Invariant Measures and Decay of Correlations for a Class of Ergodic Probabilistic Cellular Automata
aa r X i v : . [ m a t h . D S ] D ec Noname manuscript No. (will be inserted by the editor)
Coletti, Cristian F · Tisseur, Pierre
Invariant Measures and Decay ofCorrelations for a Class of ErgodicProbabilistic Cellular Automata
Received: date / Accepted: date
Abstract
We give new sufficient ergodicity conditions for two-state proba-bilistic cellular automata (PCA) of any dimension and any radius. The proofof this result is based on an extended version of the duality concept. Underthese assumptions, in the one dimensional case, we study some properties ofthe unique invariant measure and show that it is shift-mixing. Also, the decayof correlation is studied in detail. In this sense, the extended concept of du-ality gives exponential decay of correlation and allows to compute explicitilyall the constants involved.
Keywords
Probabilistic cellular automata · Invariant measures · DualityDecay of correlation
Mathematics Subject Classification (2000) · · · · Probabilistic cellular automata (PCA) are discrete time Markov processeswhich have been intensely studied since at least Stavskaja and Pjatetskii-Shapiro [15] (1968). This kind of processes have as state space a product space X = A G where A is any finite set and G is any locally finite and connectedgraph. In this work we will focus our attention on G = Z d and A = { , ..., n } for some integer n ≥
1. We may regard a PCA as an interacting particlesystem where particles update its states simultaneously and independently.Recall that a PCA is ergodic if there exists only one invariant measure µ andstarting from any initial measure µ the system converges to µ . Coletti, Cristian F. and Tisseur, PierreCentro de Matem´atica, Cogni¸c˜ao e Computa¸c˜aoUniversidade Federal do ABCSanto Andr´e S.P, BrasilE-mail: { cristian.coletti, pierre.tisseur } @ufabc.edu.br The aim of this paper is to use duality principles to study the ergodicityof two-states PCA. More precisely our work gives new sufficient ergodicityconditions for the expression of the PCA’s local transition probabilities (seeTheorem 2) and show that under these conditions the invariant measure isshift-mixing with exponential decay of correlation. Relations between thePCA and the dual process (see Lemma 4 and Lemma 2 ) also allow us togive a very simple expression of the constant of the decay of correlation asa function of the radius (of the PCA) and the transition probabilities of thePCA (see Theorem 3). Moreover the proof of Theorem 2 shows in detail howto compute the value of the invariant measure on cylinders. Results aboutthe decay of correlation is an answer to a question raised in [12].The existence of a dual process satisfying the duality equation (see Def-inition 1 and Liggett [9]) gives useful information (problems in uncountablesets can be reformulated as problems in countable sets) on the PCA but isnot always sufficient to prove that a PCA is ergodic. In [12], Lopez, Sanzand Sobottka introduced an extended concept of duality (see Definition 2)and gave general results about ergodicity (see Theorem 1). They used thispowerful general theory to give results on multi-state one-dimensional PCAof radius one and extended previous results about the
Domany-Kinzel model (see [2] for an introduction and [7] for extensions). Previously, in [8] Konnohas given ergodicity conditions for multi-state one-dimensional PCA usingself-duality equations.Even if, in some cases, the existence of null transition probabilities allowsto prove ergodicity of a certain class of PCA (see [7] and [8]), it had beenconjectured that in the one-dimensional case positive noise cellular automataare ergodic. However, P. Gacks, in 2000 , introduced a very complex coun-terexample (see [4] and [5]) for noisy deterministic cellular automata. In thatcase, the noisy one-dimensional cellular automata does not forget the pastand starting from different initial distribution, the PCA may converge todifferent invariant measures. His result can be extended to noisy PCA withpositive rates. This conjecture was formulated only in the one-dimensionalcase since in dimension 2 or higher, it is easier to show the existence of atleast two invariant measures. For example the two dimensional Ising model[5] or the Toom example (see [16]) that exhibit eroder properties. From The-orem 2 there exists a subclass of attractive PCA (class C ) where the noisyconjecture is verified ( p ( I r ) < conditions can not be applied. Moreover, for some classes of ergodic PCATheorem 3 gives greater constants for the decay of spatial correlation.This paper is organized as follows. In section 2, we present the basicdefinitions, notations and some preliminary results. In section 3, we state themain results, Theorem 2 and Theorem 3. We prove Theorem 2 in section 4.We conclude the paper in section 5 with the proof of the decay of correlation. A be a finite set and G a countable set. Let X = A G be endowedwith the product topology. A probabilistic cellular automata is a discretetime Markov process on the state space X .Let M ( X ) be the set of probability measures on X and F ( X ) the set ofreal functions on X which depend only on a finite number of coordinates of G . The evolution of probabilistic cellular automata is given through theirtransition operator. Definition 1
A local transition operator is a linear operator P : F ( X ) → F ( X ) such that P f ≥ for all f ≥ and P I = I, where I : X → X is the identityfunction. Definition 2
A local transition operator is called independent if P ( ϕφ ) = P ( ϕ ) P ( φ ) for all ϕ, φ ∈ F ( X ) such that the finite subsets of G on which they dependdo not intersect. The independent local transition operators can be defined through the values p x ( η, k ) for x ∈ G, η ∈ X and k ∈ A , as P f ( η ) = Z X f ( σ ) Y x ∈ G p x ( η, dσ ( x )) , ∀ f ∈ F ( X ) , where for all site x ∈ G , for all η, p x ( η, . ) is a probability measure on A .The value p x ( η, k ), called transition probabilities, represent the probabilitythat the sites x ∈ G takes the value k in the next transition if the presentconfiguration of the system is η . For more details see Toom et al. [17], Maesand Shlosman [13] and Lopez and Sanz [10].Let d ≥ R a finite subset of Z d of cardinality | R | and f a map from A | R | +1 to [0 , G = Z d it follows from the discussion above that the discrete time Markov process η . = { η t ( z ) ∈ A : t ∈ N , z ∈ Z d } whose evolution satisfies P [ η t +1 ( z ) = a | η t ( z + i ) = b i , ∀ i ∈ R ] = f ( a, ( b i ) i ∈ R ) , for all t ∈ N and z ∈ Z d is a well defined (discrete time) stochastic processwhich from now on will be called d -dimensional PCA. Here, P stands forthe probability measure on A Z d induced by the family of local transitionprobabilities. Also, let E be the expectation operator with respect to thisprobability measure.Let µ be the initial distribution of the PCA. For any t ≥
0, we call µ t thedistribution of the process at time t . The measure µ t is defined on cylinder u = N ( Λ, φ ) = { ξ ∈ A Z d : ξ ( x ) = φ ( x ) ∀ x ∈ Λ } for some fixed φ ∈ A Z d and Λ ⊂ Z d , | Λ | < ∞ by µ t ( u ) = X v ∈ C t µ ( v ) P η ∈ v { η t ∈ u } , where C t is the family of all cylinders of X on the coordinates of Λ ( thefinite subset of Z d used to defined u ).In this paper the notation | Λ | will represent the cardinality of Λ when Λ is a finite subset of Z d . If U = N ( Ξ, φ ) is a cylinder set, the notation | U | willrepresent the cardinality | Ξ | of the set Ξ ⊂ Z d . In the one dimensional case weadopt the following notation: For any sequence of letters U = ( u , . . . , u n ) ∈ A n +1 , the set [ U ] s = [ u . . . u n ] s := { x ∈ A Z | x ( s ) = u , . . . , x ( s + n ) = u n } will be called cylinder and | U | = n + 1.2.2 Two-state Probabilistic Cellular AutomataIn order to simplify the notation we will focus our attention on two-statePCA, that is to say PCA η . on { , } Z d . For any positive integer r , let usdefine I r := { i = ( i , . . . , i d ) ∈ Z d : − r ≤ i , . . . , i d ≤ r } . Define a family of transition probabilities { p ( J ) : J ⊂ I r } by p ( J ) := P { η t +1 ( z ) = 1 | η t ( z + j ) = 1 : j ∈ J } . Note that any PCA with state space { , } Z d is completely characterizedby a positive integer number r called the radius of the PCA and the set oftransition probabilities { p ( J ) : J ⊂ I r } .2.3 The invariant probability measure Definition 3
Let T be a measure-preserving transformation of a probabilityspace ( X, F , µ ) , where F is the σ -algebra generated by the cylinder sets on X .We say that the probability measure µ is T -mixing if ∀ U, V ∈ F lim n →∞ µ ( U ∩ T − n V ) = µ ( U ) µ ( V ) . Since the cylinder sets generate the σ -algebra F , it follows that the measure µ is T -mixing when the last relation is satisfied by any pair of cylinder sets U and V (for more details see [18]).2.4 DualityThe concept of duality is a powerful tool in the theory of interacting particlesystem. It provides relevant information about the evolution of the processunder consideration from the study of other simpler process, the dual process.The reformulated problems may be more tractable than the original problemsand some progress may be achieved. Now we give the (classical) definition ofduality taken from [9]. Definition 1
Let η . and ζ . be two Markov processes with state spaces X and Y respectively, and let H ( η, ζ ) be a bounded measurable function on X × Y .The processes η . and ζ . are said to be dual to one another with respect to H if E η [ H ( η t , ζ )] = E ζ [ H ( η, ζ t )]for all η ∈ X and ζ ∈ Y .Unfortunately, it is not true that every process has a dual. Recently, Lopezet al [12] gave a new notion of duality which extends the previous one. Moreprecisely, they gave the following definition. Definition 2
Given two discrete time Markov processes, η t with state space X and ζ t with state space Y and H : X × Y → R and D : Y → [0 , ∞ )bounded measurable functions, the process η . and ζ . are said dual to oneanother with respect to ( H, D ) if E η = x [ H ( η , y )] = D ( y ) E ζ = y [ H ( x, ζ )] . (1.1)2.5 Duality and sufficient conditions for ergodicityIn order to state our results in section 3, we need to give the spirit and someelements of the proof of the following Theorem, which appears in [12]. Theorem 1 [12] Suppose η t is a Markov process with state space X and ξ t isa markov chain with countable state space Y , which are dual to one anotherwith respect to ( H, D ) . If ≤ D ( y ) < for all y ∈ Y , then there exist astochastic process ˜ ξ t with state space ˜ Y = Y ∪ { S } with S an extra state anda bounded measurable function ˜ H : X × ˜ Y → R such that η . and ˜ ξ . are dualto one another with respect to H . Furthermore, denoting by Θ the set of allabsorbing states of ξ . , ifi) the set of linear combinations of { H ( ., y ) : y ∈ Y } is dense in C ( X ) , theset of continuous maps from X to R ;ii) D ( y ) < for any y / ∈ Θ , and D := sup y ∈ Y : D ( y ) < { D ( y ) } < ;iii) H ( ., θ ) ≡ c ( θ ) for all θ ∈ Θ with D ( θ ) = 1 ; then η . is ergodic and its unique invariant measure is determined for any y ∈ Y by ˜ µ ( y ) = X θ ∈ Θd ( θ )=1 c ( θ ) P ˜ ξ = y h ˜ ξ τ = θ i , (1.2) where τ is the hitting time of { θ ∈ Θ : D ( θ ) = 1 } ∪ { S } for ˜ ξ t and ˆ µ =lim t →∞ ˆ µ t with ˆ µ t ( y ) = Z X H ( x, y ) dµ t ( x ) . Sketch of the proof.
First recall that τ is the hitting time of the dualprocess ˜ ξ . entering an absorbing state θ ∈ ˜ Θ . If there exists a dual process ˜ ξ and a function ˜ H that satisfies the following (classical) duality equation E η = x h ˜ H ( η , y ) i = E ˜ ξ = y h ˜ H (cid:16) x, ˜ ξ (cid:17)i , (1.3)it is possible to show that ˆ µ s ( y ) = R X H ( x, y ) dµ s ( x ) = E ˜ ξ = y [ˆ µ ( ˜ ξ s )]. If P { τ < ∞} = 1, it follows thatlim s →∞ ˆ µ s ( y ) = lim s →∞ P θ ∈ ˜ Θ E ˜ ξ = y [ˆ µ ( ˜ ξ s ) | ˜ ξ t = θ, τ ≤ s ] P ˜ ξ = y { ˜ ξ t = θ, τ ≤ s } + lim s →∞ E ˜ ξ = y [ˆ µ ( ˜ ξ s ) | τ > s ] P ˜ ξ = y { τ > s } = P θ ∈ ˜ Θ ˆ µ ( θ ) P ˜ ξ = y { ˜ ξ τ = θ } . Finally, when the set of linear combinations of the set { ˜ H ( ., y ) | y ⊂ Z d } is dense in C ( X ) (the set of continuous functions from X to R ) the sequence( µ n ) n ∈ N converges in the weak* topology. Also, the limit measure µ does notdepend on the initial measure µ .Hence, we have seen that the key point is to prove that P { τ < ∞} = 1.One way to show this, is to introduce the new type of duality (see Equation1.1). If there exists a dual process ξ . with state space Y that verifies thenew concept of duality (see Equation 1.1) then we can define a standarddual process ˜ ξ . with state space ˜ Y = Y ∪ { S } and such that the set of allabsorbing states is ˜ Θ = Θ ∪ { S } where Θ denote the set of all the absorbingstates of ξ . . Here S is an extra absorbing state and the transition probabilitiesof ˜ ξ . satisfy P ˜ ξ =˜ y { ˜ ξ = ˜ y } = D (˜ y ) P ξ = ˜ y { ξ = ˜ y } , if ˜ y , ˜ y ∈ Y − D (˜ y ) , if ˜ y ∈ Y , ˜ y = S , if ˜ y = ˜ y = S . Taking ˜ H ( x, y ) = H ( x, y ) when y ∈ Y and ˜ H ( x, S ) = 0 we obtain that thedual process ˜ ξ . satisfies the standard duality equation 1.3 and that ˆ µ ( S ) =0. Note that since D = sup y ∈ Y : D ( y ) < { D ( y ) } <
1, at each iteration theprobability to enter the extra absorbing state S is positive and this impliesthe following result: Lemma 1
Under the conditions of Theorem 1, for all integer i ≥ one has P ( τ > i ) ≤ D i . Proof.
By the Markov property we have that P ˜ ξ = ˜ y { τ > i } ≤ D × P ˜ ξ = ˜ y { τ > i − } . Then, the result follows by using the mathematical induction principle. ✷ Note that Lemma 1 implies that P { τ < ∞} = 1 which finishes the proofof Theorem 1. ✷ Before stating the main results of this paper, we introduce one morepiece of notation: let ∞ ∞ denote the all one configuration, i.e. ∞ ∞ =(1 Z d ( x )) x ∈ Z d . Analougsly, ∞ ∞ denote the all zero configuration. A PCA of radius r is called attractive if for any J ⊂ I r and j ∈ I r wehave p ( J ∪ { j } ) ≥ p ( J ). We consider here the following subclass of attractivePCA. For any r ∈ N , let ℘ ( I r ) be the set of all subsets of I r . We say that atwo-state PCA of radius r belongs to C if its transition probabilities satisfy p ( J ) = P J ′ ⊂ J λ ( J ′ ) where λ is some map from ℘ ( I r ) → [0 , C is constructive. The following Proposition givessufficient conditions for an attractive PCA to belong to C . Proposition 1
A two-state probabilistic cellular automaton η . belongs to C if its transition probabilities satisfy the following set of inequalities:(a) For any i ∈ I r , p ( { i } ) ≥ p ( ∅ ) . (b) For any ≤ k ≤ | I r | − and for any j , . . . , j k ∈ I r p ( { j , . . . , j k } ) ≥ ( − k p ( ∅ ) − k − X n =0 ( − k +1 − n X { l ,...,l n }⊂{ j ,...,j k } p ( { l , . . . l n } ) . Theorem 2
Let η . be a two-states d-dimensional probabilistic cellular au-tomaton of radius r that belongs to C . If p ( I r ) < then η . is an ergodicPCA and there exists a dual process ξ which satisfy equation 1.1. Moreover,for any cylinder set U we can find ( α k ∈ Z ) k ∈ K and ( Y ( k ) ⊂ Z d ) k ∈ K with | K | < ∞ such that µ ( U ) = X k ∈ K α k ∞ X l =1 P ξ = Y ( k ) { ξ l = ∅| ξ l − = ∅} ! . Remark 1
Note that in some cases it is possible to exchange the role of thetwo states (0 ↔ in order to show ergodicity using the previous results. Corollary 1
Under the conditions in Theorem 2 we have that if λ ( ∅ ) =0 then the unique invariant measure is δ , where δ is the Dirac measureon ∞ ∞ . Analogously, we have that if λ ( ∅ ) = 1 then the unique invariantmeasure µ is δ , where δ is the Dirac measure on ∞ ∞ . Theorem 3
Let η . be a one-dimensional probabilistic cellular automaton ∈ C of radius r with p ( I r ) =: D ∈ [0 , . Then, the unique invariant measure µ is shift-mixing. Also, if D = 0 , for any pair of cylinders [ U ] = [ u . . . u k ] , [ V ] = [ v . . . v k ′ ] and t ≥ | U | + | V | we have | µ ([ U ] ∩ σ − t [ V ] ) − µ ([ U ] ) × µ ([ V ] ) | ≤ exp ( − a × t ) × K ( U, V ) , where σ is the shift on { , } Z , a = 1 / r × ln (1 / D ) and K ( U, V ) is a constantdepending only on U , V , D and r . Remark 2
This last result can be extended to d -dimensional PCA. This is a one-dimensional PCA η . of radius r = 1 introduced in [2] withtransition probabilities P { η t +1 ( z ) = 1 | η t ( z − , z, z + 1) = 000 or 010 } = p ( ∅ ) = p ( { } ) = a , P { η t +1 ( z ) = 1 | η t ( z − , z, z + 1) = 100 or 110 } = p ( {− } )= p ( {− , } ) = a , P { η t +1 ( z ) = 1 | η t ( z − , z, z + 1) = 001 or 011 } = p ( { } ) = p ( { , } ) = a and P { η t +1 ( z ) = 1 | η t ( z − , z, z + 1) = 101 or 111 } = p ( {− , } )= p ( {− , , } ) = a , where, for any subset V ⊂ Z , η ( V ) ∈ { , } V denote the restriction of aconfiguration η ∈ { , } Z to the set of positions in V .Using Proposition 1 we obtain that η . ∈ C when p ( {− , } ) ≥ p ( {− } ) + p ( { } ) − p ( ∅ ), which is equivalent to the condition a ≥ a − a . FromTheorem 2 the PCA η . is ergodic if p ( I r ) = p ( {− , , } ) = a <
1. FromTheorem 3 the unique invariant measure is shift-mixing with exponentialdecay of spatial correlation such that for any pair of cylinders [ U ] and [ V ] and for all t ≥ | U | + | V | we obtain | µ ([ U ] ∩ σ − t [ V ] ) − µ ([ U ] ) × µ ([ V ] ) | ≤ K exp ( − (1 / /a )) t ) , where K can be explicitly computed (see the end of the Proof of Theorem 3).Using Theorem 2 we can compute, for example, the measure of the cylinder[01] which is µ ([01] ) = µ ([1] ) − µ ([11] ) = ˆ µ ( { } ) − ˆ µ ( { , } )= ∞ X t =1 P ξ = { } { ξ t = ∅| ξ t − = ∅} + ∞ X t =1 P ξ = { , } { ξ t = ∅| ξ t − = ∅} , where ξ . is the associated dual process. Let η be a two-state, two-dimensional PCA of radius one. In this case I = { ( l, k ) | − ≤ l, k ≤ } is a square of 9 sites. The transition probabilities { p ( J ) | J ⊂ I } of η . are defined by p ( J ) = α P | J | k =0 C k = α × | J | where C lk are the binomial coefficients. This PCA belongs to C since for any J ⊂ I we can write λ ( J ) = α and obtain that P ( J ) = P J ′ ⊂ J λ ( J ′ ). This PCA isa kind of generalization to dimension 2 of the Domany-Kinzel model (eachsite has the same weight) with only one parameter. The sufficient ergodicitycondition is p ( I r ) < α × < α < − ) and theconstant of decay of spatial correlation is a = ln(1 / (2 × α )). In [3], Dobrushin gives sufficient ergodicity conditions for interacting par-ticule systems. Using our notation, these conditions applied to PCA can betranslated as γ < γ = X j ∈ I r sup J ⊂ I r | p ( J ∪ { j } ) − p ( J ) | . In the case of the Domany-Kinsel model, which belongs to the class C , weobtain γ = sup J ⊂ I r | p ( J ∪ {− } ) − p ( J ) | + sup J ⊂ I r | p ( J ∪ { } ) − p ( J ) | =2( a − a ) since η . ∈ C ( a ≥ a − a ). If a < a − a ) ≥ γ = α (cid:16)P k =1 k × C k (cid:17) . In this case γ > p ( I r ) and even if γ < ln(1 / ( p ( I r ))is greater than ln(1 / ( γ )), the constant of decay of correlation given in [13].More generally, if a PCA belongs to C the sufficient condition p ( I r ) < p ( I r ) = P J ⊂ I r λ ( J ) < γ = P J = ∅ , J ⊂ I r λ ( J ) × | J | < C and their Dual ProcessIn [12], the authors give sufficient ergodicity conditions for one-dimensionalmulti-state PCA of radius one using a dual process satisfying equation 1.1.Here we will use an analogous dual process to give sufficient ergodicity condi-tions for two-state, d -dimensional PCA of radius r using the following dualityequation: E η = x [ H ( η , Y )] = D ( Y ) E ξ = Y H [( x, ξ )] , (1.4)where η . is a PCA with state space { , } Z d . The state space of the dualprocess ξ . is the class of all finite subsets of Z d . As in [12] we define the function H by H ( x, Y ) = (cid:26) x ( z ) = 1 , ∀ z ∈ Y ξ t is given by ξ t +1 = ∪ z ∈ ξ t B ( z )where for any nonempty set J ⊂ I r we have P h B ( z ) = { z + j | j ∈ J } i = π ( J )and P (cid:2) B ( z ) = ∅ (cid:3) = π ( ∅ ) . Then, take the function D such that D ( Y ) = D | Y | for any finite subset Y ⊂ Z d , where D ∈ [0 , D ( ∅ ) = 1 and ∅ is the unique absorbingstate for this dual process.4.2 The functions H and ˆ µ Note that, for this particular choice of H , we haveˆ µ ( Z d ) = Z X H ( x, Z d ) dµ ( x ) = µ ( ∞ ∞ ) = 0and ˆ µ ( ∅ ) = Z X H ( x, ∅ ) dµ ( x ) = µ ( { , } Z d ) = 1 , where X = { , } Z d and ∞ ∞ is the all one configuration (1 Z d ( x )) x ∈ Z d . Thefollowing Lemma is used in the proof of Theorem 2 and Theorem 3. Lemma 2
The set of linear combinations of { H ( ., y ) | y ∈ Z d } is dense in C (cid:16) { , } Z d , R (cid:17) , the set of continuous function from { , } Z d to R . For anycylinder U = N ( Λ, ϕ ) ⊂ { , } Z d (with Λ ⊂ Z d , | Λ | < ∞ and ϕ ∈ A Z d ) wehave µ ( U ) = X Y ( i ) α i ˆ µ ( Y ( i )) , where α i ∈ Z , Y ( i ) ⊂ Z d and max {| Y ( i ) |} < ∞ . Proof.
For the sake of simplicity, we only give the proof for the two-state,one-dimensional case. The key point of the proof consists in showing that anycylinder [ U ] t := [ u . . . u n ] t , ( u i ∈ { , } and t, n ∈ N ) can be decomposedinto a non-commutative sequence of subtractions and unions of intersectionsof cylinders of the type [1] t , t ∈ Z . We denote by T ([ U ] t ) this decomposition.One way to accomplish this decomposition is to follow the following rules: T ([1] t ) = [1] t , T ([0] t ) = { , } Z d − [1] t . Then, for all t, n ∈ Z and U = u . . . u n we have T ([ U t ) = T ([ U ] t ) ∩ [1] t + n +2 . Thus, T ([ U t ) = T ([ U ] t ) − T ([ U ] t ) ∩ [1] t +2+ n . For instance, T ([100] ) = T ([10] ) − T ([101] )= ( T ([1] ) − T ([11] )) − ( T ([10] ) ∩ [1] )= [1] − [11] − (([1] − [11] ) ∩ [1] )= ([1] − [11] − ([1] ∩ [1] )) ∪ [111] . Then, note that [1000] , the characteristic function of the cylinder [1000] ,can be written as [1000] ( x ) = [1] ( x ) + [111] ( x ) − [1] ∩ [1] ( x ) − [11]] ( x )= H ( x, { } ) + H ( x, { , , } ) − H ( x, { , } ) − H ( x, { , } ) . Since for any finite subset Y ⊂ Z we have ∩ i ∈ Y [1] i ( x ) = H ( x, Y ), it followsthat for all n ∈ N , t ∈ Z and U ∈ { , } n [ U ] t = P α i H ( x, Y ( i )). This, inturn, implies that the set of linear combinations of the set { H ( ., Y ) | Y ∈ Z d } is dense in C ( { , } Z d ). We finish the proof by observing that for any cylinder[ U ] t , we have µ ([ U ] t ) = Z [ U ] t ( x ) dµ ( x )= Z X α i H ( x, Y ( i )) dµ ( x )= X α i ˆ µ ( Y ( i )) . ✷ Remark 3
Using the definition of H taken in [12] which takes into consid-eration the multi-state case, it is possible to prove Proposition 2 for moregeneral d -dimensional PCA. P ( J ) | J ∈ I r ) and the transition probabilities of the dual process(( π ( J ) | J ∈ I r )).We can rewrite the right hand of equation (1.4) to obtain E η = x [ H ( η , Y )] = P η = x { η ( z ) = 1 ∀ z ∈ Y } . Hence, using the independence property of η . we get that P η = x { η ( z ) = 1 ∀ z ∈ Y } = Y z ∈ Y P η = x { η ( z ) = 1 } . For the left hand of equation 1.4 we have E ξ = Y [ H ( x, ξ )] = P ξ = Y { x ( z ) = 1 , ∀ z ∈ ξ } . For any x ∈ { , } Z d we denote by C x the set { z ∈ Z d | x ( z ) = 1 } . Then P ξ = Y { x ( z ) = 1 ∀ z ∈ ξ } = P ξ = Y { ξ ⊂ C x } . Using the independence property of the dual process we can assert that P ξ = Y { ξ ⊂ C x } = Y z ∈ Y P { B ( z ) ⊂ C x } . Finally we can rewrite equation 1.4 as Y z ∈ Y P η = x { η ( z ) = 1 } = D | Y | Y z ∈ Y P { B ( z ) ⊂ C x } = Y z ∈ Y D × P { B ( z ) ⊂ C x } (1.5)which implies that Y z ∈ Y p ( J z ) = Y z ∈ Y D × ∆ z , (1.6)where J z = { i − z | i ∈ { C x ∩ { j + z }| j ∈ I r }} and ∆ z is given by ∆ z = π ( ∅ ) + X i ∈ I r ( x ( z + i )) × π ( { i } )+ X i,j ∈ I r { } ( x ( z + i )) × { } (( x ( z + j )) × π ( { i, j } ) + . . . + X i ,...,i k ∈ I r k Y l =1 { } ( x ( z + i k )) ! × π ( { i , . . . , i k } )+ . . . + Y i ∈ I r { } ( x ( z + i )) ! × π ( I r ) . By simplicity of notation we write π ( i , . . . i k ) and p ( i , . . . i k ) instead of π ( { i , . . . i k } ) and p ( { i , . . . i k } ).Since equation 1.6 is true for all x ∈ { , } Z d we obtain the followingequations for π ( . ), p ( ∅ ) = D π ( ∅ ) p ( i ) = D [ π ( ∅ ) + π ( i )] p ( i, j ) = D [ π ( ∅ ) + π ( i ) + π ( j ) + π ( i, j )] p ( i, j, k ) = D [ π ( ∅ ) + π ( i ) + π ( j ) + π ( i, j ) + π ( i, k ) + π ( j, k ) + π ( i, j, k )] where i, j, k ∈ I r .More generally, for any 0 ≤ k ≤ | I r | − , p ( i , ..., i k ) = D h π ( ∅ )+ k X l =0 π ( l )+ . . . + k − X i =0 X l ,...,l i ∈{ i ,...,i k } π ( l , ..., l i )+ π ( l , l , ..., l k ) i . (1.7)3 Since π ( ∅ ) + | I r | X k =0 X l ,l ,...l k ∈ I r π ( l , l , . . . l k ) + π ( I r ) = 1 , we get that D = p ( I r ).By definition, the dual process is completely determined by the parame-ters 0 ≤ π ( J ) ≤ J ⊂ I r ). From the sequence of equations 1.7 the dual pro-cess associated with the particular functions H and D exists if the transitionprobabilities of the PCA satisfy p ( J ) = D P J ⊂ I r π ( J ) with 0 < p ( I r ) ≤ I r .In this case we have that λ ( J ) = D π ( J ) and we claim that a PCA η . admits adual process that satisfies the duality equation 1.1 with particular functions H and D given in section 4.1 if and only if this PCA belongs to the class C .To show that the PCA is ergodic we need to verify the three conditions ofTheorem 1.Condition i ) is verified since from Lemma 2, the set of linear combinationsof functions belonging to { H ( ., Y ) , | Y ∈ Z d } is dense in C (cid:16) { , } Z d , R (cid:17) .Condition ii ) is satisfied since sup Y = ∅ { D ( Y ) } = D = p ( I r ) < iii ) follows from the fact that H ( ., ∅ ) = 1 and D ( ∅ ) = D |∅| = 1.Since ∅ is the only absorbing state for ξ . , using Theorem 1 (equation 1.2) weget that for any nonemptyset Y ⊂ Z d ˆ µ ( Y ) = ˆ µ ( ∅ ) P ξ = y { ξ τ = ∅} = + ∞ X t =1 P ξ = y { ξ t = ∅| ξ t − = ∅} . From Lemma 2, for any cylinder set U there exist α k ∈ R and Y ( k ) finitesubset of Z d such that µ ( U ) = P α k ˆ µ ( Y ( k )), which implies the last statementof Theorem 2. ✷ When λ ( ∅ ) = 1, starting from any initial measure µ , we obtain that µ = δ .When λ ( ∅ ) = 0, Theorem 2 and Lemma 2 imply that for each cylinder U that does not contain the point ∞ ∞ we get µ ( U ) = X α i ∞ X k =0 P Y = Y ( i ) { Y k = ∅| Y k − = ∅} ! = 0since π ( ∅ ) = λ ( ∅ ) p ( I r ) = 0. Finally we get that µ ( ∞ ∞ ) = 1 − µ ( { , } Z d − ∞ ∞ ) =1 which finishes the proof. ✷ { π ( J ) | J ⊂ I r } represent the transition probabilities of the dualprocess for all J ∈ I r one has π ( J ) ≥ Lemma 3
The transition probabilities π () of the dual process satisfy π ( ∅ ) = p ( ∅ ) D π ( i ) = p ( i ) − p ( ∅ ) D π ( i, j ) = D [ p ( i, j ) + p ( ∅ ) − p ( i ) − p ( j )] π ( i, j, k ) = D [ p ( i, j, k ) − p ( ∅ ) + p ( i ) + p ( j ) + p ( k ) − p ( i, j ) − p ( i, k ) − p ( j, k )] π ( i, j, k, l ) = D [ p ( i, j, k, l ) + p ( ∅ ) − P l ∈{ i,j,k,l } p ( l ) + P { l ,l }⊂{ i,j,k,l } p ( l , l ) − P { l ,l ,l }⊂{ i,j,k,l } p ( l , l , l )] More generally, for any ≤ k ≤ | I r | − and for any j , . . . , j k ∈ I r π ( j , . . . , j k ) = 1 D ( − k +1 p ( ∅ ) + k X j =0 ( − k − j X { l ,...,l j }⊂{ j ,...,j k } p ( l , . . . , l j ) . Proof. of Lemma 3
From the proof of Theorem 2 a PCA belongs to class C if and only if the transitions probabilities p () and π () satisfy the sequenceof equations 1.7. We use mathematical induction to solve the sequence ofequations 1.7. For the two first iterations it is easily seen that π ( ∅ ) = p ( ∅ ) D , π ( i ) = p ( i ) − p (0) D and π ( i, j ) = D [ p ( i, j ) + p (0) − p ( i ) − p ( j )]. Then supposethat the order k is true: π ( j , . . . j k ) = 1 D ( − k +1 p ( ∅ ) + k X j =0 ( − k − j X ( l ,...,l j ) ∈{ j ,...,j k } p ( l , . . . l j ) . Using equation 1.7 we obtain that π ( j , . . . , j k +1 ) equals1 D p ( j , . . . , j k +1 ) − dπ ( ∅ ) − D k X j =0 X ( l ,...,l j ) ∈{ j ,...,j k +1 } π ( l , . . . , l j ) . (1.8)Then we suppose the rank k true and use equation 1.8 to obtain that theterm in p ( ∅ ) in π ( j , . . . , j k +1 ) is − p ( ∅ ) − k X i =0 X l ,...,l i ∈{ j ,...,j k +1 } ( − i +1 p ( ∅ ) = p ( ∅ ) − − k X i =0 C k +2 i +1 ( − i +1 ! = p ( ∅ ) (cid:16) − C k +20 ( − + C k +2 k +2 ( − k +2 − (1 − k +2 (cid:17) = ( − k +2 p ( ∅ ) , where the constants C ki represent the binomial coefficients. Next we obtainthat the term in P l ∈{ j ,...,j k +1 } p ( l ) in π ( j , . . . , j k +1 ) is equal to − k X i =0 X ( l ,...,l i ) ∈{ j ,...,j k +1 } X h ∈{ l ,...,l i } p ( h ) ( − i = − X l ∈{ j ,...,j k +1 } p ( l ) k X i =0 C k +1 i ( − i ! = − X l ∈{ j ,...,j k +1 } p ( l ) (cid:0) (1 − k +1 − C k +1 k +1 ( − k +1 (cid:1) = X l ∈{ j ,...,j k +1 } p ( l )( − k +1 . Note that C k +1 i represents the number of ways to choose l , . . . , l i in j , . . . , j k +1 when we have chosen l and j . More generally, for 0 ≤ M ≤ k , the term in X ( l ,...,l M ) ∈{ j ,...,j k +1 } p ( l , . . . , l M ) in π ( j , . . . , j k +1 ) is equal to − k X i = M X ( l ,...,l i ) ∈{ j ,...,j k +1 } X ( h ,...,h M ) ∈{ l ,...,l j } p ( h , . . . , h M ) ( − i − M = − X ( l ,...,l M ) ∈{ j ,...,j k +1 } p ( l , . . . , l M ) k − M X i =0 C k +1 − Mi ( − i ! = − X ( l ,...,l M ) ∈{ j ,...,j k +1 } p ( l , . . . , l M ) (cid:0) (1 − k +1 − M − ( − k +1 − M (cid:1) = X ( l ,...,l M ) ∈{ j ,...,j k +1 } p ( l , . . . , l M )( − k +1 − M . ✷ For the sake of simplicity we study the decay of correlation for PCA withstate space { , } Z . An extension of this result to the multi-dimensional caseis straightforward but requires too much notation.5.1 Proof of Theorem 3The proof of Theorem 3 requires the following two results. The second one isnew and is a key point for the proof of Theorem 3. The first one seems to bewell known. However, its proof can not be found or at least it is quite hardto be found so we provide a proof of that result. Recall that µ stands for the unique invariant measure of an ergodic PCA. Proposition 2
Every invariant measure of an ergodic PCA is shift-invariant.
Lemma 4
Let [ U ] and [ V ] be two cylinders. If µ ([ U ] ) = P α i ˆ µ ( A i ) , µ ([ V ] )= P β i ˆ µ ( B i ) and t ≥ | U | + | V | , then µ ([ U ] ∩ σ − t [ V ] ) = µ ([ U ] ∩ [ V ] t ) = X α i ˆ µ ( A i )( ∗ , t ) X β i ˆ µ ( B i ) , where X α i ˆ µ ( A i )( ∗ , t ) X β i ˆ µ ( B i ) := X i,j α i β j ˆ µ ( A i ∪ { B i + t } ) . Proof of Theorem 3If D = 0, then p ( ∅ ) = 0. From Corollary 1, µ = δ and µ has exponentialdecay of correlation. For the remainder of this proof we therefore take 0 < D = p ( I r ) < E of Z and s ∈ Z , define E + s := { x + s : x ∈ E } .We claim that for any finite subsets E and F , if t ≥ N r + | E | + | F | we have | ˆ µ ( E ∪ { F + t } ) − ˆ µ ( E ) × ˆ µ ( F ) | ≤ D N +1 − D . The proof of this claim uses Theorem 1 and 2 which together say that forany finite subset E ⊂ Z , ˆ µ ( E ) = P η = E { η τ = ∅} . This, in turn, implies thatˆ µ ( E ) = ∞ X k =0 P η = E { τ = k } , where τ is the hitting time for the process η . . In fact, by Lemma 1, for anyinteger N > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ µ ( E ) − N X k =0 P η = E { τ = k } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ D N +1 − D . Note that if s ≥ ri + | E | + | F | , where i is any positive integer, then P η = E ∪{ F + s } { τ = i } = P η = E { τ = i } × i X j =0 P η = { F + s } { τ = j } + P η = { F + s } { τ = i } × i X j =0 P η = E ∪{ F + s } { τ = j } . It follows that if s ≥ | E | + | F | + 2 N × r , then N X i =0 P E = E ∪{ F + s } { τ = i } = N X i =0 P E = E { τ = i } × N X i =0 P E = { F + s } { τ = i } . This easily implies | ˆ µ ( E ∪ { F + s } ) − ˆ µ ( E ) × ˆ µ ( F ) | ≤ D N +1 − D , (1.9)for s ≥ | E | + | F | + 2 N × r , which proves our claim.By Lemma 2, for any pair of cylinders [ U ] and [ V ] , there exist finitesequences of sets ( A i ) and ( B i ) and finite sequences of real numbers α i and β i such that µ ([ U ] ) = X α i ˆ µ ( A i )and µ ([ V ] ) = X β i ˆ µ ( B i ) . Thus, by inequality 1.9, (cid:12)(cid:12)(cid:12) α i β j ˆ µ ( A i ∪ { B i + s } − α i ˆ µ ( A i ) × β j ˆ µ ( B j )) (cid:12)(cid:12)(cid:12) ≤ | α i β j | D N +1 − D for any pair of subsets A i and B j of Z and for any s ≥ | U | + | V | + 2 N r .It follows from this that (cid:12)(cid:12)(cid:12) X i,j α i β j ˆ µ (cid:0) A i ∪ { B i + s } (cid:1) − X i α i ˆ µ ( A i ) × X j β j ˆ µ ( B j ) (cid:12)(cid:12)(cid:12) ≤ F ( U, V ) D N , where F ( U, V ) = P i,j | α i β j | D − D .Using Lemma 4, if t ≥ | U | + | V | we obtain (cid:12)(cid:12)(cid:12) µ (cid:0) [ U ] ∩ σ − t [ V ] (cid:1) − µ ([ U ] ) × µ ([ V ] ) (cid:12)(cid:12)(cid:12) ≤ K ( U, V ) exp (cid:16) − t × ln (1 / D )2 r (cid:17) , where K ( U, V ) = F ( U, V ) D − ( | U | + | V | r ) .Finally, it follows from Proposition 2 that the invariant measure is shift-invariant and that the exponential decay of correlations of cylinders impliesthe mixing property. ✷ It is sufficient to show that for any cylinder [ U ] t , where U ∈ { , } l for some l ∈ N , we have µ ( σ − [ U ] t ) = µ ([ U ] t ) . Since µ is the invariant measure of an ergodic PCA η . , there exits a sequence( µ i ) i ∈ N which converges in the weak* topology to µ , where µ i is the distribu-tion of a PCA η . at time i starting from an initial distribution µ . It followsthat for any cylinder [ U ] t we havelim n →∞ µ n ([ U ] t ) = µ ([ U ] t ) . Since for any positive integer i we have µ i ([ U ] t ) = X V j ∈{ , } n +1+2 ir µ ([ V j ] t − ir ) P η ∈ [ V j ] t − ir { η i ∈ [ U ] t } , we can choose µ as a shift-invariant probability measure. Hence, for anypositive integer i and any cylinder [ U ] t we have µ i ([ U ] t ) = µ i ( σ − [ U ] t ) , which finishes the proof. ✷ We prove the lemma using the principle of mathematical induction. Firstwe prove it for the case | U | = 1 and | V | = 1. Note that for any finite set { j , . . . , j k } , with j , . . . , j k ∈ Z ,ˆ µ ( { j , . . . , j k } ) = Z { , } Z H ( { j , . . . , j k } , x ) dµ ( x )= µ ( {∩ [1] j | j ∈ { j , . . . , j k }} )and observe that for any k ≥ ∈ N we have µ ([1] ∩ [1] k ) = ˆ µ ( { } ∪ { k } ) . Since µ ([0] k ) = 1 − ˆ µ ( { k } ) = ˆ µ ( ∅ ) − ˆ µ ( { k } ) and µ ([1] ∩ [0] k ) = µ ([1] ) − µ ([1] ∩ [1] k ) = ˆ µ ( { } ) − ˆ µ ( { , k } ) we get, again, that µ ([1] ∩ [0] k ) = ˆ µ ( { } ∪ ∅ ) − ˆ µ ( { } ∪ { k } ) . Furthermore, we have µ ([0] ∩ [1] k ) = ˆ µ ( { } ) − ˆ µ ( { , k } ).Finally, note that µ ([0] ∩ [0] k ) = 1 − µ ([1] ∩ [1] k ) − µ ([1] ∩ [0] k ) − µ ([0] ∩ [1] k )= ˆ µ ( ∅ ) − ˆ µ ( { } ∪ { k } ) − ˆ µ ( { } ) + ˆ µ ( { } ∪ { k } ) − ˆ µ ( { k } )+ ˆ µ ( { } ∪ { k } )= ˆ µ ( ∅ ) + ˆ µ ( { } ∪ { k } ) − ˆ µ ( { } ) − ˆ µ ( { k } )= [ˆ µ ( ∅ ) − ˆ µ ( { } )] ( ∗ , t ) [ˆ µ ( ∅ ) − ˆ µ ( { k } )] , which finishes the proof in the case | U | = | V | = 1.Now, suppose that µ ([ U ] ∩ σ − t [ V ] ) = P i,j α i β j ˆ µ ( A i ∪ { B i + t } ) istrue for | U | = | V | = n . Consider t ≥ n + 1 and let [ U ] , [ V ] be twocylinders such that µ ([ U ] ) = P α i ˆ µ ( A i ) and µ ([ V ] ) = P β j ˆ µ ( B j ). Since µ ([ U ) = P α i ˆ µ ( A i ∪ {| U |} ), we get µ ([ U [ V t ) = X i,j α i β j ˆ µ ( A i ∪ { B j + t } ∪ {| U | , | V | + t } ) . Noting that µ ([ V t ) = P α i ˆ µ ( { B i + t } ∪ {| V | + t } ) we get the desired resultfor the case [ U ∩ [ V t .The result for the case [ U ∩ [ V t follows by noting that µ ([ U ∩ [ V t ) = µ ([ U ∩ [ V ] t ) − µ ([ U ∩ [ V t )= X i,j α i β j ˆ µ ( A i ∪ {| U |} ∪ { B j + t } ) − X i,j α i β j ˆ µ ( A i ∪ { B j + t } ∪ {| U | , | V | + t } )= X α i ˆ µ ( A i ∪ {| U |} )( ∗ , t ) X α i ˆ µ ( { B i + t } ) − X α i ˆ µ ( { B i + t } ∪ {| V | + t } ) . It can also be shown that µ ([ U ∩ [ V t ) = [ X α i ˆ µ ( A i ) − X α i ˆ µ ( A i ∪ {| U |} )]( ∗ , t ) [ X α i ˆ µ ( { B i + t } ∪ {| V | + t } )] . Finally, using that µ ([ U ∩ [ V t ) = µ ([ U ] ∩ [ V ] t ) − µ ([ U ∩ [ V t ) − µ ([ U ∩ [ V t ) − µ ([ U ∩ [ V t )we can show that µ ([ U ∩ [ V t ) = [ X i,j α i β j ˆ µ ( A i ∪ { B i + t } ) − X i,j α i β j ˆ µ ( A i ∪ { B i + t } ∪ {| U |} )]( ∗ , t )[ X i,j α i β j ˆ µ ( A i ∪ { B i + t } ) − X i,j α i β j ˆ µ ( A i ∪ { B i + t } ) ∪ {| V | + t } ]which finishes the proof. ✷ Final Questions i) Is there exist an ergodic PCA such that the unique invariant measure isnot shift-mixing?ii) Is there exist an ergodic PCA such that the invariant measure has non-exponential decay of correlation?
Acknowledgements
We would like to thank Marcelo Sobottka for a carefullyreading of a previous version of this work and for many comments and sugges-tions. We also thank the referees for constructive remarks. During the realizationof this work the first author received partial financial support from FAPESP, grant2006/54511-2. The second author received financial support from UFABC, grant2008.