as the "instantaneous thermalization limit" of the processes defined in (6.1) which we now briefly recall. An important example in this class is the dual KMP model, see [1,12]. An instantaneous thermalization process gives rise, for each couple of nearest neighbouring sites, to an instantaneous redistribution of the total number of particles. For each bond, the total number of particles in that bond η i +η j is redistributed according to the stationary measure of the original process at equilibrium on that bond, conditioned to the conservation of η i + η j . The generators of the instantaneous thermalization processes can thus be defined by with L θ,i,j as in (6.5). Since we have that the commutation [L θ,i,j , A ] = 0 for all i, j ∈ V , it follows that also L th θ,i,j commutes with A and then the thermalized models (6.17) are also consistent, namely [L th θ , A ] = 0. A more explicit expression of the generator is given by: (6.6). The process (6.18) is self-dual with duality function (6.8) (see Section 5 of [1]).
Also for the instantaneous thermalization models it is possible to add absorbing boundaries in such a way to preserve the consistency property, and also in this case there is duality with the system with absorbing boundaries and a system with reservoirs (if the action of reservoirs is properly chosen, see [1]).
In the next section we use consistency to obtain an expression for the n-points stationary correlation function for the system with reservoirs (6.10) in a specific setting. This result can be easily extended to the thermalized models with reservoirs, since, as we have seen here, they share the same commutation property and as a consequence consistency property.

Correlation functions in non-equilibrium steady states
Here we consider processes in the class of models introduced in the previous section, and, as in Section 5.2, we restrict to the case where V = {1, . . . , N }, with V ext = {1, N } and V abs = {0, N + 1}. In the spirit of the previous section we assign to each site coupled to a reservoir an absorbing site, namely we say that 1 * = 0 and N * = N + 1. Then we denote by {η(t), t ≥ 0} on Ω θ the process with generator: for some c , c r , ρ , ρ r ≥ 0, and by {ξ(t), t ≥ 0} its dual process with state space Ω * θ and generator: with duality function given by Hence, the formula for the non-equilibrium stationary state (6.13) becomes: where G(ξ, ·) is the generating-function defined in Section 5.2, equation (5.12). The following theorem then follows by combining (7.4) with Theorem 5.10 and expresses the difference between the expectations of the duality functions in the non-equilibrium stationary measure and their non-interacting counterparts in terms of the probabilities of κ dual particles being all absorbed at one end.
The following corollary of Theorem 7.1 specializes to expectations of products of occupation numbers at different sites in the non-equilibrium steady state.

Remark 7.3.
Specializing the corollary to n = 2 we obtain information on the covariance. We see that the explicit dependence on the boundary densities ρ and ρ r that turns out to be a quadratic function of their difference: for x = y, The covariance is exactly quadratic in (ρ − ρ r ) with a multiplying factor (i.e. the difference of the two absorption probabilities above) not depending on ρ and ρ r . This multiplying factor is non-positive for exclusion particles, θ < 0 (by Liggett's inequality [11], chapter 8) and non-negative for inclusion particles, θ > 0 (by the analogue of Ligget's inequality from [8]).

Examples
In this section we consider three concrete examples where the non-equilibrium steady state expectations are known in closed form, and we explicitly verify the recursion (5.14).
We restrict to the situation where V = {1, . . . , N } is a a one-dimensional chain and particles jump only to nearest neigbors, and interact (in a symmetric way) only if sitting in neighboring sites. More precisely we choose the function p(·, ·) in (7.1) and (7.2) as p(i, j) = 1 j=i±1 (7.12) Moreover we choose the reservoirs clocks to have rates 1:
If we now consider the corresponding system with reservoirs: we have that the stationary measure µ st 0 is a non-homogeneous product measure with Poisson-distributed marginals: x as in (7.14).
Notice that in this case, since we can easily compute the generating function G irw (·, ·), it is possible to directly verify the recursion relation (5.13). For ξ ∈ Ω * we have EJP 26 (2021), paper 125.
and then (5.13) is satisfied. Since we explicitly know both the generating function G irw (·, ·) and the stationary measure (7.16), it is possible to verify a posteriory the duality relation (7.4). It is also possible to verify that, by iterating the recursion relation in its integrated form (5.14) one can recover the generating function G irw (ξ, z) starting from the knowledge of G irw (ξ, 0). Remark 7.4. As a further application of the recursion (5.13) in the same spirit, one can easily show by induction that if the probabilities for all the particles to be absorbed at zero factorize, i.e., if G(ξ, 0) = i G(δ i , 0) ξi for all ξ, then the generating function factorizes, i.e., G(ξ, z) = i G(δ i , z) ξi and as a consequence the system has a product invariant non-equilibrium stationary measure µ st 0 .

Interacting walkers (case θ = 0)
For the interacting case, we consider two special cases. The first example is the simple exclusion process. This is the only interacting model in the class for which there is a full knowledge of the n-points correlation functions. The second example is a special case of the inclusion process for which an exact formula is known for the two-point correlations [6]. For the interacting case, specialized to nearest neighbor jumps and rate 1 reservoirs, the process has generator L res θ f (η) = Simple exclusion process (case θ = −1) For exclusion process the matrix product ansatz gives an algebraic procedure to calculate all correlation functions. This provides a recursion relation for the correlation functions (formula (A.7) in [3]  1 ≤ x 1 < x 2 < . . . < x m ≤ N and denote by q (N) ξ (k) the absorption probabilities: q (N) ξ (k) := P ξ (ξ 0 (∞) = k), k ∈ {0, 1, . . . , m} Clearly, if m = 1, namely ξ = δ x = ϕ(x) for some x ∈ V we have Thus for the exclusion process the probability generating function for the number of particles absorbed at zero can be computed by iterating (7.22).