- Email: [email protected]
Ergodic theory of the symmetric inclusion process
Available online at www.sciencedirect.com
ScienceDirect Stochastic Processes and their Applications 126 (2016) 3480–3498 www.elsevier.com/locate/spa
Ergodic theory of the symmetric inclusion process Kevin Kuoch a,∗ , Frank Redig b a Johann Bernoulli Institute, Rijksuniversiteit Groningen, Postbus 407, 9700AK Groningen, The Netherlands b Delft Institute of Applied Mathematics, TU Delft, Mekelweg 4, 2628CD Delft, The Netherlands
Received 31 August 2015; received in revised form 3 May 2016; accepted 4 May 2016 Available online 27 May 2016
Abstract We prove the existence of a successful coupling for n particles in the symmetric inclusion process. As a consequence we characterise the ergodic measures with finite moments, and obtain sufficient conditions for a measure to converge in the course of time to an invariant product measure. c 2016 Elsevier B.V. All rights reserved. ⃝
Keywords: Interacting particle systems; Simple inclusion process; Invariant measures; Coupling
1. Introduction In [12, Chapter VIII], a rather complete ergodic theory is given for the symmetric exclusion process (SEP). In particular, the only extremal invariant measures for the SEP are Bernoulli measures with constant density. This complete characterisation of the set of invariant measures is quite exceptional and follows from the fact that the SEP is self-dual. As a consequence, invariant measures can be related to bounded harmonic functions for the finite SEP. Then, by the construction of a successful coupling of the SEP with a finite number of particles, it is shown that all bounded harmonic functions are constant, i.e. depend only on the number of particles. From this in turn, one can conclude that all invariant measures for the SEP are permutation
∗ Corresponding author.
E-mail addresses: [email protected] (K. Kuoch), [email protected] (F. Redig). http://dx.doi.org/10.1016/j.spa.2016.05.002 c 2016 Elsevier B.V. All rights reserved. 0304-4149/⃝
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3481
invariant, from which one derives by the De Finetti theorem that they are convex combinations of Bernoulli measures. In [6,7] an attractive version (in the sense of having an attractive interaction between the particles) of the SEP is introduced and called the simple inclusion process (SIP). In the SIP, particles perform nearest-neighbour jumps according to a simple symmetric random walk and interact by “inclusion jumps”, where pairs of neighbouring particles jump to the same site at rate 1. This analogy between SIP (attractive interaction) and SEP (repulsive interaction) becomes even more apparent in [7], where it is shown that the SIP satisfies the analogue of Liggett’s comparison inequality [12, Chapter VIII, Proposition I.7] for the evolution of positive definite symmetric functions. The expectation at time t > 0 of such a function in the course of the evolution of n SIP-particles is larger than in the course of the evolution of n independent random walkers. In particular this implies that a certain class of product measures is mapped by the evolution under the SIP to measures with positive correlations (as opposed to negative correlations in the SEP). In this paper, we want to investigate as much as possible the invariant measures of the SIP, i.e., understand its ergodic measures and their attractors. Since the number of particles is unbounded and we want make use of self-duality, we will have to restrict to a set of measures with all moments finite. The main problem is then to construct a successful coupling for two sets of n SIP-particles initially at different locations. This coupling seems possible thanks to the fact that as long as SIP-particles do not collide, i.e., are not at neighbouring positions, they behave as independent random walkers and these can be coupled by the coordinate-wise Ornstein coupling in any dimension. The idea of the coupling of SIP-particles comes from [4] combined with [13]: in [13], it is shown that inclusion particles √ and independent random walkers can be coupled in such a way that at time t they are o( t) apart; the period of time [0, (1 − δ)t] in which coupling according to [13] is used (stage 1) is then followed by a period of time [(1−δ)t, t] (stage 2) during which the coordinate-wise Ornstein coupling of independent random walkers is used both for the independent walkers as well as for the SIP particles. The only problem for this coupling to be successful is to estimate the probability of being coupled before a collision occurs. One can understand however that such a collision event is highly improbable (as √ t → ∞), as after a long time, the independent random walkers are much farther apart (O(√ t)) than the distance between the independent random walkers and their inclusion partners (o( t)). Once one has the successful coupling of SIP-particles, and as a consequence results on the structure of the invariant measures of the SIP, all these results can be transferred without effort to corresponding results for interacting diffusion processes to which the SIP is a dual process such as the Brownian Energy Process (BEP) and the Brownian Momentum Process (BMP). For particle systems with unbounded occupation numbers, in general only very little information is available on the structure of the set of invariant measures. See [1] for a set of general results under Lipschitz conditions on the jumps rates. In this work, both the symmetry and the self-duality property are crucial. Indeed, for an asymmetric version of the SIP based on the Lie-algebraic construction, recently introduced in [3], despite self-duality, there is no characterisation of invariant measures, and the translation invariant ones are known not to be product. On the other side, the naive asymmetric version of the SIP (putting different factor for the jumps in different directions keeping the same rates) has formally the same invariant product measures, but for that process we have no self-duality and therefore no proof of existence in infinite volume. The rest of our paper is organised as follows. In Section 2, we give basic definitions and set notations up, in Section 3, we prove the successful coupling, in Section 4, we characterise the
3482
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
class of ergodic so-called tempered (with finite moments) measures and in Section 5, we give sufficient conditions for a measure to converge to an invariant product measure. 2. Notations and definitions 2.1. The symmetric inclusion process We denote by p(., .) the transition probability of a simple symmetric nearest-neighbour random walk on the lattice Zd , i.e., 1 if ∥x − y∥ = 1, p(x, y) = 2d 0 otherwise. The simple symmetric inclusion process with parameter m > 0 (denoted SIP(m)) is an interacting particle system where particles perform independent random walks moving at rate m/2 and according to the transition probabilities p(., .) while on top of that, they interact by inclusion, i.e. each particle “invites” any other particle standing at a nearest neighbour position at rate 1 to join its site (invitations are always followed up). These “invitation jumps”, or “inclusion jumps”, create an attractive interaction between the particles. This has to be compared with the interaction between particles of the symmetric exclusion process (SEP) where jumps joining two particles at the same site are forbidden. Here, on the contrary, these jumps are encouraged. d More formally, the SIP(m) on Zd is a continuous-time Markov process (ηt )t≥0 ∈ NZ whose generator L acts on local functions f , i.e. depending only on a finite number of occupation variables, as m L f (η) = p(x, y)η(x) + η(y) f (η x,y ) − f (η) , (2.1) 2 d y:|y−x|=1 x∈Z
η x,y
d NZ
d
where ∈ stands for the configuration obtained from η ∈ NZ by moving one particle x,y from x to y, i.e. η = η − δx + δ y , where δx denotes the configuration with a single particle at x and none elsewhere; | · | stands for the ℓ1 -norm. The existence of the SIP with generator (2.1) is not guaranteed by general existence criteria such as [1]. Though, its existence follows from self-duality, see Section 2.3 and [5]. 2.2. Invariant product measures d
For λ ∈ [0, 1), define the homogeneous discrete Gamma product measure νλm on NZ whose marginals are given by 1 λk Γ m2 + k m νλ η(x) = k = (2.2) Z λ,m k! Γ m2 where Γ (·) denotes the Gamma function and m/2 1 Z λ,m = 1−λ is the normalising constant. These measures νλm are reversible and ergodic for the SIP(m), see [7] for more details. One of the main questions answered in the present paper is whether these are the only ergodic measures within a certain class.
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3483
2.3. Duality A duality relation is a link between a dual process with the process of interest in such a way that it allows to perform computations for one process in terms of the other one. This link is created via the duality function. For further details see [12, Chapter II, Section 4]. Definition 2.1 (Duality Relation). Suppose (ξt )t≥0 and (ηt )t≥0 are Markov processes on S1 and S2 respectively. Let D be a bounded measurable function on S1 × S2 . The processes (ξt )t≥0 and (ηt )t≥0 are said to be dual to one another with respect to D if Eξ D(ξt , η) = Eη D(ξ, ηt ).
(2.3)
If the processes (ξt )t≥0 and (ηt )t≥0 are the same, then we call (2.3) self-duality. In that sense, the SIP(m) is self-dual (see [6]) with duality functions D(·, ·), for a process ξ with finite configuration and η an arbitrary process, D(ξ, η) = d(ξ(x), η(x)) (2.4) x∈Zd
where
d(k, l) =
Γ m2 l! m (l − k)! Γ 2 + k
for k ≤ l for k > l for k = 0.
0 1
Self-duality of the SIP(m) therefore means that SIP(m)
EηSIP(m) D(ξ, ηt ) = Eξ
D(ξt , η),
(2.5)
SIP(m)
where Eη denotes the expectation of a SIP(m) starting from an initial configuration η and where ξ is a finite configuration (i.e., having a finite number of particles). The self-duality functions D(., .) and the reference measure νλm are naturally connected via λ |ξ | , (2.6) D(ξ, η)dνλm (η) = 1−λ where |ξ | denotes the number of particles in the finite configuration ξ . We refer the reader to [6,7] for the proof of the self-duality (2.5) and further details and properties of the SIP. By duality relations, we derive as well related results for interacting diffusions that are dual to the SIP: the Brownian momentum process and the Brownian energy process, see [7] and references therein. The main advantage of self-duality is that it allows us to study the SIP with infinitely many particles by studying the SIP with a finite number of particles. Indeed, to know the timedependent expectations of the polynomials D(ξ, η) it suffices to follow the evolution of the particles in the finite configuration ξ , and the initial configuration ξ . d
Remark 2.1. If both ξ and η ∈ NZ are finite configurations, then both processes (ξt )t≥0 and (ηt )t≥0 are well-defined. Moreover, by self-duality, one has from (2.6) that Eη D ξ, ηt = Eξ D ξt , η = pt (ξ, ξ ′ )D ξ ′ , η . (2.7) ξ ′ ∈Ωn
3484
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
If ξ is finite and η is infinite, for N ∈ N, denote η(x) if |x| ≤ N , (N ) η = 0 otherwise. Hence by (2.7), N → Eη(N ) D ξ, ηt is non-decreasing and one can define Eη D ξ, ηt = sup Eη(N ) D ξ, ηt N = pt (ξ, ξ ′ )D ξ ′ , η), ξ′
provided the above series converges, that is, if sup N Eη(N ) D ξ, ηt < ∞. By elementary Gaussian bounds on pt (., .), the above series is convergent if there exist some positive constants c and k such that η(x) ≤ c|x|k ,
for all x.
For such initial configurations, for all finite ξ , the expectations Eη D(ξ, ηt ) are well-defined and from this, we can define the process (ηt )t≥0 starting from η. For more details on such construction via self-duality, we refer the reader to [5, pages 11–14]. While exploiting the self-duality property, we necessarily restrict to starting measures having finite moments. Let us denote by P the set of all probability measures on the configuration d space NZ . Definition 2.2. We then consider the class Ptemp of so-called tempered probability measures defined as follows: Ptemp := µ : µ ∈ P : ∀n ∈ N : sup D(ξ, η)dµ(η) =: cn < ∞ (2.8) |ξ |=n
where cn satisfies the Carleman condition ∞
−1/n
cn
= ∞.
(2.9)
n=1
The condition (2.9) ensures that the moments D(ξ, η)dµ(η) characterise uniquely the measure µ, i.e., if µ and ν are two probability measures in Ptemp such that for all finite configurations ξ , D ξ, η dµ(η) = D ξ, η dν(η), then µ = ν. See [9, Corollary 3] for further details. Remark that by self-duality, a tempered measure remains tempered in the course of the evolution of the SIP. Indeed, if µ ∈ Ptemp then, by conservation of the number of particles in the finite SIP, denoting sup D(ξ, η)dµ(η) =: cn , |ξ |=n
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3485
we have, for |ξ | = n and t > 0, Eη D(ξ, ηt )dµ(η) = Eξ D(ξt , η)dµ(η) ≤ cn , hence, the time-evolved measure µt is tempered for all t > 0, with the same dominating constants cn , n ∈ N as the one of the starting measure µ. We are interested in characterising the invariant measures which are ergodic for the SIP and belong to Ptemp . d
Definition 2.3. For the SIP (ηt )t≥0 on NZ , d
i. A probability measure µ on NZ is invariant if whenever η0 follows the distribution µ then ηt has this same distribution µ for all t > 0. ii. An invariant measure µ is ergodic for the SIP if the process (ηt )t≥0 starting from η0 distributed according to µ is ergodic. Note that since (ηt )t≥0 is a Markov process, the ergodicity of µ is equivalent with (a) µSt = µ for all t > 0, (b) for all invariant functions f (that is, St f = f for all t > 0), f = f dµ µ-a.s, where St stands for the semigroup of (ηt )t≥0 . Furthermore, if µ is ergodic then the Birkhoff ergodic theorem holds: for all f ∈ L 1 (µ), 1 T f (ηt )dt → f dµ PSIP µ -a.s., T 0 and as a consequence, 1 T St f → f dµ T 0
µ-a.s.
Moreover, the set of ergodic measures is the set of extreme points of the set of invariant measures. We denote by I the set of invariant probability measures for the SIP, and by Itemp the set I ∩ Ptemp of tempered invariant measures. Furthermore, we call Ie the set of extreme points of I , i.e., the set of invariant and ergodic probability measures for the SIP. For a measure µ ∈ Ptemp , we denote its D-transform by µ(ξ ˆ ) := D(ξ, η)dµ(η). (2.10) Here, ξ varies in the set of finite configurations, which we denote henceforth by Ωn if ξ is a configuration which contains n particles. Note that ξ ∈ Ωn such that n = |ξ | can be identified with an n-tuple x = (x1 , . . . , xn ) via ξ=
n
δxi .
(2.11)
i=1
As a consequence µˆ can also be viewed as a symmetric function on ∪n∈N (Zd )n . We will therefore use both notations µ(ξ ) as well as µ(x) = µ(x1 , . . . , xn ).
3486
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
The following result is then a straightforward consequence of the self-duality of the SIP. Proposition 2.1. Let µ be a tempered probability measure. Then µ is invariant if and only if its D-transform µˆ is bounded harmonic for the SIP, i.e., if and only if for all t > 0 Eξ µ(ξ ˆ t ) = µ(ξ ˆ ).
(2.12)
Proof. Suppose that µ ∈ Itemp . Since µ is a tempered measure, by Fubini’s theorem, for all t > 0, Eξ µ(ξt ) = D(ξt , η)dµ(η) dµ(ξ ) = D(ξt , η)dµ(ξ ) dµ(η) = D(ξ, η)dµ(η). Conversely, suppose (2.12) holds for all ξ . Then, letting µt := µSt at time t > 0, starting from µ at time 0, D(ξ, η)dµt (η) = D(ξ, η)dµ(η). (2.13) Hence µ = µt for all t > 0, as tempered.
As a consequence, the study of ergodic measures in Itemp is reduced to the problem of identifying the set of bounded harmonic functions for the SIP. This is done via the construction of a successful coupling, which implies that bounded harmonic functions are constant. 2.4. Bounded harmonic functions and successful coupling Via the identification (2.11), one can see the evolution of n SIP-particles initially at positions x = (x1 , . . . , xn ) as a process X S (t) = (X 1S (t), . . . , X nS (t)) on (Zd )n so that for each 1 ≤ i ≤ n, X iS (t) keeps track of the location of the SIP-particle i started from xi . We denote by PSIP x its path space measure. A coupling of two copies of the SIP starting initially at different locations x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) is then defined as usual as a process {X S (t), Y S (t), t ≥ 0} with X S (0) = x, Y S (0) = y with first (resp. second) marginals {X S (t), t ≥ 0}, the SIP starting from x (resp. {Y S (t), t ≥ 0}, the SIP starting from y). Its path space measure is denoted by PSIP x,y where the hat stands for the joint distribution in the coupling. The coupling time is defined via τ = inf{t ≥ 0 : X S (s) = Y S (s) ∀s ≥ t},
(2.14)
where by convention inf(∅) = ∞. The coupling is successful if τ < ∞ PSIP x,y -almost surely for d 2n all (x, y) ∈ (Z ) . It is well known (see e.g. [12, Chapter 2]) that the existence of a successful coupling implies that all bounded harmonic functions are constant. 2.5. Ornstein-coupling of n-dimensional random walks Consider two d-dimensional random walks X(t) = (X 1 , . . . , X d )(t) and X(t) = ( X 1, ..., X d )(t), starting from X(0) = x and X(0) = x respectively. The Ornstein-coupling works as follows. First, let the random walks evolve independently component-wise, until the first time at which (X 1 (t))t≥0 and ( X 1 (t))t≥0 coincide. From that time on, the first components perform
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3487
the same jumps hence remain identical while other components continue to evolve independently. We continue like this until the second components coincide, and so on, until all components coincide. Because continuous-time n-dimensional random walks have d independent 1-dimensional random walk components and because the difference between two independent 1-dimensional random walks is still a 1-dimensional random walk (at twice the speed), the latter is recurrent and the coupling is successful. See [10, Chapter 2, Section 3] for more background. 2.6. Diffusion processes related to the SIP The SIP is related via duality to the BEP, a system of interacting diffusions with state space d d [0, ∞)Z and to the BMP, a system of interacting diffusions with state space RZ . See [6] for more details. This implies that many results on the invariant measures and characterisation of ergodic measures can be transferred to these processes. This transference is a consequence of the following proposition. d
Proposition 2.2. Assume {ζt : t ≥ 0} is a Feller process on the state space K Z with K a Polish space. Assume that the process is dual to the SIP with duality function D(ξ, ζ ). Then, let D Ptemp := µ : µ ∈ P and ∀n ∈ N, sup D(ξ, ζ )dµ(ζ ) < ∞ |ξ |=n
d
KZ
D , denote the corresponding set of tempered probability measures, and denote for µ ∈ Ptemp D is invariant for {ζ : t ≥ 0} µ(ξ ˆ ) = D(ξ, ζ )dµ(ζ ) its D-transform. Then we have µ ∈ Ptemp t if and only if µˆ is a bounded harmonic function for the SIP.
Proof. Let ξ be a fixed finite configuration. Assume µ is bounded harmonic for the SIP, then, and by duality, D(ξ, η)dµt (η) = Eη D(ξ, ηt )dµ(η) = ESIP ξ D(ξt , η)dµ(η) = ESIP µ(ξt ) ξ = µ(ξ ) = D(ξ, η)dµ(η). D are characterised by their D-transform, µ = µ for all t. Hence, since measures in Ptemp t The converse is straightforward.
3. Successful coupling for the SIP We introduce here and onwards the following notations for sets of SIP-particles X S (t) = (X 1S (t), . . . , X nS (t)) and Y S (t) = (Y1S (t), . . . , YnS (t)). We denote by x = (x1 , . . . , xn ), resp. y = (y1 , . . . , yn ), the initial locations of X S , resp. Y S , that is, X S (0) = x, resp. Y S (0) = y. In this section, we prove: Theorem 3.1. There exists a successful coupling for the SIP with finite number of particles, n i.e., for any initial locations x, y ∈ (Zd , there exists a process {(X S (t), Y S (t)); t ≥ 0} where
3488
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
{X S (t); t ≥ 0} and {Y S (t); t ≥ 0} are SIP processes starting respectively from x and y such that PSIP x,y {τ < ∞} = 1,
(3.15)
where τ denotes the coupling time, defined in (2.14). Likewise, standing as auxiliary processes, we introduce two sets of n independent random walkers, or IRW-particles, noted X I (t) = (X 1I (t), . . . , X nI (t)) and Y I (t) = (Y1I (t), . . . , YnI (t)), starting from x and y, respectively. Prior to the proof, we define the notion of collision. We say that a collision for SIP-particles happens at time t > 0 if two SIP-particles belonging to a same set are at nearest-neighbour positions at time t, i.e., {∃i ̸= j, 1 ≤ i, j ≤ n : |X iS (t) − X Sj (t)| = 1 or |YiS (t) − Y jS (t)| = 1}. It is important to note that the event of particles X iS (t) and Y jS (t) at neighbouring positions is not considered as a collision, i.e., collisions only happen within the same set of particles. Similarly, we define a collision for IRW-particles at time t > 0 if two IRW-particles belonging to a same set are at nearest-neighbour position at time t, i.e., {∃i ̸= j, 1 ≤ i, j ≤ n : |X iI (t) − X Ij (t)| = 1 or |YiI (t) − Y jI (t)| = 1}. Proof. The proof is split into two parts, first we consider the case when d ≥ 3 and next, the case when d ≤ 2. Transient case: d ≥ 3. We start with the simplest case d ≥ 3 where a continuous-time random walk Z (t) based on p(., .) is transient. More precisely, we have Pz (|Z (t)| > 1,
∀t ≥ 0) =: H (z) > 0,
where Pz is the distribution of {Z (t), t ≥ 0} starting from site z and H (z) → 1 when z → ∞. As a consequence, by the union of events bound, with positive probability, n IRW-particles X I (t) = (X 1I (t), . . . , X nI (t)) starting from initial position x = (x1 , . . . , xn ) such that |xi − x j | > R, for all 1 ≤ i ̸= j ≤, are large enough, will not collide. If during a lapse of time no collision happens, then the IRW-particles and their corresponding coupled SIP-particles perform exactly the same jumps. It is only when IRW-particles collide that their corresponding SIP partners can behave differently. Assume all initial positions satisfying |xi − x j |, |yi − y j | > R for all 1 ≤ i ̸= j ≤ n. Then with positive probability p(R), the IRW-particles starting from x will never collide and neither will the IRW-particles starting from y. Moreover, one has p(R) → 1 as R → ∞. The two sets of IRW-particles can be coupled by the coordinate-wise Ornstein coupling. Now we couple each set of IRW-particles with a corresponding set of SIP-particles via the coupling used in [13, Theorem 3.2], i.e., both sets perform the same random walk jumps, and inclusion jumps are only performed by the SIP-particles. In this coupling, when no collision happens for both sets of IRW-particles, the corresponding two sets of SIP-particles, starting respectively from x and y as well, perform exactly the same jumps as their IRW-particle partners, since IRW- and SIP-particles only behave differently when they collide. Therefore, since sets of IRW-particles are successfully coupled via Ornstein-coupling, so are their SIP-partners. Therefore, the two sets of SIP-particles can be coupled with positive probability bounded from below by p(R). To show that they can be coupled almost surely from arbitrary initial locations x
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3489
and y, let them first move for some time T > 0 and then start the coupling just described. With probability π(T ), close to one as T → ∞, any two SIP- or IRW-particles will be at distance larger than α(T ) where α(T ) → ∞ when T → ∞, and correspondingly, from then on they can be coupled with probability p(α(T )). Therefore, the probability that they cannot be coupled is bounded from above by (1 − π(T )) + (1 − p(α(T ))) which tends to zero as T → ∞. Recurrent case: d = 1, 2 To tackle the case d = 1, we first give an outline. Note that in the two-dimensional case the same arguments hold and we therefore omit the proof for d = 2. We follow the line of thought of [4]. The coupling proceeds in two stages: 1. First stage: in the time interval [0, (1 − δ)t] (where δ ∈ (0, 1) is fixed and t will tend to infinity eventually), the two sets of IRW-particles make the same jumps and the SIP partners follow according to the coupling of [13]. After this first stage, with probability close to one (as t → ∞) any two different IRW-particles within √the same set, as well as their corresponding SIP-particles partners will be at distance O( t). The distance between the independent √ random walkers and their corresponding SIP partner, on the contrary will be of order o( t). 2. Second stage: in the time interval [(1 − δ)t, t], as long as the two sets of IRW-particles neither the two sets of SIP walkers do not collide, they are coupled via the coordinate-wise Ornstein coupling. As long as the SIP particles do not collide, this is indeed a coupling, since then the SIP particles behave as independent random walkers. If such a collision does happen, then we say that we have a failed coupling attempt. Stage 1. During the time interval [0, (1 − δ)t], couple the two sets of IRW-particles so that they perform the same jumps, thus, for any s ≤ (1 − δ)t, n n (X I − Y I )(s) = |xi − yi | =: kn . i i
(3.16)
i=1
i=1
According to the coupling of [13, Theorem 3.2], two sets of SIP-particles X S and Y S , starting respectively from x and y, are coupled to two sets of IRW-particles X I and Y I , starting respectively from x and y as well, such that their positions satisfy n
|X iS (s) − X iI (s)| + |YiS (s) − YiI (s)| ≤ ψ(s)
(3.17)
i=1
with ψ(s) lim √ = 0, s
s→∞
(3.18)
with probability one (w.r.t. coupling distribution). By way of illustration, see Fig. 1. Gathering (3.16) and (3.17), it is now straightforward to see that for any δ ∈ (0, 1), with high probability, n X S − Y S (t − δt) i i i=1
≤
n X S − X I (t − δt) + X I − Y I (t − δt) + Y I − Y S (t − δt) i i i i i i i=1
≤ kn + 2ψ(1 − δ)t.
3490
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
Fig. 1. Stage 1. SIP-particles X S (resp. Y S ) in blue and IRW-particles X I (resp. Y I ) in red both start from x (resp. y) on the same segment [A, B] of Z. Between each set of particles, the moves of the IRW-particles are coupled so that√the distance in-between is constant to |x1 − x2 | (resp. |y1 − y2 |). Each SIP-particle is distant from an IRW-particle by o( t) being coupled thanks to the coupling given by [13, Theorem 3.2]. If no collision occurs for the SIP-particles, IRW-√and SIP-particles follow the same path (in purple). At time (1 − δ)t, the distance between any pair of IRW-particle is O( t). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Stage 2. Now, couple the two sets of IRW-particles and the SIP-particles coordinate-wise during the time interval [(1 − δ)t, t] using the Ornstein coupling. If the two sets of SIP particles are coupled in this lapse of time [t −δt, t], and no collision occurred then we say that the coupling attempt is successful. After time (1 − δ)t, any pair of different SIP-particles as well as every pair of different IRW-particles of the same √ set (i.e., the ones starting from x as well as the ones starting from y) are at distance of order t with probability close to one as t → ∞. Indeed, for the IRW-particles this is clear from the invariance principle, whereas for the corresponding SIP-particles, it then follows via (3.17). Therefore, it is enough to prove that for all i, the probability that a collision occurs for the SIP-particles before X iI (t) and YiI (t) are coupled, tends to zero as t → ∞. See Fig. 2. The motion of the difference of X iI (t) − YiI (t) in the Ornstein coupling is that of a simple continuous-time random walk at rate m (in the sense of twice the speed of the underlying random walk of one SIP(m), m = 2(m/2)). So it suffices to see that for two IRW-particles moving at rate m, we have √ τ1 ≥ τ2 lim PIRW {−1,1} = 0, ψ(t), t 0
t→∞
(3.19)
IRW where Pa,b denotes the joint distribution of two IRW-particles starting respectively from lo2 cations a and b, τ01 , resp. τ{−1,1} , stands for the hitting time of 0, resp. of {−1, 1}, for some I IRW-particle X , resp. another IRW-particle Y I . This is, using standard arguments, in turn implied by
√ τ 1 ≥ τ 2 = 0. lim PIRW 0 ψ(t), t 0
t→∞
(3.20)
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3491
Fig. 2. Stage 2. While the IRW-particles X I and Y I are coupled coordinate-wise via an Ornstein coupling (there, omitted from the figure), the SIP-particles, here two of them in blue, are consequently coupled in the same way provided no collision occurs within any set of particles (that is, here, between X 1S and X 2S or Y1S and Y2S , but any other does not matter). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Recall by the reflection principle (see e.g. [11, Chapter I]), for any a ≥ 0: IRW PIRW 0 (τa ≥ t) = P0 (|Z (t)| ≤ a),
where here and in what follows Z (t) denotes a continuous-time simple random walk moving at rate m. Therefore, choosing a time scale ϕ(t) such that ϕ(t) → ∞ as t → ∞, we obtain IRW √ 1 2 2 IRW √ 1 2 2 √ τ1 ≥ τ2 ≤ P τ ≥ τ , τ ≥ ϕ(t) + P τ ≥ τ , τ ≤ ϕ(t) PIRW 0 0 0 0 0 ψ(t), t 0 ψ(t), t 0 ψ(t), t 0 IRW 1 IRW 2 ≤ Pψ(t) τ0 ≥ ϕ(t) + P√t τ0 ≤ ϕ(t) √ IRW ≤ PIRW |Z (ϕ(t))| ≤ ψ(t) + P |Z (ϕ(t))| ≥ t 0 0 |Z (ϕ(t))| |Z (ϕ(t))| = PIRW ≤ 1 + PIRW ≥1 . (3.21) √ 0 0 ψ(t) t Since |Z (ϕ(t))| ∼ ϕ(t)1/2 as t → ∞, (3.20) follows by choosing ϕ(t) = cψ(t)2 for some positive constant c so that we let t go to infinity, and then c → ∞. Indeed, as t → ∞ by the Donsker invariance principle, the first term of the r.h.s. becomes P(|N (0, c)| ≤ 1), where N (0, c) denotes a normal random variable √ with mean zero and variance c. While the sec√ ond term of the r.h.s. vanishes since ψ(t) = o( t), and hence |Z (ϕ(t))| → 0 as t → ∞. Hence t the r.h.s. of (3.21) goes to zero as t, then c go to infinity. So far, we have proved the probability that the two sets of SIP-particles are coupled in the time lapse [(1 − δ)t, t] is strictly positive, and tends to 1 as t → ∞. Then by iterating independently such coupling attempts, one sees that the probability of eventual successful coupling of the SIP particles is one.
3492
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
As a consequence, Corollary 3.1. Let µ be a tempered invariant for the SIP. Then for all n ∈ N, ∃ αn ∈ measure n [0, ∞) such that for all x = (x1 , . . . , xn ) ∈ Zd , µ(x) = αn i.e., µ(x) only depends on n = |x| but not on the precise locations x1 , . . . , xn . Proof. The proof is quite standard, using that bounded harmonic functions are constant when a successful coupling exists, but we give it here for the sake of self-consistency. Let µ be a tempered invariant measure, and put, as before, cn = supx1 ,...,xn µ(x1 , . . . , xn ). Consider two sets of SIP-particles X S (t) = (X 1S (t), . . . , X nS (t)) and Y S (t) = (Y1S (t), . . . , YnS (t)). Since there exists d n a successful coupling for the SIP-particles, for any x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Z , PSIP x,y (τ < ∞) = 1, where τ = inf t ≥ 0 : X S (s) = Y S (s) for all s ≥ t . Now since µ ∈ Itemp by Proposition 2.1, µ(x) = ESIP µ(X S (t)) x S S S SIP S S S = ESIP µ (Y (t))1{X (t) = Y (t)} + E µ (X (t))1{X (t) = ̸ Y (t)} x,y x,y SIP S SIP S = Ex,y µ(Y (t)) + Ex,y µ(X (t)) − µ(Y S (t)) 1{X S (t) ̸= Y S (t)} S S S S = µ(y) + ESIP µ (X (t)) − µ (Y (t)) 1{X (t) = ̸ Y (t)} . x,y Hence, using µ(x) ≤ cn for x ∈ (Zd )n , we obtain the estimate S S | µ(x) − µ(y)| ≤ 2cn PSIP x,y X (t) ̸= Y (t) and the result follows by letting t go to infinity.
4. Ergodic tempered measures From now on, we can characterise the ergodic tempered measures. First, in the next lemma, we prove that this set coincides with the extreme elements of the set of tempered invariant measures. Lemma 4.1. The set of tempered ergodic invariant measures coincides with the set of extreme points of the tempered invariant measures, i.e., (Itemp )e = (I ∩ Ptemp )e = Ie ∩ Ptemp . Proof. Let µ ∈ (I ∩ Ptemp ) and µ ̸∈ (I ∩ Ptemp )e , then there exist 0 < λ < 1 and µ1 , µ2 ∈ (I ∩ Ptemp ) such that µ = λµ1 + (1 − λ)µ2 , therefore µ ̸∈ Ie , and hence µ ̸∈ Ie ∩ Ptemp . Conversely, suppose that µ ∈ Ptemp is not in Ie ∩ Ptemp , then µ ̸∈ Ie and hence then there exist λ ∈ (0, 1) and µ1 , µ2 ∈ I such that µ = λµ1 + (1 − λ)µ2 .
3493
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
This equality together with the fact that µ ∈ Ptemp , and the positivity of the functions D(ξ, ·), imply that µ1 , µ2 ∈ Ptemp . Therefore µ ̸∈ (I ∩ Ptemp )e . Finally if µ is not in Ie ∩ Ptemp and µ is also not in Ptemp then trivially µ ̸∈ (I ∩ Ptemp )e . We can then characterise the ergodic tempered measures. Theorem 4.1. If µ ∈ Ie ∩ Ptemp then µ = νλm for some λ ∈ [0, 1). As a consequence (Itemp )e = {νλm : λ ∈ [0, 1)}.
(4.22)
Proof. Remark that by the multivariate version of the Carleman moment condition, within the class Ptemp , a measure µ is uniquely determined by its D-transform µ. ˆ Let µ ∈ (Itemp )e . Since µ is invariant its D-transform µ(ξ ) depends only on |ξ | so we put, with slight abuse of notation µ(ξ ˆ ) = µ(n). ˆ In order to show that µ = νλm for some λ ∈ [0, 1), it suffices now to show that µ(n) ˆ = a n for some a ≥ 0. This in turn follows if we show that µ(n ˆ + m) = µ(n) ˆ µ(m), ˆ
(4.23) T
for all n, m ∈ N. Recall St stands for the semigroup of the SIP and denote ST = T1 0 St dt. Fix ξ, ξ ′ to finite configurations, with |ξ | = n, |ξ ′ | = m. By ergodicity we have, µ-almost surely ST D(ξ, η) → µ(ξ ˆ ) = µ(|ξ ˆ |), as T → ∞ (where ST works on η). Therefore, using that µ ∈ Ptemp , by dominated convergence, D(ξ ′ , η)ST D(ξ, η)dµ(η) → µ(ξ ˆ ′ )µ(ξ ˆ )= µ(m) µ(n).
(4.24)
(4.25)
On the other hand, by self-duality 1 T ST D(ξ, η) = pt (ξ, ξ ′′ )dt D(ξ ′′ , η). T 0 ′′ ξ ∈Ω
(4.26)
n
As a finite number of SIP-particles eventually spread out all over the lattice Zd , for large T , the main contribution of the sum over ξ ′′ in the r.h.s. of (4.26) is from configurations ξ ′′ in which there are no particles at locations occupied by particles in ξ ′ (let us denote this property by ξ ′ ⊥ ξ ′′ ). If ξ ′ ⊥ ξ ′′ , then D(ξ ′′ , η)D(ξ ′ , η) = D(ξ ′′ + ξ ′ , η). More precisely, consider n-SIP particles (X 1S (t), . . . , X nS (t)). By translation invariance and irreducibility, for all i = 1, . . . , n, X i (t) drifts to infinity as t goes to infinity. Hence, let n n ξ ′ = i=1 δ yi and ξ = i=1 δxi , so that pt (ξ, ξ ′′ ) ≤ PSIP ∃i, j = 1, . . . , n : X iS (t) = y j ξ ′′ ⊥ξ ′
≤
PSIP X iSIP (t) = y j
i, j
which tends to zero when t goes to infinity.
3494
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
Therefore, 1 T ′ D(ξ , η)St D(ξ, η)dµ(η) = pt (ξ, ξ ′′ )D(ξ ′′ , η)D(ξ ′ , η)dµ(η) T 0 ′′ ξ ∈Ωn 1 T = pt (ξ, ξ ′′ )D(ξ ′ + ξ ′′ , η)dµ(η) + bT , T 0 ξ ′′ ∈Ω ,ξ ′′ ⊥ξ ′ n
where bT vanishes as T goes to infinity. Indeed, since µ is a tempered measure, 1 T bT ≤ pt (ξ, ξ ′′ ) sup D(ξ ′ , η)D(ξ ′′ , η)dµ(η) T ′′ ′ ′′ 0 ξ :|ξ |=|ξ | ξ ′′ ∈Ωn ,ξ ′′ ̸⊥ξ ′ T 1 ≤C· pt (ξ, ξ ′′ ) , T 0 ξ ′′ ∈Ω ,ξ ′′ ̸⊥ξ ′ n
where, using (2.8), C = C(|ξ |, |ξ ′ |) is a constant. This can be seen as follows. Since d(k, ℓ)2 is a polynomial of degree 2k, it is a combination of d( j, .) with j ≤ 2k, i.e., d(k, ℓ)2 = 2k j=0 α(k, j)d( j, ℓ), for some constants α(., .). Therefore if µ ∈ Ptemp , then for |ξ | = n, one has n 2k 2 n . D(ξ, η) dµ(η) ≤ c2n |α(k, j)| := C j=0
Therefore by Cauchy–Schwarz, one can choose |ξ | C |ξ ′ | . C= C Therefore, aT + µ(n) ˆ µ(m) ˆ =
D(ξ ′ , η)ST D(ξ, η)dµ(η) 1 T = pt (ξ, ξ ′′ )dt D(ξ ′′ + ξ ′ , η)dµ(η) + bT T 0 ξ ′′ ∈Ωn ,ξ ′′ ⊥ξ ′ 1 T ′′ pt (ξ, ξ )dt µ(n ˆ + m) + bT = T 0 ξ ′′ ∈Ωn ,ξ ′′ ⊥ξ ′ 1 T = pt (ξ, ξ ′′ )dt µ(n ˆ + m) + bT + cT T 0 ξ ′′ ∈Ω n
= µ(n ˆ + m) + bT + cT
(4.27)
where aT → 0 by (4.25) and as explained before bT , cT → 0. Letting now T → ∞ give (4.23). Now combining this with Lemma 4.1, and the fact that all νλm are elements of Ie ∩ Ptemp , i.e., are ergodic under the SIP dynamics (see e.g. [7]) gives the result (4.22). Remark 4.1. As a consequence of Proposition 2.2 this result can be transferred to the BEP, showing that its tempered invariant ergodic measures are product of Gamma distributions, and to the BMP, showing that its tempered invariant ergodic measures are product of mean zero Gaussians.
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3495
Next, we show that all tempered invariant measures satisfy a correlation inequality of the type derived in [7]. n Proposition 4.1. Let µ ∈ Itemp , then for all (x1 , . . . , xn ) ∈ Zd , n n D δxi , η dµ(η). (4.28) δxi , η dµ(η) ≥ D i=1
i=1
Proof. Because every element of Itemp can be decomposed into extreme elements, we have µ = νλm dΛ(λ), for some probability measure Λ on [0, 1) and as a consequence, denoting λ ρ(λ) = 1−λ we have n n m δxi , η dνλ (η)dΛ(λ) = ρ(λ)n dΛ(λ) δxi , η dµ(η) = D D i=1
i=1
≥
ρ(λ) dΛ(λ)
n
=
n
D(δxi , η)dµ(η).
i=1
5. Convergence to ergodic product measures In this section, we give sufficient criteria for a starting measure µ to converge in the course of time to one of the product measures νλm . To this purpose, we introduce the following notions of d asymptotic independence and homogeneity. We call a function f : NZ → R local if it is a finite linear combination of the functions D(ξ, ·). Definition 5.1. A measure µ is asymptotically homogeneous (AH), if there exists ρ > 0 such that (5.29) lim sup Ez D(δ Z (t) , η)dµ(η) − ρ = 0, t→∞ z
where Ez denotes the expectation w.r.t. a simple random walk {Z (t), t ≥ 0} starting from z. Notice that every translation invariant measure with finite moments is trivially AH. Definition 5.2. A measure µ is asymptotically independent (AI), if for all n and for all choices of local functions f 1 , . . . , f n , n n lim τ yi f i dµ − τ yi f i dµ = 0. |yi −y j |→∞
i=1
i=1
Then we have the following result, Theorem 5.1. Let µ be tempered, AH and AI, then m µSt → νλ(ρ) ,
with λ(ρ) =
ρ 1+ρ ,
as t → ∞
where ρ is the constant defined as the limit in (5.29).
3496
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
Here, the convergence holds in the sense of convergence of all multivariate moments, that is, m D(ξ, η)dµSt (η) → D(ξ, η)dνλ(ρ) (η), as t → ∞ for all finite configurations ξ . Proof. We have to prove that for all x1 , . . . , xn ∈ Zd , n n Eη D δxi , η dµ(η) = lim D δxi , η dνλ(ρ) (η). t→∞
i=1
(5.30)
i=1
Remark the r.h.s is equal to ρ n thanks to (2.6). Now, dealing with the l.h.s., rewrite n n Eη D δxi , η dµ(η) = ESIP δ X S (t) , η dµ(η) x D i=1
i=1
i
n Using [7, Lemma 1] for x ∈ Zd , it is equal to n ESIP D(δ X S (t) , η)dµ(η) + o(t). x i
i=1
Indeed, after an arbitrary large time, with probability close to one, the n SIP-particles initially at x will have spread out and will be at different locations. By the Markov property, for any time scale ψ(t) ≤ t and such that ψ(t)/t 1/4 → 0 as t → ∞, n n SIP SIP D(δ X S (ψ(t)) , η)dµ(η) ESIP D(δ , η)dµ(η) = E E xi x x X S (t−ψ(t)) i=1
i=1
IRW = ESIP x EX S (t−ψ(t))
n i=1
i
D(δ X I (ψ(t)) , η)dµ(η) + R(t). i
For the first equality, we used invariance of µ, the Markov property and self-duality. For the second equality, we used the fact that after the large time √ span t − ψ(t), the SIP-particles are with probability close to one at distance of the order√of t from each other and therefore, in the remaining time ψ(t) they will not come closer than t − t 1/8 to each other, i.e., are still far apart and therefore will move as if they are IRW-particles. We denoted R(t) to be the contribution of the complementary event. Let us denote by Q the set of configurations with n particles where at least two are at the same position, and A(ψ(t)) the event that ξ(s) is in Q for some s ∈ (t − ψ(t), t). We then have, using Cauchy–Schwarz and the same argument as in the proof of Theorem 4.1, based on the fact that µ is tempered, PSIP (A(ψ(t))) sup D(ξ, η)2 dµ(η) R(t) ≤ ESIP x X S (t−ψ(t)) |ξ |=n
which tends to zero as t → ∞. Next because µ is AI, the product over i and the integral over µ can be exchanged at the price of another o(t) term. Once the product is out of the integral, it trivially also comes out of the IRW expectation, and therefore, it is equal to n IRW ESIP E D(δ X I (ψ(t)) , η)dµ(η) + o(t). x X I (t−ψ(t)) i=1
i
i
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3497
Use the AH property (5.29) of µ to conclude that this expression in turn is equal to ρ n + o(t), we conclude by letting t → ∞.
Remark 5.1. We can replace the AH and AI assumptions in Theorem 5.1 by the following assertion: lim pt (x, y)η(y) = ρ, (5.31) t→∞
y
where the limit is in µ-probability and pt (., .) stands for a simple random walk transition probability. Indeed, assuming (5.31) we can write n n IRW SIP ESIP E D(δ , η)dµ(η) = E pt−s (X iI (s), yi )ηs (yi )dµ(η) I x x X (t−s) X I (s) i
i=1
i=1 yi
ρn
whose r.h.s. converges to as t, then s, go to infinity, by (5.31) and dominated convergence (because µ is tempered by assumption). Examples where (5.31) is satisfied are measures µ which are absolutely continuous with dµ , which is a local function of the respect to reversible product measures νρ with density dν ρ configuration η, i.e., depending on a finite number of coordinates η(x). We conclude with two additional remarks: Remark 5.2. The fact that p is nearest-neighbour can be replaced without any difficulty by a finite range kernel (with the same proof, adapting the definition of collision). Presumably it is enough that p is translation invariant and has a finite second moment. Remark 5.3. Related to the SIP are the dual Kipnis–Marchioro–Presutti (KMP) process and its generalised so-called “thermalised” SIP [2], where when particles are at nearest neighbour positions, several particles can jump at the same time. However, if in this thermalised SIP model all the particles are separated (i.e., at distance >1), they behave exactly as independent random walkers, and therefore a finite number of them can be successfully coupled just as SIP particles can. This implies that for the thermalised SIP we have the same set of ergodic tempered measures. The thermalised SIP in turn is the dual process of a mass-redistribution model, called the thermalised BEP in [2] which generalises the KMP process [8]. Hence, the only ergodic tempered measures of this generalised KMP process are also products of Gamma distributions. Acknowledgements We thank Pablo A. Ferrari for pointing us to [4] and further valuable comments and motivating discussions. We thank an anonymous referee for careful reading and many useful comments. References [1] E. Andjel, Invariant measures for the zero range process, Ann. Probab. 10 (3) (1982) 525–547. [2] G. Carinci, C. Giardin`a, C. Giberti, F. Redig, Duality for stochastic models of transport, J. Stat. Phys. 152 (4) (2013) 657–697.
3498
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
[3] G. Carinci, C. Giardin`a, F. Redig, T. Sasamoto, Asymmetric transport models with Uq (su(1, 1)) symmetry, J. Stat. Phys. 163 (2) (2016) 239–279. [4] A. De Masi, E. Presutti, Probability estimates for symmetric simple exclusion random walks, Ann. Inst. Poincar´e 19 (1) (1983) 71–85. [5] A. De Masi, E. Presutti, Mathematical Methods for Hydrodynamic Limits, Springer-Verlag Berlin, Heidelberg, 1991. [6] C. Giardin`a, J. Kurchan, F. Redig, Duality and hidden symmetries in interacting particle systems, J. Stat. Phys. 135 (1) (2009) 25–55. [7] C. Giardin`a, F. Redig, K. Vafayi, Correlation inequalities for interacting particle systems with duality, J. Stat. Phys. 141 (2) (2010) 242–263. [8] C. Kipnis, C. Marchioro, E. Presutti, Heat flow in an exactly solvable model, J. Stat. Phys. 27 (1) (1982) 65–74. [9] C. Kleiber, J. Stoyanov, Multivariate distributions and the moment problem, J. Multivariate Anal. 113 (2013) 7–18. [10] T. Lindvall, Lectures on Coupling Method, Dover, 1992. [11] G.M. Lawler, V. Limic, Random Walk: A Modern Introduction, Cambridge University Press, 2010. [12] T.M. Liggett, Interacting Particle Systems, Springer-Verlag Berlin, 2005. [13] A. Opoku, F. Redig, Coupling and hydrodynamic limit for the inclusion process, J. Stat. Phys. 160 (3) (2015) 532–547.
ScienceDirect Stochastic Processes and their Applications 126 (2016) 3480–3498 www.elsevier.com/locate/spa
Ergodic theory of the symmetric inclusion process Kevin Kuoch a,∗ , Frank Redig b a Johann Bernoulli Institute, Rijksuniversiteit Groningen, Postbus 407, 9700AK Groningen, The Netherlands b Delft Institute of Applied Mathematics, TU Delft, Mekelweg 4, 2628CD Delft, The Netherlands
Received 31 August 2015; received in revised form 3 May 2016; accepted 4 May 2016 Available online 27 May 2016
Abstract We prove the existence of a successful coupling for n particles in the symmetric inclusion process. As a consequence we characterise the ergodic measures with finite moments, and obtain sufficient conditions for a measure to converge in the course of time to an invariant product measure. c 2016 Elsevier B.V. All rights reserved. ⃝
Keywords: Interacting particle systems; Simple inclusion process; Invariant measures; Coupling
1. Introduction In [12, Chapter VIII], a rather complete ergodic theory is given for the symmetric exclusion process (SEP). In particular, the only extremal invariant measures for the SEP are Bernoulli measures with constant density. This complete characterisation of the set of invariant measures is quite exceptional and follows from the fact that the SEP is self-dual. As a consequence, invariant measures can be related to bounded harmonic functions for the finite SEP. Then, by the construction of a successful coupling of the SEP with a finite number of particles, it is shown that all bounded harmonic functions are constant, i.e. depend only on the number of particles. From this in turn, one can conclude that all invariant measures for the SEP are permutation
∗ Corresponding author.
E-mail addresses: [email protected] (K. Kuoch), [email protected] (F. Redig). http://dx.doi.org/10.1016/j.spa.2016.05.002 c 2016 Elsevier B.V. All rights reserved. 0304-4149/⃝
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3481
invariant, from which one derives by the De Finetti theorem that they are convex combinations of Bernoulli measures. In [6,7] an attractive version (in the sense of having an attractive interaction between the particles) of the SEP is introduced and called the simple inclusion process (SIP). In the SIP, particles perform nearest-neighbour jumps according to a simple symmetric random walk and interact by “inclusion jumps”, where pairs of neighbouring particles jump to the same site at rate 1. This analogy between SIP (attractive interaction) and SEP (repulsive interaction) becomes even more apparent in [7], where it is shown that the SIP satisfies the analogue of Liggett’s comparison inequality [12, Chapter VIII, Proposition I.7] for the evolution of positive definite symmetric functions. The expectation at time t > 0 of such a function in the course of the evolution of n SIP-particles is larger than in the course of the evolution of n independent random walkers. In particular this implies that a certain class of product measures is mapped by the evolution under the SIP to measures with positive correlations (as opposed to negative correlations in the SEP). In this paper, we want to investigate as much as possible the invariant measures of the SIP, i.e., understand its ergodic measures and their attractors. Since the number of particles is unbounded and we want make use of self-duality, we will have to restrict to a set of measures with all moments finite. The main problem is then to construct a successful coupling for two sets of n SIP-particles initially at different locations. This coupling seems possible thanks to the fact that as long as SIP-particles do not collide, i.e., are not at neighbouring positions, they behave as independent random walkers and these can be coupled by the coordinate-wise Ornstein coupling in any dimension. The idea of the coupling of SIP-particles comes from [4] combined with [13]: in [13], it is shown that inclusion particles √ and independent random walkers can be coupled in such a way that at time t they are o( t) apart; the period of time [0, (1 − δ)t] in which coupling according to [13] is used (stage 1) is then followed by a period of time [(1−δ)t, t] (stage 2) during which the coordinate-wise Ornstein coupling of independent random walkers is used both for the independent walkers as well as for the SIP particles. The only problem for this coupling to be successful is to estimate the probability of being coupled before a collision occurs. One can understand however that such a collision event is highly improbable (as √ t → ∞), as after a long time, the independent random walkers are much farther apart (O(√ t)) than the distance between the independent random walkers and their inclusion partners (o( t)). Once one has the successful coupling of SIP-particles, and as a consequence results on the structure of the invariant measures of the SIP, all these results can be transferred without effort to corresponding results for interacting diffusion processes to which the SIP is a dual process such as the Brownian Energy Process (BEP) and the Brownian Momentum Process (BMP). For particle systems with unbounded occupation numbers, in general only very little information is available on the structure of the set of invariant measures. See [1] for a set of general results under Lipschitz conditions on the jumps rates. In this work, both the symmetry and the self-duality property are crucial. Indeed, for an asymmetric version of the SIP based on the Lie-algebraic construction, recently introduced in [3], despite self-duality, there is no characterisation of invariant measures, and the translation invariant ones are known not to be product. On the other side, the naive asymmetric version of the SIP (putting different factor for the jumps in different directions keeping the same rates) has formally the same invariant product measures, but for that process we have no self-duality and therefore no proof of existence in infinite volume. The rest of our paper is organised as follows. In Section 2, we give basic definitions and set notations up, in Section 3, we prove the successful coupling, in Section 4, we characterise the
3482
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
class of ergodic so-called tempered (with finite moments) measures and in Section 5, we give sufficient conditions for a measure to converge to an invariant product measure. 2. Notations and definitions 2.1. The symmetric inclusion process We denote by p(., .) the transition probability of a simple symmetric nearest-neighbour random walk on the lattice Zd , i.e., 1 if ∥x − y∥ = 1, p(x, y) = 2d 0 otherwise. The simple symmetric inclusion process with parameter m > 0 (denoted SIP(m)) is an interacting particle system where particles perform independent random walks moving at rate m/2 and according to the transition probabilities p(., .) while on top of that, they interact by inclusion, i.e. each particle “invites” any other particle standing at a nearest neighbour position at rate 1 to join its site (invitations are always followed up). These “invitation jumps”, or “inclusion jumps”, create an attractive interaction between the particles. This has to be compared with the interaction between particles of the symmetric exclusion process (SEP) where jumps joining two particles at the same site are forbidden. Here, on the contrary, these jumps are encouraged. d More formally, the SIP(m) on Zd is a continuous-time Markov process (ηt )t≥0 ∈ NZ whose generator L acts on local functions f , i.e. depending only on a finite number of occupation variables, as m L f (η) = p(x, y)η(x) + η(y) f (η x,y ) − f (η) , (2.1) 2 d y:|y−x|=1 x∈Z
η x,y
d NZ
d
where ∈ stands for the configuration obtained from η ∈ NZ by moving one particle x,y from x to y, i.e. η = η − δx + δ y , where δx denotes the configuration with a single particle at x and none elsewhere; | · | stands for the ℓ1 -norm. The existence of the SIP with generator (2.1) is not guaranteed by general existence criteria such as [1]. Though, its existence follows from self-duality, see Section 2.3 and [5]. 2.2. Invariant product measures d
For λ ∈ [0, 1), define the homogeneous discrete Gamma product measure νλm on NZ whose marginals are given by 1 λk Γ m2 + k m νλ η(x) = k = (2.2) Z λ,m k! Γ m2 where Γ (·) denotes the Gamma function and m/2 1 Z λ,m = 1−λ is the normalising constant. These measures νλm are reversible and ergodic for the SIP(m), see [7] for more details. One of the main questions answered in the present paper is whether these are the only ergodic measures within a certain class.
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3483
2.3. Duality A duality relation is a link between a dual process with the process of interest in such a way that it allows to perform computations for one process in terms of the other one. This link is created via the duality function. For further details see [12, Chapter II, Section 4]. Definition 2.1 (Duality Relation). Suppose (ξt )t≥0 and (ηt )t≥0 are Markov processes on S1 and S2 respectively. Let D be a bounded measurable function on S1 × S2 . The processes (ξt )t≥0 and (ηt )t≥0 are said to be dual to one another with respect to D if Eξ D(ξt , η) = Eη D(ξ, ηt ).
(2.3)
If the processes (ξt )t≥0 and (ηt )t≥0 are the same, then we call (2.3) self-duality. In that sense, the SIP(m) is self-dual (see [6]) with duality functions D(·, ·), for a process ξ with finite configuration and η an arbitrary process, D(ξ, η) = d(ξ(x), η(x)) (2.4) x∈Zd
where
d(k, l) =
Γ m2 l! m (l − k)! Γ 2 + k
for k ≤ l for k > l for k = 0.
0 1
Self-duality of the SIP(m) therefore means that SIP(m)
EηSIP(m) D(ξ, ηt ) = Eξ
D(ξt , η),
(2.5)
SIP(m)
where Eη denotes the expectation of a SIP(m) starting from an initial configuration η and where ξ is a finite configuration (i.e., having a finite number of particles). The self-duality functions D(., .) and the reference measure νλm are naturally connected via λ |ξ | , (2.6) D(ξ, η)dνλm (η) = 1−λ where |ξ | denotes the number of particles in the finite configuration ξ . We refer the reader to [6,7] for the proof of the self-duality (2.5) and further details and properties of the SIP. By duality relations, we derive as well related results for interacting diffusions that are dual to the SIP: the Brownian momentum process and the Brownian energy process, see [7] and references therein. The main advantage of self-duality is that it allows us to study the SIP with infinitely many particles by studying the SIP with a finite number of particles. Indeed, to know the timedependent expectations of the polynomials D(ξ, η) it suffices to follow the evolution of the particles in the finite configuration ξ , and the initial configuration ξ . d
Remark 2.1. If both ξ and η ∈ NZ are finite configurations, then both processes (ξt )t≥0 and (ηt )t≥0 are well-defined. Moreover, by self-duality, one has from (2.6) that Eη D ξ, ηt = Eξ D ξt , η = pt (ξ, ξ ′ )D ξ ′ , η . (2.7) ξ ′ ∈Ωn
3484
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
If ξ is finite and η is infinite, for N ∈ N, denote η(x) if |x| ≤ N , (N ) η = 0 otherwise. Hence by (2.7), N → Eη(N ) D ξ, ηt is non-decreasing and one can define Eη D ξ, ηt = sup Eη(N ) D ξ, ηt N = pt (ξ, ξ ′ )D ξ ′ , η), ξ′
provided the above series converges, that is, if sup N Eη(N ) D ξ, ηt < ∞. By elementary Gaussian bounds on pt (., .), the above series is convergent if there exist some positive constants c and k such that η(x) ≤ c|x|k ,
for all x.
For such initial configurations, for all finite ξ , the expectations Eη D(ξ, ηt ) are well-defined and from this, we can define the process (ηt )t≥0 starting from η. For more details on such construction via self-duality, we refer the reader to [5, pages 11–14]. While exploiting the self-duality property, we necessarily restrict to starting measures having finite moments. Let us denote by P the set of all probability measures on the configuration d space NZ . Definition 2.2. We then consider the class Ptemp of so-called tempered probability measures defined as follows: Ptemp := µ : µ ∈ P : ∀n ∈ N : sup D(ξ, η)dµ(η) =: cn < ∞ (2.8) |ξ |=n
where cn satisfies the Carleman condition ∞
−1/n
cn
= ∞.
(2.9)
n=1
The condition (2.9) ensures that the moments D(ξ, η)dµ(η) characterise uniquely the measure µ, i.e., if µ and ν are two probability measures in Ptemp such that for all finite configurations ξ , D ξ, η dµ(η) = D ξ, η dν(η), then µ = ν. See [9, Corollary 3] for further details. Remark that by self-duality, a tempered measure remains tempered in the course of the evolution of the SIP. Indeed, if µ ∈ Ptemp then, by conservation of the number of particles in the finite SIP, denoting sup D(ξ, η)dµ(η) =: cn , |ξ |=n
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3485
we have, for |ξ | = n and t > 0, Eη D(ξ, ηt )dµ(η) = Eξ D(ξt , η)dµ(η) ≤ cn , hence, the time-evolved measure µt is tempered for all t > 0, with the same dominating constants cn , n ∈ N as the one of the starting measure µ. We are interested in characterising the invariant measures which are ergodic for the SIP and belong to Ptemp . d
Definition 2.3. For the SIP (ηt )t≥0 on NZ , d
i. A probability measure µ on NZ is invariant if whenever η0 follows the distribution µ then ηt has this same distribution µ for all t > 0. ii. An invariant measure µ is ergodic for the SIP if the process (ηt )t≥0 starting from η0 distributed according to µ is ergodic. Note that since (ηt )t≥0 is a Markov process, the ergodicity of µ is equivalent with (a) µSt = µ for all t > 0, (b) for all invariant functions f (that is, St f = f for all t > 0), f = f dµ µ-a.s, where St stands for the semigroup of (ηt )t≥0 . Furthermore, if µ is ergodic then the Birkhoff ergodic theorem holds: for all f ∈ L 1 (µ), 1 T f (ηt )dt → f dµ PSIP µ -a.s., T 0 and as a consequence, 1 T St f → f dµ T 0
µ-a.s.
Moreover, the set of ergodic measures is the set of extreme points of the set of invariant measures. We denote by I the set of invariant probability measures for the SIP, and by Itemp the set I ∩ Ptemp of tempered invariant measures. Furthermore, we call Ie the set of extreme points of I , i.e., the set of invariant and ergodic probability measures for the SIP. For a measure µ ∈ Ptemp , we denote its D-transform by µ(ξ ˆ ) := D(ξ, η)dµ(η). (2.10) Here, ξ varies in the set of finite configurations, which we denote henceforth by Ωn if ξ is a configuration which contains n particles. Note that ξ ∈ Ωn such that n = |ξ | can be identified with an n-tuple x = (x1 , . . . , xn ) via ξ=
n
δxi .
(2.11)
i=1
As a consequence µˆ can also be viewed as a symmetric function on ∪n∈N (Zd )n . We will therefore use both notations µ(ξ ) as well as µ(x) = µ(x1 , . . . , xn ).
3486
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
The following result is then a straightforward consequence of the self-duality of the SIP. Proposition 2.1. Let µ be a tempered probability measure. Then µ is invariant if and only if its D-transform µˆ is bounded harmonic for the SIP, i.e., if and only if for all t > 0 Eξ µ(ξ ˆ t ) = µ(ξ ˆ ).
(2.12)
Proof. Suppose that µ ∈ Itemp . Since µ is a tempered measure, by Fubini’s theorem, for all t > 0, Eξ µ(ξt ) = D(ξt , η)dµ(η) dµ(ξ ) = D(ξt , η)dµ(ξ ) dµ(η) = D(ξ, η)dµ(η). Conversely, suppose (2.12) holds for all ξ . Then, letting µt := µSt at time t > 0, starting from µ at time 0, D(ξ, η)dµt (η) = D(ξ, η)dµ(η). (2.13) Hence µ = µt for all t > 0, as tempered.
As a consequence, the study of ergodic measures in Itemp is reduced to the problem of identifying the set of bounded harmonic functions for the SIP. This is done via the construction of a successful coupling, which implies that bounded harmonic functions are constant. 2.4. Bounded harmonic functions and successful coupling Via the identification (2.11), one can see the evolution of n SIP-particles initially at positions x = (x1 , . . . , xn ) as a process X S (t) = (X 1S (t), . . . , X nS (t)) on (Zd )n so that for each 1 ≤ i ≤ n, X iS (t) keeps track of the location of the SIP-particle i started from xi . We denote by PSIP x its path space measure. A coupling of two copies of the SIP starting initially at different locations x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) is then defined as usual as a process {X S (t), Y S (t), t ≥ 0} with X S (0) = x, Y S (0) = y with first (resp. second) marginals {X S (t), t ≥ 0}, the SIP starting from x (resp. {Y S (t), t ≥ 0}, the SIP starting from y). Its path space measure is denoted by PSIP x,y where the hat stands for the joint distribution in the coupling. The coupling time is defined via τ = inf{t ≥ 0 : X S (s) = Y S (s) ∀s ≥ t},
(2.14)
where by convention inf(∅) = ∞. The coupling is successful if τ < ∞ PSIP x,y -almost surely for d 2n all (x, y) ∈ (Z ) . It is well known (see e.g. [12, Chapter 2]) that the existence of a successful coupling implies that all bounded harmonic functions are constant. 2.5. Ornstein-coupling of n-dimensional random walks Consider two d-dimensional random walks X(t) = (X 1 , . . . , X d )(t) and X(t) = ( X 1, ..., X d )(t), starting from X(0) = x and X(0) = x respectively. The Ornstein-coupling works as follows. First, let the random walks evolve independently component-wise, until the first time at which (X 1 (t))t≥0 and ( X 1 (t))t≥0 coincide. From that time on, the first components perform
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3487
the same jumps hence remain identical while other components continue to evolve independently. We continue like this until the second components coincide, and so on, until all components coincide. Because continuous-time n-dimensional random walks have d independent 1-dimensional random walk components and because the difference between two independent 1-dimensional random walks is still a 1-dimensional random walk (at twice the speed), the latter is recurrent and the coupling is successful. See [10, Chapter 2, Section 3] for more background. 2.6. Diffusion processes related to the SIP The SIP is related via duality to the BEP, a system of interacting diffusions with state space d d [0, ∞)Z and to the BMP, a system of interacting diffusions with state space RZ . See [6] for more details. This implies that many results on the invariant measures and characterisation of ergodic measures can be transferred to these processes. This transference is a consequence of the following proposition. d
Proposition 2.2. Assume {ζt : t ≥ 0} is a Feller process on the state space K Z with K a Polish space. Assume that the process is dual to the SIP with duality function D(ξ, ζ ). Then, let D Ptemp := µ : µ ∈ P and ∀n ∈ N, sup D(ξ, ζ )dµ(ζ ) < ∞ |ξ |=n
d
KZ
D , denote the corresponding set of tempered probability measures, and denote for µ ∈ Ptemp D is invariant for {ζ : t ≥ 0} µ(ξ ˆ ) = D(ξ, ζ )dµ(ζ ) its D-transform. Then we have µ ∈ Ptemp t if and only if µˆ is a bounded harmonic function for the SIP.
Proof. Let ξ be a fixed finite configuration. Assume µ is bounded harmonic for the SIP, then, and by duality, D(ξ, η)dµt (η) = Eη D(ξ, ηt )dµ(η) = ESIP ξ D(ξt , η)dµ(η) = ESIP µ(ξt ) ξ = µ(ξ ) = D(ξ, η)dµ(η). D are characterised by their D-transform, µ = µ for all t. Hence, since measures in Ptemp t The converse is straightforward.
3. Successful coupling for the SIP We introduce here and onwards the following notations for sets of SIP-particles X S (t) = (X 1S (t), . . . , X nS (t)) and Y S (t) = (Y1S (t), . . . , YnS (t)). We denote by x = (x1 , . . . , xn ), resp. y = (y1 , . . . , yn ), the initial locations of X S , resp. Y S , that is, X S (0) = x, resp. Y S (0) = y. In this section, we prove: Theorem 3.1. There exists a successful coupling for the SIP with finite number of particles, n i.e., for any initial locations x, y ∈ (Zd , there exists a process {(X S (t), Y S (t)); t ≥ 0} where
3488
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
{X S (t); t ≥ 0} and {Y S (t); t ≥ 0} are SIP processes starting respectively from x and y such that PSIP x,y {τ < ∞} = 1,
(3.15)
where τ denotes the coupling time, defined in (2.14). Likewise, standing as auxiliary processes, we introduce two sets of n independent random walkers, or IRW-particles, noted X I (t) = (X 1I (t), . . . , X nI (t)) and Y I (t) = (Y1I (t), . . . , YnI (t)), starting from x and y, respectively. Prior to the proof, we define the notion of collision. We say that a collision for SIP-particles happens at time t > 0 if two SIP-particles belonging to a same set are at nearest-neighbour positions at time t, i.e., {∃i ̸= j, 1 ≤ i, j ≤ n : |X iS (t) − X Sj (t)| = 1 or |YiS (t) − Y jS (t)| = 1}. It is important to note that the event of particles X iS (t) and Y jS (t) at neighbouring positions is not considered as a collision, i.e., collisions only happen within the same set of particles. Similarly, we define a collision for IRW-particles at time t > 0 if two IRW-particles belonging to a same set are at nearest-neighbour position at time t, i.e., {∃i ̸= j, 1 ≤ i, j ≤ n : |X iI (t) − X Ij (t)| = 1 or |YiI (t) − Y jI (t)| = 1}. Proof. The proof is split into two parts, first we consider the case when d ≥ 3 and next, the case when d ≤ 2. Transient case: d ≥ 3. We start with the simplest case d ≥ 3 where a continuous-time random walk Z (t) based on p(., .) is transient. More precisely, we have Pz (|Z (t)| > 1,
∀t ≥ 0) =: H (z) > 0,
where Pz is the distribution of {Z (t), t ≥ 0} starting from site z and H (z) → 1 when z → ∞. As a consequence, by the union of events bound, with positive probability, n IRW-particles X I (t) = (X 1I (t), . . . , X nI (t)) starting from initial position x = (x1 , . . . , xn ) such that |xi − x j | > R, for all 1 ≤ i ̸= j ≤, are large enough, will not collide. If during a lapse of time no collision happens, then the IRW-particles and their corresponding coupled SIP-particles perform exactly the same jumps. It is only when IRW-particles collide that their corresponding SIP partners can behave differently. Assume all initial positions satisfying |xi − x j |, |yi − y j | > R for all 1 ≤ i ̸= j ≤ n. Then with positive probability p(R), the IRW-particles starting from x will never collide and neither will the IRW-particles starting from y. Moreover, one has p(R) → 1 as R → ∞. The two sets of IRW-particles can be coupled by the coordinate-wise Ornstein coupling. Now we couple each set of IRW-particles with a corresponding set of SIP-particles via the coupling used in [13, Theorem 3.2], i.e., both sets perform the same random walk jumps, and inclusion jumps are only performed by the SIP-particles. In this coupling, when no collision happens for both sets of IRW-particles, the corresponding two sets of SIP-particles, starting respectively from x and y as well, perform exactly the same jumps as their IRW-particle partners, since IRW- and SIP-particles only behave differently when they collide. Therefore, since sets of IRW-particles are successfully coupled via Ornstein-coupling, so are their SIP-partners. Therefore, the two sets of SIP-particles can be coupled with positive probability bounded from below by p(R). To show that they can be coupled almost surely from arbitrary initial locations x
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3489
and y, let them first move for some time T > 0 and then start the coupling just described. With probability π(T ), close to one as T → ∞, any two SIP- or IRW-particles will be at distance larger than α(T ) where α(T ) → ∞ when T → ∞, and correspondingly, from then on they can be coupled with probability p(α(T )). Therefore, the probability that they cannot be coupled is bounded from above by (1 − π(T )) + (1 − p(α(T ))) which tends to zero as T → ∞. Recurrent case: d = 1, 2 To tackle the case d = 1, we first give an outline. Note that in the two-dimensional case the same arguments hold and we therefore omit the proof for d = 2. We follow the line of thought of [4]. The coupling proceeds in two stages: 1. First stage: in the time interval [0, (1 − δ)t] (where δ ∈ (0, 1) is fixed and t will tend to infinity eventually), the two sets of IRW-particles make the same jumps and the SIP partners follow according to the coupling of [13]. After this first stage, with probability close to one (as t → ∞) any two different IRW-particles within √the same set, as well as their corresponding SIP-particles partners will be at distance O( t). The distance between the independent √ random walkers and their corresponding SIP partner, on the contrary will be of order o( t). 2. Second stage: in the time interval [(1 − δ)t, t], as long as the two sets of IRW-particles neither the two sets of SIP walkers do not collide, they are coupled via the coordinate-wise Ornstein coupling. As long as the SIP particles do not collide, this is indeed a coupling, since then the SIP particles behave as independent random walkers. If such a collision does happen, then we say that we have a failed coupling attempt. Stage 1. During the time interval [0, (1 − δ)t], couple the two sets of IRW-particles so that they perform the same jumps, thus, for any s ≤ (1 − δ)t, n n (X I − Y I )(s) = |xi − yi | =: kn . i i
(3.16)
i=1
i=1
According to the coupling of [13, Theorem 3.2], two sets of SIP-particles X S and Y S , starting respectively from x and y, are coupled to two sets of IRW-particles X I and Y I , starting respectively from x and y as well, such that their positions satisfy n
|X iS (s) − X iI (s)| + |YiS (s) − YiI (s)| ≤ ψ(s)
(3.17)
i=1
with ψ(s) lim √ = 0, s
s→∞
(3.18)
with probability one (w.r.t. coupling distribution). By way of illustration, see Fig. 1. Gathering (3.16) and (3.17), it is now straightforward to see that for any δ ∈ (0, 1), with high probability, n X S − Y S (t − δt) i i i=1
≤
n X S − X I (t − δt) + X I − Y I (t − δt) + Y I − Y S (t − δt) i i i i i i i=1
≤ kn + 2ψ(1 − δ)t.
3490
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
Fig. 1. Stage 1. SIP-particles X S (resp. Y S ) in blue and IRW-particles X I (resp. Y I ) in red both start from x (resp. y) on the same segment [A, B] of Z. Between each set of particles, the moves of the IRW-particles are coupled so that√the distance in-between is constant to |x1 − x2 | (resp. |y1 − y2 |). Each SIP-particle is distant from an IRW-particle by o( t) being coupled thanks to the coupling given by [13, Theorem 3.2]. If no collision occurs for the SIP-particles, IRW-√and SIP-particles follow the same path (in purple). At time (1 − δ)t, the distance between any pair of IRW-particle is O( t). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Stage 2. Now, couple the two sets of IRW-particles and the SIP-particles coordinate-wise during the time interval [(1 − δ)t, t] using the Ornstein coupling. If the two sets of SIP particles are coupled in this lapse of time [t −δt, t], and no collision occurred then we say that the coupling attempt is successful. After time (1 − δ)t, any pair of different SIP-particles as well as every pair of different IRW-particles of the same √ set (i.e., the ones starting from x as well as the ones starting from y) are at distance of order t with probability close to one as t → ∞. Indeed, for the IRW-particles this is clear from the invariance principle, whereas for the corresponding SIP-particles, it then follows via (3.17). Therefore, it is enough to prove that for all i, the probability that a collision occurs for the SIP-particles before X iI (t) and YiI (t) are coupled, tends to zero as t → ∞. See Fig. 2. The motion of the difference of X iI (t) − YiI (t) in the Ornstein coupling is that of a simple continuous-time random walk at rate m (in the sense of twice the speed of the underlying random walk of one SIP(m), m = 2(m/2)). So it suffices to see that for two IRW-particles moving at rate m, we have √ τ1 ≥ τ2 lim PIRW {−1,1} = 0, ψ(t), t 0
t→∞
(3.19)
IRW where Pa,b denotes the joint distribution of two IRW-particles starting respectively from lo2 cations a and b, τ01 , resp. τ{−1,1} , stands for the hitting time of 0, resp. of {−1, 1}, for some I IRW-particle X , resp. another IRW-particle Y I . This is, using standard arguments, in turn implied by
√ τ 1 ≥ τ 2 = 0. lim PIRW 0 ψ(t), t 0
t→∞
(3.20)
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3491
Fig. 2. Stage 2. While the IRW-particles X I and Y I are coupled coordinate-wise via an Ornstein coupling (there, omitted from the figure), the SIP-particles, here two of them in blue, are consequently coupled in the same way provided no collision occurs within any set of particles (that is, here, between X 1S and X 2S or Y1S and Y2S , but any other does not matter). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Recall by the reflection principle (see e.g. [11, Chapter I]), for any a ≥ 0: IRW PIRW 0 (τa ≥ t) = P0 (|Z (t)| ≤ a),
where here and in what follows Z (t) denotes a continuous-time simple random walk moving at rate m. Therefore, choosing a time scale ϕ(t) such that ϕ(t) → ∞ as t → ∞, we obtain IRW √ 1 2 2 IRW √ 1 2 2 √ τ1 ≥ τ2 ≤ P τ ≥ τ , τ ≥ ϕ(t) + P τ ≥ τ , τ ≤ ϕ(t) PIRW 0 0 0 0 0 ψ(t), t 0 ψ(t), t 0 ψ(t), t 0 IRW 1 IRW 2 ≤ Pψ(t) τ0 ≥ ϕ(t) + P√t τ0 ≤ ϕ(t) √ IRW ≤ PIRW |Z (ϕ(t))| ≤ ψ(t) + P |Z (ϕ(t))| ≥ t 0 0 |Z (ϕ(t))| |Z (ϕ(t))| = PIRW ≤ 1 + PIRW ≥1 . (3.21) √ 0 0 ψ(t) t Since |Z (ϕ(t))| ∼ ϕ(t)1/2 as t → ∞, (3.20) follows by choosing ϕ(t) = cψ(t)2 for some positive constant c so that we let t go to infinity, and then c → ∞. Indeed, as t → ∞ by the Donsker invariance principle, the first term of the r.h.s. becomes P(|N (0, c)| ≤ 1), where N (0, c) denotes a normal random variable √ with mean zero and variance c. While the sec√ ond term of the r.h.s. vanishes since ψ(t) = o( t), and hence |Z (ϕ(t))| → 0 as t → ∞. Hence t the r.h.s. of (3.21) goes to zero as t, then c go to infinity. So far, we have proved the probability that the two sets of SIP-particles are coupled in the time lapse [(1 − δ)t, t] is strictly positive, and tends to 1 as t → ∞. Then by iterating independently such coupling attempts, one sees that the probability of eventual successful coupling of the SIP particles is one.
3492
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
As a consequence, Corollary 3.1. Let µ be a tempered invariant for the SIP. Then for all n ∈ N, ∃ αn ∈ measure n [0, ∞) such that for all x = (x1 , . . . , xn ) ∈ Zd , µ(x) = αn i.e., µ(x) only depends on n = |x| but not on the precise locations x1 , . . . , xn . Proof. The proof is quite standard, using that bounded harmonic functions are constant when a successful coupling exists, but we give it here for the sake of self-consistency. Let µ be a tempered invariant measure, and put, as before, cn = supx1 ,...,xn µ(x1 , . . . , xn ). Consider two sets of SIP-particles X S (t) = (X 1S (t), . . . , X nS (t)) and Y S (t) = (Y1S (t), . . . , YnS (t)). Since there exists d n a successful coupling for the SIP-particles, for any x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Z , PSIP x,y (τ < ∞) = 1, where τ = inf t ≥ 0 : X S (s) = Y S (s) for all s ≥ t . Now since µ ∈ Itemp by Proposition 2.1, µ(x) = ESIP µ(X S (t)) x S S S SIP S S S = ESIP µ (Y (t))1{X (t) = Y (t)} + E µ (X (t))1{X (t) = ̸ Y (t)} x,y x,y SIP S SIP S = Ex,y µ(Y (t)) + Ex,y µ(X (t)) − µ(Y S (t)) 1{X S (t) ̸= Y S (t)} S S S S = µ(y) + ESIP µ (X (t)) − µ (Y (t)) 1{X (t) = ̸ Y (t)} . x,y Hence, using µ(x) ≤ cn for x ∈ (Zd )n , we obtain the estimate S S | µ(x) − µ(y)| ≤ 2cn PSIP x,y X (t) ̸= Y (t) and the result follows by letting t go to infinity.
4. Ergodic tempered measures From now on, we can characterise the ergodic tempered measures. First, in the next lemma, we prove that this set coincides with the extreme elements of the set of tempered invariant measures. Lemma 4.1. The set of tempered ergodic invariant measures coincides with the set of extreme points of the tempered invariant measures, i.e., (Itemp )e = (I ∩ Ptemp )e = Ie ∩ Ptemp . Proof. Let µ ∈ (I ∩ Ptemp ) and µ ̸∈ (I ∩ Ptemp )e , then there exist 0 < λ < 1 and µ1 , µ2 ∈ (I ∩ Ptemp ) such that µ = λµ1 + (1 − λ)µ2 , therefore µ ̸∈ Ie , and hence µ ̸∈ Ie ∩ Ptemp . Conversely, suppose that µ ∈ Ptemp is not in Ie ∩ Ptemp , then µ ̸∈ Ie and hence then there exist λ ∈ (0, 1) and µ1 , µ2 ∈ I such that µ = λµ1 + (1 − λ)µ2 .
3493
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
This equality together with the fact that µ ∈ Ptemp , and the positivity of the functions D(ξ, ·), imply that µ1 , µ2 ∈ Ptemp . Therefore µ ̸∈ (I ∩ Ptemp )e . Finally if µ is not in Ie ∩ Ptemp and µ is also not in Ptemp then trivially µ ̸∈ (I ∩ Ptemp )e . We can then characterise the ergodic tempered measures. Theorem 4.1. If µ ∈ Ie ∩ Ptemp then µ = νλm for some λ ∈ [0, 1). As a consequence (Itemp )e = {νλm : λ ∈ [0, 1)}.
(4.22)
Proof. Remark that by the multivariate version of the Carleman moment condition, within the class Ptemp , a measure µ is uniquely determined by its D-transform µ. ˆ Let µ ∈ (Itemp )e . Since µ is invariant its D-transform µ(ξ ) depends only on |ξ | so we put, with slight abuse of notation µ(ξ ˆ ) = µ(n). ˆ In order to show that µ = νλm for some λ ∈ [0, 1), it suffices now to show that µ(n) ˆ = a n for some a ≥ 0. This in turn follows if we show that µ(n ˆ + m) = µ(n) ˆ µ(m), ˆ
(4.23) T
for all n, m ∈ N. Recall St stands for the semigroup of the SIP and denote ST = T1 0 St dt. Fix ξ, ξ ′ to finite configurations, with |ξ | = n, |ξ ′ | = m. By ergodicity we have, µ-almost surely ST D(ξ, η) → µ(ξ ˆ ) = µ(|ξ ˆ |), as T → ∞ (where ST works on η). Therefore, using that µ ∈ Ptemp , by dominated convergence, D(ξ ′ , η)ST D(ξ, η)dµ(η) → µ(ξ ˆ ′ )µ(ξ ˆ )= µ(m) µ(n).
(4.24)
(4.25)
On the other hand, by self-duality 1 T ST D(ξ, η) = pt (ξ, ξ ′′ )dt D(ξ ′′ , η). T 0 ′′ ξ ∈Ω
(4.26)
n
As a finite number of SIP-particles eventually spread out all over the lattice Zd , for large T , the main contribution of the sum over ξ ′′ in the r.h.s. of (4.26) is from configurations ξ ′′ in which there are no particles at locations occupied by particles in ξ ′ (let us denote this property by ξ ′ ⊥ ξ ′′ ). If ξ ′ ⊥ ξ ′′ , then D(ξ ′′ , η)D(ξ ′ , η) = D(ξ ′′ + ξ ′ , η). More precisely, consider n-SIP particles (X 1S (t), . . . , X nS (t)). By translation invariance and irreducibility, for all i = 1, . . . , n, X i (t) drifts to infinity as t goes to infinity. Hence, let n n ξ ′ = i=1 δ yi and ξ = i=1 δxi , so that pt (ξ, ξ ′′ ) ≤ PSIP ∃i, j = 1, . . . , n : X iS (t) = y j ξ ′′ ⊥ξ ′
≤
PSIP X iSIP (t) = y j
i, j
which tends to zero when t goes to infinity.
3494
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
Therefore, 1 T ′ D(ξ , η)St D(ξ, η)dµ(η) = pt (ξ, ξ ′′ )D(ξ ′′ , η)D(ξ ′ , η)dµ(η) T 0 ′′ ξ ∈Ωn 1 T = pt (ξ, ξ ′′ )D(ξ ′ + ξ ′′ , η)dµ(η) + bT , T 0 ξ ′′ ∈Ω ,ξ ′′ ⊥ξ ′ n
where bT vanishes as T goes to infinity. Indeed, since µ is a tempered measure, 1 T bT ≤ pt (ξ, ξ ′′ ) sup D(ξ ′ , η)D(ξ ′′ , η)dµ(η) T ′′ ′ ′′ 0 ξ :|ξ |=|ξ | ξ ′′ ∈Ωn ,ξ ′′ ̸⊥ξ ′ T 1 ≤C· pt (ξ, ξ ′′ ) , T 0 ξ ′′ ∈Ω ,ξ ′′ ̸⊥ξ ′ n
where, using (2.8), C = C(|ξ |, |ξ ′ |) is a constant. This can be seen as follows. Since d(k, ℓ)2 is a polynomial of degree 2k, it is a combination of d( j, .) with j ≤ 2k, i.e., d(k, ℓ)2 = 2k j=0 α(k, j)d( j, ℓ), for some constants α(., .). Therefore if µ ∈ Ptemp , then for |ξ | = n, one has n 2k 2 n . D(ξ, η) dµ(η) ≤ c2n |α(k, j)| := C j=0
Therefore by Cauchy–Schwarz, one can choose |ξ | C |ξ ′ | . C= C Therefore, aT + µ(n) ˆ µ(m) ˆ =
D(ξ ′ , η)ST D(ξ, η)dµ(η) 1 T = pt (ξ, ξ ′′ )dt D(ξ ′′ + ξ ′ , η)dµ(η) + bT T 0 ξ ′′ ∈Ωn ,ξ ′′ ⊥ξ ′ 1 T ′′ pt (ξ, ξ )dt µ(n ˆ + m) + bT = T 0 ξ ′′ ∈Ωn ,ξ ′′ ⊥ξ ′ 1 T = pt (ξ, ξ ′′ )dt µ(n ˆ + m) + bT + cT T 0 ξ ′′ ∈Ω n
= µ(n ˆ + m) + bT + cT
(4.27)
where aT → 0 by (4.25) and as explained before bT , cT → 0. Letting now T → ∞ give (4.23). Now combining this with Lemma 4.1, and the fact that all νλm are elements of Ie ∩ Ptemp , i.e., are ergodic under the SIP dynamics (see e.g. [7]) gives the result (4.22). Remark 4.1. As a consequence of Proposition 2.2 this result can be transferred to the BEP, showing that its tempered invariant ergodic measures are product of Gamma distributions, and to the BMP, showing that its tempered invariant ergodic measures are product of mean zero Gaussians.
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3495
Next, we show that all tempered invariant measures satisfy a correlation inequality of the type derived in [7]. n Proposition 4.1. Let µ ∈ Itemp , then for all (x1 , . . . , xn ) ∈ Zd , n n D δxi , η dµ(η). (4.28) δxi , η dµ(η) ≥ D i=1
i=1
Proof. Because every element of Itemp can be decomposed into extreme elements, we have µ = νλm dΛ(λ), for some probability measure Λ on [0, 1) and as a consequence, denoting λ ρ(λ) = 1−λ we have n n m δxi , η dνλ (η)dΛ(λ) = ρ(λ)n dΛ(λ) δxi , η dµ(η) = D D i=1
i=1
≥
ρ(λ) dΛ(λ)
n
=
n
D(δxi , η)dµ(η).
i=1
5. Convergence to ergodic product measures In this section, we give sufficient criteria for a starting measure µ to converge in the course of time to one of the product measures νλm . To this purpose, we introduce the following notions of d asymptotic independence and homogeneity. We call a function f : NZ → R local if it is a finite linear combination of the functions D(ξ, ·). Definition 5.1. A measure µ is asymptotically homogeneous (AH), if there exists ρ > 0 such that (5.29) lim sup Ez D(δ Z (t) , η)dµ(η) − ρ = 0, t→∞ z
where Ez denotes the expectation w.r.t. a simple random walk {Z (t), t ≥ 0} starting from z. Notice that every translation invariant measure with finite moments is trivially AH. Definition 5.2. A measure µ is asymptotically independent (AI), if for all n and for all choices of local functions f 1 , . . . , f n , n n lim τ yi f i dµ − τ yi f i dµ = 0. |yi −y j |→∞
i=1
i=1
Then we have the following result, Theorem 5.1. Let µ be tempered, AH and AI, then m µSt → νλ(ρ) ,
with λ(ρ) =
ρ 1+ρ ,
as t → ∞
where ρ is the constant defined as the limit in (5.29).
3496
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
Here, the convergence holds in the sense of convergence of all multivariate moments, that is, m D(ξ, η)dµSt (η) → D(ξ, η)dνλ(ρ) (η), as t → ∞ for all finite configurations ξ . Proof. We have to prove that for all x1 , . . . , xn ∈ Zd , n n Eη D δxi , η dµ(η) = lim D δxi , η dνλ(ρ) (η). t→∞
i=1
(5.30)
i=1
Remark the r.h.s is equal to ρ n thanks to (2.6). Now, dealing with the l.h.s., rewrite n n Eη D δxi , η dµ(η) = ESIP δ X S (t) , η dµ(η) x D i=1
i=1
i
n Using [7, Lemma 1] for x ∈ Zd , it is equal to n ESIP D(δ X S (t) , η)dµ(η) + o(t). x i
i=1
Indeed, after an arbitrary large time, with probability close to one, the n SIP-particles initially at x will have spread out and will be at different locations. By the Markov property, for any time scale ψ(t) ≤ t and such that ψ(t)/t 1/4 → 0 as t → ∞, n n SIP SIP D(δ X S (ψ(t)) , η)dµ(η) ESIP D(δ , η)dµ(η) = E E xi x x X S (t−ψ(t)) i=1
i=1
IRW = ESIP x EX S (t−ψ(t))
n i=1
i
D(δ X I (ψ(t)) , η)dµ(η) + R(t). i
For the first equality, we used invariance of µ, the Markov property and self-duality. For the second equality, we used the fact that after the large time √ span t − ψ(t), the SIP-particles are with probability close to one at distance of the order√of t from each other and therefore, in the remaining time ψ(t) they will not come closer than t − t 1/8 to each other, i.e., are still far apart and therefore will move as if they are IRW-particles. We denoted R(t) to be the contribution of the complementary event. Let us denote by Q the set of configurations with n particles where at least two are at the same position, and A(ψ(t)) the event that ξ(s) is in Q for some s ∈ (t − ψ(t), t). We then have, using Cauchy–Schwarz and the same argument as in the proof of Theorem 4.1, based on the fact that µ is tempered, PSIP (A(ψ(t))) sup D(ξ, η)2 dµ(η) R(t) ≤ ESIP x X S (t−ψ(t)) |ξ |=n
which tends to zero as t → ∞. Next because µ is AI, the product over i and the integral over µ can be exchanged at the price of another o(t) term. Once the product is out of the integral, it trivially also comes out of the IRW expectation, and therefore, it is equal to n IRW ESIP E D(δ X I (ψ(t)) , η)dµ(η) + o(t). x X I (t−ψ(t)) i=1
i
i
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
3497
Use the AH property (5.29) of µ to conclude that this expression in turn is equal to ρ n + o(t), we conclude by letting t → ∞.
Remark 5.1. We can replace the AH and AI assumptions in Theorem 5.1 by the following assertion: lim pt (x, y)η(y) = ρ, (5.31) t→∞
y
where the limit is in µ-probability and pt (., .) stands for a simple random walk transition probability. Indeed, assuming (5.31) we can write n n IRW SIP ESIP E D(δ , η)dµ(η) = E pt−s (X iI (s), yi )ηs (yi )dµ(η) I x x X (t−s) X I (s) i
i=1
i=1 yi
ρn
whose r.h.s. converges to as t, then s, go to infinity, by (5.31) and dominated convergence (because µ is tempered by assumption). Examples where (5.31) is satisfied are measures µ which are absolutely continuous with dµ , which is a local function of the respect to reversible product measures νρ with density dν ρ configuration η, i.e., depending on a finite number of coordinates η(x). We conclude with two additional remarks: Remark 5.2. The fact that p is nearest-neighbour can be replaced without any difficulty by a finite range kernel (with the same proof, adapting the definition of collision). Presumably it is enough that p is translation invariant and has a finite second moment. Remark 5.3. Related to the SIP are the dual Kipnis–Marchioro–Presutti (KMP) process and its generalised so-called “thermalised” SIP [2], where when particles are at nearest neighbour positions, several particles can jump at the same time. However, if in this thermalised SIP model all the particles are separated (i.e., at distance >1), they behave exactly as independent random walkers, and therefore a finite number of them can be successfully coupled just as SIP particles can. This implies that for the thermalised SIP we have the same set of ergodic tempered measures. The thermalised SIP in turn is the dual process of a mass-redistribution model, called the thermalised BEP in [2] which generalises the KMP process [8]. Hence, the only ergodic tempered measures of this generalised KMP process are also products of Gamma distributions. Acknowledgements We thank Pablo A. Ferrari for pointing us to [4] and further valuable comments and motivating discussions. We thank an anonymous referee for careful reading and many useful comments. References [1] E. Andjel, Invariant measures for the zero range process, Ann. Probab. 10 (3) (1982) 525–547. [2] G. Carinci, C. Giardin`a, C. Giberti, F. Redig, Duality for stochastic models of transport, J. Stat. Phys. 152 (4) (2013) 657–697.
3498
K. Kuoch, F. Redig / Stochastic Processes and their Applications 126 (2016) 3480–3498
[3] G. Carinci, C. Giardin`a, F. Redig, T. Sasamoto, Asymmetric transport models with Uq (su(1, 1)) symmetry, J. Stat. Phys. 163 (2) (2016) 239–279. [4] A. De Masi, E. Presutti, Probability estimates for symmetric simple exclusion random walks, Ann. Inst. Poincar´e 19 (1) (1983) 71–85. [5] A. De Masi, E. Presutti, Mathematical Methods for Hydrodynamic Limits, Springer-Verlag Berlin, Heidelberg, 1991. [6] C. Giardin`a, J. Kurchan, F. Redig, Duality and hidden symmetries in interacting particle systems, J. Stat. Phys. 135 (1) (2009) 25–55. [7] C. Giardin`a, F. Redig, K. Vafayi, Correlation inequalities for interacting particle systems with duality, J. Stat. Phys. 141 (2) (2010) 242–263. [8] C. Kipnis, C. Marchioro, E. Presutti, Heat flow in an exactly solvable model, J. Stat. Phys. 27 (1) (1982) 65–74. [9] C. Kleiber, J. Stoyanov, Multivariate distributions and the moment problem, J. Multivariate Anal. 113 (2013) 7–18. [10] T. Lindvall, Lectures on Coupling Method, Dover, 1992. [11] G.M. Lawler, V. Limic, Random Walk: A Modern Introduction, Cambridge University Press, 2010. [12] T.M. Liggett, Interacting Particle Systems, Springer-Verlag Berlin, 2005. [13] A. Opoku, F. Redig, Coupling and hydrodynamic limit for the inclusion process, J. Stat. Phys. 160 (3) (2015) 532–547.