Chains with complete connections and one-dimensional Gibbs measures

We discuss the relationship between discrete-time processes (chains) and one-dimensional Gibbs measures. We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specifications). Furthermore, we discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. As an auxiliary result we prove a (re)construction theorem for specifications starting from single-site conditioning, which applies in a more general setting (general spin space, specifications not necessarily Gibbsian).


Introduction
One dimensional systems are simultaneously the object of the theory of stochastic processes and the theory of Gibbs measures.The complemen-tarity of both approaches has yet to be fully exploited.Stochastic processes are defined on the basis of transition probabilities.A consistent chain is one for which these probabilities are a realization of the single-site conditional probabilities given the past.A Gibbs measure is defined in terms of specifications, which determine its finite-volume conditional probabilities given the exterior of the volume.In one dimension this implies conditioning both the past and the future.In this paper we study conditions under which a stochastic process defines, in fact, a Gibbs measure and, in the opposite direction, when a Gibbs measures can be seen as a stochastic process.
This type of questions has been completely elucidated for Markov processes and fields.See, for instance, Chapter 11 of the treatise by Georgii (1988).The equivalence, however, is obtained by eigenvalue-eigenvector considerations which are not readily applicable to non-Markovian processes.The Gibbsian character of processes with exponentially decreasing continuity rate is also known.It follows from Bowen's characterization of Gibbs measures (Theorem 5.2.4 in Keller, 1998, for instance).No result seems to be available on the opposite direction, namely on the characterization of a onedimensional Gibbs measure for an exponentially summable interaction as a stochastic process.
In our paper we present both a generalization and an alternative to this previous work.We directly establish consistency-preserving maps between specifications and transition probabilities.More precisely, these applications are between specifications and their analogous for stochastic processes, which we call left-interval specifications (LIS).The description in terms of LIS is equivalent to that in terms of transition probabilities, but it offers a setting that mirrors the statistical mechanical setting of Gibbs measures.In fact, the use of LIS allows us to "import", in a painless manner, concepts and results from statistical mechanics into the theory of stochastic processes.This will be further exploited in a companion paper (Fernández and Maillard, 2003).
We consider systems with a finite alphabet, possibly with a grammar, that is, with exclusion rules such that the non-excluded configurations form a compact set.We do not assume translation invariance either of the kernels or of the consistent measures.The main limitation of our results is that, in order to insure that the necessary limits are uniquely defined, specifications and processes are required to satisfy a strong uniqueness condition called hereditary uniqueness condition (HUC).A second property, called good future (GF) is demanded for stochastic processes to guarantee some control of the conditioning with respect to the future.HUC is verified, for instance, by specifications satisfying Dobrushin and boundary-uniformity criteria (reviewed below).Both GF and HUC are satisfied by a large family of processes, for instance by the chains with summable variations studied by Harris (1955), Ledrappier (1974), Walters (1975), Lalley (1986), Berbee (1987), Bressaud et al (1999), . . . .Our results show that under these hypotheses there exist: (i) a map that to each LIS associates a specification such that the process consistent with the former is a Gibbs measure consistent with the latter (Theorem 4.12), and (ii) a map that to each specification associates a LIS such that the Gibbs measure consistent with the former is a process consistent with the latter (Theorem 4.16).If domain and image match, these maps are inverses of each other.This happens, in particular, in the case of exponentially decreasing continuity rates (Theorem 4.19).As part of the proofs, we obtain estimates linking the continuity rates of LIS and specifications related by these maps (Theorem 4.18).We also show that the validity of the Dobrushin and boundary-uniformity criteria for the specification implies the validity of analogous criteria for the associated stochastic process (Theorem 4.17).Finally, in Appendix A we show that a system of single-site normalized kernels, satisfying order-consistency and boundedness properties with respect to an a-priori measure, can be extended, in a unique manner, to a full specification.This generalizes the reconstruction Theorem 1.33 in Georgii (1988).As this theorem may be of independent interest, we have stated it in rather general terms, for arbitrary spin space and non-necessarily Gibbsian kernels (Theorem A.4).

Notation and preliminary definitions
We consider a finite alphabet A endowed with the discrete topology and σ-algebra, and Ω a compact subset of A Z .The space Ω is endowed with the projection F of the product σ-algebra associated to A Z .The space Ω represents admissible "letter configurations", where the admissibility is defined, for instance by some exclusion rule as in Ruelle (1978) or by a "grammar" (subshift of finite type) as in Walters (1975).For each Λ ⊂ Z, and each configuration σ ∈ A Z we denote σ Λ its projection on Λ, namely the family (σ i ) i∈Λ ∈ A Λ .We denote We denote F Λ the corresponding sub-σ-algebra of F .When Λ is an interval, Λ = [k, n] with −∞ ≤ k ≤ n ≤ +∞, we shall use the "sequence" notation: , etc.The notation ω Λ σ ∆ , where Λ ∩ ∆ = ∅, indicates the configuration on Λ ∪ ∆ coinciding with ω i for i ∈ Λ and with σ i for i ∈ ∆.In particular, ω n k σ m n+1 = ω k , . . ., ω n , σ n+1 , . . ., σ m .For ω, σ ∈ A Z , we note ("σ equal to ω off j").We denote S the set of finite subsets of Z and S b the set of finite intervals of Z.For every Λ ∈ S b we denote l Λ min Λ and The expression lim Λ↑V will be used in two senses.For kernels associated to a LIS (defined below), lim Λ↑V f Λ is the limit of the net {f Λ , {Λ} Λ∈S b , Λ⊂V , ⊂}, for V an infinite interval of Z.For kernels associated to a specification , lim Λ↑V γ Λ is the limit of the net {γ Λ , {Λ} Λ∈S, Λ⊂V , ⊂}, for V an infinite subset of Z.To lighten up formulas involving probability kernels, we will freely use ρ(h) instead of E ρ (h) for ρ a measure on Ω and h a F -measurable function.Also ρ(σ Λ ) will mean ρ({ω ∈ Ω : We start by briefly reviewing the well known notion of specification. (2.4) The specification is: Property c) is usually referred to as consistency.There and in the sequel we adopt the standard notation for composition of probability kernels (or of a probability kernel with a measure).For instance, (2.4) is equivalently to for each F -measurable function h and configuration ω ∈ Ω. Remarks 2.6 A Markov specification of range k corresponds to the particular case in which the applications (2.5) are in fact F ∂ k Λ -measurable, where ∂ k Λ = {i ∈ Λ c : |i − j| ≤ k for some j ∈ Λ}.

2.7
In the sequel, we find useful to consider also the natural extension of the kernels γ Λ to functions ∈ Ω.We shall not distinguish notationally both types of kernels. (2.9) The family of these measures will be denoted G(γ).
Remarks 2.10 A Markov field of range k is a measure consistent with a Markov specification of range k.
2.11 A Gibbs measure on (Ω, F ) is a measure µ consistent with a specification that is continuous and non-null on Ω.The SRB measures (Bowen, 1975) are particular one-dimensional examples.
We now introduce the analogous notion for processes.Due to the nature of the defining transition probabilities, the corresponding finite-region kernels must apply to functions measurable only with respect to the region and its past.Furthermore, finite intervals already suffice.
The LIS is: (i) Continuous on Ω if for all Λ ∈ S b and all σ Λ ∈ Ω Λ the functions

2.16
As for specifications, in the sequel we shall not distinguish notationally the kernels f Λ from their extensions on Definition 2.17 (2.18) The family of these measures will be denoted G(f ).
Remarks 2.19 A Markov chain of range k is a measure consistent with a Markov LIS of range k.

2.20
Measures consistent with general, non-necessarily Markovian LIS were initially called Chains with complete connections by Onicescu and Mihoc (1935).These objects have been reintroduced several times in the literature under a variety of names: chains of infinite order (Harris, 1955), g-measures (Keane, 1972), uniform martingales (=random Markov processes) (Kalikow, 1990), . . . .Finally, we introduce a strong notion of uniqueness needed in the sequel.
Definition 2.21 1) A specification γ satisfies a hereditary uniqueness condition (HUC) for a family H of subsets of Z if for all (possibly infinite) sets V ∈ H and all configurations ω ∈ Ω, the specification γ (V,ω) defined by admits a unique Gibbs measure.The specification satisfies a HUC if it satisfies a HUC for H = P(Z).
2) A LIS f satisfies a hereditary uniqueness condition (HUC) if for all intervals of the form V = [i, +∞[, i ∈ Z, or V = Z, and all configurations ω ∈ Ω, the LIS f (V,ω) defined by (2.23) admits a unique consistent chain.

Preliminary results
Let us summarize a number of useful properties of LIS and specifications.First we introduce functions associated to LIS singletons.For a LIS f and a configuration ω ∈ Ω, let In the shift-invariant case, the function f 0 is a g-function in the sense of Keane (1972).
The following theorem expresses the equivalence between the description in terms of LIS and the usual description in terms of transition probabilities (=LIS singletons).
Theorem 3.2 (singleton consistency for chains) Let (g i ) i∈Z be a family of measurable functions over (Ω, F ) which enjoy the following properties Then there exists a unique left interval-specification f (f Λ ) Λ∈S b such that f i = g i , for all i in Z. Furthermore: (ii) f is non-null on Ω if, and only if, so are the functions g i , that is, if and only if g i (ω) > 0 for each i ∈ Z and each ω ∈ Ω i −∞ .
(iii) f is weakly non-null on Ω if, and only if, so are the functions g i , that is, if and only if for each i ∈ Z there exists The proof of this result is rather simple (it is spelled up in Fernández and Maillard, 2003).The following theorem is the analogous result for specifications.Let us consider the following functions associated to an specification γ: Theorem 3.6 (singleton consistency for Gibbs measure) Let (ρ i ) i∈Z be a family of measurable functions over (A Z , F ) which enjoys the following properties (b) Order-consistency on Ω: for every i, j ∈ Z and ω ∈ Ω, (3.7) (c) Normalization on Ω: for every i ∈ Z and ω ∈ Ω, Then there exists a unique specification γ on (Ω, F ) such that γ i (ω) = ρ i (ω), for all i in Z. Furthermore, (ii) γ satisfies an order-independent prescription: For each Λ, Γ ∈ S with This result will be proved in Appendix A in a more general setting.
For completeness, we list now several, mostly well known, sufficient conditions for hereditary uniqueness.They refer to different ways to bound continuity rates of transition kernels.We start with the relevant definitions.Definition 3.10 (i) The k-variation of a F {i} -measurable function f i is defined by (ii) The interdependence coefficients for a family of probability kernels π = π {i} i∈Z , π {i} : for all i, j ∈ Z.Here we use the variation norm and A LIS f on (Ω, F ) satisfies a HUC if it satisfies one of the following conditions: • Johansson and Öberg (2002): The LIS f is stationary, non-null and k≥0 var 2 k (log f 0 ) < +∞.
• One-sided Dobrushin (Fernàndez and Maillard, 2003): • One-sided boundary-uniformity (Fernàndez and Maillard, 2003): There exists a constant K > 0 so that for every cylinder set A = {x m l } ∈ Ω m l there exists an integer n such that The last two conditions, proven in a companion paper (Fernández and Maillard, 2003), are in fact adaptations of the following well known criteria for specifications.
• Georgii (1974) boundary-uniformity: There exists a constant K > 0 so that for every cylinder set A ∈ F there exists Λ ∈ S b such that We remark that the conditions involve no non-nullness assumption.

Main results
For a LIS f on Ω let us denote, for each and Similarly, for a specification γ on Ω, let us denote, for each ω ∈ Ω and each k, j ∈ Z c ω j (γ k ) min for each k ≥ m Λ .
(ii) A LIS f on Ω is said to have an exponentially-good future (EGF) if it is non-null on Ω and there exists a real a > 1 such that lim sup for all j ∈ Z.
(iii) A specification γ on Ω is said to have an exponentially-good future (EGF) if it is non-null on Ω and there exists a real a > 1 such that lim sup for all j ∈ Z.
Definition 4.9 Let us introduce the following sets.
Θ {LIS f continuous and non-null on Ω} Π {specifications γ continuous and non-null on Ω}, We remark that each of the LIS or specifications of any of the preceding sets has at least one consistent measure.This is because the (interesting part of) the configuration space is compact and the LIS or specifications are assumed to be continuous.Indeed, as the space of probability measures on a compact space is weakly compact, every sequence of measures γ Λn ( • | ω {n} ) or f Λn ( • | ω {n} ), for (Λ n ) an exhausting sequence of regions and (ω {n} ) a sequence of configurations, has a weakly convergent subsequence.By continuity of the transitions the limit is respectively a Gibbs measure or a consistent chain.

Consider the function
for all Λ ∈ S, n ≥ m Λ and ω ∈ Ω.The continuity of f implies that the functions F Λ,n (ω Λ | • ) are continuous on Ω Λ c for each ω Λ ∈ Ω Λ .We use these functions to introduce the map for all Λ ∈ S and ω ∈ Ω.
Theorem 4.12 (LIS specification) 1) The map b is well defined.That is, for f ∈ Θ 1 (a) the limit (4.11) exists for all Λ ∈ S and ω ∈ Ω.

Remark 4.13
Since for all k ≥ 1, var k (g) ≥ δ k (g), Θ 1 includes the set of stationary non-null LIS with summable variation.
Consider now the map for all Λ ∈ S b , A ∈ F Λ , and ω ∈ Ω for which the limit exists.
Theorem 4.16 (specification LIS) 1) The map c is well defined.That is, for γ ∈ Π 2 (a) the limit (4.15) exists for all , where µ γ is the only Gibbs measure consistent with γ.

(b)
The map c is one-to-one.
In addition a LIS of the form f γ satisfies the following properties.
(a) If γ satisfies Dobrushin uniqueness condition, then so does f γ .
(b) If γ satisfies the boundary-uniformity uniqueness condition, then so does f γ .
Under suitable conditions the maps b and c are reciprocal.
(c) b and c establish a one-to-one correspondence between Θ 3 and Π 3 that preserves the consistent measure.
We remark that Θ 3 includes the well studied processes with Holdërian transition rates (see, for instance, Lalley, 1986, or Keller, 1998).Part (c) of the theorem shows, in particular, the equivalence between such processes and Bowen's Gibbs measures.

Proofs
We start with a collection of results used for several proofs.
The last three displays imply In particular, LIS and specifications are completely defined by the families of their restrictions.
To prove (5.4) we use definition (4.10) and the factorization We then apply inequalities (5.3) to bound each of the factors by similar factors with conditioning configuration σ.
To obtain (5.5) we apply the LIS-reconstruction formula (3.4) with m = n + 1 which yields In the denominator, only We use inequalities (5.3) for these.
To prove item 1) (b), we observe that γ f Λ (A | • ) is clearly F Λ c -measurable for every Λ ∈ S and every A ∈ F .Moreover condition (b) of Definition 2.12 together with the presence of the indicator function for every Λ ∈ S and every B ∈ F Λ c .Therefore it suffices to show that for each Λ, ∆ ∈ S such that Λ ⊂ ∆ and each ω ∈ Ω.Let us denote, for each (5.9) Using the reconstruction property (3.4) of LIS with l = l ∆ , n = l Λ − 1 and m = n, we obtain and (5.10) Identity (5.8) follows from (5.9) and (5.10).
We proceed with item 1) (c).By (5.7) and the summability of the bound Let us fix k 0 such that ǫ k < 1 for k ≥ k 0 .By (4.6) and the lower bound in (5.5) The right-hand side is strictly positive on Ω due to the non-nullness of F Λ,k 0 and the summability of the ǫ k .Hence γ f Λ is non-null on Ω.To prove assertion 2)(a) we consider µ ∈ G f (V,ω) and denote By a straightforward extension of (5.9), the dominated convergence theorem and the consistency of µ with respect to G Applying the consistency hypothesis a second time we obtain µ γ f Λ = µ.Assertion 2)(b) is an immediate consequence of 2) (a) and of the fact that |G(γ)| = 1 for all γ ∈ Π 1 .
Finally we prove 2) (c).Let f 1 and f 2 be two LIS on (Ω, F ), both in b −1 (Π 1 ), and such that γ The non-nullness of f 1 and f 2 on Ω implies that µ charges all open sets in Ω.Therefore, f 1 Λ and f 2 Λ coincide, on Ω, with the unique continuous realization of

Specification LIS
Let us introduce the spread of a (bounded) function h on Ω: Lemma 5.11 1) Let γ be a specification on Ω. Proof We proof part 1), the proof of 2) is similar.The obvious spreadreducing relation valid for every bounded measurable function h on Ω and every configuration ω ∈ Ω, plus the consistency condition (2.4) imply that the sequence {sup(γ Λn h)} is decreasing (and bounded below by inf h), while the sequence {inf(γ Λn h)} is increasing (and bounded above by sup h).
for every n, which yields for every n.This proves item 1)(a).
Our last auxiliary result refers to the following notion.
Definition 5.17 Proposition 5.18 Let γ be a continuous specification over (Ω, F ) which satisfies a HUC.Then γ can be extended into a continuous global specification such that for every subset for all continuous functions h ∈ F and all ω ∈ Ω. Moreover for all V ⊂ Z and all ω ∈ Ω, (5.20) Georgii (1988) gives a proof of this proposition in the Dobrushin regime (Theorem 8.23).The same proof extends, with minor adaptations, under a HUC (see Fernández and Pfister, 1997).
Proof of Theorem 4.16 Items 1) (a)-(b) are proven in Proposition (5.18).There are three things to prove regarding 1) (c): (i) Continuity of f γ .This is, in fact, an application of Proposition (5.18).(ii) Non-nullness of f γ .Consider Λ ∈ S, ω ∈ Ω, n ≥ m Λ and k ≥ 0. By the non-nullness and the continuity of γ and the compactness of Ω Λ c , there Therefore by the consistency of γ As, by hypothesis, each specification γ (V,ω) admits an unique Gibbs measure, it follows from lemma 5.11 1) (b) that This proves that G f γ (V,ω) = 1 by lemma 5.11 2) (a).The uniqueness part of assertion 2) (a) is contained in the just proven hereditary uniqueness.To show that µ γ ∈ G(f γ ), consider Λ ∈ S b and h a continuous F ≤m Λ -measurable function.By the dominated convergence theorem The consistency of µ γ with respect to γ implies, hence, that µ γ ∈ G(f γ ).
To prove assertion 2) (b), let γ 1 and γ 2 such that The non-nullness of γ 1 and γ 2 implies that µ charges all open sets on Ω. Therefore for each Λ ∈ S, γ 1 Λ and γ 2 Λ coincide with the unique continuous realization of Proof of Theorem 4.17 To prove item (a), let us recall one of the equivalent definitions of the variational distance between probability measures over (Ω i , F i ) For a proof of this result see for example Georgii (1988) (section 8.1).By the consistency of Therefore, by dominated convergence, (5.21) Since γ is continuous, we can do an infinite telescoping of (5.21) to obtain To show assertion (b), consider γ ∈ Π 2 for which there exists a constant K > 0 such that for every cylinder set A = {x m l } ∈ Ω m l there exist integers n, p satisfying Hence, by consistency of γ, we have that for some fixed σ ∈ Ω and for each k ≥ 0 In a similar way we obtain We conclude that for each k ≥ 0 Letting k → ∞ we obtain, due to definition (4.15), that
To prove assertion 2), let k, j ∈ Z such that j < k and consider ω, σ ∈ Ω such that ω =j = σ.As a direct consequence of definitions 4.3-4.4we have that, for all i ≥ k, (5.23) By the specification reconstruction formula (3.9) with Λ = {n + 1} and Γ = [l Λ , n] we have Using (5.22) and (5.23) it is easy to show, by induction over n ≥ m Λ + 1, that for all ξ, η ∈ Ω : Taking the limit when n tends to infinity, we obtain 2).

Proof of Theorem 4.19
For the proof of item (a) we consider γ ∈ Π 2 such that f γ ∈ Θ 1 and fix Λ ∈ S and ω ∈ Ω.By definition of the maps b and c [see (4.10)-(4.11)and (4.15)], we have that The consistency of γ Λ and γ [l Λ ,n+k] implies (5.25) By continuity of γ Λ (ω Λ | • ) we have that, for each ε > 0, for n large enough uniformly in k.Combining this with (5.24)-(5.25)we conclude that (5.26) A.10 This theorem is a strengthening of the reconstruction result given by Theorem 1.33 in Georgii (1988).In the latter, the order-consistency condition (A.9) is replaced by the requirement that the singletons come from a pre-existing specification (which the prescription reconstructs).
For finite E, Nahapetian and Dachian (2001) have presented an alternative approach where (A.9) is replaced by a more detailed pointwise condition.Their non-nullness hypotheses are also different from ours.
A.11 Identity (ii) can be used, in fact, to inductively define the family ρ by adding one site at a time.In fact, this is what is done in the proof below.The inequalities (iv) relate the non-nullness properties of ρ to those of the original family {ρ i } i∈Z d .
A.12 In the case E countable, λ i =counting measure, the order-consistency requirement (A.9) is automatically verified if the singletons are defined through a measure µ on F in the form for an exhausting sequence of volumes {V n }.Indeed, a simple computation shows that the last two terms in (A.9) coincide with lim n→∞ µ(ω Vn ) µ(ω Vn\{i,j} ) .
Proof In the following all functions are defined on Ω or on a projection of Ω over a subset of Z d .Initially we define ρ by choosing a total order for Z d and prescribing, inductively, that for each Λ ∈ S with |Λ| ≥ 2 and each ω where k = max Λ and Λ * k = Λ \ {k}.For each Λ, Γ ∈ S such that Γ ⊂ Λ c , we will prove, by induction over |Λ ∪ Γ|, that the functions so defined satisfy the following properties: Let us first comment why these properties imply the theorem.It is clear that properties (I3)-(I5), together with the deterministic character of (λ Λ ) on So the construction is unique.
(I1) Assume first that |Γ| = 1 and let k = max Λ. Combining the definition (A.13) and the property (A.3), we obtain If |Γ| ≥ 2 we consider l max Γ and apply the definition (A.13) to obtain We can now apply the inductive hypothesis (I1) to the right-hand side of (A.14) and (A.15) to prove (I1) at the next inductive level.
(I2) The argument is symmetric in Λ and Γ, so we can assume without loss that k = max(Λ ∪ Γ) belongs to Λ.If |Λ| = 1 (I2) is just the definition (A.13) applied to Λ ∪ Γ.We assume, hence, that |Λ| ≥ 2 and consider j ∈ Λ such that j = k.By the inductive assumption (I2) we have We now use this relation together with the first identity in (A.16) to conclude that We iterate this formula Λ * j − 1 times and we arrive to which is precisely ρ Λ∪Γ according to our definition (A.13).
(I3) We assume that |Λ| ≥ 2, otherwise (I3) is just the normalization hypothesis (A.6).Let k = max Λ. Definition A.13 and property A.3 yield where the last identity follows from the inductive hypothesis (I3).But, as in (A.18), (I4) To avoid a triviality we assume that |Λ| ≥ 2. Let µ be a probability measure on (Ω, F ) such that µ (ρ i λ i ) = µ for all i ∈ Λ.Consider k = max Λ and a measurable function h.By the factorization property (A.3) of λ Λ and the definition (A.13) of ρ Λ , we have By the inductive hypothesis (I4) µ is consistent with ρ Λ * k λ Λ * k and with ρ k λ k , thus But, in the right-hand side, the two innermost integrals with respect to λ k commute with the external one, so we have which proves (I4).

( a )
If there exists an exhausting sequence of regions Λ n ⊂ Z such that lim n→+∞ Spr (γ Λn h) = 0 (5.12) for each continuous F -measurable function h, then |G(γ)| ≤ 1.(b) If γ is continuous and |G(γ)| ≤ 1, then (5.12) holds for all exhausting sequences of regions Λ n ⊂ Z and all continuous F -measurable function h. 2) Let f be a LIS on Ω (a) If for each i ∈ Z and each continuous F ≤i -measurable continuous function h lim n→+∞ Spr f [i−n,i] h = 0 , (5.13) then |G(f )| ≤ 1.(b) If f is continuous and |G(f )| ≤ 1, then (5.13) is verified for all i ∈ Z and all continuous F ≤i -measurable continuous function h.