About the Long Range Exclusion Process - CiteSeerX

Markov Processes Relat. Fields 10, 457–476 (2004)

Markov MP Processes &R F and Related Fields c Polymat, Moscow 2004

About the Long Range Exclusion Process H. Guiol Institut de Math´ ematiques, EPFL, 1015 Lausanne, Suisse. E-mail: [email protected] TIMC-TIMB Facult´ e de M´ edecine, Pav. D, 38706 La Tronche cedex, France E-mail: [email protected] Received January 26, 2004

Abstract. Introduced by Spitzer [23] and studied by Liggett [14] the Long Range Exclusion Process (LREP ) is an interacting particle system with truly long range interaction. Informally speaking: each particle on a lattice hops at independent random times following instantaneously a random dynamic on the lattice until finding a vacant site (if any). These instantaneous, potentially long jumps prevent the process to have the Feller property. In this paper we review the main results about the LREP including recent developments obtained in [11,24] and [4]. New results on Feller approximations and about the regularity set of the LREP are also provided. Finally we briefly discuss some connections of the LREP with the discrete Hammersley process introduced in [8] and the sandpile process in infinite volume developed in [18] and [17]. Keywords: infinite particle systems, non-Feller process, long range exclusion, invariant measures, formal generator AMS Subject Classification: 60K35

1. Introduction In a famous paper Spitzer [23] initiated the following program: exhibit and study “Markovian particle motions (of necessity with interaction between particles) under which point processes other than the Poisson point process are invariant ”. For such a purpose he described several models of interacting particle systems among which the zero range process: see [3]; the simple exclusion process: for an overview see [16] and [15]; and a variation of these two with long range interaction. In contrast to the previous two, Spitzer only defined this long

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range model on a finite set and pointed out the difficulty of constructing it, and therefore studying it, in infinite volume. This difficulty was overcome by the use of coupling techniques in a very nice and deep paper by Liggett [14] where the main general properties and the name Long Range Exclusion Process (LREP ) were popularized. A list of challenging open problems were then stated: see Section 6 in [14]. Some answers have been given in the following papers: Zheng [24], who studied a generalization of the (LREP ) when the underlying transition matrix is positive recurrent; Guiol [11] characterized all the invariant and translation invariant measures for the process when the transition matrix is an irreducible random walk on Zd . In a more recent paper Andjel and Guiol [4] provide a general criterion linking the invariant measures with the pseudo generator and apply it to characterize all invariant measures for some random walks on Z. The plan of the paper is the following: in Section 2 we review and discuss Spitzer’s model in finite volume, including Zheng’s result for positive recurrent transition matrices (also valid in infinite volume). Section 3 will provide three approaches for the construction of the LREP in infinite volume: the first is based on coupling, the two others are based on approximations through Feller processes. The first two are due to Liggett [14], the last one is original. Section 4 is devoted to the regularity set of the process: the main properties are due to Liggett [14], but Theorem 4.3 and its proof are new. Equilibrium results due to Liggett [14], Guiol [11] and Andjel and Guiol [4] are discussed in Section 5. Finally in Section 6 we discuss some links with two other processes: the discrete Hammersley process: see [8] and the one-dimensional infinite-volume sandpile process: see [18]. 2. Spitzer’s finite model Let S be a finite set, called the set of sites, and consider the motion of n particles, 0 ≤ n ≤ |S|, of an incompressible gas on S. Let P = p(x, y) x,y∈S be an arbitrary irreducible doubly stochastic matrix on S. Suppose that all particles of the gas are of the same type (no spin), multi-occupation of sites is prohibited (i.e. at most one particle per site) and there are no external forces acting on the system. The model: (i) Each site of S is endowed with a random clock which rings according to a Poisson process, with mean one, independently from the other clocks. (ii) Each time a particle stands on a site for which the corresponding clock rings the particle enters in motion. According to Spitzer’s description: instantaneously the “particle which is in motion is so strongly repelled by occupied sites that it keeps on going (according to P ) until, for the first time, it reaches an

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unoccupied site (which may be the site it started from)” and stands there until this new site’s clock rings, and so on. Let X = {0, 1}S be the state space of the process. For η ∈ X and x ∈ S we interpret η(x) = 1 (resp. 0) as the presence (resp. absence) of a particle at site x. In the sequel ξtη ∈ X will denote the state of the process at time t starting originally from state η ∈ X, i.e. ξ0η ≡ η. As S is finite ξtη is completely determined by the finite subset At = {x ∈ S : η ξt (x) = 1} ⊂ S. Thus it is enough to study the Markov process (At )t≥0 and since we consider a fixed numberPK ≤ |S| of particles one can restrict the state space to XK = {η ∈ X : |η| := x∈S η(x) = K}. Using the ergodicity of such a process, Spitzer showed Theorem 2.1 (Spitzer [23]). Suppose P is irreducible and doubly stochastic on S. Then for any η, ζ ∈ XK lim P ξtη = ζ =

t→∞

−1 |S| . K

Remark 2.1. Let S = TdN = (Z/N Z)d be the discrete d-dimensional torus with N d points. To fix ideas 1/N represents the distance between neighboring sites on the macroscopic scale (the unit d-dimensional torus). With the same hypothesis on P as in Spitzer’s result, we can easily deduce that for each 0 ≤ K ≤ N d there exists a unique invariant measure νK,N concentrated on XK (known also as the canonical measure) with d N −1 K K −1 EνK,N (η(x)) = d = d ∀x ∈ TdN . N N K Observe that those measures have nice limiting properties: EνK,N η(x1 ) · · · η(xj ) − K/N d j = O (N 2d−1 )−1 .

This in particular implies that

lim

N →∞ K/N d →α

EνK,N (f ) = Eνα (f )

for every bounded cylinder function, where να is a product probability measure d on {0, 1}Z with density α. Provided that the limiting process corresponds to the desired infinite-volume process and passing over all the technical difficulties this might be seen as an heuristic argument to see that the να ’s are invariant for the

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infinite-volume dynamics. But this reasoning is far from being unconditionally correct. For instance the full configuration 1 (with one particle at each site) is not always invariant in infinite volume: when P corresponds to a transient birth and death chain Theorem 1.4. of [14] showed that this is not the case. This leads us to conclude that taking the classical thermodynamic limit is not necessarily the right technique and that a more careful study is needed. For finite S and more general transition matrix P , a complete result, attributed to Waymire, can be obtained. Theorem 2.2 (Corollary 3.16 in [14]). For any irreducible transition matrix P on the finite set S the product probability measure on X with marginal νπ η(x) = 1 =

π(x) , 1 + π(x)

(2.1)

x ∈ S, is invariant for the process ξt· whenever πP = π. The finite-volume version of the following result is just a corollary of the preceding result. Theorem 2.3 ((3) of Theorem 1.2 in [24]). Let P be a positive definite irreducible chain on a finite or at most countable S. Denote by νK the restriction of νπ to the set XK , i.e. νπ (· | XK ). By convention ν∞ ≡ ν1 is the Dirac measure on the full configuration. For any µ ∈ P(X) and any bounded continuous function f on X lim

t→∞

Z

E f (ξtη ) dµ(η) =

|S| X

K=0

µ(XK )

Z

f (ξ) dνK (ξ).

Comments. Theorem 2.2 may not surprise the reader familiar with Zero Range Processes (ZRP ) since in finite volume the dynamics of the LREP can be seen as a degenerate version of the zero range dynamics: in a ZRP multi-occupancy of site is allowed and the rate at which particles leave a site is a function g of the number of particles on that site. If we consider the degenerate case where g(k) = 0, if k = 0; 1 if k = 1 and +∞ for k ≥ 2, then the dynamics corresponds to the LREP . For ZRP it is known that πP = π implies that νeπ is invariant for the process, where νeπ is the product probability measure on ZS + with marginals νeπ η(x) = k =

1 π(x)k , Z(π(x)) g(1) · · · g(k)

where Z(·) is the normalizing partition function. However to construct a uniquely defined Markov process corresponding to the ZRP in infinite volume the rate function g must satisfy some regularity


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properties (which implies g(k) < ∞ for all k): see e.g. Section 1 of [3]. The invariance of the measures νeπ remains true for the ZRP in infinite volume but this is not the case for LREP : i) When P corresponds to a transient birth and death chain none of the νπ is invariant when πP = π (see Theorem 5.2 in [14]); ii) When P is a random walk on Zd , νπ is invariant if and only if π is constant (see Theorem 5.2 below). 3. The construction for |S| = ∞ From now on S is supposed to be a countable set. 3.1. Monotonicity The construction of the LREP in infinite volume is due to Liggett [14], for the convenience of the reader we recall here the approach given in [4]. Let any site x ∈ S independently be equipped with: i) a Poisson process (Nx (t))t≥0 and ii) for each n ∈ N, (Xkn,x )k∈Z+ a Markov chain with transition matrix p(x, y) such that X0n,x ≡ x. Assume that all these Poisson processes and Markov chains are defined on the same probability space Ω and are independent. Let Xf be the set of finite initial configurations: n o X Xf = η ∈ X : η(x) < ∞ . x∈S

The pathwise construction of the process starting from any η ∈ Xf is then provided: a particle at x ∈ S wait until the Poisson process Nx (t) jumps. If this jump is the nth jump of Nx (t), then the particle jumps to Xτn,x (ζ) where ζ is the configuration of the process just before the jump and τ (ζ) = inf r ≥ 1 : Xrn,x = x or ζ(Xrn,x ) = 0 .

Call ξsη the random configuration obtained at time s when the initial configuration was η. From the property of Poisson processes it follows that there exists a subset Ω0 of Ω with probability 1 on which for any initial η ∈ Xf 1) there are no simultaneous jumps of particles, 2) the total number of jumps of ξsη is finite on any finite time interval, 3) ξ0η ≡ η.

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To extend the construction to configurations with an infinite number of particles we will need the following partial order in X: we say that η ≤ ζ if η(x) ≤ ζ(x), for all x ∈ S. It is also easy to verify that the construction satisfies the following property: 4) if η, ζ ∈ Xf are such that η ≤ ζ, then on Ω0 we have ξsη ≤ ξsζ , for all s ≥ 0. For an arbitrary initial configuration ζ ∈ X and s ≥ 0 we define on Ω0 : ξsζ (x) =

lim

η↑ζ, η∈Xf

ξsη (x).

This limit exists by monotonicity and defines ξsζ for any initial ζ ∈ X. With this construction property 4) remains true for η, ζ ∈ X. Hence all ξsη are defined on the same probability space for all s ≥ 0 and all η ∈ X. The coupled process starting from (η, ζ) is defined as the process (ξsη , ξsζ ). In the rest of this paper P and E denote respectively the probability and the expectation operator of the space Ω0 . When computations involve auxiliary Markov chains or random variables, P and E denote respectively the probability and the expectation operator which apply to them. Finally, when we need to emphasize the initial condition of one of these chains we will write Px and Ex . Let S(t) be the semi-group, acting on the set of bounded measurable functions on X, given by: S(t)f (η) = Ef (ξtη ) and for any probability measure µ on X let µS(t) be the unique probability measure such that Z Z f dµS(t) = S(t)f dµ, for all bounded measurable f . Denote by I = {µ : µS(t) = µ for all t > 0} the set of invariant probability measures on X for the LREP . 3.2. Feller approximations 3.2.1. Liggett’s k-step exclusion processes Let k ∈ N. To approximate the LREP Liggett [14] introduced the following sequence of non-conservative processes whose generator applied to a bounded cylinder function f reads X Lk f (η) = qk (x, y, η) η(x)[1 − η(y)] [f (η x,y ) − f (η)] x,y∈S

+

X

x∈S

δk (x, η) η(x)[f (ηx ) − f (η)]

(3.1)


463

where η x,y (x) = η(y), η x,y (y) = η(x) and η x,y (z) = η(z) otherwise; η(z) if z 6= x, ηx (z) = 0 if z = x; Y σy qk (x, y, η) = Ex η(Xn ), σy ≤ σx , σy ≤ k ; n=1

δk (x, η) = E

x

Y k

η(Xn ), σx > k

n=1

and σy = inf{n ≥ 1 : Xn = y}. In words: a particle at site x waits a random exponentially distributed time with parameter one. Then it jumps and, as in the LREP dynamics, it is instantaneously repelled by occupied sites, except that only a maximum of k successive instantaneous repulsions is permitted (site x itself is considered vacant during the motion). If no vacant site has been found after k successive attempts, then the particle simply disappears from the system. For all k ∈ N the construction of these processes is classic (see Chapter I of [16]): The closure of Lk is the generator of the semigroup Tk (t) which has the Feller property (i.e. Tk (t) maps C(X), the set of continuous functions on X, into C(X)) provided that X sup p(x, y) < ∞ (3.2) y∈S x∈S

which we will call the non-explosive condition for P , i.e. it guarantees a finite rate (for the discrete chain associated to P ) for arrivals at any site. Observe that this is automatically verified for doubly stochastic P (and thus for any translation invariant P ). The links between Lk and Tk are provided via the Hille – Yosida theorem (see e.g. Theorem 2.9 in Chapter I of [16]). To state the approximation result we need some notation. For η, ξ ∈ X we write η ≤ ξ if the two configurations verify η(x) ≤ ξ(x) for all x ∈ S. Let M be the set of monotone bounded functions f on X such that η ≤ ξ implies f (η) ≤ f (ξ);

and

f (ξ) =

lim

η↑ξ, η∈Xf

f (η).

The semigroups {Tk (t)}k∈N have the following nice monotonic behavior. Theorem 3.1 ([14], Theorem 3.9). For all t > 0, (a) Tk (t)f ≤ Tk+1 (t)f ≤ S(t)f (b)

lim Tk (t)f = S(t)f

k→+∞

for any f ∈ M ∩ C(X);

for any f ∈ C(X).

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Remark 3.1. In particular this shows that when a particle fails to find an empty site (in infinite volume) it simply disappears for the LREP dynamics. Unfortunately this approximation cannot be carried out easily at the level of infinitesimal generators. Taking the limit (in k) of the generators Lk gives the following formal generator L: For any bounded cylinder function f and η ∈ X X X Lf (η) = q(x, y, η)η(x)[1 − η(y)] [f (η xy ) − f (η)] + δ(x, η)[f (ηx ) − f (η)] x,y∈S

x∈S

(3.3) where for x 6= y q(x, y, η) = Ex

σY y −1 i=1

δ(x, η) = E

Y ∞ x i=0

η(Xi ), σy < σx , σy < ∞ ;

η(Xi ), σx = ∞ .

Note that the operator L is only a “pseudo” infinitesimal generator. The series above might not converge for some values of η even for a nice bounded cylinder function and a simple transition matrix P : take e.g. S = Z, p(x, x + 1) = 1, η ∈ X such that there exists y ∈ S such that η(x) = 1 for all x < y and η(y) = 0. Applying (3.3) to fy (η) := η(y) gives +∞ in the first summation. Furthermore the semi-group S(t) has not, in general, the Feller property (see the discussion after Theorem 4.4 below). This ruins the use of Hille – Yosida theory (or the equivalent Martingale problem approach) to associate the semigroup S(t) with the pseudo generator L. The following results are due to Liggett (see (a) of Theorem 3.17 and (c) of Theorem 3.19 in [14]). Part (a) is crucial for the proof of part (b) which provides a weak relation between the semigroup S(t) and the “formal generator” L. For any x ∈ S we denote by fx the function on X such that fx (η) = η(x) for all η ∈ X. Theorem 3.2. Suppose πP = π. Then (a) νπ S(t) is non-increasing in t, Z Z d (b) S(t)fx dνπ = Lfx dνπ S(t); where νπ is the product measure dedt fined by (2.1). Comments. (i) Note that part (b) is not enough to prove that L is a generator in L2 (νπ ).


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(ii) When k = 1 and P is the simple random walk on S = Zd Liggett’s 1-step exclusion process corresponds to the coalescing random walk process (see [7]). To our knowledge the case k ≥ 2 has not been investigated. It might be of interest to study its asymptotic properties for fixed k ≥ 2. However one has to note that the nice duality property, available when k = 1, between coalescing random walk and the voter model does not hold for k ≥ 2. 3.2.2. k-step exclusion processes k-step exclusion processes are conservative versions of Liggett’s k-step exclusion processes, in the sense that there is no loss of particles. The description of the process is the same as before with the notable exception that when no vacant site has been found after k attempts then the particle remains at its original position instead of disappearing. Under condition (3.2), one can construct a Markov semigroup Sk (t) with Feller property associated with the infinitesimal generator X Lk f (η) = qk (x, y, η)η(x)[1 − η(y)] [f (η x,y ) − f (η)] x,y∈S

where f is any bounded cylinder function on X. When k = 1 it corresponds to the classic simple exclusion process (see [16] and [15]). The cases k ≥ 2 have been studied in [12]. Remark 3.2. For the readers who are familiar with hydrodynamic we mention an interesting fact about these processes: For a given k ≥ 2 the corresponding flux function (i.e. the mean of the instantaneous flux of particles through the origin) is a non-convex non-concave function of the density. This provides entropic solutions of the associated hydrodynamic equation of different types than for the simple exclusion process (see [5, 9]). Theorem 3.1 implies that the limit in k (if this limit exists) of the k-step exclusion processes is stochastically larger than the LREP constructed by Liggett. Denote by ν1 the point mass measure on the full configuration 1. The measure ν1 is trivially invariant for all k-step exclusion processes but as quoted in Remark 2.1 in Section 2 this is not true in general for the LREP (see also (ii) in Remark 4.1 in Section 4). Furthermore Sk is not monotonic in k (see e.g. Remark 2.4 in [12]). Therefore obtaining a nice approximation result such as in the previous section runs into problems. However when P is a random walk on Zd , denoting by S the set of translation invariant probability measures on X, we obtain easily the following weak result. Theorem 3.3. Suppose p(y −x) = p(x, y) for all x, y ∈ Zd and let µ ∈ S. Then lim µSk (t) = µS(t).

k→∞

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Proof. For any transition matrix P (satisfying the non explosive condition) we have the obvious inequality Sk (t)f ≥ Tk (t)f for all t ≥ 0 and monotone f . Using then (b) of Theorem 3.1 one gets (a) of Lemma 3.1. For all t ≥ 0, (a) lim inf Sk (t)f ≥ S(t)f

for all

k

f ∈ M ∩ C(X).

(b) lim inf µSk (t) ≥ µS(t). k

Part (b) is an immediate consequence of µSk (t) ≥ µTk (t). Using that both P and µ are translation invariant we find µSk (t){η(x) = 1} = µ{η(x) = 1} = µS(t){η(x) = 1}.

(3.4)

The first equality is due to the translation invariance of µSk (t), the second equality is proved in Lemma 4.3 below. Now apply the classical Lemma 3.2 (Corollary 2.8, p.75 in [16]). Let µ1 ≥ µ2 denote two probability measures on X such that for all x ∈ Zd , µ1 {η(x) = 1} = µ2 {η(x) = 1}. Then µ1 = µ2 . This, part (b) of Lemma 3.1 and (3.4) imply Theorem 3.3.

2

In combination with part (a) of Lemma 3.1 and Fatou’s Lemma this gives immediately the following d

Corollary 3.1. Let µ be a translation invariant probability measure on {0, 1}Z . Then for any f ∈ C(X) lim Sk (t)f (η) = S(t)f (η)

µ -a.s.

k

Comments. (i) Observe that for more general probability measures µ this is not true: take for instance S = Z and P totally asymmetric (say to the right) and let µ0,1 be the product measure with marginal µ0,1 {η(x) = 1} = IN (x). Then obviously µ0,1 Sk (t) = µ0,1 > µ0,1 S(t) → ν0 for any t > 0 and k ≥ 1. (ii) In dimension 1, when P is nearest neighbor one should take advantage of the pushing effect (see e.g. [5] and [9] see also [2] in a different setting) of the k-step exclusion process. A monotonicity argument (in k) can be derived on the order of particles which is preserved under the pushing interpretation. This leads, under suitable conditions on the set of initial configurations, to a limit process which also corresponds to the LREP . This was observed in a private conversation with T. Sepp¨ al¨ ainen. This would permit to prove the hydrodynamic behavior of the one-dimensional totally asymmetric LREP through the equivalent result for the k-step exclusion processes (see also the discussion in Subsection 6.1).


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4. The regularity set Lemma 4.1 (Lemma 2.2 of [14]). (a) Let T f (η) = lim t↓0 S(t)f (η); this limit exists for all f ∈ C(X) and η ∈ X; (b) T f ≥ f for all f ∈ M ∩ C(X); (c) T f (1) = f (1) for all f ∈ C(X). The operator T permits to define a “nice” set of configurations, denoted by D, that will be called the regularity set of the LREP , D := {η ∈ X : T f (η) = f (η)

∀f ∈ C(X)}.

Lemma 4.2. D = {η ∈ X : T fx (η) = fx (η) ∀x ∈ S}, where fx (η) = η(x). Proof. One inclusion is obvious. For the other one, first observe that each f ∈ C(X) is the uniform limit of finite combinations of cylinder functions fR (η) = Q x∈R η(x), where R is a finite subset of S. From (b) of Lemma 4.1 one gets T fR ≥ fR . Now by Cauchy – Schwarz inequality S(t)f{x,y} (η) = E ξtη (x)ξtη (y) 1/2 1/2 1/2 ≤ E ξtη (x) E ξtη (y) = S(t)fx (η) S(t)fy (η) ; which in turn gives

1/2 1/2 T f{x,y}(η) ≤ T fx (η) T fy (η) = fx (η)fy (η) = f{x,y}(η)

where we have used the hypothesis in the first equality. An inductive argument gives T fR ≤ fR which in turn implies the lemma. 2 As observed by Liggett points in B := X \ D are similar to the branch points that appear for Ray processes (see Chapter I of [22] for an overview on Ray processes). But the LREP also fails to have the Ray property: it is not even right continuous. The main properties of D are given by Theorem 4.1 ([14]). (a) lim t↓0 P(ξtη (x) 6= η(x)) = 0 for each x ∈ S if and only if η ∈ D; (b) 1 ∈ D; (c) if η ≤ ζ ∈ D and ζ 6= 1, then η ∈ D;

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(d) P(ξt1 ∈ D) = 1 for all t; (e) if η ∈ D, then P(ξtη ∈ D), except possibly for countably many values of t; (f) if η ∈ B, then P(ξtη ∈ A) = P(ξt1 ∈ A) for all t > 0 and all measurable A. The importance of the set D, in terms of measure, is given by the two following results due to Liggett (see Theorem 3.18 in [14]). Since the proofs are short, we repeat the arguments for the convenience of the reader. Theorem 4.2. If µS(t) is non-increasing in t, then µ(D) = 1. R R Proof. By hypothesis S(t)f dµ ≤ Rf dµ for any R f ∈ M ∩ C(X). Thus from (a) and (b) of Lemma 4.1 we deduce T f dµ = f dµ. Therefore µ(D) = 1. 2 Combining this with (a) of Theorem 3.2 gives the first part of the following result; (b) is an immediate consequence of the previous result. Corollary 4.1. (a) If πP = P , then νπ (D) = 1 where νπ is the product measure with marginals as in (2.1). (b) If µ ∈ I, then µ(D) = 1. The following result is new: we prove that if P is a random walk on Zd , then D carries full measure for any translation invariant measure. Theorem 4.3. If p(y − x) = p(x, y) for all x, y ∈ Zd , then for all µ ∈ S = the set of translation invariant probability measures on X, µ(D) = 1. Recall that 1 denotes the full configuration (i.e. 1(x) = 1 for all x). We begin by showing that, starting from a translation invariant measure, the LREP preserves the density. Observe that this is not obvious since in the LREP dynamics particles may disappear at infinity in finite time (see Remark 3.1). Lemma 4.3. Suppose p(y − x) = p(x, y) for all x, y ∈ Zd . Let µ ∈ S, then µ {η(x) = 1} = µS(t) {η(x) = 1} , for all t ≥ 0. Proof. We begin by proving µS(t){η(x) = 1} ≤ µ{η(x) = 1}, for all t ≥ 0.

(4.1)


469

Using Hille – Yosida theorem, (3.1) and µ ∈ S one has Z Z Z d Tk (t)η(x) dµ = Lk η(x) dµTk (t) = − η(x)δk (x, η) dµTk (t). dt

(4.2)

Since 0 ≤ η(x)δk (x, η) and is monotone in η, part (a) of Theorem 3.1 gives Z Z 0 ≤ η(x)δk (x, η) dµTk (t) ≤ η(x)δk (x, η) dµS(t). (4.3) Writing the integral form of Hille – Yosida theorem for Tk (t) one has Z

Tk (t)η(x) dµ = µ{η(x) = 1} +

Z Zt

d Tk (s)η(x) ds dµ. ds

0

Hence by the Fubini theorem (since |Lk η(x)| is bounded) and by (4.2) we derive Z

Tk (t)η(x) dµ = µ{η(x) = 1} −

Zt Z 0

η(x)δk (x, η) dµTk (s) ds.

(4.4)

From this it is easy to see that µTk (t){η(x) = 1} ≤ µ{η(x) = 1}. Thus taking the limit gives (4.1) by part (b) of Theorem 3.1. We now prove the converse inequality. Using (4.3) and (4.4) one has µTk (t){η(x) = 1} ≥ µ{η(x) = 1} −

Zt Z

η(x)δk (x, η) dµS(s) ds.

0

Hence using again part (b) of Theorem 3.1 and the monotone convergence theorem we obtain µS(t){η(x) = 1} ≥ µ{η(x) = 1} −

Zt Z 0

η(x)δ∞ (x, η) dµS(s) ds

Q∞ where δ∞ (x, η) = Ex n=1 η(Xn ), σx = ∞ . As 1(x) ≥ η(x)δ∞ (x, η) for any η ∈ X and x ∈ Zd one has µS(t){η(x) = 1} ≥ µ{η(x) = 1} −

Zt 0

for all t ≥ 0 and µ ∈ S.

µS(s){1} ds,

(4.5)

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H. Guiol

Let f (t) := µS(t){η(x) = 1} and g(t) := µS(t){1}. Since µ ∈ S, we claim that there exists µ′ ∈ S such that µ′ {1} = 0 and µS(t) = g(t)ν1 + [1 − g(t)] µ′ for any t ≥ 0. Since S(·) is a semi-group and ν1 ∈ I by Corollary 4.4 of [14], one can rewrite the preceding equality as µS(t + s) = g(t)ν1 + [1 − g(t)] µ′ S(s)

(4.6)

for all s, t ≥ 0. Applying this to configuration 1 we get g(t + s) = g(t) + [1 − g(t)] µ′ S(s){1},

(4.7)

which proves that g(·) and µ′ S(·){1} are non-decreasing. Applying (4.6) to the event {η(x) = 1} gives f (t + s) = g(t) + [1 − g(t)] µ′ S(s){η(x) = 1}, for all s, t ≥ 0. Then applying (4.5) to µ′ and using this in the previous equality gives Zs ′ ′ f (t + s) ≥ g(t) + [1 − g(t)] µ {η(x) = 1} − µ S(u){1} du 0

for all s, t ≥ 0. Since f (t) = g(t) + [1 − g(t)] µ′ {η(x) = 1} and µ′ S(u){1} is non decreasing on [0, s], f (t + s) ≥ f (t) − [1 − g(t)] sµ′ S(s){1} = f (t) − s[g(t + s) − g(t)], for all s, t ≥ 0. This leads to f (s) ≥ f (0) for all s ≥ 0 which is the required inequality. This ends the proof of Lemma 4.3. 2 Proof of Theorem 4.3. By Lemma 4.2 it is enough to show that µT fx = µfx , where fx (η) = η(x). From Lemma 4.3 we deduce Z Z fx dµ = lim S(t)fx dµ. t↓0

By Fatou’s lemma the right hand side is bigger or equal to Z Z Z lim S(t)fx dµ := T fx dµ ≥ fx dµ. t↓0

The last inequality holds by (b) of Lemma 4.1 since fx is monotone continuous. 2


471

To conclude this section we invoke once again some results borrowed from Liggett showing that under some mild conditions D is a proper subset of X and the process does not have the Feller property. Theorem 4.4 (Theorem 3.28 in [14]). Suppose ν1 is invariant for the process

(4.8)

and P is such that for every x ∈ S X Py (σx < σy ) = +∞.

(4.9)

y∈S

Then if η has a finite positive number of holes, i.e. 0 < then η ∈ / D.

P

x∈S [1

− η(x)] < ∞,

Following Liggett we now recall why the conclusion of this theorem implies that the process ξt· does not have the Feller property. Suppose (4.8) P and(4.9) hold, and take a sequence {ηn }n∈N of configurations such that 0 < x∈S [1 − ηn (x)] < ∞ for all n and such that ηn (x) → 0 when n → ∞ for each x ∈ S. Then from the previous theorem and item (f) of Theorem 4.1 for all n ∈ N P ξtηn (x) = 1 = P ξt1 (x) = 1 = 1 ∀t > 0.

But of course P(ξt0 (x) = 1) = 0, where 0 denotes the empty configuration. This proves that S(t)fx is not in C(X) thus the process is not Feller.

Remark 4.1. (i) Observe that (4.8) and(4.9) are satisfied when P is an arbitrary random walk on Zd or is a recurrent birth and death chain on Z+ with inf x p(x, x + 1) > 0. (ii) If P is transient, Liggett [14] has shown that if the series in (4.9) are finite then ν1 is not invariant for the process. P Open questions. (a) When does x [1 − η(x)] = +∞ imply η ∈ D? (b) Find (or prove one cannot find) η ∈ / D such that there exist t, s > 0 for which ξtη ∈ D η and ξt+s ∈ / D. 5. Equilibrium results 5.1. General results For a Markov semi-group with the Feller property, like e.g. Sk (t), with associated generator Lk , it is well-known R that a necessary and sufficient condition for a measure µQto be invariant is Lk fR dµ = 0 for any finite subset R of S where fR (η) = x∈R η(x). Unfortunately this result is not available for the LREP . However recently we proved the following

472

H. Guiol

Theorem 5.1 ([4]). Suppose that P is irreducible and ν1 ∈ I. Let Qµ ∈ I be such that µ({1}) = 0 and for any finite subset R ⊂ S let fR (η) = x∈R η(x). Then, the series defining LfR (η) converges µ-a.e. Moreover LfR (η) ∈ L1 (µ) and R LfR dµ = 0.

Remark 5.1. (i) The converse of this theorem is a challenging question (see open question (a) in Subsection 5.3 below). (ii) The assumption ν1 ∈ I is always verified for a general random walk on Zd . This is just a technical hypothesis, the result can be proved without it (Andjel, private communication). However it is believed that ν1 S(t) → ν0 when ν1 ∈ / I (see open problem (e) in Section 6 and Example 3.5 (a) in [14]) thus one would have I = {ν0 } in this case. There are two kinds of other general results available: i) The ones that indicate when ν1 is or is not invariant (we have already seen some of them in the previous sections, see also Theorem 3.1 in [14]). ii) The ones that show that some νπ , with πP = π, are not invariant (see (b) of Corollary 3.4 and Corollary 3.26 of [14]). 5.2. The random walk case Let thus S = Zd and P be a random walk on Zd in this subsection. The starting point is the observation that the product measures with constant density are invariant. Theorem 5.2 (Theorem 4.2 and Corollary 4.4 [14]). Suppose p(x, y) = p(y − x) for all x, y ∈ Zd and πP = P . Then νπ ∈ I if and only if π is constant and in particular ν1 ∈ I. Denote by (νρ )ρ∈[0,1] the one parameter family of Bernoulli product measures, i.e. νρ (η(x) = 1) = ρ for any x ∈ S = Zd . The following two theorems require that S is an integer lattice and p(x, y) is the transition matrix of an irreducible random walk on that lattice. Theorem 5.3 (Th´ eor` eme 3.2, [11]). Suppose p(x, y) is the transition matrix of an irreducible random walk on Zd . Then the set of invariant and translation invariant measures coincides with the closed convex hull of {νρ : 0 ≤ ρ ≤ 1}. This result was first obtained in [11] relying essentially on the approximations with Liggett’s k-exclusion processes. However, thanks to the coupled version of Theorem 5.1 a more natural proof can be derived as explained in [4]. Applying once again the coupled version of Theorem 5.1 it is possible to show that in some cases all the invariant measures are convex combinations of the νρ ’s (0 ≤ ρ ≤ 1).


473

Theorem 5.4 ([4]). Suppose X = {0, 1}Z and either i) P is the transition P matrix of an irreducible random walk on Z such that P x |x|p(x) < ∞ and x xp(x) = 0; or ii) p(x, x + 1) := p = 1 − p(x, x − 1) for all x ∈ Z. Then the set of invariant measures coincides with the closed convex hull of {νρ : 0 ≤ ρ ≤ 1}. Remark 5.2. Unlike the simple exclusion process, all the invariant measures of the one dimensional nearest neighbor LREP are translation invariant. 5.3. Some open questions R (a) Show a converse of Theorem 5.1: under what condition(s) Lf dµ = 0 does imply µ ∈ I? This would permit to solve the following question: (b) Prove that να is invariant for the infinite-volume dynamics when P is doubly stochastic (this is open problem (c) in Section 6 of [14]). (c) Does P recurrent implies ν1 invariant? If furthermore πP = π, is νπ invariant? (these are open problems (b) and (c) in Section 6 of [14]). Zheng [24] has answered positively to these conjectures when P is positive recurrent. 6. Links with other processes 6.1. The discrete Hammersley process Ferrari [8] introduced the following discrete version of the so called Hammersley process (see [1] or/and [21] for the main properties of the continuous process). Let η ∈ Y = {0, 1}Z and f be a local function. Define Hf (η) =

X

x,y∈Z

1y