On Null-Homology and Stationary Sequences

Abstract

The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measure-preserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (Cocycles on ergodic transformation groups. Macmillan lectures in mathematics, vol 1, Macmillan Company of India, Ltd., Delhi, 1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same null-homology equivalence class. We also discuss various aspects of null-homology within the class of Markov random walks and compare null-homology with a formally stronger notion which we call strict-sense null-homology. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner.

1 Introduction and Motivation

The notion homology arises in various branches of mathematics. It was originally developed in algebraic topology in order to associate a sequence of algebraic objects. A typical fundamental question is the following: When does an n-cycle of a (simplical) complex form the boundary of an $(n+1)$-chain, or equivalently, when is its fundamental class a boundary for the singular homology? If such a requirement is fulfilled, the cycle is said to be homologous to 0 or null-homologous. In ergodic theory, the notion appears in the context of measure-preserving transformations arising from natural group actions on a complete and separable metric space. A prominent result in this context, due to Schmidt [36, Theorem 11.8] and sometimes called the coboundary theorem, states that a random walk or cocycle generated by a measure-preserving transformation of $\mathbb {G}=\mathbb {R}^{d}$ or a closed subgroup of it is tight iff it is null-homologous (and then called a coboundary). The result was extended to locally compact second countable Abelian groups $\mathbb {G}$ with no compact subgroup in [32], and generalized to Polish topological groups $\mathbb {G}$ in [1]. In probabilistic language, the coboundary theorem provides a characterization of tightness for a random walk with stationary increments. It will be stated below both in the ergodic-theoretic and the probabilistic framework. A similar framework was also used by Bradley who, by adapting Schmidt’s arguments, proved the result for random walks with nonstationary $\mathbb {G}$-valued increments; first for $\mathbb {G}=\mathbb {R}$ in [13], then for $\mathbb {G}$ a separable Banach space in [15] and for products of upper triangular random matrices in [14].

In order to motivate the present work, let us note that a random walk with stationary increments can be viewed as a particular instance of a so-called Markov random walk or Markov-additive process with stationary recurrent driving chain. A precise definition will be given shortly. Our purpose is to give a new and relatively short proof of the aforementioned coboundary theorem (see Theorem 2.2), discuss various aspects of null-homology within the class of Markov random walks (in Sect. 3.1), compare null-homology with a formally stronger notion we call strict-sense null-homology (cf. Theorem 3.1) and provide an extension of the coboundary theorem to the case when the driving chain is null-recurrent (cf. Theorem 3.2. Finally, we also provide further background information and applications (cf. Section 5) so as to sustain the relevance of these results in probability theory where the notion of null-homology has remained relatively unnoticed, at least until its appearance in recent progress made in the strong coupling limit of the Polaron path measures [33] (see Sect. 6) and in stochastic homogenization (see Sect. 7) under the name increment-stationarity, see, e.g., Blanc et al. [11], Gloria and Otto [21] and especially Gloria [20].

2 The Coboundary Theorem, Revisited

We start with the ergodic-theoretic framework, where the coboundary theorem is usually cast in terms of a measure-preserving transformation and a cocycle. In this framework, we give a new and short proof of the above result with the help of commuting maps and Schauder’s fixed-point theorem.

We proceed with a definition of null-homology in terms of probability measures rather than random variables. Let $\Omega =(\mathbb {R}^{d})^{\otimes \mathbb {Z}}$ be the space of doubly infinite sequences $\textbf{x}=(x_{n})_{n\in \mathbb {Z}}$ endowed with the Borel $\sigma$-field and $T:\Omega \rightarrow \Omega$ the (left) shift operator on $\Omega$, viz.

$$\begin{aligned} \textbf{x}\ =\ (\ldots ,x_{-1},x_{0},x_{1},\ldots )\ \mapsto \ (\ldots ,x_{0},x_{1},x_{2},\ldots ). \end{aligned}$$

The coordinate mappings on $\Omega$ are denoted $X_{n}$ for $n\in \mathbb {Z}$, and we let $S_{n}$ be the mapping $\textbf{x}\mapsto s_{n}$ on $\Omega$ for $n\in \mathbb {Z}$, where

$$\begin{aligned} s_{n}\ =\ {\left\{ \begin{array}{ll} x_{1}+\cdots +x_{n}&{}\text {if }n\geqslant 1,\\ 0&{}\text {if }n=0,\\ -(x_{-n+1}+\cdots +x_{0})&{}\text {if }n\leqslant -1. \end{array}\right. } \end{aligned}$$

(1)

Next, let $\mathscr {M}(\Omega )$ denote the locally convex vector space of finite signed measures on $\Omega$ endowed with the topology of weak convergence and $\mathscr {M}_{T}(\Omega )$ its subset of T-invariant probability measures. Defining the map $D:\Omega \rightarrow \Omega$ by

$$\begin{aligned} \textbf{x}\ \mapsto \ T\textbf{x}-\textbf{x}\ =\ (\ldots ,x_{0}-x_{-1},x_{1}-x_{0},x_{2}-x_{1},\ldots ), \end{aligned}$$

we obviously have that T and D commute and therefore that $\mathbb {P}\in \mathscr {M}_{T}(\Omega )$ implies $\mathbb {P}D^{-1}:=\mathbb {P}(D\in \cdot )\in \mathscr {M}_{T}(\Omega )$. Null-homology for elements of $\mathscr {M}_T(\Omega )$ can now be defined as follows.

Definition 2.1

A T-invariant probability measure $\mathbb {P}\in \mathscr {M}_T(\Omega )$ is called null-homologous if $\mathbb {P}=\mathbb {Q}D^{-1}$ for some $\mathbb {Q}\in \mathscr {M}_T(\Omega )$.

Theorem 2.2

Given any $\mathbb {P}\in \mathscr {M}_{T}(\Omega )$, the following assertions are equivalent:

• (a) $\{\mathbb {P}S_{n}^{-1}:n\geqslant 0\}$ is tight.

• (b) $\mathbb {P}$ is null-homologous.

Proof

It suffices to show that (a) implies (b) for which we consider the bivariate mappings

$$\begin{aligned}&\Lambda _{k}:\Omega \rightarrow \Omega \times \Omega ,\\&\quad \textbf{x}\ \mapsto \ \big (T^{k}{} \textbf{x},\textbf{x}+\ldots +T^{k-1}\textbf{x}\big )\ =\ \big ((x_{n+k})_{n\in \mathbb {Z}},(x_{n}+\cdots +x_{n+k-1})_{n\in \mathbb {Z}}\big ) \end{aligned}$$

for $k\in \mathbb {N}$ and point out that (a) entails the tightness of the family

$$\begin{aligned} \mathscr {P}\ =\ \left\{ \mathbb {P}\Lambda _{k}^{-1}:k\in \mathbb {N}\right\} . \end{aligned}$$

We can lift the shift T as well as the projection $X_{n}$ for each n in a canonical way to a mapping on $\Omega \times \Omega$, namely $(\textbf{x},\textbf{y})\mapsto (T\textbf{x},T\textbf{y})$ and $(\textbf{x},\textbf{y})\mapsto x_{n}$, and, by slight abuse of notation, call these mappings again T and $X_{n}$. The projections on the y-components, namely $(\textbf{x},\textbf{y})\mapsto y_{n}$ if $\textbf{y}=(y_{k})_{k\in \mathbb {Z}}$, are denoted $Y_{n}$ for $n\in \mathbb {Z}$. Then, the T-invariance of $\mathbb {P}$ implies the very same for the elements of $\mathscr {P}$.

Now let $\mathscr {D}$ be the closed convex hull of all weak limit points of $\mathscr {P}$ which forms a compact convex subset of $\mathscr {M}(\Omega \times \Omega )$. Consider the map

$$\begin{aligned} S:\Omega \times \Omega \rightarrow \Omega \times \Omega ,\quad \big (\textbf{x},\textbf{y}\big )\ \mapsto \ \big (T\textbf{x},\textbf{x}+\textbf{y}\big ) \end{aligned}$$

which is linear, continuous, commutes with T, i.e., $S\circ T=T\circ S$, and satisfies further $S\circ \Lambda _{n}=\Lambda _{n+1}$, thus $\Gamma _{n}S^{-1}=\Gamma _{n+1}$ for all $n\in \mathbb {N}$, where $\Gamma _{n}:=\mathbb {P}\Lambda _{n}^{-1}$. Then, the last property entails that the set $\mathscr {D}$ is S-invariant which in turn, by invoking Schauder’s fixed-point theorem, allows us to conclude that S has a fixed point, say $\Gamma$, in $\mathscr {D}$. This means that $\Gamma S^{-1}=\Gamma$ or, equivalently, that $\Gamma$ is S-invariant.

Finally, by considering the map

$$\begin{aligned} G=(X_{0},Y_{0}):\Omega \times \Omega \rightarrow \mathbb {R}^{d}\times \mathbb {R}^{d},\quad \big (\textbf{x},\textbf{y}\big )\ \mapsto \ (x_{0},y_{0}) \end{aligned}$$

we have that $(X_{n}',Y_{n}'):=G\circ S^{n}=(X_{n},Y_{n}\circ S^{n})$, $n\geqslant 0$, is stationary under $\Gamma$ and satisfies:

• (i) $(X_{n}')_{n\geqslant 0}=(X_{n})_{n\geqslant 0}$ and has law $\mathbb {P}$ under $\Gamma$, because this is the case under any element of $\mathscr {D}$.

• (ii) $Y_{n+1}'=Y_{0}+X_{0}+\cdots +X_{n}=Y_{n}'+X_{n}'$, thus

  $$\begin{aligned} X_{n}\ =\ X_{n}'\ =\ Y_{n+1}'-Y_{n}' \end{aligned}$$

  for all $n\geqslant 0$.

Since $(Y_{n}')_{n\geqslant 0}$ is stationary under $\Gamma$, (a) follows. $\square$

Remark 2.3

As mentioned earlier, the coboundary theorem was earlier proved by Schmidt [36, Theorem 11.8] using ergodic decomposition of “skew-products.” The technique there relied on showing that a cocycle is a coboundary if and only if its skew product decomposes into finite measure-preserving transformations. The above proof of Theorem 2.2 is quite different (and a bit shorter) compared to the arguments in [36, Ch. 11].

3 Markov Random Walks, Null-Homology and Strict-Sense Null-Homology

Let ${\varvec{X}}=(X_{n})_{n\in \mathbb {Z}}$ denote a doubly infinite stationary sequence of random variables which are defined on a probability space with underlying probability measure $\mathbb {P}$ and take values in a complete separable metric space $\mathscr {S}$. Stationarity means that, for all $n\in \mathbb {N}$ and $m\in \mathbb {Z}$,

$$\begin{aligned} \mathbb {P}\big ((X_{1},\dots , X_{n}) \in \cdot \big )\ =\ \mathbb {P}\big ((X_{m+1},\dots , X_{m+n})\in \cdot \big ). \end{aligned}$$

In other words, the joint law of $(X_{1},\dots , X_{n})$ for any n coincides with the law of any of its “shifts” under the action of the additive group $\mathbb {Z}$ on the space of doubly infinite sequences $\mathscr {S}^{\times \mathbb {Z}}$. Equivalently, putting ${\varvec{X}}_{n}:=(X_{n+k})_{k\in \mathbb {Z}}$ for $n\in \mathbb {Z}$, the law of ${\varvec{X}}_{n}$ is the same for each $n\in \mathbb {Z}$. The notion of homology now arises naturally from this group action, given measurable functions $F, G:\mathscr {S}^{\times \mathbb {Z}}\rightarrow \mathbb {R}^{d}$. Following Lalley [28, p. 197], we say that F is homologous to G (with respect to ${\varvec{X}}$ and $\mathbb {P}$) and write $F\sim G$ if there exists a function $\xi : \mathscr {S}^\mathbb {Z}\rightarrow \mathbb {R}^{d}$ such that

$$\begin{aligned} F({\varvec{X}}_{1}) - G({\varvec{X}}_{1})\ =\ \xi ({\varvec{X}}_{1})- \xi ({\varvec{X}}_{0})\quad \mathbb {P}\text {-a.s.} \end{aligned}$$

(2)

Then, $\sim$ is an equivalence relation, and F is called null-homologous if $F\sim 0$, thus

$$\begin{aligned} F({\varvec{X}}_{1})\ =\ \xi ({\varvec{X}}_{1})- \xi ({\varvec{X}}_{0})\quad \mathbb {P}\text {-a.s.} \end{aligned}$$

(3)

Observe that, given any stationary sequence ${\varvec{X}}$ and null-homologous function F, the process $(F({\varvec{X}}_{n}))_{n\in \mathbb {Z}}$ is not only stationary as well (and thus also tight), but in fact the incremental sequence of another stationary process, viz. $(\xi ({\varvec{X}}_{n}))_{n\in \mathbb {Z}}$. In view of this in fact equivalent definition of null-homology, the coboundary theorem answers the natural question which stationary processes are of that “incremental” type and therefore allowing a representation with respect to a null-homologous function.

3.1 Markov Random Walks

In [28], Lalley considered random walks with increments from a fairly general class of integrable stationary sequences. As a main result, he proved a Blackwell-type renewal theorem for which it was necessary to rule out a certain “lattice-type” behavior which to define requires the notion of null-homology. In the following, we give a brief introduction of this notion within the more general framework of Markov random walks.

Let $(\mathscr {S},\mathfrak {S})$ be a measurable space and $\mathscr {B}(\mathbb {R}^{d})$ the Borel $\sigma$-field on $\mathbb {R}^{d}$. A sequence $(M_{n},X_{n})_{n\geqslant 0}$ taking values in $(\mathscr {S}\times \mathbb {R}^{d},\mathfrak {S}\otimes \mathscr {B}(\mathbb {R}^{d}))$ is called Markov-modulated if

• $(M_{n})_{n\geqslant 0}$ forms a temporally homogeneous Markov chain,

• $X_{0},X_{1},\ldots$ are conditionally independent given $(M_{n})_{n\geqslant 0}$, and

• the conditional laws of $X_{0}$ and $X_{n}$ for $n\geqslant 1$ given that $M_{0}=s_{0},M_{1}=s_{1},\ldots$ are given by $P_{0}(s_{0},\cdot )$ and $P((s_{n-1},s_{n}),\cdot )$, respectively, for suitable stochastic kernels $P_{0}$ and P.

It follows that $(M_{n},X_{n})_{n\geqslant 0}$ forms a temporally homogeneous Markov chain such that the conditional law of $(M_{n},X_{n})$ given the history up to time $n-1$ is a function of $M_{n-1}$ only (instead of $(M_{n-1},X_{n-1})$) for each $n\geqslant 1$. This can in fact be shown to be an equivalent property if $\mathfrak {S}$ is countably generated. We make the additional assumption that $(M_{n})_{n\geqslant 0}$ has a stationary distribution $\pi$ and is ergodic under $\mathbb {P}_{\pi }:=\int _{\mathscr {S}}\mathbb {P}(\cdot |M_{0}=s)\,\pi (\textrm{d}s)$. By ergodic decomposition, the latter assumption does not entail any loss of generality regarding our results. Due to the special Markovian structure, this entails the same for $(M_{n},X_{n})_{n\geqslant 0}$. Let $\mu$ be the pertinent stationary law and put $\mathbb {P}_{\mu }:=\int _{\mathscr {S}\times \mathbb {R}^{d}}\mathbb {P}(\cdot |M_{0}=s,X_{0}=x)\,\mu (\textrm{d}s\times \textrm{d}x)$. Notice that the laws of $(M_{n},X_{n})_{n\geqslant 1}$ under $\mathbb {P}_{\mu }$ and $\mathbb {P}_{\pi }$ are identical.

Defining $S_{0}:=0$ and $S_{n}:=\sum _{i=1}^{n}X_{i}$ for $n\geqslant 1$, the bivariate sequence $(M_{n},S_{n})_{n\geqslant 0}$ and also $(S_{n})_{n\geqslant 0}$ alone are called Markov random walk (MRW) and $(M_{n})_{n\geqslant 0}$ its driving or modulating chain. Whenever these objects are studied in stationary regime, that is, under $\mathbb {P}_{\mu }$, we may also consider a doubly infinite stationary extension $(M_{n},X_{n})_{n\in \mathbb {Z}}$ with associated doubly infinite random walk

$$\begin{aligned} S_{n}\ =\ {\left\{ \begin{array}{ll} \sum _{i=1}^{n}X_{i}&{}{}\text{ if } n\geqslant 1,\\ 0&{}{}\text{ if } n=0,\\ -\sum _{i=n+1}^{0}X_{i}&{}{}\text{ if } n<0. \end{array}\right. } \end{aligned}$$

(4)

In this framework, we call both $(M_{n},S_{n})_{n\in \mathbb {Z}}$ and $(M_{n},X_{n})_{n\in \mathbb {Z}}$

• null-homologous if there exists a stationary sequence $(Y_{n})_{n\in \mathbb {Z}}$ such that

  $$\begin{aligned} X_{n}\ =\ Y_{n}-Y_{n-1}\quad \mathbb {P}_{\mu }\text {-a.s.} \end{aligned}$$

  and thus

  $$\begin{aligned} S_{n}\ =\ \textrm{sign}(n)(Y_{n}-Y_{0})\quad \mathbb {P}_{\mu }\text {-a.s.} \end{aligned}$$

  for all $n\in \mathbb {Z}$;

• strict-sense null-homologous if, furthermore, there exists a measurable function $\xi :\mathscr {S}\rightarrow \mathbb {R}^{d}$ such that $Y_{n}=\xi (M_{n})$ $\mathbb {P}_{\mu }$-a.s. for all $n\geqslant 0$.

For the unilateral sequences $(M_{n},S_{n})_{n\geqslant 0}$ and $(M_{n},X_{n})_{n\geqslant 0}$, the stated conditions must naturally hold for all $n\geqslant 1$. On the other hand, these conditions persist under doubly infinite stationary extension whence it actually makes no difference whether the unilateral or the doubly infinite framework is adopted. Regarding the relation between null-homology and the formally stronger strict-sense null-homology, it is not obvious and therefore shown as part of our main result, Theorem 3.1, that they are actually equivalent.

We further note that the function $\xi$ must be unique up to translation by some $c\in \mathbb {R}^{d}$. Namely, if $X_{n}=\zeta (M_{n})-\zeta (M_{n-1})$ $\mathbb {P}_{\mu }$-a.s. for all $n\geqslant 1$ and another function $\zeta :\mathscr {S}\rightarrow \mathbb {R}^{d}$, then

$$\begin{aligned} \xi (M_{0})-\zeta (M_{0})\ =\ \xi (M_{1})-\zeta (M_{1})\ =\ \cdots \quad \mathbb {P}_{\mu }\text {-a.s.} \end{aligned}$$

In other words, $\xi -\zeta$, viewed as a function on $\mathscr {S}^{\mathbb {N}_{0}}$, is a.s. shift-invariant with respect to the law of $(M_{n})_{n\geqslant 0}$ under $\mathbb {P}_{\mu }$ and therefore as claimed equal to some $c\in \mathbb {R}^{d}$ by ergodicity.

Null-homology arises quite naturally in connection with the lattice-type of one-dimensional MRW’s. Let $(M_{n})_{n\geqslant 0}$ be ergodic with unique stationary law $\pi$. Following Shurenkov [37], the MRW $(M_{n},S_{n})_{n\geqslant 0}$ is called d-arithmetic if d is the maximal positive number such that

$$\begin{aligned} \mathbb {P}_{\pi }\big (X_{1}\in \xi (M_{1})-\xi (M_{0})+d\mathbb {Z}\big )\ =\ 1 \end{aligned}$$

for a suitable function $\xi :\mathscr {S}\rightarrow [0,d)$, called shift function. If no such d exists, it is called nonarithmetic. Equivalently, $(M_{n},S_{n})_{n\geqslant 0}$ is d-arithmetic if $d>0$ is the maximal number such that $(M_{n},X_{n}-X_{n}')_{n\in \mathbb {Z}}$ is Markov-modulated and strict-sense null-homologous for a sequence of $d\mathbb {Z}$-valued random variables $(X_{n}')_{n\in \mathbb {Z}}$. Namely, with $\xi$ denoting the shift function,

$$\begin{aligned} X_{n}':=\,X_{n}-\xi (M_{n})+\xi (M_{n-1}) \end{aligned}$$

for $n\in \mathbb {Z}$.

Since, given an arbitrary stationary sequence $(X_{n})_{n\in \mathbb {Z}}$, any of

$$\begin{aligned} M_{n}\,=\ (X_{n+i})_{i\in \mathbb {Z}},\ (X_{n-i})_{i\leqslant 0},\ \text {or }(X_{n+i})_{i\geqslant 0},\quad n\in \mathbb {Z}\end{aligned}$$

constitutes a modulating stationary Markov chain on a Polish state space (ergodic iff $(X_{n})_{n\in \mathbb {Z}}$ is ergodic [16, Prop. 6.31], and also Fellerian), we see that null-homology for stationary processes can also be studied within the framework of Markov-modulation, with some freedom to choose the driving chain. Moreover, the conditional law of each $X_{n}$ given the driving chain is always degenerate under the above choices. As for the coordinate mappings $X_{n}$ considered in Sect. 2, they form a stationary sequence with associated MRW $(M_{n},S_{n})_{n\in \mathbb {Z}}$ under any T-invariant probability measure on $\Omega$, with $M_{n}:=T^{n}$ and $S_{n}$ defined by (1), and the null-homology of $\mathbb {P}$ in Definition 2.1 is equivalent to the null-homology of the MRW $(M_{n},S_{n})_{n\in \mathbb {Z}}$ under $\mathbb {P}$.

3.2 Main Results Relating Strict-Sense Null-Homology and Tightness

By definition, null-homology implies stationarity of the sequence $(S_{n}+Y_{0})_{n\in \mathbb {Z}}$ and thus “almost stationarity” as well as tightness of the random walk $(S_{n})_{n\in \mathbb {Z}}$ itself. Regarding tightness, the converse is established by the next theorem together with the nonobvious fact that null-homology and strict-sense null-homology are in fact the same. This is our main result.

Theorem 3.1

Let $(M_{n},X_{n})_{n\in \mathbb {Z}}$ be a doubly infinite stationary Markov-modulated sequence of $\mathscr {S}\times \mathbb {R}^{d}$-valued random variables with ergodic driving chain. Then, the following assertions are equivalent for the associated MRW $(M_{n},S_{n})_{n\in \mathbb {Z}}$ defined by (4):

• (a) $(M_{n},S_{n})_{n\in \mathbb {Z}}$ is strict-sense null-homologous.

• (b) $(M_{n},S_{n})_{n\in \mathbb {Z}}$ is null-homologous and $X_{n}=g(M_{n-1},M_{n})$ a.s. for each $n\in \mathbb {Z}$ and some measurable $g:\mathscr {S}\times \mathscr {S}\rightarrow \mathbb {R}^{d}$.

• (c) $(M_{n},S_{n})_{n\in \mathbb {Z}}$ is null-homologous.

• (d) $(S_{n})_{n\in \mathbb {Z}}$ is tight.

• (e) $(S_{n})_{n\geqslant 0}$ is tight.

As strict-sense null-homology by definition ensures $X_{n}=\xi (M_{n})-\xi (M_{n-1})$ a.s. for some $\xi$, we see that (a) trivially implies (b) and particularly that the $X_{n}$ given their driving chain are all a.s. constant. In other words, the modulation is rigid in the sense of no extra randomness beyond the contribution of the driving chain. Our result may therefore be summarized by saying that this rigidity in combination with null-homology is necessary and sufficient for the tightness and thus atypical behavior of a MRW.

3.3 Null-Recurrent Driving Chain

For a MRW $(M_{n},S_{n})_{n\geqslant 0}$ with null-recurrent driving chain, the notion of null-homology can be generalized as follows: Let $\pi$ denote the essentially unique $\sigma$-finite stationary measure of $(M_{n})_{n\geqslant 0}$ and $\mathfrak {S}_{\pi }=\{A\in \mathfrak {S}:0<\pi (A)<\infty \}$. Putting $\pi _{A}:=\pi (\cdot \cap A)/\pi (A)$, it is well known that the driving chain is cycle-stationary under $\mathbb {P}_{\pi _{A}}$ for any $A\in \mathfrak {S}_{\pi }$ in the sense that the sequence of cycles

$$\begin{aligned} (M_{\tau _{n}(A)},\ldots ,M_{\tau _{n+1}(A)-1}),\quad n\geqslant 0 \end{aligned}$$

is stationary under $\mathbb {P}_{\pi _{A}}$ in the ordinary sense. Here the $\tau _{n}(A)$ for $n\geqslant 0$ denote the successive return times to A of the chain with $\mathbb {P}_{\pi _{A}}(\tau _{0}(A)=0)=1$. By Markov modulation, the sequence $(M_{n},X_{n})_{n\in \mathbb {Z}}$ is then cycle stationary under $\mathbb {P}_{\mu _{A}}$, where $\mu _{A}$ equals the law of $(M_{\tau _{1}(A)},X_{\tau _{1}(A)})$ under $\mathbb {P}_{\mu _{A}}$. Null-homology and strict-sense null-homology for $(M_{n},X_{n})_{n\geqslant 0}$ and its associated MRW are now defined exactly as before, but with $(Y_{n})_{n\geqslant 0}$ being cycle-stationary under $\mathbb {P}_{\mu _{A}}$ for any $A\in \mathfrak {S}_{\pi }$. The following result provides the counterpart of Theorem 3.1 in the null-recurrent situation.

Theorem 3.2

Let $(M_{n},X_{n})_{n\geqslant 0}$ be a Markov-modulated sequence with null-recurrent driving chain and associated MRW $(M_{n},S_{n})_{n\geqslant 0}$. Then, the following assertions are equivalent:

• (a) $(M_{n},S_{n})_{n\geqslant 0}$ is strict-sense null-homologous.

• (b) $(M_{n},S_{n})_{n\geqslant 0}$ is null-homologous and $X_{n}=g(M_{n-1},M_{n})$ $\mathbb {P}_{\mu _{A}}$-a.s. for each $n\in \mathbb {N}$, $A\in \mathfrak {S}_{\pi }$ and some measurable $g:\mathscr {S}\times \mathscr {S}\rightarrow \mathbb {R}^{d}$.

• (c) $(M_{n},S_{n})_{n\geqslant 0}$ is null-homologous.

• (d) $(S_{\tau _{n}(A)})_{n\geqslant 0}$ is tight under $\mathbb {P}_{\mu _{A}}$ for any $A\in \mathfrak {S}_{\pi }$.

Provided they hold and $S_{n}=\xi (M_{n})-\xi (M_{0})$ $\mathbb {P}_{\mu }$-a.s. for all n, the random walk $(S_{n})_{n\geqslant 0}$ itself is tight under $\mathbb {P}_{\mu _{A}}$ for all $A\in \mathfrak {S}_{\pi }$ iff the pre-image of the complement of some $[-a,a]^{d}$, $a\geqslant 0$, under $\xi$ has finite mass with respect to $\pi$, the stationary measure of the driving chain.

We have organized the rest of the article as follows. The proof of Theorem 3.1 together with further results such as an $L^{p}$-version of the coboundary theorem and another characterization of strict-sense null-homology are the content of Sect. 4. A collection of instances where null-homology arises as a relevant issue are discussed in Sects. 5–7. In particular, we discuss in Sect. 6 another characterization of null-homology (cf. Theorem 6.1) and show in Sect. 7 that the so-called corrector, which comes into play when aiming at an almost sure central limit theorem (CLT), is strict-sense null-homologous. We also explain that this does not contradict a seemingly different statement by Gloria [20] about the same issue.

4 Further Results and Proofs of Theorems 3.1 and 3.2

Adopting the notation from the previous section and additionally assuming $(\mathscr {S},\mathfrak {S})$ to be Polish, the following theorem, stated for one-sided sequences, may be viewed as a more elaborate formulation of the equivalence between (c) and (e) in Theorem 3.1. Its proof, given after one more theorem, is probabilistic and relatively short. Since assertions about the additive part of a given MRW $(M_{n},S_{n})_{n\geqslant 0}$ on $\mathscr {S}\times \mathbb {R}^{d}$ are not affected by the choice of the driving chain, the Polish state space assumption does not constitute a real restriction because we can always choose as driving chain $M_{n}=(X_{n+k})_{k\geqslant 0}$ for $n\geqslant 0$ which has Polish state space $((\mathbb {R}^{d})^{\times \mathbb {N}_{0}},\mathscr {B}(\mathbb {R}^{d})^{\times \mathbb {N}_{0}})$ and also the Feller property.

Theorem 4.1

Let $(M_{n},X_{n})_{n\geqslant 0}$ be a Markov-modulated sequence in a standard model with state space $\mathscr {S}\times \mathbb {R}^{d}$, ergodic stationary distribution $\mu$ and associated zero-delayed MRW $(M_{n},S_{n})_{n\geqslant 0}$, thus $S_{0}=0$ and $S_{n}=X_{1}+\ldots +X_{n}$ for $n\geqslant 1$. Let further $Y_{n}=X_{0}+S_{n}$ for $n\geqslant 0$. Provided that $(\mathscr {S},\mathfrak {S})$ is Polish, the following assertions are equivalent:

• (a) $(S_{n})_{n\geqslant 0}$ is tight under $\mathbb {P}_{\mu }$.

• (b) There exists a law $\nu$ on $\mathscr {S}\times \mathbb {R}^{d}$ such that the sequences $(M_{n},X_{n})_{n\geqslant 1}$ and $(M_{n},X_{n+1},Y_{n})_{n\geqslant 0}$ are both stationary under $\mathbb {P}_{\nu }$. Moreover,

  $$\begin{aligned} \mathbb {P}_{\nu }((M_{n},Y_{n})\in \cdot )\,=\,\nu \quad \text {and}\quad \mathbb {P}_{\nu }((M_{n+1},X_{n+1})\in \cdot )\,=\,\mu \end{aligned}$$

  for all $n\geqslant 0$.

• (c) $(M_{n},S_{n})_{n\geqslant 0}$ is null-homologous.

If these assertions hold, the $Y_{n}$ can be chosen such that

$$\begin{aligned} Y_{n}\ =\ f(X_{n+1},X_{n+2},\ldots )\quad \text {a.s.} \end{aligned}$$

(5)

for all $n\in \mathbb {Z}$ and some measurable function $f:\mathbb {R}^{\times \mathbb {Z}}\rightarrow \mathbb {R}^{d}$.

In part (b) the stationarity of $(M_{n},X_{n})_{n\geqslant 1}$ is in fact a consequence of the stationarity of $(M_{n},X_{n+1},Y_{n})_{n\geqslant 0}$. Moreover, it should be observed that $\nu$ constitutes a stationary law for the Markov chain $(M_{n},S_{n})_{n\geqslant 0}$.

Our last theorem is concerned with another characterization of strict-sense null-homology that arises in connection with the so-called corrector in stochastic homogenization (see Sect. 7). Its proof is postponed to the end of this section. Obviously, if there exists a measurable function $\xi :\mathscr {S}\rightarrow \mathbb {R}^{d}$ such that

$$\begin{aligned} S_{n}\ =\ \xi (M_{n})-\xi (M_{0})\quad \mathbb {P}_{s}\text {-a.s.~for all }s\in \mathscr {S}\text { and }n\geqslant 0, \end{aligned}$$

(6)

then the closed-loop condition

$$\begin{aligned} \mathbb {P}(S_{n}=0|M_{0}=M_{n})\,=\,1\quad \mathbb {P}_{s}\text {-a.s.~for all }s\in \mathscr {S}\end{aligned}$$

(7)

holds which means that, whenever the driving chain returns to its initial state and thus completes a loop, then so does the modulated random walk. For the case when $(M_{n})_{n\geqslant 0}$ is an irreducible, but not necessarily recurrent Markov chain on a discrete state space, the subsequent theorem asserts that the converse is also true.

Theorem 4.2

Let $(M_{n},S_{n})_{n\geqslant 0}$ be a MRW such that its driving chain $(M_{n})_{n\geqslant 0}$ is irreducible with discrete state space $\mathscr {S}$. Then, the following assertions are equivalent:

• (a) $(M_{n},S_{n})_{n\geqslant 0}$ satisfies the closed-loop condition

  $$\begin{aligned} \mathbb {P}(S_{n}=0|M_{0}=M_{n})\,=\,1\quad \mathbb {P}_{s}\text {-a.s.~all }s\in \mathscr {S}. \end{aligned}$$

• (b) $(M_{n},S_{n})_{n\geqslant 0}$ is strict-sense null-homologous in the sense that $g(s,t)=\xi (t)-\xi (s)$ for all $s,t\in \mathscr {S}$ and some $\xi :\mathscr {S}\rightarrow \mathbb {R}^{d}$.

Proof of Theorem 4.1

“(a)$\Rightarrow$(b)” Defining $\widehat{M}_{n}:=(M_{n+k})_{k\geqslant 0}$ and $\widehat{X}_{n},\widehat{Y}_{n}$ similarly, consider the Feller chain $(\widehat{M}_{n},\widehat{X}_{n},\widehat{Y}_{n})_{n\geqslant 0}$ with state space $\mathbb {X}:=\mathscr {S}^{\times \mathbb {N}_{0}}\times (\mathbb {R}^{d})^{\times \mathbb {N}_{0}}\times (\mathbb {R}^{d})^{\times \mathbb {N}_{0}}$ and generated by the continuous map $\Psi :\mathbb {X}\rightarrow \mathbb {X}$,

$$\begin{aligned} ((s_{k})_{k\geqslant 0},(x_{k})_{k\geqslant 0},(y_{k})_{k\geqslant 0})\mapsto ((s_{k})_{k\geqslant 1},(x_{k})_{k\geqslant 1},(y_{k-1}+x_{k})_{k\geqslant 1}) \end{aligned}$$

(it is here where the additional assumption on $(\mathscr {S},\mathfrak {S})$ to be Polish enters to render the continuity). Under $\mathbb {P}_{\mu }$, the chain is stationary in the first two coordinates and therefore tight as a whole if (a) holds. By Prokhorov’s theorem, the relative compactness of the distributional time averages

$$\begin{aligned} Q_{n}\ :=\ \frac{1}{n}\sum _{j=1}^{n}\mathbb {P}_{\mu }\big (\big (\widehat{M}_{j},\widehat{X}_{j},\widehat{Y}_{j}\big )\in \cdot \big ),\quad n\geqslant 1 \end{aligned}$$

follows and thus the existence of $1\leqslant n_{1}<n_{2}<\cdots$ such that, as $k\rightarrow \infty$, both $Q_{n_{k}}$ and $Q_{n_{k}+1}$ converge weakly to a probability law on $\mathbb {X}$ which constitutes a stationary distribution of $(\widehat{M}_{n},\widehat{X}_{n},\widehat{Y}_{n})_{n\geqslant 0}$ (here the Feller property enters, again rendered by the assumption on $(\mathscr {S},\mathfrak {S})$ to be Polish). In particular, the trivariate Markov chain $(M_{n},X_{n},Y_{n})_{n\geqslant 0}$ has a stationary law $\Lambda$ satisfying $\Lambda (\cdot \times \cdot \times \mathbb {R}^{d})=\mu$. Moreover, its transition kernel, say $K((s,x,y),\cdot )$, does not depend on x because $Y_{n}=Y_{n-1}+X_{n}$ for each $n\geqslant 1$ and by the Markov-modulated structure of $(M_{n},X_{n})_{n\geqslant 0}$. Namely, with $\mathbb {P}_{s,x,y}:=\mathbb {P}(\cdot |M_{0}=s,X_{0}=x,Y_{0}=y)$, we infer

$$\begin{aligned} K((s,x,y),A\times B\times C)\ {}&=\ \mathbb {P}_{s,x,y}(M_{1}\in A,X_{1}\in B,Y_{0}+X_{1}\in C)\\&=\ \mathbb {P}_{s,x,y}(M_{1}\in A,X_{1}\in B,y+X_{1}\in C)\\&=\ \mathbb {P}_{s}(M_{1}\in A,X_{1}\in B,y+X_{1}\in C) \end{aligned}$$

for all $A\in \mathfrak {S}$ and $B,C\in \mathscr {B}(\mathbb {R}^{d})$. After these observations define

$$\begin{aligned} \nu :=\,\Lambda (\cdot \times \mathbb {R}^{d}\times \cdot ) \end{aligned}$$

and recall that $Y_{0}=X_{0}$. It follows that

$$\begin{aligned} \mathbb {P}_{\nu }((M_{1},X_{1})\in \cdot )\ {}&=\ \int _{\mathscr {S}\times \mathbb {R}^{d}}\mathbb {P}_{s}((M_{1},X_{1})\in \cdot )\ \nu (\textrm{d}s\times \textrm{d}x)\\&=\ \int _{\mathscr {S}}\mathbb {P}_{s}((M_{1},X_{1})\in \cdot )\ \pi (\textrm{d}s)\\&=\ \mathbb {P}_{\pi }((M_{1},X_{1})\in \cdot )\ =\ \mu . \end{aligned}$$

and therefore that $(M_{n},X_{n})_{n\geqslant 1}$ is stationary under $\mathbb {P}_{\nu }$ as claimed. Finally, as $(M_{n},X_{n+1},Y_{n})_{n\geqslant 0}$ is stationary under $\mathbb {P}_{\Lambda }$ (recall that $\Lambda$ equals the stationary distribution of $(M_{n},X_{n},Y_{n})_{n\in \mathbb {Z}}$), the same holds true under $\mathbb {P}_{\nu }$. Namely, by another use of the fact that $K((s,x,y),\cdot )$ does not depend on x,

$$\begin{aligned} \mathbb {P}_{\nu }((M_{0},X_{1},Y_{0})\in A\times B\times C)\ {}&=\ \int _{A\times C}\mathbb {P}_{s}(X_{1}\in B)\ \nu (\textrm{d}s\times \textrm{d}y)\\&=\ \int _{A\times \mathbb {R}^{d}\times C}\mathbb {P}_{s}(X_{1}\in B)\ \Lambda (\textrm{d}s\times \textrm{d}x\times \textrm{d}y)\\&=\ \mathbb {P}_{\Lambda }((M_{0},X_{1},Y_{0})\in A\times B\times C)\\&=\ \mathbb {P}_{\Lambda }((M_{n},X_{n+1},Y_{n})\in A\times B\times C) \end{aligned}$$

for all $n\geqslant 1$ and A, B, C as before. By choosing $B=\mathbb {R}^{d}$, we particularly find that the law of $(M_{n},Y_{n})$ under $\mathbb {P}_{\nu }$ equals $\nu$ for all n. This completes the proof of (b).

“(b)$\Rightarrow$(c)” Since the laws of $(M_{n},S_{n})_{n\geqslant 0}$ under $\mathbb {P}_{\nu }$ and $\mathbb {P}_{\mu }$ are the same (as under $\mathbb {P}_{\pi }$), the null-homology follows from $Y_{n}=X_{0}+S_{n}=Y_{0}+S_{n}$ and thus $S_{n}=Y_{n}-Y_{0}$ $\mathbb {P}_{\nu }$-a.s. for all $n\geqslant 1$.

Left with the proof of (5), we begin with some preliminary remarks. If $\mathscr {I}$ denotes the invariant $\sigma$-field of $\widehat{Y}=(Y_{n})_{n\geqslant 0}$, which is generated by the random variables $g(\widehat{Y})$ for shift-invariant functions $g:(\mathbb {R}^{d})^{\times \mathbb {N}}\rightarrow \mathbb {R}^{d}$, then any $\widehat{Y}'=(Y_{n}')_{n\geqslant 0}$ with $Y_{n}'=Y_{n}+Z$ for an $\mathscr {I}$-measurable $\mathbb {R}^{d}$-valued Z can be used instead of $(Y_{n})_{n\geqslant 0}$ in the sense that $(M_{n},X_{n+1},Y_{n}')_{n\geqslant 0}$ is still stationary under $\mathbb {P}=\mathbb {P}_{\nu }$ and

$$\begin{aligned} X_{n}\,=\,Y_{n}'-Y_{n-1}'\quad \mathbb {P}\text {-a.s.~for all }n\geqslant 1. \end{aligned}$$

(8)

This observation allows us to pick the $Y_{n}'$ in such a way that their conditional median given $\mathscr {I}$ is 0. Namely, let $F(\omega ,\cdot ):=\mathbb {P}(Y_{0}\leqslant \cdot |\mathscr {I})(\omega )$ be a regular version of the conditional distribution function of $Y_{0}$ and thus of each $Y_{n}$ given $\mathscr {I}$. Their conditional median $R_{0}$ is defined as

$$\begin{aligned} R_{0}(\omega )\,=\ \sup \left\{ r\in \mathbb {R}:F(\omega ,r)< 1/2\right\} . \end{aligned}$$

and $\mathscr {I}$-measurable. Consequently, a sequence with the desired property is obtained by replacing $Y_{n}$ with $Y_{n}-R_{0}$ for $n\geqslant 0$.

Proceeding with the proof of (5), we may now assume $R_{0}=0$ a.s. and also $d=1$, for instance by considering the d components of $X_{n}$ separately. The following argument is similar to the one given by Bradley in [13] for the nonstationary situation, but it is simpler and shorter because the sequence $(Y_{n})_{n\geqslant 0}$ he needs to define is here already given and stationary. In particular, we do not require Komlós’ law of large numbers.

Put $S_{k,n}:=S_{k+n}-S_{k}$ and note that $Y_{k+n}=Y_{k}+S_{k,n}$ for all $k,n\in \mathbb {N}_{0}$. Birkhoff’s ergodic theorem ensures that, as $n\rightarrow \infty$,

$$\begin{aligned} F_{k,n}(\omega ,r)\,=\ \frac{1}{n}\sum _{j=1}^{n}\textbf{1}_{(-\infty ,r]}(Y_{k+n}(\omega )) \end{aligned}$$

converges to $F(\omega ,r)$ for all $(k,r)\in \mathbb {N}_{0}\times \mathbb {R}$ and $\mathbb {P}_{\nu }$-almost all $\omega \in \Omega$, say $\omega \in \Omega _{r}$. By a standard selection argument (used in a similar manner for the proof of the Helly–Bray theorem, see, e.g., [16, p. 161]), we can also determine a set $\Omega '$ of probability one such that $\Omega '\subset \Omega _{r}$ for all rational r and

$$\begin{aligned} F(\omega ,r-)\ \leqslant \ \liminf _{n\rightarrow \infty }F_{k,n}(\omega ,r)\ \leqslant \ \limsup _{n\rightarrow \infty }F_{k,n}(\omega ,r)\ \leqslant \ F(\omega ,r) \end{aligned}$$

for all $(\omega ,r)\in \Omega '\times \mathbb {R}\backslash \mathbb {Q}$. Moreover $R_{0}=0$ on $\Omega '$. Next observe that

$$\begin{aligned} G_{k,n}(r)\,=\ \frac{1}{n}\sum _{j=1}^{n}\textbf{1}_{(-\infty ,r]}(S_{k,j}) \end{aligned}$$

satisfies

$$\begin{aligned} G_{k,n}(\omega ,r)\ =\ F_{k,n}(\omega ,r+Y_{k}(\omega )) \end{aligned}$$

for all $k,n\in \mathbb {N}_{0}$. It follows for all $\omega \in \Omega '$ and $k\in \mathbb {N}_{0}$ that

$$\begin{aligned} \lim _{n\rightarrow \infty }G_{k,n}(\omega ,r)\ =\ F(\omega ,r+Y_{k}(\omega )) \end{aligned}$$

if $r+Y_{k}(\omega )$ is rational, and

$$\begin{aligned} F(\omega ,r+Y_{k}(\omega )-)\ {}&\leqslant \ \liminf _{n\rightarrow \infty }G_{k,n}(\omega ,r)\\&\leqslant \ \limsup _{n\rightarrow \infty }G_{k,n}(\omega ,r)\ \leqslant \ F(\omega ,r+Y_{k}(\omega )) \end{aligned}$$

for all $r\in \mathbb {R}$. Finally defining the stationary sequence

$$\begin{aligned} Z_{k}\ :=\ \sup \left\{ r\in \mathbb {Q}:\limsup _{n\rightarrow \infty }G_{k,n}(r)<\frac{1}{2}\right\} \end{aligned}$$

for $k\in \mathbb {N}_{0}$, we have $Z_{k}=f(X_{k+1},X_{k+2},\ldots )$ for some measurable function $f:\mathbb {R}^{\times \mathbb {N}}\rightarrow \mathbb {R}\cup \{\pm \infty \}$. Furthermore, $Z_{k}=-Y_{k}$ a.s. because, if $\omega \in \Omega '$, then

$$\begin{aligned} \limsup _{n\rightarrow \infty }G_{k,n}(\omega ,r-Y_{k}(\omega ))\ =\ \limsup _{n\rightarrow \infty }F_{k,n}(\omega ,r)\ \geqslant \ F(r-)\ \geqslant \ \frac{1}{2} \end{aligned}$$

for $r>0\equiv R_{0}$, and

$$\begin{aligned} \limsup _{n\rightarrow \infty }G_{k,n}(\omega ,r-Y_{k}(\omega ))\ =\ \limsup _{n\rightarrow \infty }F_{k,n}(\omega ,r)\ \leqslant \ F(r)\ <\ \frac{1}{2} \end{aligned}$$

for $r<0$. Hence $Y_{k}=f(X_{k+1},X_{k+2},\ldots )$ a.s. and f is real-valued. $\square$

Before proceeding to the remaining proofs, we insert the $L^{p}$-version of the coboundary theorem, with a short proof and for all $p>0$. It goes back to Leonov [29] for $p=2$ and even wide-sense stationary sequences, to Lalley [28] for $p=1$ (as already mentioned), and it is stated by Aaronson and Weiss [1] for general $p\geqslant 1$.

Corollary 4.3

In the situation of the previous theorem, let $|\cdot |$ denote Euclidean norm on $\mathbb {R}^{d}$. Then, the following assertions are equivalent for any $p>0$:

• (a) $\sup _{n\geqslant 0}\mathbb {E}_{\mu }|S_{n}|^{p}<\infty$.

• (b) $\mathbb {E}_{\nu }|Y_{0}|^{p}=\int |x|^{p}\,\nu (dx)<\infty$.

We call $(S_{n})_{n\geqslant 0}$ $L^{p}$-null-homologous under these conditions.

Proof

Plainly, we must only verify that (a) implies (b). Returning to the proof of Theorem 4.1, note that $\sup _{n}\mathbb {E}_{\mu }|S_{n}|^{p}<\infty$ implies

$$\begin{aligned} \sup _{n\geqslant 1}\int |x|^{p}\ Q_{n}(dx)\ <\ \infty . \end{aligned}$$

Choosing the $n_{k}$ such that $Q_{n_{k}}{\mathop {\rightarrow }\limits ^{w}}\nu$, we can always define, on a possibly enlarged probability space, a sequence $(W_{k})_{k\geqslant 0}$ such that $\mathbb {P}_{\mu }(W_{0}\in \cdot )=\nu$, $\mathbb {P}_{\mu }(W_{k}\in \cdot )=Q_{n_{k}}$ for $k\geqslant 1$ and $W_{k}\rightarrow W_{0}$ $\mathbb {P}_{\mu }$-a.s. Now

$$\begin{aligned} \mathbb {E}_{\nu }|Y_{0}|^{p}\ =\ \mathbb {E}_{\mu }|W_{0}|^{p}\ \leqslant \ \liminf _{k\rightarrow \infty }\mathbb {E}_{\mu }|W_{k}|^{p}<\infty \end{aligned}$$

by Fatou’s lemma. $\square$

Proof of Theorem 3.1

Since “(a)$\Rightarrow \ldots \Rightarrow$(e)” is straightforward and “(e)$\Rightarrow$(c)” is ensured by Theorem 4.1, only “(c)$\Rightarrow$(a)” remains to be verified. By the aforementioned result, we have that $S_{n}=Y_{n}-Y_{0}$ a.s. for all $n\in \mathbb {Z}$ and a stationary sequence $(Y_{n})_{n\in \mathbb {Z}}$ satisfying $Y_{n}=f(X_{n+1},X_{n+2},\ldots )$ a.s. for all n and some measurable $f:(\mathbb {R}^{d})^{\times \mathbb {N}}\rightarrow \mathbb {R}^{d}$. We recall the assumption that the driving chain $(M_{n})_{n\in \mathbb {Z}}$ is ergodic and also exclude the trivial case when all $X_{n}$ and thus all $Y_{n}$ vanish almost surely.

As a first step, we show that the conditional law of $S_{n}$ given $M_{0},M_{n}$ must be a.s. degenerate for each $n\in \mathbb {N}$, thus $S_{n}=g_{n}(M_{0},M_{n})$ a.s. for some measurable $g_{n}:\mathscr {S}\times \mathscr {S}\rightarrow \mathbb {R}^{d}$.

For $x,y\in \mathscr {S}$, $n\in \mathbb {Z}$ and $t\in \mathbb {R}^{d}$, let

$$\begin{aligned} \varphi ^{x}(t)=\mathbb {E}(e^{itY_{0}}|M_{0}=x)\quad \text {and}\quad \psi _{n}^{x,y}(t)=\mathbb {E}(e^{itS_{n}}|M_{0}=x,M_{n}=y) \end{aligned}$$

be the conditional Fourier transforms (FT) of $Y_{0}$ given $M_{0}=x$ and of $S_{n}$ given $M_{0}=x,\,M_{n}=y$, respectively. Using the Markov-modulated structure, we infer that the conditional FT of $S_{kn}$ given $M_{0},M_{kn}$ satisfies

$$\begin{aligned} \psi _{kn}^{M_{0},M_{kn}}(t)\ =\ \mathbb {E}\left[ \prod _{j=1}^{k}\psi _{n}^{M_{(j-1)n},M_{jn}}(t)\bigg |M_{0},M_{kn}\right] \quad \text {a.s.} \end{aligned}$$

for all $k,n\in \mathbb {N}$. Next, if there exist $m\in \mathbb {N}$ and a subset C of $\mathscr {S}^{2}$ such that $\mathbb {P}((M_{0},M_{m})\in C)>0$ and the conditional law of $S_{n}$ given $M_{0}=x,M_{m}=y$ is nondegenerate for $(x,y)\in C$, then C may in fact be chosen in such a way that, for some $t_{0}>0$,

$$\begin{aligned} |\psi _{m}^{M_{0},M_{m}}(t_{0})|\,<\,1\quad \text {a.s. on }\{(M_{0},M_{m})\in C\} \end{aligned}$$

holds together with $\mathbb {E}|\varphi ^{M_{0}}(t_{0})|>0$. Birkhoff’s ergodic theorem then further implies that

$$\begin{aligned} \frac{1}{k}\sum _{j=1}^{k}\log |\psi _{m}^{M_{(j-1)m},M_{jm}}(t_{0})|\ \xrightarrow {k\rightarrow \infty }\ \mathbb {E}\log |\psi _{m}^{M_{0},M_{m}}(t_{0})|\ <\ 0\quad \text {a.s.} \end{aligned}$$

Consequently,

$$\begin{aligned} |\psi _{km}^{M_{0},M_{km}}(t_{0})|\ {}&=\ \left| \mathbb {E}\left[ \prod _{j=1}^{k}\psi _{m}^{M_{(j-1)m},M_{jm}}(t_{0})\Bigg |M_{0},M_{km}\right] \right| \\&\leqslant \ \mathbb {E}\left[ \exp \left( \sum _{j=1}^{k}\log |\psi _{m}^{M_{(j-1)m},M_{jm}}(t_{0})|\right) \Bigg |M_{0},M_{km}\right] \\&=\ \mathbb {E}\left[ \exp \Big (k\,\mathbb {E}\log |\psi _{m}^{M_{0},M_{m}}(t_{0})|(1+o(1))\Big )\Bigg |M_{0},M_{km}\right] \\&\xrightarrow {k\rightarrow \infty }\ 0\quad \text {a.s.} \end{aligned}$$

On the other hand, use $Y_{0}=Y_{n}-S_{n}$ and the conditional independence of $S_{n}$ and $Y_{n}=f(X_{n+1},X_{n+2},\ldots )$ given $M_{0},M_{n}$ to obtain the equation

$$\begin{aligned} \varphi ^{M_{0}}(t)\ =\ \mathbb {E}\Big [\varphi ^{M_{n}}(t)\overline{\psi _{n}^{M_{0},M_{n}}(t)}\Big |M_{0}\Big ]\quad \text {a.s.} \end{aligned}$$

for any $n\in \mathbb {N}$ (with $\overline{z}$ denoting the complex conjugate of z). But for $t=t_{0}$, this provides with the help of the dominated convergence theorem that

$$\begin{aligned} 0\ <\ \mathbb {E}|\varphi ^{M_{0}}(t_{0})|\ {}&=\ \lim _{k\rightarrow \infty }\mathbb {E}\Big |\varphi ^{M_{km}}(t_{0})\overline{\psi _{km}^{M_{0},M_{km}}(t_{0})}\Big |\\&\leqslant \ \mathbb {E}\left[ \lim _{k\rightarrow \infty }\Big |\psi _{km}^{M_{0},M_{km}}(t_{0})\Big |\right] \ =\ 0 \end{aligned}$$

which is impossible.

Having verified that $S_{n}=g_{n}(M_{0},M_{n})$ a.s. for all $n\in \mathbb {N}$ and a suitable function $g_{n}$, we particularly infer that $X_{1}=S_{1}=g_{1}(M_{0},M_{1})$, thus $X_{n}=g_{1}(M_{n-1},M_{n})$ a.s. for all $n\in \mathbb {N}$, and finally

$$\begin{aligned} Y_{n}\ =\ h(M_{n},M_{n+1},\ldots )\quad \text {a.s.} \end{aligned}$$

for all n and $h:\mathscr {S}^{\times \mathbb {N}}\rightarrow \mathbb {R}^{d}$ defined by

$$\begin{aligned} h(x_{0},x_{1},\ldots )\ =\ f(g_{1}(x_{0},x_{1}),g_{1}(x_{1},x_{2}),\ldots ). \end{aligned}$$

It follows that

$$\begin{aligned} h(M_{0},M_{1},\ldots )\ =\ Y_{0}\ =\ Y_{n}-S_{n}\ =\ H_{n}(M_{0},M_{n},M_{n+1},\ldots )\quad \text {a.s.} \end{aligned}$$

where $H_{n}(x,y_{1},y_{2},\ldots ):=f(y_{1},y_{2},\ldots )-g_{n}(x,y_{1})$, that is, $h(M_{0},M_{1},\ldots )$ is a.s. constant in $M_{1},\ldots ,M_{n-1}$. But this being true for each $n\geqslant 2$ means that $h(M_{0},M_{1},\ldots )$ must be a.s. constant in $M_{1},M_{2},\ldots$ and thus a.s. equal to $\xi (M_{0})$ for some $\xi :\mathscr {S}\rightarrow \mathbb {R}^{d}$. This completes the proof. $\square$

Proof of Theorem 3.2

We must only prove “(d)$\Rightarrow$(a)” as one can readily see. Put $(M_{n}^{A},X_{n}^{A}):=(M_{\tau _{n}(A)},S_{\tau _{n}(A)}-S_{\tau _{n-1}(A)})$ for $n\geqslant 1$ and $(M_{0}^{A},X_{0}^{A}):=(M_{0},X_{0})$. For $A\in \mathfrak {S}_{\pi }$, this sequence is Markov-modulated, ergodic with unique stationary law $\mu _{A}$ and thus stationary under $\mathbb {P}_{\mu _{A}}$. The assumed tightness of the associated MRW $(S_{\tau _{n}(A)})_{n\geqslant 0}$ implies by Theorem 3.1 that it is strict-sense null-homologous. So $X_{n}^{A}=\xi (M_{n}^{A})-\xi (M_{n-1}^{A})$ $\mathbb {P}_{\mu _{A}}$-a.s. for all $n\geqslant 1$ and a suitable function $\xi _{A}:A\rightarrow \mathbb {R}^{d}$ which is unique up to a translation by some $c\in \mathbb {R}^{d}$ as noted after the definition of strict-sense null-homology.

Since $\pi$ is $\sigma$-finite, we can pick a strictly increasing sequence $(\mathscr {S}_{m})_{m\geqslant 1}$ such that $\mathscr {S}_{m}\in \mathfrak {S}_{\pi }$ for each m and $\bigcup _{m}\mathscr {S}_{m}=\mathscr {S}$. As just pointed out, we have $S_{n}^{\mathscr {S}_{m}}=\xi _{m}(M_{n}^{\mathscr {S}_{m}})-\xi _{m}(M_{0}^{\mathscr {S}_{m}})$ $\mathbb {P}_{\mu _{\mathscr {S}_{m}}}$-a.s. for all $m,n\geqslant 1$ and functions $\xi _{m}:\mathscr {S}_{m}\rightarrow \mathbb {R}^{d}$, and we show now that these functions can be chosen such that the restriction of $\xi _{m+1}$ to $\mathscr {S}_{m}$ equals $\xi _{m}$ for each m. First note that, by the nested structure of the $\mathscr {S}_{m}$, the $S_{n}^{\mathscr {S}_{m}}$, $n\geqslant 1$, form a subsequence of $(S_{n}^{\mathscr {S}_{m+1}})_{n\geqslant 1}$. Consequently,

$$\begin{aligned} S_{n}^{\mathscr {S}_{m}}\ =\ \xi _{m}(M_{\tau _{n}(\mathscr {S}_{m})})-\xi _{m}(M_{0})\ =\ \xi _{m+1}\big (M_{\tau _{n}(\mathscr {S}_{m})}\big )-\xi _{m+1}(M_{0}) \end{aligned}$$

and thus

$$\begin{aligned} \xi _{m+1}(M_{0})-\xi _{m}(M_{0})\ =\ \xi _{m+1}\big (M_{n}^{\mathscr {S}_{m}}\big )-\xi _{m}\big (M_{n}^{\mathscr {S}_{m}}\big ) \end{aligned}$$

holds $\mathbb {P}_{\mu _{\mathscr {S}_{m}}}\text {-a.s.}$ for all $m,n\in \mathbb {N}$. This further yields

$$\begin{aligned} \xi _{m+1}(M_{0})-\xi _{m}(M_{0})\ =\ \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=1}^{n}\Big (\xi _{m+1}\big (M_{n}^{\mathscr {S}_{m}}\big )-\xi _{m}\big (M_{n}^{\mathscr {S}_{m}}\big )\Big ) \end{aligned}$$

$\mathbb {P}_{\mu _{\mathscr {S}_{m}}}\text {-a.s.}$ for any m. But the right-hand limit is measurable with respect to the invariant $\sigma$-field of the ergodic and $\mathbb {P}_{\mu _{\mathscr {S}_{m}}}$-stationary Markov chain $(M_{n}^{\mathscr {S}_{m}})_{n\geqslant 0}$, hence $\mathbb {P}_{\mu _{\mathscr {S}_{m}}}\text {-a.s.}$ equal to some $c_{m}\in \mathbb {R}^{d}$. But the $\xi _{m}$ being unique only up to translation, they may be chosen such that, for all $m\geqslant 1$, $\xi _{m+1}(s)=\xi _{m}(s)$ a.s. with respect to $\mathbb {P}_{\mu _{\mathscr {S}_{m}}}(M_{0}\in \cdot )=\pi _{\mathscr {S}_{m}}$. In other words, there exists a mapping $\xi :\mathscr {S}\rightarrow \mathbb {R}^{d}$ that $\mathbb {P}_{\mu }$-a.s. equals $\xi _{m}$ on $\mathscr {S}_{m}$ for any m and

$$\begin{aligned} S_{\tau _{1}(\mathscr {S}_{m})}\ =\ S_{1}^{\mathscr {S}_{m}}\ =\ \xi \big (M_{\tau _{1}(\mathscr {S}_{m})}\big )-\xi (M_{0})\quad \mathbb {P}_{\mu }\text {-a.s.} \end{aligned}$$

But $\tau _{1}(\mathscr {S}_{m})\rightarrow 1$ $\mathbb {P}_{\mu }$-a.s. as $m\rightarrow \infty$ finally proves $X_{1}=\xi (M_{1})-\xi (M_{0})$ and thus, by stationarity, $X_{n}=\xi (M_{n})-\xi (M_{n-1})$ $\mathbb {P}_{\mu }$-a.s. for any n. $\square$

Proof of Theorem 4.2

We must only prove “(a)$\Rightarrow$(b).” Put $p_{xy}^{n}:=\mathbb {P}_{x}(M_{n}=y)$ and let $\psi _{n}^{x,y}$ be as in the previous proof for $x,y\in \mathscr {S}$ and $n\geqslant 0$. Since $(M_{n})_{n\geqslant 0}$ is irreducible, $p_{xy}^{n(x,y)}>0$ for all $x,y\in \mathscr {S}$ and some minimal integer n(x, y). The closed-loop condition implies, with $n=n(x,y)+n(y,x)$,

$$\begin{aligned} \mathbb {P}(S_{n}=0|M_{0}=M_{n}=x)\,=\,1 \end{aligned}$$

and thereupon

$$\begin{aligned} 1\ = \ \psi _{n}^{x,x}(t)\ = \mathbb {E}\Big (\psi _{n(x,y)}^{x,M_{n(x,y)}}(t)\psi _{n(y,x)}^{M_{n(x,y)},x}(t)\Big |M_{0}=M_{n}=x\Big ) \end{aligned}$$

which in turn yields $\psi _{n(x,y)}^{x,s}(t)\psi _{n(y,x)}^{s,x}(t)=1$ for all $s\in \mathscr {S}$ (including $s=y$) such that $\mathbb {P}(M_{n(x,y)}=s|M_{0}=M_{n}=x)>0$. This shows that

$$\begin{aligned} \psi _{n(x,y)}^{x,y}(t)=e^{itg(x,y)}=e^{-itg(y,x)}=\overline{\psi _{n(y,x)}^{y,x}(t)} \end{aligned}$$

for all $t\in \mathbb {R}^{d}$ and some $g:\mathscr {S}\times \mathscr {S}\rightarrow \mathbb {R}^{d}$ which satisfies $g(x,y)=-g(y,x)$. In particular, $X_{n}=g(M_{n-1},M_{n})$ $\mathbb {P}_{s}\text {-a.s.}$ for all $s\in \mathscr {S}$ and $n\in \mathbb {N}$. Next, a similar argument using a path $x\rightarrow y\rightarrow z\rightarrow x$ of length $n=n(x,y)+n(y,z)+n(z,x)$ provides us with

$$\begin{aligned} g(x,y)+g(y,z)+g(z,x)\ =\ g(x,y)+g(y,z)-g(x,z)\ =\ 0 \end{aligned}$$

so that, fixing a reference state x and defining $\xi (y):=g(x,y)$ for all $y\in \mathscr {S}$ (thus $\xi (x)=0$), we arrive at the desired conclusion $g(y,z)=\xi (z)-\xi (y)$ for all $y,z\in \mathscr {S}$. $\square$

5 Null-Homology in Applications

This section is devoted to a small collection of instances where null-homology arises as a relevant or even crucial issue.

5.1 Fluctuation Theory for Markov Random Walks

This section aims to extend results about the fine structure of ordinary random walks with i.i.d. increments (like fluctuation-type, recurrence versus transience, ladder variables, arcsine law) to Markov random walks [4,5,6,7]. For instance, the well-known trichotomy that an ordinary random walk $(S_{n})_{n\geqslant 0}$ must be either positive divergent $(S_{n}\rightarrow \infty \text { a.s.})$, negative divergent $(S_{n}\rightarrow -\infty \text { a.s.})$, or oscillating $(\liminf _{n\rightarrow \infty }S_{n}=-\infty$ and $\limsup _{n\rightarrow \infty }S_{n}=\infty \text { a.s.})$ embarks on the simple but basic fact that the only exceptional case is the trivial one when the increments are a.s. zero. In the simplest Markov-modulated case, where the driving chain is positive recurrent on a finite or countable state space, the same trichotomy only holds when ruling out the class of all strict-sense null-homologous MRWs, see [6, Sect. 5, especially Prop. 5.4]. In other words, the exceptional class is no longer just a singleton.

Null-homology also arises in connection with the ladder epochs of a MRW $(M_{n},S_{n})_{n\geqslant 0}$. To be more precise, let the driving chain be positive Harris recurrent with stationary law $\pi$ and the stationary drift $\mathbb {E}_{\pi }S_{1}$ of the random walk be positive. Define $\sigma _{0}:=0$ and

$$\begin{aligned} \sigma _{n}:=\,\inf \{k>\sigma _{n-1}:S_{k}>S_{\sigma _{n-1}}\} \end{aligned}$$

for $n\geqslant 1$. Then, $(M_{\sigma _{n}},\sigma _{n})_{n\geqslant 0}$ is also a MRW with the same lattice-span as the MRW $(M_{n},n)_{n\geqslant 0}$ that has deterministic additive part [4, Thm. 1(ii)]. Moreover, if $(M_{n})_{n\geqslant 0}$ is d-periodic for some $d>0$ with $\mathbb {P}_{\pi }$-a.s. unique cyclic classes $\mathscr {S}_{0},\ldots ,\mathscr {S}_{d-1}$ (indexed in correct transitional order), then $(M_{n},n)_{n\geqslant 0}$ is d-arithmetic with shift function

$$\begin{aligned} \gamma :\mathscr {S}\rightarrow \{0,\ldots ,d-1\},\quad x\ \mapsto \ \sum _{r=0}^{d-1}(d-r)\textbf{1}_{\mathscr {S}_{r}}(x), \end{aligned}$$

cf. [4, Sect. 4]. In other words, the ladder epochs $\sigma _{n}$ are d-arithmetic up to the null-homologous sequence $(\gamma (M_{\sigma _{n}})-\gamma (M_{\sigma _{n-1}}))_{n\geqslant 1}$.

5.2 Null-Homologous $L^{2}$, but Not $L^{2}$ Null-Homologous

A strongly mixing stationary sequence of nondegenerate, zero-mean and pairwise uncorrelated random variables such that the associated RW fails to satisfy the central limit theorem (CLT) was given by Herrndorf [25] who states as part of his result that this RW is tight and $\gamma :=\min _{n\geqslant 1}\mathbb {P}(S_{n}=0)>0$. So it is null-homologous, and if $S_{n}=Y_{n}-Y_{0}$ a.s. for stationary $Y_{0},Y_{1},\ldots$, then furthermore $\mathbb {P}(Y_{n}=Y_{0})\geqslant \gamma$ for all $n\geqslant 1$. On the other hand, since $\mathbb {E}S_{n}^{2}=n\,\mathbb {E}X_{1}^{2}>0$ for all n and thus $\sup _{n\geqslant 1}\mathbb {E}S_{n}^{2}=\infty$, the $Y_{n}$ cannot be square-integrable by Corollary 4.3. This was also pointed out in [13] and shows that an $L^{2}$-sequence can be null-homologous, but not $L^{2}$ null-homologous.

5.3 Poisson Equation

Let $(M_{n})_{n\geqslant 0}$ be an ergodic Markov chain with transition kernel P and stationary law $\pi$. Then, any pair (f, g) of real-valued $L^{1}(\pi )$-functions satisfying the Poisson equation $g=f+Pg$ can be associated with a RW, namely

$$\begin{aligned} S_{n}(f)\ :=\ \sum _{k=1}^{n}f(M_{k}),\quad n\geqslant 0, \end{aligned}$$

(9)

Under $\mathbb {P}_{\pi }$, this RW is also a martingale with stationary increments up to a null-homologous sequence. To see this, just notice that $S_{n}(f)=W_{n}+R_{n}$ with martingale part

$$\begin{aligned} W_{n}\,:=\,\sum _{k=1}^{n}\big (g(M_{k})-Pg(M_{k-1})\big ) \end{aligned}$$

and null-homologous part

$$\begin{aligned} R_{n}\,:=\,Pg(M_{0})-Pg(M_{n}) \end{aligned}$$

for $n\geqslant 0$. Provided that $f,g\in L^{2}(\pi )$, Gordin and Lifšic [23] showed that this allows to derive a CLT for $n^{-1/2}S_{n}(f)$. Namely, since null-homology implies $n^{-1/2}R_{n}\rightarrow 0$ in probability by Theorem 4.1, the problem reduces to a CLT for the martingale part which may be found, e.g., in [24]. The same approach was used by Benda [9] and by Wu and Woodroofe [40] to establish CLT’s for certain contractive iterated function systems. It was pointed out by Woodroofe [39] that, given $f\in L^{2}(\pi )$, a solution g to Poisson’s equation does indeed exist if

$$\begin{aligned} g_{n}:=\,\sum _{k=1}^{n}P^{k}f\ \xrightarrow {n\rightarrow \infty }\ g\quad \text {in }L^{2}(\pi ), \end{aligned}$$

and that this condition is also necessary if the doubly infinite extension of $(M_{n})_{n\geqslant 0}$ has trivial left tail $\sigma$-field.

When there is no solution g, a perturbed version of the Poisson equation, viz. $(1+\varepsilon )g_{\varepsilon }=f+Pg_{\varepsilon }$ for $\varepsilon >0$, can be considered instead and has unique solution

$$\begin{aligned} g_{\varepsilon }\,=\,\sum _{n\geqslant 1} \frac{P^{n-1}f}{(1+\varepsilon )^{n}}\,\in \, L^{2}(\pi ) \end{aligned}$$

if $f\in L^{2}(\pi )$. This was done by Kipnis and Varadhan [26] (for reversible P, see also the next subsection) and by Maxwell and Woodroofe [31]. Further extensions of the CLT for RWs $S_{n}(f)$ as defined in (9) were obtained by Derriennic and Lin [17,18,19] by building upon a fractional version of the Poisson equation which actually also leads to a fractional notion of null-homology (by them called fractional coboundary). We refrain from giving further details.

6 Null-Homology in the Polaron Problem

Let $\Omega _0$ denote the space of continuous functions $\omega :\mathbb R\rightarrow \mathbb R^d$ vanishing at the origin. For any $t\in \mathbb {R}$, let $\theta _t:\Omega _0\rightarrow \Omega _0$ be the shift defined by $(\theta _t\omega )(\cdot )=\omega (t+\cdot )-\omega (t)$ and denote by ${\mathcal {M}}_{\textrm{si}}(\Omega _0)$ the space of $\theta _{t}$-invariant probability measures on $\Omega _0$, or the space of processes with stationary increments. There is also an action on $\Omega _0\otimes \mathbb R^{d}$, by slight abuse of notation again denoted $\theta _{t}$ and defined by $\theta _t(\omega ,x)=(\omega (t+\cdot )-\omega (t), x+ \omega (t))$. Then, let $\mathcal M_{\textrm{s}}(\Omega _0\otimes \mathbb {R}^d)$ denote the space of $\theta _t$-invariant probability measures on $\Omega _0\otimes \mathbb {R}^d$, or the space of stationary processes. Given any $\mathbb Q\in \mathcal M_{\mathrm s}(\Omega _0\otimes \mathbb {R}^d)$, its “first marginal" $\mathbb Q^{{\scriptscriptstyle {({1}})}}$ can be defined as follows: Let $\Omega =\{\omega : \mathbb {R}\rightarrow \mathbb {R}^d: \omega (\cdot ) \,\,\textrm{continuous}\}$ stand for all $\mathbb {R}^d$-valued continuous functions on $\mathbb {R}$, which, equipped with the topology of uniform convergence on bounded intervals, forms a complete separable metric space. The Borel $\sigma$-field over $\Omega$, denoted by $\mathcal F$, is generated by the sets $\{\omega (t): -\infty<t<\infty \}$. Then, $\Omega$ can be identified with $\Omega _0\otimes \mathbb R^d$ by mapping

$$\begin{aligned} \Omega \ni \omega \leftrightarrow (\omega ^\prime ,a), \quad \text{ where }\quad a=\omega (0)\text { and }\omega ^\prime (t)=\omega (t)-\omega (0). \end{aligned}$$

Thus, any probability measure $\mathbb Q \in \mathcal M_{\textrm{s}}(\Omega )$ on $\Omega$ can be viewed as a measure on $\Omega _0\otimes \mathbb R^d$ and will then have marginals ${\mathbb Q}^{{\scriptscriptstyle {({1}})}} \in \mathcal M_{\textrm{si}}(\Omega _0)$, ${\mathbb Q}^{{\scriptscriptstyle {({2}})}} \in \mathcal M_{1}(\mathbb R^d)$, respectively. The first marginal ${\mathbb Q}^{{\scriptscriptstyle {({1}})}}$ is now just the law of the increments of a process that has distribution $\mathbb Q$ on $\mathcal F$.

In this context, the issue of null-homology appeared in [33] while identifying the strong coupling limit of Polaron path measures. The following result is quoted from there (in slightly adapted form) and provides an integral criterion for null-homology in the sense that it characterizes all $\mathbb Q'\in \mathcal M_{\textrm{si}}(\Omega _0)$ which are not the increment law of some $\mathbb Q\in \mathcal M_{\textrm{s}}(\Omega _0\otimes \mathbb R^d)$. We stress that this criterion does not need any tightness and is formulated in terms of convergence of integrals of continuous functions vanishing at infinity w.r.t. measures on the function space $\Omega _0$. Therefore, the criterion may be combined with our Theorem 3.1 to prove tightness where this cannot be verified directly.

Theorem 6.1

[33, Theorem 3.1] Let $\mathbb Q'$ be the law of an ergodic process with stationary increments, i.e., $\mathbb Q'\in \mathcal M_{\textrm{si}}(\Omega _0)$ is a $\theta _{t}$-invariant and ergodic probability law on $\Omega _{0}$. Then, either

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbb E^{\beta }\bigg [\varepsilon \int _0^\infty \mathrm e^{-\varepsilon t} V(\omega (t)-\omega (0)) \textrm{d}t\bigg ]=0 \end{aligned}$$

(10)

for all continuous functions $V: \mathbb R^d\rightarrow \mathbb {R}$ with $\lim _{|x|\rightarrow \infty } |V(x)|=0$, or there is a $\theta _{t}$-invariant probability law $\mathbb Q\in \mathcal M_{\mathrm s}(\Omega _0\otimes \mathbb {R}^d)$ such that $\mathbb Q'={\mathbb Q}^{{\scriptscriptstyle {({1}})}}$, i.e., $\mathbb Q$ is the law of a null-homologous stationary process on $\Omega$ with associated increment law $\mathbb Q'$.

We finally note that, since $\Omega _0$ is not even locally compact, there is no ordinary notion of vague convergence of measures on this space (determined by convergence of integrals w.r.t. continuous functions vanishing at infinity). A notion of wea-gue convergence on measures on $\Omega _0\otimes \mathbb R^d$ was therefore formulated in [33, Sect. 2.1] that is conceptually important for the proof of the above result.

7 Stochastic Homogenization in the Random Conductance Model

Part of the subsequent considerations may be viewed as a special instance of what has just been discussed before.

The notion of a corrector plays an important rôle in the context of stochastic homogenization of a random media. We will describe the setup and how null-homology comes into play for a particular instance of a random walk in random environment (RWRE) in the reversible setup, known as the random conductance model. Let

$$\begin{aligned} E_d\ =\ \big \{(x,y):|x-y|=1, \, x,y \in \mathbb {Z}^{d}\big \} \end{aligned}$$

be the set of nearest neighbor bonds in $\mathbb {Z}^{d}$ and $\Omega = [a,b]^{E_d}$ for any two fixed numbers $0<a<b$. We assume that $\Omega$ is equipped with the product $\sigma$-field $\mathcal B$ and carries a probability measure $\mathbb {P}$. For simplicity, we also assume that the canonical coordinates are i.i.d. variables under $\mathbb {P}$. Note that any $x\in \mathbb {Z}^{d}$ acts on $(\Omega ,\mathcal B,\mathbb {P})$ as a $\mathbb {P}$-preserving and ergodic transformation $\tau _{x}$, defined as the canonical translation

$$\begin{aligned} \Omega \ \ni \ \omega (\cdot )\,\mapsto \,\omega (x+\cdot ). \end{aligned}$$

For any $\omega \in \Omega$ and an associate family of probability measures $(\mathbb {P}_{\omega ,x})_{x\in \mathbb {Z}^{d}}$, let then $(S_{n})_{n\geqslant 0}$ be the RWRE on $\mathbb {Z}^{d}$ which is defined as a Markov chain such that $\mathbb {P}_{\omega ,x}(S_{0}=x)=1$ and transition probabilities are given by

$$\begin{aligned} \begin{aligned}&\mathbb {P}_{\omega ,x}(S_{n+1}=y+e|S_{n}=y)\ =\ \pi _\omega (y,y+e)\\&\quad :=\ \frac{\omega ((y,y+e))}{\sum _{|e^\prime |=1} \omega ((y,y+e^\prime ))}\ =\ \pi _{\tau _{y}\omega }(0,e) \end{aligned} \end{aligned}$$

(11)

for any e with $|e|=1$ and $x\in \mathbb {Z}^{d}$. Furthermore, the sequence

$$\begin{aligned} M_{n}:=\,\tau _{S_{n}}\omega \quad \text {for }n\geqslant 0, \end{aligned}$$

with initial state $\omega$ and taking values in the “environment space” $\Omega$, is also a Markov chain which drives the $S_{n}$, i.e., $(M_{n},S_{n})_{n\geqslant 0}$ constitutes a MRW. It has transition kernel P defined by

$$\begin{aligned} (Pf)(\omega )\,=\ \sum _{|e|=1} \pi _\omega (0,e) f(\tau _{e} \omega ) \end{aligned}$$

for all measurable, bounded f, is called the environment seen from the moving particle, or just the environmental process, and is particularly useful in the following scenario: Suppose there exists a probability density $\phi \in L^1(\mathbb {P})$ (i.e., $\phi \geqslant 0$ and $\int \phi \,\textrm{d}\mathbb {P}=1$) such that $\mathbb {Q}=\phi \,\mathbb {P}$ is P-invariant, i.e.,

$$\begin{aligned} \langle Pf, \phi \rangle _{L^{2}(\mathbb {P})}=\langle f, \phi \rangle _{L^{2}(\mathbb {P})}, \end{aligned}$$

(12)

for all bounded and measurable f or, equivalently,

$$\begin{aligned} L^\star \phi \,=\,0,\quad L^{\star }\text { the dual of }L\,=\,\textrm{Id}-P. \end{aligned}$$

It can be shown, see [26, 27, 34] and also [12, Theorem 1.2], that such an invariant density $\phi$, if it exists, is necessarily unique. Moreover, $\mathbb {P}$ and $\mathbb {Q}$ are then equivalent measures (that is, having the same null sets) and $(M_{n})_{n\geqslant 0}$ as well as $(M_{n},X_{n})_{n\geqslant 1}$ (by Markov-modulation) are ergodic processes in equilibrium (under initial law $\mathbb {Q}$), where as usual $X_{n}=S_{n}-S_{n-1}$.

In the random conductance model with transition probabilities (11), the invariant density $\phi$ can actually be found by solving the detailed balance equations (reversibility), viz.

$$\begin{aligned} \phi (\omega )\,=\,\frac{1}{C} \sum _{|e|=1} \omega ((0,e)),\quad \text {where}\quad C\,=\,\int \sum _{|e|=1}\omega ((0,e))\ \mathbb {P}(\textrm{d}\omega ). \end{aligned}$$

Reversibility also provides that P is self-adjoint on $L^{2}(\mathbb {Q})$, that is

$$\begin{aligned} \langle f, P g\rangle _{L^{2}(\mathbb {Q})}\ =\ \langle P f,g\rangle _{L^{2}(\mathbb {Q})} \end{aligned}$$

(13)

for all bounded and measurable functions f, g.

Returning to the RWRE $(S_{n})_{n\geqslant 0}$ under $\mathbb {P}_{\omega ,0}$, the ergodicity of the Markov-modulated sequence $(M_{n},X_{n})_{n\geqslant 0}$ under $\mathbb {Q}$ easily provides that $S_{n}/n \rightarrow 0$ $\mathbb {P}_{\omega ,0}$-a.s. for $\mathbb {P}$-almost all $\omega$. To see this, let

$$\begin{aligned} \textrm{d}(\omega ,x)\ =\ \mathbb {E}_{\omega ,x}X_{1}\ =\ \sum _{|e|=1}e\pi _{\omega }(x,x+e)\ =\ \textrm{d}(\tau _{x}\omega ,0) \end{aligned}$$

denote the local drift at x under $\mathbb {P}_{\omega ,0}$. As $(S_{n})_{n\geqslant 0}$ has stationary ergodic increments under initial law $\mathbb {Q}$, Birkhoff’s ergodic theorem implies

$$\begin{aligned} \frac{S_{n}}{n}\ \xrightarrow {n\rightarrow \infty }\ \int \mathbb {E}_{\omega ,0}X_{1}\ \mathbb {Q}(\textrm{d}\omega )\ =\ \int \textrm{d}(\omega ,0)\ \mathbb {Q}(\textrm{d}\omega )\ =\ 0 \end{aligned}$$

for $\mathbb {Q}$-almost all and thus $\mathbb {P}$-almost all $\omega$ (as $\mathbb {P},\mathbb {Q}$ are equivalent), the right-hand side being 0 by reversibility (recall (13)) and the definition of $\mathbb {Q}$.

As will be explained next, stochastic homogenization comes into play and leads to the notion of a corrector, when turning to the derivation of an almost sure CLT (or an invariance principle) for the law of $S_{n}$ under the quenched measure $\mathbb {P}_{\omega ,0}$. Note that $Z_{n}= S_{n}-S_{0}-\sum _{j=0}^{n-1}\textrm{d}(\omega ,S_{j})$, $n\geqslant 0$, is a $\mathbb {P}_{\omega ,0}$-martingale with bounded increments (uniformly in $\omega$). Moreover, the local drift $\textrm{d}$ is bounded and thus particularly $\in L^{2}(\mathbb {P})$. For any fixed $\varepsilon >0$, let $g_{\varepsilon }=\sum _{n\geqslant 1} \frac{P^{n-1}\textrm{d}}{(1+\varepsilon )^{n}}$ be the $L^{2}(\mathbb {P})$-solution to the perturbed Poisson equation $\big ((1+\varepsilon )\text {Id}-P\big )g_{\varepsilon }=\textrm{d}$ that was also mentioned at the end of Sect. 5.3. Putting

$$\begin{aligned} G_{\varepsilon }(\omega ,e):=\,(\nabla _e g_{\varepsilon })(\omega )=g_{\varepsilon }(\tau _e\omega )-g_{\varepsilon }(\omega ) \end{aligned}$$

for any e with $|e|=1$, Kipnis and Varadhan [26, Theorem 1.3] showed that

$$\begin{aligned} G_{\varepsilon }(\cdot ,e)\circ \tau _{x}\ \xrightarrow {L^{2}(\mathbb {P})}\ G(\cdot ,e)\circ \tau _{x}\quad \text {as }\varepsilon \downarrow 0 \end{aligned}$$

for any $x\in \mathbb {Z}^{d}$, where G is a (divergence free) gradient field, i.e., it satisfies the closed loop condition

$$\begin{aligned} \sum _{j=0}^{n-1}G(\tau _{s_{j}}\omega , {s_{j+1}-s_{j}})\ =\ 0\quad \mathbb {P}\text {-a.s.} \end{aligned}$$

(14)

for any closed path $s_{0}\rightarrow s_{1}\rightarrow \cdots \rightarrow s_{n}=s_{0}$ in $\mathbb {Z}^{d}$. The last property allows us to define the corrector corresponding to G as

$$\begin{aligned} V_{G}(\omega ,x)\,:=\,\sum _{j=0}^{n-1} G(\tau _{s_{j}}\omega , {s_{j+1}-s_{j}}) \end{aligned}$$

(15)

along any path $0\rightarrow s_{1}\rightarrow \cdots \rightarrow s_{n-1}\rightarrow s_{n}=x$, the particular choice of the path being irrelevant because of (14). It also follows that $V_{G}$ has stationary and $L^{2}$-bounded gradient in the sense that

$$\begin{aligned} V_{G}(\omega ,y)-V_{G}(\omega ,x)\ =\ V_{G}(\tau _{x}\omega ,y-x)\quad \text {for all }x,y\in \mathbb {Z}^{d} \end{aligned}$$

and

$$\begin{aligned} \sup _{x\in \mathbb {Z}^{d}}\Vert V_{G}(\cdot ,x+e)-V_{G}(\cdot ,x)\Vert _{L^{2}(\mathbb {P})}\,<\,C, \end{aligned}$$

respectively. Furthermore, fixing any $\omega \in \Omega$, the sequence

$$\begin{aligned} (M_{n},V_{G}(\omega ,S_{n}))_{n\geqslant 0} \end{aligned}$$

forms a MRW under $\mathbb {P}_{\omega ,0}$ whose driving chain is irreducible on the discrete state space $\Omega _{\omega }=\{\omega (x+\cdot ):x\in \mathbb {Z}^{d}\}$. The Markov-additive structure can be assessed by using (15), which provides

$$\begin{aligned} V_{G}(\omega ,S_{n})\ =\ \sum _{j=1}^{n}G(\tau _{S_{j-1}}\omega ,X_{j})\ =\ \sum _{j=1}^{n}G(M_{j-1},X_{j}) \end{aligned}$$

for each $n\geqslant 0$, and by (14) it also shows validity of the closed-loop condition. Hence, by invoking Theorem 4.2, we infer that

$$\begin{aligned} V_{G}(\omega ,S_{n})\ =\ \xi (M_{n})-\xi (M_{0})\ =\ \xi (\tau _{S_{n}}\omega )-\xi (\omega )\quad \mathbb {P}_{\omega ,0}\text {-a.s.} \end{aligned}$$

(16)

for all $n\geqslant 0$ and a function $\xi :\Omega _{\omega }\rightarrow \mathbb {R}^{d}$. But this being true for each $\omega$, the function $\xi$ can be defined on the whole set $\Omega$ (in a measurable way) giving that (16) holds $\mathbb {Q}$-a.s. In other words, the contractor is strict-sense null-homologous and thus tight, and it has a stationary version under $\mathbb {Q}$, namely $\xi (\omega )+V_{G}(\omega ,S_{n})$ for $n\geqslant 0$. Although pointed out by Gloria [20, p. 4] that stationary correctors in $L^{2}$ do not exist in dimension $d=1$ and $d=2$, this does not contradict our assertion. It rather implies that $\xi (\omega )$ cannot be square-integrable under $\mathbb {Q}$. In dimension $d\geqslant 3$, $L^{2}$-stationary correctors may exist under some additional conditions. Also the tightness is known in that case, see [8, 22].

Now use that $x\mapsto V_{G}(\omega ,x)+x$ is harmonic with respect to the transition probabilities (11) for $\mathbb {P}$-almost all $\omega$ to infer that $(S_{n}+V_{G}(\cdot ,S_{n}))_{n\geqslant 0}$ is a martingale with respect to $\mathbb {P}_{\omega ,0}$. The corrector $V_{G}$ therefore expresses the “distance” (or the deformation) of the martingale from the random walk $(S_{n})_{n\geqslant 0}$ itself. The tightness ensures that the contribution of this deformation grows at most sub-linearly at large distances (i.e., $\sup _{|x|\le n} n^{-1} V_{G}(x,\cdot )\xrightarrow {n\rightarrow \infty } 0$ a.s.) whence, by the martingale CLT, the laws $\mathbb {P}_{\omega ,0}(S_{n}/\sqrt{n}\in \cdot )$ converge weakly to a Gaussian law for almost every $\omega$, see [10, 30, 38] for a detailed recount of the substantial progress made in this direction.