On Null-homology and stationary sequences
aa r X i v : . [ m a t h . P R ] O c t ON NULL-HOMOLOGY AND STATIONARY SEQUENCES
By Gerold Alsmeyer ∗ and Chiranjib Mukherjee ∗ University of M¨unster
The concept of homology, originally developed as a useful tool inalgebraic topology, has by now become pervasive in quite differentbranches of mathematics. The notion particularly carries over quitenaturally to the setup of measure-preserving transformations arisingfrom various group actions or, equivalently, the setup of stationarysequences considered in this paper. Our main result provides a sharpcriterion which determines (and rules out) when two stationary pro-cesses belong to the same null-homology equivalence class . We alsodiscuss some concrete cases where the notion of null-homology turnsup in a relevant manner.
1. Introduction and motivation.
Homology is a notion that arisesin various branches of mathematics. It was originally developed in algebraictopology in order to associate a sequence of algebraic objects. A typicalfundamental question is the following: When does a n -cycle of a (simplical)complex form the boundary of a ( n + 1)-chain, or equivalently, when is itsfundamental class a boundary for the singular homology? If such a require-ment is fulfilled, the cycle is said to be homologous to null-homologous .In the present article, we provide a suitable criterion for null-homology in adifferent context, namely measure-preserving transformations arising fromnatural group actions on any complete and separable metric space. To for-mulate the question precisely, we recall some basic definitions.Let X = ( X n ) n ∈ Z be a sequence of random variables defined on a proba-bility space with underlying probability measure P and such that the X n ’stake values in a complete separable metric space S . Note that X forms astationary stochastic process if, for all n ∈ N and m ∈ Z , P (cid:0) ( X , . . . , X n ) ∈ · (cid:1) = P (cid:0) ( X m +1 , . . . , X m + n ) ∈ · (cid:1) . In other words, the joint law of ( X , . . . , X n ) for any n coincides with thelaw of any of its “shifts” under the action of the additive group Z on the ∗ Funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s ExcellenceStrategy EXC 2044–390685587, Mathematics M¨unster: Dynamics–Geometry–Structure.
MSC 2010 subject classifications:
Primary 28D05; secondary 60G10,
Keywords and phrases: stationary process, null-homology, Markov random walk,lattice-type, stochastic homogenization, random conductance model, Schauder’s fixed-point theorem G. ALSMEYER AND C. MUKHERJEE space of doubly-infinite sequences S Z . There is a natural notion of homology ,first coined by Lalley [L86, p. 197] in this setup, that arises from the groupaction. Indeed, given any stationary sequence X and measurable functions F, G : S Z → R d , we say that F is homologous to G (with respect to X and P ) and write F ∼ G if there exists a function ξ : S Z → R d such that(1) F ( X ) − G ( X ) = ξ ( X ) − ξ ( X ) P -a.s.Then ∼ is an equivalence relation, and if F ∼
0, thus(2) F ( X ) = ξ ( X ) − ξ ( X ) P -a.s. , we say that F is null-homologous . Now observe that, given any stationaryprocess X and a null-homologous function F , the process ( F ( X n )) n ∈ Z is notonly also stationary but in fact the incremental sequence of another station-ary process, viz. ( ξ ( X n )) n ∈ Z . In view of this, the converse question whichstationary processes are of this “incremental” type and therefore allowing arepresentation with respect to a null-homologous function appears to be nat-ural. The main goal of the present article is to provide a sharp criterion forthis fundamental property which is of interest for various reasons as will alsobe explained. Indeed, mere tightness of the partial sums S n = X + · · · + X n , n ∈ N , associated with the stationary process X turns out to be the neces-sary and sufficient condition, see Theorem 2.2. The proof, which does noteven require ergodicity, is quite simple and relies on the construction of some commutative maps in a proper setup and an application of Schauder’s fixedpoint theorem. To put our work into context, we first discuss some concretecases where null-homology turns up in a relevant way.1.1. Markov random walks.
In the [L86], Lalley considered random walkswith increments from a fairly general class of stationary sequences, albeitrestricted to the integrable set up , see Remark 2.3. As a main result, heproved a Blackwell-type renewal theorem for which it was necessary to ruleout a certain “lattice-type” behavior which is intimately connected to thenotion of null-homology. In the following, we give a brief introduction ofthis notion within the framework of
Markov random walks which are alsocalled
Markov-additive processes and indeed comprise random walks withstationary increments as explained below.Let ( S , S ) be an arbitrary measurable space and B ( R m ) the Borel σ -field on R m for m >
1. Suppose that ( M n , X n ) n > is a Markov-modulatedsequence of S × R d -valued random variables, where S × R d is endowed withthe product σ -field S ⊗ B ( R d ). This means that X , X , . . . are conditionally ULL HOMOLOGY AND STATIONARY SEQUENCES independent given the driving chain ( M n ) n > and P ( X i ∈ B i , k n | M j = s j , j >
0) = P ( s , B ) n Y i =1 P (( s i − , s i ) , B i )for all n ∈ N , s , . . . , s n ∈ S , measurable B , . . . , B n ⊂ R d and suitablekernels P and P which describe the conditional laws of X given M andof X n given ( M n − , M n ) for n >
1, respectively. We make the additionalassumption that ( M n ) n > is ergodic with unique stationary distribution µ .Defining S := 0 and S n := n X i =1 X i , n = 1 , , . . . , the bivariate sequence ( M n , S n ) n > and also ( S n ) n > are called Markov ran-dom walk (MRW) and ( M n ) n > its driving or modulating chain . For ourpurposes, it is enough to study these objects in stationary regime, that is,under P µ := R S P ( ·| M = s ) µ (d s ). We may then further assume the exis-tence of a doubly infinite stationary extension ( M n , X n ) n ∈ Z with associateddoubly infinite random walk S n = P ni =1 X i if n > , , if n = 0 , − P i = n +1 X i if n < . In this context, both ( M n , S n ) n ∈ Z and ( M n , X n ) n ∈ Z are called null-homolo-gous if there exists a measurable function ξ : S → R d such that X n = ξ ( M n ) − ξ ( M n − ) P µ -a.s.(3)and thus S n = ξ ( M n ) − ξ ( M ) P µ -a.s.(4)for all n ∈ Z . The reader should note that the latter implies the stationarityof ( S n + ξ ( M )) n ∈ Z and thus the “almost stationarity” of the random walk( S n ) n ∈ Z itself, in particular its tightness.Now let ( X n ) n ∈ Z be any doubly infinite stationary sequence of R m -valuedrandom variables and put M n := ( X i ) i n for n ∈ Z . Observe that ( M n ) n ∈ Z constitutes a stationary Markov chain(ergodic iff ( X n ) n ∈ Z is ergodic) and ( M n , X n ) n ∈ Z a Markov-modulated se-quence. This shows that null-homology for stationary processes may indeed G. ALSMEYER AND C. MUKHERJEE be viewed as a special instance of the very same notion within the frameworkof Markov-modulation under stationarity.Null-homology arises also quite naturally in connection with the lattice-type of MRW’s. As before, let ( M n ) n > be ergodic with unique stationary law µ . Following Shurenkov [S84], the MRW ( M n , S n ) n > is called d -arithmeticif d is the maximal positive number such that P µ (cid:0) X ∈ ξ ( M ) − ξ ( M ) + d Z (cid:1) = 1for a suitable function ξ : S → [0 , d ), called shift function . If no such d exists,it is called nonarithmetic. Equivalently, ( M n , S n ) n > is d -arithmetic if d > M n , X n − X ′ n ) n ∈ Z is Markov-modulatedand null-homologous for a sequence of d Z -valued random variables ( X ′ n ) n ∈ Z .Namely, with ξ denoting the shift function, X ′ n := X n − ξ ( M n ) + ξ ( M n − )for n ∈ Z .1.2. Stochastic homogenization.
The notion of a corrector plays an im-portant rˆole in the context of stochastic homogenization of a random media.We will describe the setup and how null-homology comes into play for a par-ticular instance of a random walk in random environment (in the reversiblesetup) known as the random conductance model . Let E d = (cid:8) ( x, y ) : | x − y | = 1 , x, y ∈ Z d (cid:9) be the set of nearest neighbor bonds in Z d and Ω = [ a, b ] E d for any two fixednumbers 0 < a < b . We assume that Ω is equipped with the product σ -field B and carries a probability measure P . For simplicity, we also assume thatthe canonical coordinates are i.i.d. variables under P . Note that any x ∈ Z d acts on (Ω , B , P ) as a P -preserving and ergodic transformation τ x , definedas the canonical translationΩ ∋ ω ( · ) ω ( x + · ) . For any ω ∈ Ω, we then have a Markov chain ( S n ) n > on Z d under a family ofprobability measures ( P π,ωx ) x ∈ Z d such that P π,ωx ( S = x ) = 1 and transitionprobabilities are given by(5) P x,ω ( S n +1 = y + e | S n = y ) = π ω ( y, y + e ):= ω (( y, y + e )) P | e ′ | =1 ω (( y, y + e ′ )) = π τ y ω (0 , e ) ULL HOMOLOGY AND STATIONARY SEQUENCES for any e with | e | = 1 and x ∈ Z d . Furthermore, the sequence b ω n def = τ S n ω for n >
0, with initial state ω and taking values in the “environment space”Ω, is also a Markov chain with transition kernel b Π defined by( b Π f )( ω ) := X | e | =1 π ω (0 , e ) f ( τ e ω )for all bounded and measurable f . It is called the environment seen from themoving particle , or simply the environmental process , and particularly usefulin the following scenario: Suppose there is a probability density φ ∈ L ( P )(i.e. φ > R φ d P = 1) such that d Q = φ d P is b Π-invariant, i.e. h b Π f, φ i L ( P ) = h f, φ i L ( P ) , (6)for all bounded and measurable f or, equivalently, L ⋆ φ = 0 if L = Id − b Π . It can be shown, see [PV81, K85, KV86] and also [BS02, Theorem 1.2],that such an invariant density φ if it exists is necessarily unique. Moreover, P and Q are then equivalent measures and ( b ω n ) n > an ergodic process inequilibrium (under initial law Q ).In the random conductance model with transition probabilities (5), theinvariant density φ can easily be found by reversibility (solving the detailedbalance equations), viz. φ ( ω ) = 1 C X | e | =1 ω ((0 , e )) , where C = Z X | e | =1 ω ((0 , e )) P (d ω ) . Reversibility further implies that b Π is self-adjoint on L ( Q ), that is(7) h f, b Π g i L ( Q ) = h b Π f, g i L ( Q ) for all bounded and measurable functions f, g .Returning to the Markov chain ( S n ) n > under P ,ω , the ergodicity of( b ω n ) n > fairly easily provides a strong law of large numbers, viz. S n /n → P ,ω -a.s. for P -almost all ω . To see this, letd( x, ω ) = E π,ωx [ S − S ] = X | e | =1 eπ ω ( x, x + e ) = d(0 , τ x ω ) G. ALSMEYER AND C. MUKHERJEE denote the local drift at x under P ,ω . As ( b ω n ) n > is ergodic under initiallaw Q , Birkhoff’s ergodic theorem implies n − n − X j =0 d( S j , ω ) = n − n − X j =0 d(0 , b ω j ) n →∞ −−−→ Z d(0 , ω ′ ) Q (d ω ′ ) = 0for Q -almost all and thus P -almost all ω (as P , Q are equivalent), the right-hand side being 0 by reversibility (recall (7)) and the definition of Q . Nowobserve that Z n = S n − S − P n − j =0 d( S j , ω ), n >
0, is a P ,ω -martingalewith bounded (uniformly in ω ) increments and therefore satisfies, by theAzuma-Hoeffding inequality, P ,ω (cid:0) n − Z n > n − + ε (cid:1) exp( − Cn ε )for any ε > C > ω ). Finally, by an appealto the Borel-Cantelli lemma, we infer that Z n /n → P -a.s., and since P n − j =0 d( S j , ω ) = o ( n ) a.s., it follows that S n /n → P -a.s., too.As will be explained next, stochastic homogenization comes into playwhen turning to the more ambitious aim of deriving an almost sure centrallimit theorem (or an invariance principle) for the distribution (under thequenched measure P π,ω ) of the random walk ( S n ) n > , and it leads to thenotion of a corrector . Note that the local drift d is bounded and thereforeparticularly ∈ L ( P ). For any fixed ε >
0, let g ε ∈ L ( P ) be a solution tothe Poisson equation ((1 + ε )Id − b Π) g ε = dThe solution is well-defined and in fact given by the Neumann series g ε = d + X n > b Π n d (1 + ε ) n . Putting G ε ( ω, e ) := ( ∇ e g ε ) ω ) = g ε ( τ e ω ) − g ε ( ω ) for any e with | e | = 1, wethen have the result G ε ( · , e ) ◦ τ x L ( P ) −−−→ G ( · , e ) ◦ τ x for any x ∈ R d , see [KV86, Theorem 1.3], where G is a (divergence free) gradient field , i.e., it satisfies the closed loop condition (8) n X j =1 G ( τ x j ω, x j +1 − x j ) = 0 a.s. ULL HOMOLOGY AND STATIONARY SEQUENCES for any closed path x → x → · · · → x n = x in R d . The last propertyallows us to define the corrector corresponding to G as(9) V G ( x, ω ) := n X j =1 G ( τ x j ω, x j +1 − x j )along any path 0 → x → . . . → x n − → x n = x , the particular choice of thepath being irrelevant because of (8). It also follows that V G has stationaryand L -bounded gradient in the sense that V G ( x, ω ) − V G ( y, ω ) = V G ( x − y, τ y ω ) for all x, y ∈ Z d and sup x ∈ Z d k V G ( x + e, · ) − V G ( x, · ) k L ( P ) < C, respectively. Even more importantly, the mapping x V G ( x, ω ) + x is har-monic with respect to the transition probabilities (5) for P -almost all ω .This means that ( S n + V G ( S n , · )) n > is a martingale with respect to P π,ω sothat the corrector V G expresses the “distance” (or the deformation ) of themartingale from the random walk ( S n ) n > itself. One can show that the con-tribution of this deformation grows at most sub-linearly at large distances(i.e. sup | x |≤ n n − V G ( x, · ) n →∞ −−−→ P π,ω ( S n / √ n ∈ · ) converge weakly to a Gaussian lawfor almost every ω , see [SS04, BB07, MP07] for a detailed recount of thesubstantial progress made in this direction.In order to finally make a connection with the notion of null-homology, letus note that the result just mentioned does not rule out the possibility thatthe corrector grows stochastically to infinity. Namely, although the gradientof V G is stationary and thus tight as pointed out above, the latter propertymay naturally fail for V G itself. On the other hand, a tight corrector meansthat the above martingale is just a “negligible” perturbation of the randomwalk ( S n ) n > itself which is a much stronger statement than the above cen-tral limit theorem. Our main result, Theorem 2.2 below, establishes, as afurther information, the equivalence of this property with the null-homologyof its stationary gradient, under no extra assumptions. Now, for the randomconductance model in dimension d >
3, the tightness of V G has indeed beenshown to hold, see [GO15] and [AKM17].
2. The main result.
We proceed with a description of the setup thatallows us to define null homology in terms of probability measures ratherthan random variables. This appears to be more convenient to state andprove our main result.
G. ALSMEYER AND C. MUKHERJEE
Without any loss of generality, we work with S = R d and write Ω =( R d ) ⊗ Z for the space of doubly infinite sequences x = ( x n ) n ∈ Z endowed withthe Borel σ -field and T : Ω → Ω the (left) shift operator on Ω, viz. x = ( . . . , x − , x , x , . . . ) ( . . . , x , x , x , . . . ) . The coordinate mappings on Ω are denoted X n for n ∈ Z , and we let S n bethe mapping x s n on Ω for n ∈ Z , where s n = x + . . . + x n if n > , , if n = 0 , − ( x − n +1 + . . . + x ) if n − . So ( X n ) n ∈ Z forms a stationary sequence with associated random walk ( S n ) n ∈ Z under any T -invariant probability measure on Ω.Next, we denote by M (Ω) the locally convex vector space of finite signedmeasures on Ω endowed with the topology of weak convergence and furtherby M T (Ω) its subsets of T -invariant probability measures. Defining the map D : Ω → Ω by x T x − x = ( . . . , x − x − , x − x , x − x , . . . ) , we obviously have that P ∈ M T (Ω) implies P D − = P ( D ∈ · ) ∈ M T (Ω).Null homology for elements of M T (Ω) can now be defined as follows. Definition . Any T -invariant probability measure P ∈ M T (Ω) iscalled null-homologous if P = Q D − for some Q ∈ M T (Ω) . Plainly, null homology of P is equivalent to the null homology of ( X n ) n ∈ Z under P .The subsequent Theorem 2.2 provides a characterization of this propertyin terms of the laws of the S n under P , thus { P S − n : n ∈ Z } . Theorem . Given any P ∈ M T (Ω) , the following assertions areequivalent:(a) P is null-homologous.(b) { P S − n : n > } is tight.(c) { P S − − n : n > } is tight. Remark . In [L86, Proposition 6], Lalley provided a criterion fornull-homology within the subclass of integrable stationary sequences, called
ULL HOMOLOGY AND STATIONARY SEQUENCES L -null-homology . It enabled him to rule out a certain lattice-type behaviorfor the derivation of a renewal theorem for certain stationary processes. Infact, he showed that L -null-homology is equivalent to the L -boundednessof the partial sums of the stationary sequence, i.e., of the associated randomwalk. Naturally, this is a much stronger requirement than the tightness ap-pearing in our theorem above. Also, the proof of our result, which is basedon an application of the Schauder fixed-point theorem in an appropriatecontext, differs entirely from the arguments used in [L86]. Remark . Note that our criterion for null-homology holds for any T -invariant measure P ∈ M T (Ω) and is not restricted to the the ergodicones, i.e., extremal points of M T (Ω).In the above context of T -invariant probability measures, we say that P ∈ M T (Ω) as well as the coordinate sequence ( X n ) n ∈ Z (under P ) are L p -null-homologous if they are null-homologous and E | X | p < ∞ . Before givingthe proof of Theorem 2.2, we provide as an immediate consequence thefollowing corollary which characterizes L p -null-homology for any p > p = 1 and p = 2. Corollary . Given any P ∈ M T (Ω) , the following assertions areequivalent for any p > :(a) P is L p -null-homologous.(b) ( S n ) n > is L p -bounded.(c) ( S − n ) n > is L p -bounded. Proof of Theorem 2.2.
Obviously, it suffices to show that (b) implies(a). To this end, we consider the bivariate mappingsΛ k : Ω → Ω × Ω , x (cid:0) T k x , x + . . . + T k − x (cid:1) = (cid:0) ( x n + k ) n ∈ Z , ( x n + . . . + x n + k − ) n ∈ Z (cid:1) for k ∈ N and point out that (b) entails the tightness of the family P = { P Λ − k : k ∈ N } . We can lift the shift T as well as the projections X n in a canonical way tomappings on Ω × Ω and, by slight abuse of notation, may call these mappingsagain T and X n . The projections on the y -components, namely ( x , y ) y n if y = ( y k ) k ∈ Z , are denoted Y n for n ∈ Z . Then the T -invariance of P impliesthe very same for the elements of P . G. ALSMEYER AND C. MUKHERJEE
Now let D be the closed convex hull of all weak limit points of P whichforms a compact convex subset of M (Ω × Ω). Consider the map S : Ω × Ω → Ω × Ω , (cid:0) x , y (cid:1) (cid:0) T x , x + y (cid:1) which is linear, continuous, commutes with T , i.e. S ◦ T = T ◦ S , and satisfiesfurther S ◦ Λ n = Λ n +1 , thus Γ n S − = Γ n +1 for all n ∈ N , where Γ n := P Λ − n .Then the last property entails that the set D is S -invariant which in turn,by invoking Schauder’s fixed point theorem, allows us to conclude that S has a fixed point, say Γ, in D . This means that Γ S − = Γ or, equivalently,that Γ is S -invariant.Finally, by considering the map G = ( X , Y ) : Ω × Ω → R d × R d , (cid:0) x , y (cid:1) ( x , y )we have that ( X ′ n , Y ′ n ) := G ◦ S n = ( X n , Y n ◦ S n ), n >
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Theory Probab. Appl. , , 247–265, 1984. G. AlsmeyerInst. Math. Stochastics,Department of Mathematicsand Computer ScienceUniversity of M¨unsterOrl´eans-Ring 10, D-48149M¨unster, GermanyE-mail: [email protected]