A driven tagged particle in asymmetric exclusion processes

We consider the asymmetric exclusion process with a driven tagged particle on Z which has different jump rates from other particles. When the non-tagged particles have non-nearest-neighbor jump rates , we show that the tagged particle can have a speed which has a different sign from the mean derived from its jump rates. We also show the existence of some non-trivial invariant measures for the environment process viewed from the tagged particle. Our arguments are based on coupling, martingale methods, and analyzing currents through ﬁxed bonds.

grows linearly in t when the mean z z · q(z) is positive, and p(.) is non-nearest-neighbor in d = 1 or general in d ≥ 2. This conjecture is partially verified when d ≥ 3 and p(.) is symmetric [19], and it remains open for most of the other cases. When q(.) is close to p(.), one can show the displacement D t grows linearly in t with a corresponding Einstein relation [19]. However, the speed of the tagged particle is unknown because there is no explicit formula for the invariant measures. For a mixing dynamical environment with a positive spectral gap, Komorowski and Olla [14] obtained a full expansion of invariant measures, and showed the explicit speed and the corresponding Einstein relation.
Another approach is to study the currents through a fixed bond in the one-dimensional asymmetric exclusion process (AEP) with coupling arguments. The current describes the average number of particles across a site, and it is a natural object to study especially when z z ·p(z) is non-zero. Liggett [16,17] computed the currents and limiting measure in AEP explicitly for a class of general initial measures by couplings. For a more general class of asymmetric conservative particle systems with a blockage, one can show a hydrodynamic limit result with a coupling argument different from Liggett's [2]. In these systems, the current across the blockage is a key quantity in the hydrodynamic limit because it describes the densities near the blockage. Although this second type of couplings is different from Liggett's [16,17], it is available in the one-dimensional AEP case. When jump rates p(.) satisfy certain monotonicity conditions, Ferrari, Lebowitz, and Speer [6] showed a coupling of two AEPs and applied this coupling to prove the existence of blocking measures. In the case of our model, when a driven tagged particle is present, we can also consider the current across a particular site, the (moving) tagged particle, and obtain estimates of currents by coupling different AEPs with a driven tagged particle. This article will consider the case where d = 1 and p(.) is non-nearest-neighbor and asymmetric with a positive mean z · p(z) > 0. The main tools are the couplings and martingale arguments. There are two types of couplings similar to those in [6,16,17]. These two types of couplings allow us to compare currents in different processes and obtain estimates of currents. With martingale arguments, we can relate estimates of currents to estimates of the displacement D t and some invariant measures. In the end, we will show that the displacement D t grows linearly in t in three scenarios (Theorems 2.1, 2.2, 2.3). These results suggest behavior of the tagged particle depends on jump rates p(.), q(.) and the initial measure in a nontrivial way. By characterizing some nontrivial invariant measure, we will show that the tagged particle can have a positive speed in AEP even when it has a negative drift, z z · q(z) < 0 (Theorem 2.3). We will make some mild assumptions in the next section.

Notation and results
In this section, we first introduce the problem and describe the environment process viewed from the tagged particle; next we describe the assumptions and introduce some notation; and lastly we state the main results and provide an outline of the proofs.
A configuration ξ(.) on Z \ {0} indicates which sites are occupied relative to the tagged particle: ξ(x) = 1 if site x is occupied, and ξ(x) = 0 otherwise. The collection of all configurations X = {0, 1} Z\{0} forms a state space for the environment process ξ t .
Local functions on X are functions of the form g(ξ(x 1 ), . . . , ξ(x n )) for some finite integer n, such that g : {0, 1} n → R. We will use C to denote the space of local functions on Z \ {0} and M 1 to denote the space of probability measures on X. Examples of local functions are: ξ A (ξ) = x∈A ξ(x), A is a finite subset of Z \ {0}.
(2.1) When A = {x} for some integer x, we abuse the notation and write it as ξ x . We will always use subscript to stress that ξ x is a local function. We also hope this will not cause confusion when ξ t is the configuration at time t, as we will see in a moment.
The environment process ξ t with respect to the simple exclusion process is a Feller process. Starting from any initial configuration in X, ξ t is described by its generator L = L ex + L sh . The action of L on any local function f is given by: where ξ x,y represents the configuration after exchanging particles at sites x and y of ξ, (2.3) and θ z ξ represents the configuration shifted by −z unit due to the jump of the tagged particle to an empty site z, The generator L ex corresponds to the motion of red particles, while the generator L sh corresponds to the motion of the tagged particle.
Denote by P η,q the probability measure on the space of càdlàg paths on X starting from a deterministic configuration ξ 0 = η, and let P ν0,q = P η,q dν 0 (η) when the initial configuration ξ 0 is distributed according to some measure ν 0 on X. We also denote by E ν0,q the expectation with respect to P ν0,q . A special initial measure is the step measure µ 1,0 , which concentrates on the configuration ξ, with ξ(x) = 1, for x < 0, and ξ(x) = 0, for x > 0. Also, we use P ν0,0 and E ν0,0 in the case when q(.) is a zero function.
Lastly, we will denote by D t the displacement of the green tagged particle up to time t. Initially, D 0 = 0 a.s. When q(.) is nearest-neighbor, we can represent D t as the difference of numbers of right and left jumps, see (3.9) in section 3. The main problem is to investigate the long time behavior of D t when q(.) is different from p(.).
It turns out that these assumptions can be generalized and we can get similar results. We will mention them and give outlines of their proofs after the proofs of the main results. A'2 (Positive Mean) p(k) ≥ p(−k) for all k > 0, and p(k) > p(−k) for some k, A'3 (Finite-range) there is an R > 0 such that p(x) = 0 for |x| > R.
Our main results are the ballistic behavior of a driven tagged particle in asymmetric exclusion processes under different assumptions. The first result is the most natural one.
When the initial measure is the step measure µ 1,0 and the tagged particle has only pure left jump rates, it has a ballistic behavior towards left, i.e. Dt t has a strictly negative asymptotic upper bound. Consider the AEP with a driven tagged particle. Let the jump rates p(.) for the red particles satisfy A1 and A2, and the jump rates q(.) be supported on negative axis with q(−1) > 0. Then, starting from the step initial measure µ 1,0 , there exists a negative constant c such that lim sup t→∞ D t t ≤ c < 0, P µ1,0,q − a.s.
When the tagged particle can jump in both directions, we can also obtain ballistic behavior. In the case when p(.) = q(.), and the initial measure is the Bernoulli measure µ ρ , for some 0 < ρ < 1, the tagged particle has a speed (1−ρ) z z ·p(z), see [18]. Now, if we change the jump rate q(.) such that the drift z z · q(z) is greater than z z · p(z), we expect its mean displacement to have the same asymptotic lower bound, (1−ρ) z z ·p(z). The second result confirms that under some conditions on p(.) and q(.), the displacement D t has an asymptotic lower bound (1 − ρ) z z · p(z). For the second result, we make the following assumptions on jump rates p(.), q(.).

Theorem 2.2. (Ballistic Behavior of a Fast Tagged Particle in AEP)
Consider the AEP with a driven tagged particle. Let the jump rates p(.), q(.) satisfy assumptions A"1, A"2, and A"3. Then, starting from a Bernoulli product measure µ ρ with ρ ∈ (0, 1) (on {0, 1} Z\{0} ), we have a. the tagged particle has a negative drift: −q(−1) + q(1) < 0, b. the tagged particle has a positive speed under P νe,q , that is, In these results, we have shown ballistic behavior of the tagged particle in AEP. With arguments to be introduced in section 3, the ballistic behavior (with estimates) implies the existence of some non-trivial invariant measures, measures other than µ 0 or µ 1 , for the environment process viewed from the tagged particle. In the driven tagged particle problem, the invariant measure is in general impossible to compute due to the break of symmetry. The Bernoulli product measure is no longer invariant. In principle, there could be multiple invariant measures, which makes the behavior of the tagged particle hard to predict.
To get these three results, we use similar ideas. The proofs of Theorem 2.1 and Theorem 2.3 are similar, and they are in section 7. The proof of Theorem 2.2 is in section 8. We will mainly discuss the approach to Theorem 2.3. It consists of three parts.
We first start from any initial measure ν 0 and obtain a candidateν for the invariant measure in Theorem 2.3 and some estimates of the displacement D t . Let N t be the number of red particles which initially start from the left of the tagged particle and move to the right of the tagged particle by time t. By standard martingale arguments and an algebraic identity, we can see that, up to an error of q(1) − q(−1), a multiple of E ν0,q Nt t is a lower bound for E ν0,q Dt t . This is done in section 3. On the other hand, we will show that the speed of the tagged particle is E ν0,q Dt t if ν 0 is ergodic. This is done in section 7.
Next, we want to prove a positive lower bound for E ν0,q Nt t for some ν 0 , and we use two steps. The first step is to obtain an estimate for E ν0,q [N t ] − E ν0,0 [N t ], which allows us to consider the case where the tagged particle does not move. This estimate indicates that the case when the tagged particle is moving slowly can be viewed as a pertubation of the case when the tagged particle is fixed. This estimate requires a coupling result, which is the main subject in section 4. The existence of coupling requires mainly assumptions A'1 and A'3, and we will show it in Appendix A.
The second step is to prove a positive current E ν0,0 Nt t for some initial measure ν 0 .
When the tagged particle does not move, the environment process evolves as the AEP with a blockage at site 0. A blockage is simply a site which particles are not allowed to jump to. We consider the case where ν 0 is the step measure µ 1,0 and prove that the current is strictly positive by contradiction. The idea is to consider the limiting measure of an invariant measure under translations {τ xν } in the Cesàro sense. We will get an estimate for this limiting measure by comparing this process with another process called asymmetric exclusion process (AEP) on the half-line even though with creation and annihilation. The analysis of the latter process requires a second coupling argument, and follows results and ideas of Liggett, [16,17]. The second step is done in section 5 and section 6.
We end this section with some remarks on the coupling result to be introduced in section 4 and the current through a fixed bond.

Invariant measure and the lower bound for the displacement of a tagged particle
In this section, we will assume that q(.) is nearest-neighbor and p(2) = p(−2) > 0, and assumptions A2, A1 are in force. This simplifies the computation, for some generalization, see Remark 7.1. We construct a candidate invariant measure by using the empirical measures. We also relate the displacement D t of the tagged particle to the current through bond (−1, 1). Most results in this section are shown by standard martingale arguments.
We start with a tightness result on M 1 with weak topology. Since X = {0, 1} Z d \{0} is equipped with the product topology, it is compact. By Prokhorov's Theorem, M 1 is precompact with the weak topology.
Define the (random) empirical measure µ t for process ξ t and its mean ν t by their actions on local functions: and ν t , f : for all f in C and t > 0. We also have continuity at t = 0, ν 0 = lim t↓0 ν t . By precompactness of M 1 , we can obtain a measureν as the weak limit of a subsequence ν Tn . It is an invariant measure by Theorem B7 [15].
Let F t := σ(ξ s : s ≤ t) and let N t be the net number of the red particles moving from the left of the tagged particle to the right of the tagged particle up to time t (or the integrated current through bond (-1,1)). Since the tagged particle has only nearestneighbor jumps, the jumps of the tagged particle do not change the value of N t and N t is the difference of two numbers: Under P ξ,q , R t has (varying) jump rates λ 1 (ξ t ) = p(2)(1 − ξ t (1))ξ t (−1), and L t has (varying) jump rates λ 2 (ξ t ) = p(−2)(1 − ξ t (−1))ξ t (1). By using P ν0,q -martingales, and uniform integrability, we can obtain the following result: A driven tagged particle in AEP Lemma 3.1. Let p(.) satisfy assumptions A2 and A1. For a sequence of T n ↑ ∞, ν = lim n→∞ ν Tn exists, andν is an invariant measure for the environment process ξ t .
We have we also have Proof. We write two P ν0,q -martingales These two martingales are generalizations of the classical martingale for a Poisson process n t with a rate λ, n t − λt. Combining them, we can get a P ν0,q − martingale, For more details, see Chapter 6.2 [13]. Taking expectation with respect to P ν0,q , we Passing through the weak limit, we get the equation (3.4). As L t and R t are both dominated by a Poisson Process with rate 1, { Mt t } t>1 and {M t t } t>1 are uniformly integrable. Using p(2) = p(−2), we get (3.6) from (3.4), (3.5).
We can also write the displacement of the tagged particle D t as the difference of two numbers, r t and l t , the numbers of right jumps and left jumps of the tagged particle: With a similar argument, we see the displacement D t has a lower bound which is a multiple of C 0 , up to an error (the difference of q(−1) and q(1)): Lemma 3.2. Let jump rates p(.) satisfy assumptions A2 and A1, and q(.) be nearestneighbor. There is a sequence of T n ↑ ∞ such thatν = lim n→∞ ν Tn exists and it is invariant, and D t has an estimate: (3.10) Furthermore, if (3.5) holds with C 0 > 0, then Proof. It is almost the same as that of Lemma 3.1. We notice that, l t − q(−1) t 0 (1 − ξ s (−1)) ds and r t − q(1) t 0 (1 − ξ s (1)) ds are P ν0,q -martingales, and that the left hand side of (3.10) can be rewritten as We use lim inf in (3.10) to emphasize that the initial measure ν 0 is arbitrary, and D t may not satisfy a law of large numbers. From the estimate (3.11) in Lemma 3.2, we can get a positive mean for the displacement when the tagged particle has almost symmetric jump rates, i.e. when q(−1) − q(1) is small, and C 0 is positive. For the next three sections, we will show how to get a positive C 0 with (3.5) in Lemmas 3.1 and 3.2 for some ν 0 .

An error estimate and couplings of particles on Z
The main result of this section is Theorem 4.4, which gives an estimate of the error This estimate allows us to consider the problem with a fixed tagged particle instead of a moving tagged particle. The proof relies on couplings of two auxiliary processes, which is the main tool in this section. The couplings are similar to those in [2,6]. Under the couplings of two auxiliary processes, we will have one auxiliary process which moves "faster" than the other process. We will order particles in increasing order, and compare the positions of particles in two processes in pairs. Typically, the "faster" process has particles with larger coordinates relative to their paired particles in the "slower" process. By coupling jumps of particles, we can preserve the relative orders of paired particles in both processes for all time t ≥ 0. Next, we introduce some notions, and show the proof of Theorem 4.4 at the end of this section.

Auxiliary processes
We can view the environment process ξ t of the asymmetric exclusion process with a tagged particle in another way. We can label all red particles according to the initial configuration in an ascending order, and track their relative positions with respect to the tagged particle.
To extend to the case when there are finitely many particles to the right or the left of zero, it is also convenient for us to add particles at +∞ and −∞, and therefore, we would enlarge the state space toX = (Z \ {0} {−∞, ∞}) Z . For example, for the step measure µ 1,0 , we can label particles as: Also, there is no particular rule for the choice of X 0 with respect to the tagged particle.
For each initial configuration X 0 satisfying (4.1), there is a Markov process X t with generatorL corresponding to the process ξ t with initial configuration ξ. In particular, we will introduce many re-labelings to keep X t satisfying (4.1) for any t > 0. There are two types of jumps for the auxiliary process, corresponding to jumps (2.3) and (2.4). The first occurs when the i-th red particle jumps to an empty target site X i + z; the second occurs when the tagged particle jumps to an empty target site z. Due to nearest-neighbor jump rates q(.), a jump of the tagged particle does not result in change of labels, while a jump of a red particle requires re-labelings of particles between the particle and its target site so that (4.1) holds. Note that there are multiple X 0 corresponding to ξ, so to the process ξ t there correspond multiple processes X t .
Let T i,z X and Θ z X represent the configurations after these two jumps respectively.
See (4.2), (4.4), (4.6) below for their expressions. We can see two examples for these two types of jumps in Figure 1 and Figure 2. For any z = 0, we have Θ z X as, For z > 0, we denote the index of the right-most particle to the left of site X i + z by I i,z ( X), I i,z ( X) = max{k : X k ≤ X i + z}. When a positive jump is possible for the i-th particle, we have the new configuration described by . (4.4) The conditions for these two types of jumps to occur are For negative jumps z < 0, we can think of the dynamics by reversing the lattice Z. That is, with a change of variable, Y = {Y i } i∈Z = R( X), we have For z = 0, we take T i,0 as the identity map and I i,0 ( X) = i. Therefore, we can write down the generatorL for the auxiliary process X t by its action on local functions F :X −→ R (i.e. F ( X) depends on a finite set {X i }) as: The transition rates are p(x, y) = p(y − x) if x, y = 0, ±∞, and p(x, y) = 0 otherwise.

Shifts of labels
In the environment process, a jump of the tagged particle influences coordinates of all red particles (and does not change labels of particles), while jumps of red particles influence only finitely many coordinates (and change labels of particles). In order to couple jumps of the tagged particle with jumps of red particles and preserve the order of the two processes, we use shifts of labels to offset the global effect on coordinates from jumps of the tagged particle.
For couplings, we also consider two other versions of auxiliary processes with shifts of labels, which correspond to the same process ξ t . Let S z X represents the configuration after shifting labels by z, (S z X) j = X j+z .

Figure 3: Tagged Particle Jumps -2 Units with Labels Shifted
In addition to shifting configurations when a tagged particle jumps, we can also shift labels after shifting the configurations. See Figure 3. We obtain the first version by adding a shift of labels by z after the tagged particle has a left jump with z units, that is, Similarly, we can have the second version by shifting labels after the tagged particle takes a right jump. See Figure 4 below for an example when the tagged particle has a right jump with size 1.

Figure 4: Tagged Particle Jumps 1 Unit with Labels Shifted
We will use X t = ( X 0 , G, p, q) to denote the auxiliary process with X 0 as the initial configuration, and generator G. In particular, G is one of the forms (4.8),(4.11), and (4.10) with p, q as parameters. And we use P ( X0,G,p,q) or P Xt to denote the corresponding probability measure on the space of càdlàg paths onX. X 0 can also be random.

Couplings of auxiliary processes and error estimates
There is a natural partial order on the setX: With this partial order,we can define that two auxiliary processes X t = ( X 0 , G, p, q) and Y t = ( Y 0 , G , p , q ) are coupled by stochastic ordering. Definition 4.1. We denote X t Y t , if two auxiliary processes X t and Y t can be coupled: that is, there exists a joint process Z t = ( W t , V t ), with a joint generator Ω on the space of local functions F :X ×X → R, such that Our main step towards Theorem 4.4 is the existence of couplings of auxiliary processes. The construction of the couplings is done in Appendix A. Theorem 4.2. Let p(.) satisfy assumption A1 and two initial configurations satisfy X 0 ≥ Y 0 . For any q(.), we can couple below two pairs of auxiliary processes: respectively, and we can estimate the error by the number of jumps of the tagged particle.
To get (4.16) and (4.17), we can consider the following. For any non-zero configuration ξ in X, we can label the particles as and equality occurs if both sides are ∞ or −∞. By Theorem 4.2, from the same initial configuration ξ, we have two couplings with X 0 = Y 0 = Z 0 , Therefore, we get, under two joint distributions (one for the coupling X t Y t , and the other for the coupling Y t Z t ) and Notice that the derivation of (4.20) and (4.21) does not require q(.) to be nearestneighbor.
On the other hand, when q(.) is nearest-neighbor, jumps of the tagged particle do not move particles between positive and negative axes, but they may shift labels. See Figure  4. For X t , we can use a decomposition similar to those in (3.3) and (3.9), and see that the change in the label of the right-most particle on the negative axis by time t comes from three sources: jumps of red particles through bond (−1, 1) (N X (t)), right jumps of the tagged particle (r X (t)) and left jumps of the tagged particle (l X (t)). In particular, each first type of jump contributes 1 to the change, each second type of jump contributes 1 to the change, and each third type of jump contributes 0 to the change. Therefore, we (4.22) where N X (t), r X (t) are the same as N t , r t for the corresponding environment process ξ t . Similarly, we obtain two identities for processes Y t , Z t , where l Z (t) is the same as l t for process Z(t).

Remark 4.5.
We can obtain further results with similar proofs of Theorem 4.4. We will assume that p(·) satisfies assumption A1 so that couplings in Theorem 4.2 are possible.
We mention these results without giving detailed proofs.
1. When q(.) is non-nearest-neighbor and finite-range, we can find an estimate similar to (4.15): there is a C R > 0 depending on the range R of q(.) such that, (4.25) We outline the proof of (4.25): we can first obtain couplings (4.18) by Theorem 4.2. Then, we can apply couplings (4.18) to the decreasing function F ( X) = max{i : X i ≤ −1} and get (4.20) and (4.21). Each side of (4.20) is the same as the integrated current N t through the bond (−1, 1) for the corresponding environment process ξ t up to a term corresponding to the shift of labels. For example, we can take the auxiliary process X t = ( X 0 ,L R , p, q). Every jump changing the value of the integrated current N t also changes the value F ( X t ) by the same amount, except for right jumps of the tagged particle. A right jump of size z decreases F ( X t ) by an additional amount z, so we can get (4.22) by interpreting N X (t) as the integrated current N t through the bond (−1, 1) for its corresponding environment process, and r X (t) as the sum of right jump sizes of the tagged particle. Similarly, we can derive (4.23) and (4.24) with new interpretations. Therefore, we can get (4.25) by taking expectations and the fact that 2. From the coupling, we can use Kingman Subadditive Ergodic Theorem to show the convergence of Nt t when the initial measure is the step measure µ 1,0 , and the tagged particle does not move, q = 0: See Remark 2.4 and Lemma 4.8 in [2], or see (7.6) in the proof of Theorem 2.1.

Current in AEP with a blockage
In this section, we will show the current in AEP with a blockage at the origin has a positive lower bound (Theorem 5.5).The existence of a positive lower bound helps us to show that the tagged particle has a positive speed under P νe,q , for some small q(.) and some ergodic measure ν e for the environment process.

Currents and densities in equilibrium
In sections 5, 6, we make the following assumptions on p(., .). Let p(., .) be jump rates for a continuous-time random walk on Z with the following conditions: for all k > 0, and a strict inequality holds for some k.
3. p(., .) has a finite jump range R > 1: Notice that the assumptions A1, A2 are sufficient for the above assumptions, but not necessary. Also, we don't need A1 or A'1, which is the main condition for the existence of couplings in section 4; instead, the second assumption above is the main condition for this section. It enables us to construct an increasing sequence G i , which will be important in the proof of Lemma 5.3.
We will consider a process, the AEP on lattice Z with a blockage at the origin, i.e., the AEP with a tagged particle when q = 0, and quantities C x,y that are currents through bond (x, y).
The AEP on lattice Z with a blockage at the origin has a generator L defined by its action on a local function f , which is the same as (2.2) when q = 0. Assume the initial configuration is the step measure µ 1,0 for the rest of this section. Recall that C −1,1 was defined in Lemma 3.1. In general, for any x < y, we can define the current C x,y through bond (x, y) as: Theorem 5.5 is the main result for the next two sections. Before its statement and proof, we shall see three lemmas on invariant measures with respect to L, and currents C x,y . The first two lemmas are direct consequences of translation invariance and finite range of p(., .) and they are standard. In the third lemma, we will need the second condition on p(., .). The first lemma says the mean of current C x,x+1 is constant in x with respect to an invariant measure.
Proof. The change of density at site x is due to the difference between currents through bonds (x − 1, x) and (x, x + 1). Computing Lη x for x = −1, 0, 1, we get We show the first one, and the next two are similar: for any x = −1, 0, 1, we have On the other hand, we can check where interchanging i and j in the third last line results in a change of sign.
Taking expectation with respect to the invariant measureν, we get (5.3).

Consider translation operators
We define translations on local functions f and on measures ν by In particular, we have that τ i η j = η i+j , τ i ν, η j = ν, η i+j .
The second lemma says that any weak limit ν * of the Cesàro means ofν under translation is a mixture of Bernoulli measures µ ρ , 0 ≤ ρ ≤ 1. This is because ν * is translation invariant and invariant with respect to the generator L 0 for AEP. Recall that the generator L 0 acts on a local function f by, see [11], is translation invariant and invariant with respect to the generator L 0 for AEP. That is, for any local function f , where L 0 is translation invariant. In particular, there is a probability measure w ρ on [0, 1], such that Proof. By Theorem VIII.3.9 [18], we only need to show translation invariance and invariance ((5.8), (5.9)) to get (5.10). The proofs for both are similar.
For any local function f , which is a bounded function on {0, 1} Z depending on finitely many ξ x , Also, asν is invariant with respect to L and L 0 τ i = τ i L 0 , we can compare (5.1) with (5.6) and get, In the last line, since f is local, (L 0 − L)(τ i f ) is non-zero for finitely many i. Taking limits as n k → ∞, we get (5.8) and (5.9).
The third lemma says if an invariant measureν has a current with a zero mean and some weak limit ν * of its Cesàro means under translation is a Bernoulli measure µ 0 with density 0, the densities of positive sites are identically 0 forν.

Lemma 5.3.
Letν be an invariant measure with respect to the generator L, and ν * be a weak limit of its Cesàro means defined in (5.7). If ν, C −1,1 = 0 and ν * , η x = 0 for some x (which implies for all x since ν * is translation invariant), we have ν, η x = 0 for all x > 0.
Proof. We will divide the proof into 3 steps. S1. Define a quantity G i : Therefore, by Lemma 5.1, we have, for i ≥ R, ν, C i,i+1 = 0, and x≤i,i+1≤y The choice for i ≥ R is to avoid x, y = 0 for any term inside the sum.
Notice that there is some symmetry on the left hand side of (5.11), which allows us to rewrite (5.11) as a backward difference for some sequence (G i ) i≥R x≤i,i+1≤y We will prove (5.12). Indeed, we can expand the left hand side of (5.11), and rearrange terms according to ν, η i+j , for j = −(R − 1), −(R − 2), . . . , R. We will get 2R terms with coefficients b j , x≤i,i+1≤y The coefficients b j are "odd" in the sense that b −(j−1) = −b j , for j = 1, . . . , R. (5.14) From (5.14), we can find 2R + 1 "even" numbers with boundary conditions a R = a −R = 0, a −j = a j , for j = 0, 1, . . . , R, (5.15) and rewrite b j as a (negative) forward difference We can also express a j in terms of p(.) explicitly as ., for |j| = 0, 1, . . . , R. (5.17) One can see (5.16) by working on an example. For example, when R = 2, we have and we can find 5 "even terms" 0, b 2 , b 2 + b 1 , b 2 , 0 and write the 4 odd terms as In fact, (5.16) is a direct consequence of the symmetry (5.14), and it does not rely on the explicit expressions (5.13), (5.17). From (5.16), we can apply the summation by parts formula to the left hand side of (5.11) and get (5.12), is unique up to a constant, and we can use the last equality of (5.18) and express G i in the matrix form, By the assumption p(k) ≥ p(−k) for k > 0, we have the right hand side of (5.11) is positive. Also, (5.19) implies that G i is bounded uniformly for i ≥ R. Therefore, we get the monotone convergence of (G i ) i≥R : We should notice that to write G i in forms of (5.19), we need i ≥ R. It is because we don't want terms involving p(0, x) or p(x, 0). This condition holds for sites sufficiently right to the origin. We will see similar conditions in Theorem 5.4 and Lemma 6.1 involved.

Proof of positive currents in AEP with a blockage
The theorem below will be proved in section 6.3. It says, if the initial configuration has no particles after some point x > 0, ν * is dominated by µ 1 2 , in the sense of (5.21).
Let's recall from section 3 that the mean of empirical measures ν t is defined by its action on local functions ν t , f = 1 t E ν0,0 t 0 f (η s ) ds for some initial measure ν 0 . Theorem 5.4. Consider the AEP on lattice Z with a blockage at the origin and p(.) has a positive mean z · p(z) > 0. Letν be a weak limit of the mean of empirical measures ν Tn , and ν * be defined via a subsequence mentioned in (5.7). If there is an x > R such that ν 0 , η y = 0 for all y ≥ x, we will have, for any finite set A ⊂ Z, Proof. See Corollary 6.4.
Theorem 5.5 is the main result of sections 5, 6. It says the current through bond (−1, 1) is strictly positive for the AEP on Z when the initial measure is the step measure µ 1,0 . We will prove it by contradiction.
For the AEP on lattice Z with a blockage at the origin, there is a lower bound C 1 > 0 for the current through bond (−1, 1), Proof. Let N t be the (net) number of particles jumping through bond (−1, 1) by time t, which is the same as (3.3) when the tagged particle is not moving. Under the initial measure µ 1,0 , there are no particles on the positive axis, we can see that N t is the same as the number of particles on the positive axis at time t, and therefore N t ≥ 0. Together with the fact that N t − t 0 C −1,1 (η s ) ds is a P µ1,0,0 -martingale (see Chapter 6.2 [13]), we get that Suppose C 1 = 0. By tightness, there is an invariant measureν with a zero current ν, C −1,1 = 0. By Lemma 5.1, ν, C x,x+1 = 0, for x ≥ R. We have Then, for any weak limit ν * = lim k→∞ ν * n k = lim k→∞ On the other hand, by Lemma 5.2, ν * is a mixture of Bernoulli measures (on {0, 1} Z ), that is, ν * = µ ρ dw ρ for some probability measure w ρ . A computation shows (p(i, j) − p(j, i)) = ρ(1 − ρ)w, (5.24) where w is the mean drift for p(., .) w = i≤R,j≥R+1 (p(i, j) − p(j, i)) = k k · p(k). (5.25) By assumption 2 for p(., .), this sum (5.24) is strictly positive unless ρ = 0 or 1. As a consequence, ν * is a convex combination of µ 0 and µ 1 ν * = c 0 µ 0 + c 1 µ 1 ,

AEP on half-line with creation and annihilation
To show Theorem 5.4, we will consider an auxiliary process: the AEP on the half-line with creation and annihilation. This model has a long history and was studied by Liggett in [16] and [17]. We will use some results from [16] and [17] to show the estimate (5.21) in Theorem 5.4.

Comparison between AEP on half-line with creation and AEP with a blockage
We first describe the AEP on the half-line with only creation formally as follows.
Particles move according to asymmetric exclusion process on half-line [1, ∞) with jump rates p(x, y) = p(y − x). If a positive site y > 0 is vacant, a particle is created at y with a rate x≤0 p(y − x). Also, no particles are allowed to jump out of the positive half-line. Alternatively, if we consider the AEP on Z with an immediate creation of particles on (−∞, 0] when sites are vacant, the dynamic restricted to the positive axis is the same as the dynamic of the AEP on the half-line with creation.
The first lemma connects the AEP with a blockage at site 0 with the AEP on the half-line with creation. Denote by η t the AEP with a blockage at site 0, which has a probability measure P ; denote by ζ t the AEP on the half-line with creation, which has a probability measure Q. Lemma 6.1. Suppose AEP with a blockage at site 0 starts from the initial measure µ 1,0 and the AEP on the half-line with creation starts from the Bernoulli measure µ 0 on positive axis. Then, for any finite subset A ⊂ Z + , and any t ≥ 0, where R is the range of jump rates p(.) as defined at the beginning of section 5. We use R to avoid sites too close to the origin.
Proof. In the AEP with a blockage, we use independent exponential clocks with rates p(x, x + z) to indicate times of potential jumps from a site x to a site x + z. These clocks also help us to interpret movements of holes. When a potential jump from site x to x + z occurs, a hole at site x can interchange with another hole at site x + z (even though the interchanging doesn't affect the configuration), but its jump to a site x + z occupied by a particle is suppressed. Then, we can obtain an intermediate process φ t by labeling holes and particles in the AEP with a blockage as different classes of "particles" and suppressing certain jumps. In this intermediate process, there are three classes of particles, we label each class by 1, 2, or 3. Holes and particles in the AEP with a blockage are labeled according to the following rules: a. a particle in the AEP with a blockage is always a first-class particles and labeled "1" in the intermediate process; b. a hole in the AEP with a blockage at any time is either a second-class particle or a third-class particle; c. a hole becomes a second-class particle once it visits or starts from a site on (−∞, R], and its label becomes "2"; d. a hole is always a third-class particle if it never visits or starts from a site on (−∞, R], and its label stays "3". We will also suppress a jump from site x to x + z (in addition to those jumps suppressed due to the target site x + z already occupied by a particle in the P -process) if x has a third-class particle and x + z has a second-class or third-class particle.
(6.2) (6.2) does not affect the P -process because both the second-class and third-class "particles" are holes, but under (6.2), only jumps from a site with a particle of a larger label to a site with a particle of a smaller label is allowed. We will denote byP the probability measure corresponding to φ t . See Figure 5 for an example. φ t0 is the a configuration at time t 0 > 0 with a specific labeling of three classes of particles. In particular, the hole at the site 4 is labeled a second-class particle. φ t1 is the configuration after a (first-class) particle jumps from −1 to 1, a (second-class) particle jumps from 2 to 3, and a (second-class) particle jumps from 4 to 6 in φ t0 ; φ t2 is a configuration at a general time t 2 . The intermediate process φ t connects both the P -process and Q-process. On one hand, it follows from the rules that the first-class particles in φ t correspond to particles in the P -process. We have that for any finite subset A ⊂ Z + , t ≥ 0 P (φ t (x) = 1, for all x ∈ A + R) = P (η t (x) = 1, for all x ∈ A + R). On the other hand, the dynamics of the third-class particles (on (R, ∞)) in φ t are identical to the dynamics of holes in the Q-process (on (0, ∞)) when the Q-process has an initial measure µ 0 . We can see this because a third-class particle is not created; a third-class particle is affected by either being moved from a site y > R to a new site x > R, when the site x is occupied previously by a non-third-class particle and a potential jump from x to y occurs, or being removed from the system due to a jump from a site x ≤ R to y. This is the same as a hole at a site y − R in the Q-process: a hole at site y − R is affected by either being moved to a new site x − R > 0, when the site x is occupied previously by a particle, and a jump from site x − R to y − R occurs, or a hole is affected by being removed from the system due to a jump from a site x − R ≤ 0 to y. Therefore, together with the initial measure µ 1,0 for the P -process, we can get that for any finite subset A ⊂ Z + , t ≥ 0P As a consequence of (6.3) and (6.4),

Couplings in the AEP with creation and annihilation
By the above lemma, we can study the asymptotic behavior of the AEP on half-line with only creation. The main theorem of this section is Theorem 6.3. The proof of Theorem 6.3 can be derived from results in [17], with stochastic orderings (couplings). We start with some notion and results from [16] and [17]. We first define partial orders on the space of configurations X −∞,∞ η ≥ ξ ⇔ η(x) ≥ ξ(x) for all x ∈ Z. Then we can define partial orders on the space of probability measures via stochastic ordering: ν ≥ µ ⇔ ν, f ≥ µ, f for all f increasing (with respect to (6.7)). (6.8) We will consider the AEP with creation and annihilation on both a finite system and an infinite system. The former is a process on X m,n with a generator Ω λ,ρ m,n and a semigroup S λ,ρ m,n . Ω λ,ρ m,n acts on a local function f by Ω λ,ρ m,n f (η) = x<m,y∈Dm,n (p(x, y)λ (1 − η(y)) + p(y, x)(1 − λ)η(y)) (f (η y ) − f (η)) + x∈Dm,n,y>n And the latter is a process on X m,∞ with a generator Ω λ m,∞ and a semigroup S λ m,∞ .
Ω λ m,∞ acts on a local function f by (6.10)

Liggett's results and their consequences
Below are results from [16] and [17]. Particularly, the monotonicity in the first part of Lemma 6.2 guarantees interchanging of limits. Recall µ m,n ρ is a Bernoulli measure on X m,n with density ρ. In the sense of (6.8), the probability measure ν λ,ρ m,n (t) is increasing in parameters m, n, t, λ and ρ.

Proofs of Theorem 2.1 and Theorem 2.3
In this section, we prove Theorems 2.1 and 2.3. Let's start with the proof of Theorem 2.3, and we will see that the proof of Theorem 2.1 follows similar arguments.
Proof. (Theorem 2.3) We divide the proof into two steps.
On the other hand, the collection of invariant measures satisfying (7.1) forms a nonempty closed convex compact set by tightness. Then, there is an extremal point ν e , which is ergodic for the environment process ξ t , and ν e also satisfies (7.1) ν e , f ≥ q(1) p(2) C 0 − (q(−1) − q(1)) > 0. (7.2) Step2. The positive speed of the tagged particle: We can use P νe,q − martingales,(see (3.7),(3.8)) where M t is a martingale with quadratic variance of order t. As ν e is invariant and ergodic for the environment process ξ t , we apply Ergodic Theorem, and get lim t→∞ D t t = ν e , f > 0, P νe,q − a.s.
In the case when the tagged particle only has pure left jumps, following arguments in Step 2 of the above proof, we only need to show that lim inf t→∞ Then under some joint distribution, we have (4.20) for the decreasing function F ( X) = max{i : X i ≤ −1}. When the tagged particle does not jump to the right, each side of (4.20) is identical to the number of red particles through bond (−1, 1) by time t (integrated current through bond (-1,1)). The above inequality (4.20) is equivalent to N X (t) ≥ N Y (t) a.s., (7.4) which implies that lim inf It is worth noting that due to non-nearest-neighbor jumps of the tagged particle, N X (t) in (7.4) is different from the N X (t) described before (4.22) because a left jump of the tagged particle can increase N X (t) when there is a red particle between the tagged particle and its target site (see Figure 3 for instance). For more details on N X (t), see point 1 in Remark 4.5.
We can use the the Kingman Subadditive Ergodic Theorem and Theorem 5.5 to get that the right hand side of (7.5) is a positive constant C 1 , lim inf Indeed, due to the step initial measure µ 1,0 , we can label particles initially At any fixed time t, we can get a new (random) configuration Y t ≥ Y t from Y t by "increasing" all the red particles in Y t that are on the positive axis to +∞, and "increasing" all the other red particles to fill the "rightmost" holes on the negative axis. More precisely, It is also immediate that Y t is identical to the initial configuration Y 0 = (Y i ) i∈Z , but with N Y (t) (random) shifts of labels. Hence, we have Then by Theorem 4.2, we can couple two auxiliary processes Z s , Y t+s with initial configu- Applying the argument for (7.4), we can get the subadditivity for N Y , for any s > 0, s., (7.9) where N Z (s) has the same distribution as N Y (s) because Z 0 and Y t are the same up to N Y (t) (random) shifts of labels, and N Z (s), N Y (s) are differences of labels, see arguments before (7.4). From (7.8) and (7.9), we can apply the Kingman Subadditive Ergodic Theorem to get the convergence in (7.6), and identify the limit by Theorem 5.5. This is also a proof for the second point in Remark 2.4. On the other hand, for the environment process of AEP with a driven tagged particle, we can compute Lξ −1 by (2.2) and the fact that p(.) is supported on [−2, 2], and q(.) is supported on the negative axis. We can bound it above by Also, we can computeĈ −1,1 by adding an extra term to (5.2), which corresponds to the jump of the tagged particle, Notice that the negative term in the last inequality of (7.10) is the same as the first term p(2)ξ −1 (1 − ξ 1 ) on the right hand side of (7.11). Therefore, we can bound the sum Lξ −1 +Ĉ −1,1 by summing the other positive terms in (7.10), (7.11), and bound the sum by a multiple of f = z<0 z · q(z) (1 − ξ z ) , f, (7.12) where C 4 = q(−1) + z p(z). We can use three P µ1,0,q − martingales, (see (3.7), (3.8), and Chapter 6.2 [13]) f (ξ s ) ds, (7.13) which all have quadratic variance of order t. Dividing by t and taking limits, we see from (7.5) and |ξ t (−1)| ≤ 1 that, P µ1,0,q -a.s., lim inf Together with (7.12), we get P µ1,0,q − a.s.
lim sup f (ξ s ) ds ≤ c < 0, P µ1,0,q − a.s. The proof is similar to that of Theorem 2.1. Once we've shown (7.5) (by the same argument), we can use an inequality similar to (7.12), see (7.20) below, to get (7.15). Indeed, we will have three P µ1,0,q − martingales similar to (7.13), f (ξ s ) ds, (7.16) whereĈ −1,1 is almost the same as C −1,1 from (5.2), except for an extra term due to left jumps of the tagged particle, (7.17) and f (ξ) = z q(z)z (1 − ξ z ). We can write C −1,1 as a difference (7.18) and compute L 0<z<R ξ −z by different jumps due to the tagged particle and red particle, By comparing the positive terms of (7.18) and the last negative term in (7.19), we can bound the positive terms of C −1,1 by the negative terms of L 0<z<R ξ −z in absolute value. Therefore,Ĉ −1,1 + L 0<z<R ξ −z is bounded above by the sum of the positive terms in (7.17) and (7.19), Since z<z ≤−1 ξ z ≤ R − 2, and 0<k<R ξ −k+z ≤ R − 1 for all −R <z ≤ −1, we can get an upper bound for the above inequalitŷ f, (7.20) where C 5 = (2R − 3) · max z q(z) + k p(k). (7.20) is an analogue of (7.12), and we can use a similar argument as (7.14) to get (7.15) for some c < 0.
2. To generalize Theorem 2.3, p(.) are under additional assumption that p(−k) = p(k) for 2 ≤ k ≤ R, and p(1) > p(−1). Then there exists jump rates q(.) with a negative drift z q(z) < 0 and an ergodic measure ν e , such that the speed of the tagged particle is positive under P νe,q .
The proof is also similar to the proof of Theorem 2.3 and we outline it below in a different order. Due to the assumptions on the jump rates p(.), we can use Theorem 5.5 to get a positive lower bound for lim inf Then we will choose R = R−1, and construct jump rates q(.) supported on [−R , R ].
We can observe the following facts: (a) When z q(z) is small enough, we can use (4.25) in the first point of Remark 4.5 to get a positive lower bound for lim inf By using a P µ1,0,q -martingale N t − t 0Ĉ −1,1 (ξ s ) ds, we can get a lower bound for lim inf (7.21) where C −1,1 is the current through bond (−1, 1) given by formula (5.2), and C −1,1 differs fromĈ −1,1 by a term of size at most C R z q(z). (7.22) where by "odd" we mean b −i = −b i , which is differnt from (5.14).
(c) When q(.) is the sum of a multiple of bz z z and an error term (e(z)) z , for 1 ≤ |z| ≤ R q(z) = c · b z z + e(z) (7.23) for some positive c > 0, by (7.22), the function f = z zq(z) (1 − η z ) is cC −1,1 up to an error of size at most |f − cC −1,1 | ≤ z |ze(z)| , (7.24) and the drift for jump rates q(.) is z z · q(z) = z z · e(z) (7.25) Therefore, by (7.21),(7.24),(7.25), we can choose positive c, (e(z)) z with z z·e(z) < 0 so that q(.) of the form (7.23) has a negative drift w = z z · q(z) = z z · e(z), and there is an invariant measureν for the environment process ξ t , such that The invariant measureν can be obtained as the weak limit of the mean ν tn of the empirical measure,see (3.2), along some sequence (t n ). We can also obtain an ergodic measure ν e which also satisfies (7.26). Then, by the step 2 of the proof of Theorem 2.3, we get that under P νe,q , the tagged particle has a positive speed.

Ballistic behavior of a fast tagged particle in AEP
In this section, we will prove Theorem 2.2. In this case, both the green tagged particle and red particles have non-nearest-neighbor jump rates and the means of jump rates are positive. This is a scenario different from Theorem 2.3. In particular, the jump rate β = z q(z) of the tagged particle can be larger than the jump rate λ = z p(z) of red particles.
We briefly discuss the steps of the proof. We will modify the auxiliary process introduced in section 4. Instead of labeling red particles and considering their positions relative to the tagged particle, we will also label the tagged particle, and keep track of its label (see (8.1)). With this modified auxiliary process, we can couple the ordered particles (including the tagged particle) in the AEP with the ordered particles in the usual AEP. By investigating the change in labels of tagged particles in both processes, we can compare their positions. We obtain a lower bound for the driven tagged particle in AEP by estimates from the usual AEP.

Assumptions and labels of the tagged particle
Let's recall assumptions A"1,A"2, and A"3 on jump rates p(.), q(.).
A"1 (Supports) p(.) has a support on −2, −1, 1; q(.) has a support on −1, 1, 2, These conditions imply that the tagged particle moves "faster" than a red particle, and that red particles starting from the left of the tagged particle always remain to the left of the tagged particle. We will explain their roles in Remark 8.2. Consider an AEP with a driven tagged particle, we label particles in an ascending order and also keep track of the label I t of tagged particle. We get a modified auxiliary process ( X t , I t ) = ( X 0 , p, q, I t ) = (X i (t)) i∈Z , I t . Its generatorL p,q is given by its action on a local function F , L p,q F ( X, I) = i =I,z∈Z p(z)1 Ai,z ( X) F (T i,z X,Î i,z ( X, I)) − F ( X, I) + z q(z)1 A I,z ( X) F (T I,z X, I I,z ( X)) − F ( X, I) (8.1) where T i,z X is defined by (4.4),(4.6), andÎ i,z ( X, I) is defined aŝ For an AEP with a usual tagged particle,i.e., p(.) = q(.), we get a second auxiliary process ( Y t , i t ) = ( Y 0 , p, p, i t ) with a generatorL p,p . For convenience, we let initial configuration be the same for both processes, and label the tagged particles with 0, ie.
The first lemma says that we can couple two modified auxiliary processes ( X t , I t ) and ( Y t , i t ) in the sense similar to Definition 4.1., for all t ≥ 0, , for all i in Z. (8.4) Lemma 8.1. Suppose p(.), q(.) satisfy A"1, A"2 and A"3. For two modified auxiliary processes ( X t , I t ) and ( Y t , i t ) with generatorsL p,q andL p,p , there is a joint Ω, such that if (8.4) holds for t = 0, we have (8.4) holds for all t > 0, and the marginal condition holds ΩF 1 X, I, Y , i =L p,q H 1 X, I , for any local functions F 1 X, I, Y , i = H 1 X, I and F 2 X, I, Y , i = H 2 Y , i .
Proof. This is proved in Corollary A.5. In this case, we have R = 2.

Remark 8.2.
A special case is when both processes have exactly one tagged particle and no red particles. This is a degenerate case because the tagged particles follow continuous time random walks with jump rates p(.), q(.), and I t = i t = 0 for all t ≥ 0.
Assumptions A"2, and A"3 guarantee that we can couple these two random walks with X 0 (t) ≥ Y 0 (t), for any t ≥ 0 (without Lemma 8.1). These two assumptions also allow us to generalize the coupling of random walks to other cases described by Lemma 8.1, so that (8.4) holds for all t ≥ 0. However, (8.4) is only useful if we know the labels I t , i t of the tagged particles or their differences I t − i t . The assumption A"1 does not affect the couplings of two modified auxiliary processes; instead, this assumption implies that I t is increasing in time t. Together with a law of large number for i t , we can get the Lemma 8.3 below which implies the signs of the I t − i t asymptotically.
The second lemma gives estimates of I t and i t with respect to the Bernoulli initial measure µ ρ . Lemma 8.3. Suppose p(.), q(.) satisfy A"1, A"2 and A"3. Let I 0 = i 0 = 0, and X 0 correspond to the initial Bernoulli product measure µ ρ . The labels I t , i t of the tagged particles in the modified processes ( X t , I t ) = ( X 0 , p, q, I t ) and ( Y t , i t ) = ( X 0 , p, p, i t ) satisfy, lim inf t→∞ I t t ≥ 0, P µρ,q − a.s. and lim t→∞ i t t = 0, P µρ,p − a.s.
Proof. Notice that i t is identical to the integrated current −N t through bond (−1, 1) in the environment process ξ t . For a general jump rateq(.) supported on [−2, 2], we can obtain the currentĈ −1,1 by considering the jumps of the red and the tagged particles, and modifying (5.2). Notice that a jump of the tagged particle to the site −2 (relative to the tagged particle) increases the integrated currents N t by one if there is a particle at the site −1 (relative to the tagged particle), and that a jump to the site 2 decreases N t by one if there is a particle at the site 1. Therefore,Ĉ −1,1 iŝ (8.5) which is the compensator of the integrated current N t . Similar to (3.7) and (3.8), N t − t 0Ĉ −1,1 (ξ s ) ds is a P µρ,q -martingale, and we can obtain When we takeq(.) = p(.), the Bernoulli measure µ ρ is ergodic for ξ t , and by (8.5), the expectation in the last integral is 0. Therefore, we have that lim t→∞ i t t = E µρ,p i t t = 0, P µρ,p − a.s.
For I t , since q(−k) = 0 for all k ≥ 2, the tagged particle cannot jump from the right side of a red particle to its left side, and I t does not decrease due to jumps of the tagged particle. Also, that p(k) = 0 for all k ≥ 2 implies that no red particle can jump from the left side of the tagged particle to its right side, which also means, I t does not decrease due to jumps of the red particles. Therefore, we have that I t is increasing in time t,

Proof of Theorem 2.2
Now we can prove Theorem 2.2.
Proof. (Theorem 2.2) By Lemma 8.1, there is a joint distribution P, and we have X t ≥ Y t , P − a.s. In particular, X It ≥ Y It , P − a.s.
On the other hand, by Lemma 8.3, under the joint distribution P, which has marginal distributions P µρ,q and P µρ,p , lim inf Therefore, for any fixed δ > 0, Since the Bernoulli product measure µ ρ is an ergodic measure for the environment process, Y it − Y it−δ·t is dominated by the sum of δ · t independent geometric random variables with parameter ρ. For each fixed k > 0, we can get a sequence (t n,k ) n = ( n Then, we can use a standard interpolation argument to replace "t n,k ↑ ∞" by "t ↑ ∞". With the law of large numbers for the displacement of a tagged particle in the usual AEP , i.e., when q(.) = p(.), lim t→∞ where w = z z · p(z) > 0. This is sufficient to get Theorem 2.2 since X It ≥ Y It .
We can also extend Theorem 2.2 to the case with more general jump rates p(.), q(.).

Remark 8.4.
To generalize Theorem 2.2, p(.) satisfies assumptions A'1, A'3 and an additional assumption that for all k ≥ 2, It is immediate that under (8.8), red particles starting from the left of the tagged particle always remain to the left of the tagged particle. The proof will be almost the same: The rest of the proof follows the same arguments after (8.7).

A Appendix
The generator for the coupled process in Theorem 4.2 is long and consists of several parts. The first lemma allows us to consider different parts separately, and then combine them to get the joint generator. The second lemma provides us some convenient inequalities. The last lemma, Lemma A.3, provides us most parts of the generator, and it is the building block for the construction of the coupling.
Firstly, we observe that these are jump processes. Because the generators are sums of terms corresponding to different jumps for the same type of G =L,L L , orL R , we can combine two pairs of coupled processes, in the sense of adding their generators, to obtain a new pair of coupled processes. The main requirement is that couplings exist for any ordered deterministic initial configurations.
Lemma A.1. Let Ω 1 , Ω 2 be two joint generators for two pairs of auxiliary processes. Suppose that these two pairs of auxiliary processes are coupled via Ω 1 , Ω 2 (see Definition 4.1) for any (deterministic) W 0 ≥ X 0 . That is, Then, the combined auxiliary processes U t and V t , starting from W 0 ≥ X 0 , are also coupled via the joint generator Ω = Ω 1 + Ω 2 . That is, We can use either p(.) or p(., .) in this context, and generators G, G can be the same.
Proof. By assumption, the condition for the marginals is immediate from the forms of the generators (4.8) (4.11) and (4.10). We need to check the first condition.
By arguments in the proof of Theorem 2.5.2 [11], to show U t ≥ V t , we need to show the closed set F 0 = {( U , V ) : U ≥ V } is an absorbing set, which can be checked via showing: Usual interpolation arguments allow us to get P (F igure3 U t ≥ V t , for all t) = 1.
Lastly, by the assumption that two pairs of auxiliary processes are coupled via Ω 1 , Ω 2 for any W 0 ≥ X 0 , we get that (without any computation) and Ω1 F0 ( W 0 , X 0 ) is a sum of differences, which have the same sign as Secondly, we observe four monotone functions on the configuration space by comparing the configurations before and after the tagged particle jump with shifts of labels or not. See Fig. 2,3 for examples.
Lemma A.2. Let z>0. If a jump of the tagged particles by z or −z is possible, we have As a consequence, there are two generators Ω 0,R and Ω 0,L , such that for any X 0 ≥ Y 0 , we can couple X t = ( X 0 ,L R , 0, q) Proof. We will prove equations (A.2) and define Ω 0,R , via which we can couple two auxiliary processes X t = ( X 0 ,L R , 0, q) Y t = ( Y 0 ,L R , 0, 0) for any initial X 0 ≥ Y 0 . The other case is similar.
By (4.2) and (4.9), we check coordinates, Then it is immediate to see that the generator Ω 0,R defined below works, since under this generator, X t is increasing in t while Y t is constant in t, Thirdly, we see that given X ≥ Y , whenever the i-th particle in Y jumps by z > 0, we can move the i-th particle in X by z ≥ 0, such that ordering is preserved after relabeling, T i,z X ≥ T i,z Y . This is the primary step for constructing couplings in Theorem A.4,and we will prove this in the next lemma. Once we can couple positive jumps of the i-th particle in the slower process by positive jumps of its corresponding particle in the faster process, we only need to assign jump rates according to different pairs z, z . The assignment is possible by Assumptions A'1,A'3.. See (A.17),(A.18) in Theorem A.4 for assignment in detail.
The choice of z can be made so that every nonzero z corresponds to a unique z in (0, R] satisfying Y ∈ A i,z . Proof. We first describe how to find z , and then we show (A.6) by considering a simple case and the general case. Without losing generality, we assume that i = 0 in figures below. Suppose there are exactly k holes in Y between Y i and Y i + R: H 1 , . . . , H k . We label them in a descending order: Step1. Define z l , l = 1, 2, . . . , k inductively by, That is, for l > 1, if z l > 0, H l = X i + z l is the right-most hole in X which is to the left of both H l−1 in X and H l in Y . (H l might equal H l , but H l < H l−1 .) See Figure 6 for an example. In this example, i = 0, R = 8, z 1 = 7, z 2 = 3, Step2. We consider a simple case first. We assume that X i = Y i and the numbers of particles on the fixed interval [X i , X i + R] in both X, Y are identical. In view of (4.3), and let we have Then, it is immediate to see that the numbers of holes on [X i , X i + R] in both X and Y are the same as k = R − I 1 + i. Following (A.7), (A.8), (A.9), we actually label all the holes in X in the descending order, and pair holes in X, Y with H l ≤ H l , for l = 1, . . . , k.
We emphasize that, when the numbers of particles on [X i , X i + R] in X, Y are the same, because X ≥ Y , (A.11) is equivalent to "holes are paired via a vertical line or a southwest line", and (A.11) is also equivalent to X j ≤ Y j for all X j , Y j on [X i , X i + R]. See Figure 7. In this example, R = 8, i = 0, I 1 = I 2 = 5.
After a jump, there is a relabeling of holes according to the previous rule. Hence (A.11) is preserved. Indeed, after the jumps to Y i + z and X i + z , we delete a line connecting Y i + z and X i + z , and add a vertical line connecting the initial positions of X i and Y i , see Figure 7. Since only particles on [X i , X i + R] are affected by the jumps, and the number of particles on [X i , X i + R] are the same for T i,z Y , T i,z X, we can conclude that T i,z Y ≤ T i,z X from the "new" (A.11).
Step3. For the general case, we can assume that X i ≤ Y i + R. Otherwise, if Y i + R < X i , we can easily find that z = 0 and T i,z Y ≤ X = T i,0 X.
To get (A.13) for all i ≤ j ≤ I 2 , we can add artificial particles and holes on [Y i , Y i + R + I 1 − I 2 ] for X and Y as follows to get two new configurations X and Y (restricted to this interval) with the same number of particles.
(a) Replace all particles on [Y i , X i ) in X with holes. Move the i-th particle in X from X i to Y i .
(b) Replace all holes on (Y i + R, Y i + R + I 1 − I 2 ] with particles for X.
(c) Replace all particles on (Y i + R, Y i + R + I 1 − I 2 ] with holes for Y .
The new configurations on [Y i , Y i + R + I 1 − I 2 ] have the same numbers of holes, too. We can pair holes in X and Y in the descending order (uniquely). Holes on (Y i +R, X i +R] in Y are the only additional holes added to Y , and they are added to match the number of holes in X and Y . We don't need to consider these additional holes and their corresponding holes in X , and therefore, we can keep the labels of the original holes in Y and label the additional holes as j-th holes, with non-positive indices 0 ≥ j ≥ I 2 −I 1 +1. We denote by H j = Y i + z j the j-th corresponding hole in X , including the ones with Then we define the generator Ω + by its action on a local function F by: if X ≥ Y , (b) for X Y , it's obvious that Ω + 1 F0 ( X, Y ) ≥ 0 since each term is nonnegative.
We only need to show the sum is finite. Notice that only finitely many terms in (A.22) are positive. Since T i,s changes finitely many X i , if one term T i,s X ≥ Y holds while X Y , T i ,s X ≥ Y holds for finitely many pairs i , s . Similarly, only finitely many terms in (A.21) are positive. Therefore, Ω + 1 F0 ( X, Y ) ≥ 0.
To show the marginal conditions, we will check for F 2 ( X, Y ) = H 2 ( Y ), and the other follows directly from T i,0 X = X, (A.17) and (A.18). On F c 0 = {( X, Y ) : X Y )}, clearly Ω + H 2 ( X, Y ) =LH 2 ( Y ). We only need for every X ≥ Y , Ω + F 2 ( X, Y ) = i∈Z,0<z≤R,0≤s≤R The fourth equality is due to Lemma A.3, which implies that there is exactly one s in [0, R] such that s = C( X, Y , i, R, z).
2. The second part is an application of Lemma A.1, the change of variable argument in (4.5), (4.6), (4.7),and the first part.
3. This is a consequence of the second part, Lemma A.1 and Lemma A.2. Take Ω R = Ω 0,R + Ω, and Ω L = Ω 0,L + Ω. We will show the first case, and the other is similar: By the second part of Theorem A.4, we have a generator Ω, via which we can couple auxiliary processes X t = ( X 0 ,L, p c , 0) ( Y 0 ,L, p c , 0) = Y t , for any X 0 ≥ Y 0 . By Lemma A.2, we can also find a generator Ω 0,R to couple auxiliary processes W t = ( X 0 ,L R , 0, q) ( Y 0 ,L, 0, 0) = Z t , for any X 0 ≥ Y 0 . Notice that X t is also ( X 0 ,L R , p c , 0). By Lemma A.1, we can use generator Ω R = Ω 0,R + Ω to couple U t = ( X 0 ,L R , p c , q) ( Y 0 ,L R , p c , 0) = V t for any X 0 ≥ Y 0 .
In They require Lemma A.2 and assumption A*3. The Lemma A.2 depends on the finite range R and X ≥ Y , while the latter is an assumption on the jump rates. We can easily modify p i,s,z , p i,s,0 and Ω + to couple two modified auxiliary processes defined in section 8. Then we can find a joint generatorΩ to couple modified auxiliary processes ( X t , I t ) = ( X 0 , p, q, I t ) and ( Y t , i t ) = ( Y 0 , p, p, i t ) for any initial condition X 0 ≥ Y 0 , in the sense X t ≥ Y t , for all t ≥ 0, ΩF 1 X, I, Y , i =L p,q H 1 X, I , ΩF 2 X, I, Y , i =L p,p H 2 Y , i , for any local functions F 1 X, I, Y , i = H 1 X, I and F 2 X, I, Y , i = H 2 Y , i .
Proof. We will give the joint generatorΩ =Ω + +Ω − by writing outΩ + andΩ − , which will have the same form in terms of p j,s,z . The rest is to check conditions, which follows almost the same arguments as those in the first part of Theorem A.4, and we will omit it.
We first define the modifiedΩ + by modifying p j,s,z , p j,s,0 from (A. 17 We can see bothΩ + andΩ − have the same form in terms ofp j,s,z . Then, we obtainΩ bỹ Ω =Ω + +Ω − .