Analysis of Correlation Bounds for Uniformly Expanding Maps on [0, 1]

Abstract

In this paper, we provide the decay of correlations for random dynamical systems. Precisely, we consider the uniformly ${\mathcal{C}}^2$ piecewise expanding maps defined on the unit interval satisfying $\lambda \left(T_{\omega }^{\prime }\right)=inf\left|T_{\omega }^{\prime }\right|>2$. As a principal tool of these studies, we use a coupling method for analyzing the coupling time of observables with bounded variation.

1. Introduction

Random dynamical systems (RDSs) are very important and are used to model many phenomena in economics (the evolution of economic processes subject to exogenous shocks [1,2]), in climatology (the modelling of ocean systems and weather [3]), in physics (multi-body celestial mechanics [4]), in biology (gene expression can be modeled using random dynamical systems [5]), etc.

In this current paper, we consider random dynamical systems generated by a finite family ${\left\{T_{\omega }\right\}}_{\omega \in \Omega }$ of uniformly ${\mathcal{C}}^2$ piecewise expanding maps on $[0,1]$ into itself, satisfying the condition $\lambda \left(T_{{\omega }_i}^{\prime }\right)=inf\left|T_{{\omega }_i}^{\prime }\right|>2$, where $(\Omega ,\mathbb{P})$ is a probability space.

Our goal is to provide the quenched decay of correlations for our random dynamical systems. Precisely, we want to analyze and estimate the following quantities:

$$\forall n\ge 0,\parallel {\mathcal{L}}_{{\omega }_n}\dots {\mathcal{L}}_{{\omega }_1}({\varphi }_1-{\varphi }_2){\parallel }_{L^1\left(m\right)},$$

(1)

$$\begin{array}{c}\hfill \forall n\ge 0,|\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}d\nu |,\end{array}$$

(2)

$$\forall n\ge 0,|\int g_1·{\mathcal{Q}}^n{g}_{2}d\mu -\int g_{1}d\mu \int g_{2}d\mu |,$$

(3)

where ${\varphi }_1$, ${\varphi }_2$ are two probability density functions belonging to the $BV$ space, $g_1\in BV$, $g_2\in L^{\infty }\left(m\right)$ and $\nu =g_{1}m$ is a measure of probability according to a Lesbuge measure m. The analysis and estimation of quantities in Equations (1)–(3) represent the analysis of the random compositions $T_{\omega }^n:=T_{{\theta }^{n-1}\omega }\circ \cdots \circ T_{\theta \omega }\circ T_{\omega }$, where $\theta$ is a shift map from ${\Omega }^{\mathbb{N}}$ into itself given by the following: for all $\underline{\omega }=({\omega }_0,{\omega }_1,{\omega }_2,\dots )\in {\Omega }^{\mathbb{N}}$, $\theta ({\omega }_0,{\omega }_1,{\omega }_2,\dots )=({\omega }_1,{\omega }_2,\dots )$. Our analysis is based on the duality relation between our random dynamical system $T_{\omega }$ and the transfer operator ${\mathcal{L}}_{\omega }$ given by

$$\int {\varphi }_1({\varphi }_2\circ T_{\omega })dm=\int \left({\mathcal{L}}_{\omega }{\varphi }_1\right){\varphi }_{2}dm,\forall {\varphi }_1\in L^1\left(m\right),{\varphi }_2\in L^{\infty }\left(m\right),$$

and the Lasota–Yorke inequality given by the following: for all $n\ge 1$, there exist $n\ge 1$, $\rho <1$ and a constant $C>0$

$$\parallel {\mathcal{L}}^n{\phi \parallel \le \rho \parallel \phi \parallel +C\parallel \phi \parallel }_{L^1\left(m\right)},\varphi \in \mathcal{B}$$

where $(\mathcal{B},\parallel .\parallel )$ is a Banach space. This inequality is very important and used to prove the existence of invariant measures for RDSs (see [6]). Also, we use a practical method called the coupling method. It is easy to use for studying the statistical properties of our random dynamical system. We start with two arbitrary density probabilities belong to the cone

$${\mathcal{H}}_K^{″}=\left\{\varphi :[0,1]\to {\mathbb{R}}_{+}\in \mathrm{BV},{\int }_0^1\varphi \left(x\right)dx=1,\mathit{TV}\left(\varphi \right)\le K^{″}\right\},$$

and we apply ${\mathcal{L}}_{\omega }^n:={\mathcal{L}}_{{\omega }_n}\circ \dots \circ {\mathcal{L}}_{{\omega }_1}$ to ${\varphi }_1$ and ${\varphi }_2$. Next, we couple the $\kappa$-fractions ${\mathcal{L}}_{\omega }^n{\varphi }_1$ and ${\mathcal{L}}_{\omega }^n{\varphi }_2$, and these remain bounded by 2$\kappa$. In addition, we continue the coupling and treat $({\mathcal{L}}_{\omega }^n{\varphi }_1-\kappa )/(1-\kappa )$ and $({\mathcal{L}}_{\omega }^n{\varphi }_2-\kappa )/(1-\kappa )$ as the initial densities belonging to the cone ${\mathcal{H}}_K^{\prime }$, where $K^{\prime }$ is a positive constant. This method is used to find the statistical properties for random dynamical systems (for more details concerning the coupling method, see [7,8,9]).

In this work, our contribution is providing the upper bounds of the tree quantities in Equations (1)–(3) that are given by the following: for all $n\ge 1$,

$$\begin{array}{cc}\hfill \parallel {\mathcal{L}}_{{\omega }_n}\dots {\mathcal{L}}_{{\omega }_1}({\varphi }_1-{\varphi }_2){\parallel }_{L^1\left(m\right)}& \le C\left(\omega \right)max\left(\parallel {\varphi }_1{\parallel }_{BV},{\parallel {\varphi }_2\parallel }_{BV}\right){\alpha }^n,\hfill \\ \hfill |\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}d\nu |& \le C\left(\omega \right)\parallel g_1{\parallel }_{BV}{\parallel f\parallel }_{BV}{\parallel g\parallel }_{\infty }{\alpha }^n,\hfill \\ \hfill |\int g_1·{\mathcal{Q}}^n{g}_{2}d\mu -\int g_{1}d\mu \int g_{2}d\mu |& \le {C\left(\omega \right)\parallel f\parallel }_{BV}{\parallel g\parallel }_{\infty }{\alpha }^n.\hfill \end{array}$$

Many scientific papers deal with the statistical properties for random dynamical systems, like the central limit theorem (CLT) [10,11,12,13,14], the almost-sure invariance principle (ASIP) [15], the large deviations principle (LDP) [16], the law of the iterated logarithm (LIL) [17], memory loss (ML) [18] and decay of correlations (DC). For example, Liverani studied the DC for piecewise expanding maps in [19]. Also, Buzzi proved that the DC is an exponential decay for random Lasota–Yorke maps in [20]. In addition, Sulku used the coupling method to prove an explicit correlation bounds for expanding circle maps in [9]. Finally, Stenlund and Sulku analyzed the random circle maps expanding on the average using the coupling method. They proved two statistical properties: (ASIP) and (DC) for the ${\mathcal{C}}^2$ non-uniformly expanding maps for observables belonging to ${\mathcal{C}}^{\alpha }$, where $\alpha \in (0,1)$.

Our paper is organized as follows. In Section 2, we present our random dynamical system and define mathematical preliminaries like the Markov chain, the skew product, the transfer and the Markov operators. In Section 3, we state our main results. In Section 4, we establish the regularity and take some proprieties of the push-forward densities. In Section 5, we use the coupling argument for analyzing the coupling time. In Section 6, we discuss our results and give some comparisons with other preceding results. In Section 7, we give the proofs of our main results. We finish with some conclusions about our work and some perspectives about future work.

2. Random Dynamical Systems

In this section, we introduce our random dynamical system given by $T_{\omega }$. Then, we define the transfer operator and the Markov operator corresponding to the Markov chain $X_n$. Finally, we finish with the definition of the skew product.

In what follows, the Lebesgue measure on $\mathbb{R}$ is denoted by m. On a finite probability space $(\Omega ,\mathbb{P})$, we consider the following:

• $\mathbb{P}={\left\{p_{\omega }\right\}}_{\omega \in \Omega }$ as a probability vector;
• ${\left\{T_{\omega }\right\}}_{\omega \in \Omega }$ as a finite family of uniformly ${\mathcal{C}}^2$ piecewise expanding maps $T_{\omega }:[0,1]\to [0,1]$, such that the derivatives satisfy

  $$\lambda \left(T_{{\omega }_i}\right):=\underset{x\in [0,1]}{inf}\left|T_{{\omega }_i}^{\prime }\right|>2.$$

The transfer of the operator ${\mathcal{L}}_{{\omega }_i}$ into $L^1\left(m\right)$ to itself is given by

$${\mathcal{L}}_{{\omega }_i}g\left(x\right)=\sum _{y\in T_{{\omega }_i}^{-1}\left\{x\right\}}\frac{g\left(y\right)}{|T_{{\omega }_i}^{\prime }\left(y\right)|},g\in L^1\left(m\right).$$

Let $\mathcal{E}$ be a set defined by

$$\mathcal{E}:=\{T_{{\omega }_i}:[0,1]\to [0,1],\mathrm{s}.\mathrm{t}.sup|T_{{\omega }_i}^{\prime }\left(x\right)|<\infty \}.$$

We define the following quantities:

$$\left\{\begin{array}{c}A=\underset{x\in [0,1]}{sup}\frac{T_{{\omega }_i}^{″}\left(x\right)}{|T_{{\omega }_i}^{\prime }{\left(x\right)|}^2}+2\underset{I\in {\mathcal{A}}_0}{sup}\frac{{sup}_{x\in I}{\left|T_{{\omega }_i}^{\prime }\left(x\right)\right|}^{-1}}{\left|I\right|}\hfill \\ \eta \left(T_{{\omega }_i}\right)=\frac{2}{\lambda \left(T_{{\omega }_i}\right)}\hfill \end{array}\right.$$

where ${\mathcal{A}}_0$ is a finite partition of $[0,1]$$\left(mod0\right)$.

Consider

$$T_{\underline{\omega }}^n=T\left(\underline{\omega }\right)=T({\omega }_1,{\omega }_2,\dots )=T_{{\omega }_n}\circ \dots \circ T_{{\omega }_1}.$$

${\left(X_n\right)}_{n\in \mathbb{N}}$ is the corresponding Markov chain given by

$$X_0\stackrel{d}{=}\mu ,X_{n+1}=T_{{\omega }_{n+1}}\left(X_n\right),$$

where $\mu$ is probability measure on $[0,1]$ and ${\left({\omega }_n\right)}_n$ is a sequence of i.i.d. random variables with common law $\mathbb{P}$. The Markov operator $\mathcal{Q}$ associated to ${\left(X_n\right)}_{n\in \mathbb{N}}$ is given by

$$\mathcal{Q}f\left(x\right)={\int }_{\Omega }f\left(T_{{\omega }_1}x\right)d\mathbb{P}\left({\omega }_1\right),f\in L^{\infty }\left(m\right).$$

The stationarity property between $X_n$ and $\mu$ is given by

$$\int \mathcal{Q}fd\mu =\int fd\mu ,\forall f\in L^{\infty }\left(m\right).$$

In this case, the distribution of $X_n$ follows the measure $\mu$ for all $n\ge 0$.

The skew product is a deterministic dynamical system defined by

$$\begin{array}{ccccc}S& :& {\Omega }^{\mathbb{N}}\times [0,1]& \to & {\Omega }^{\mathbb{N}}\times [0,1]\\ & & (\underline{\omega },x)& \mapsto & (\theta \underline{\omega },T_{{\omega }_1}x)\end{array}$$

and related to $X_n$, where the shift map $\theta$ is given by

$$\begin{array}{ccccc}\theta & :& {\Omega }^{\mathbb{N}}& \to & {\Omega }^{\mathbb{N}}\\ & & ({\omega }_0,{\omega }_1,{\omega }_2,\dots )& \mapsto & ({\omega }_1,{\omega }_2,\dots )\end{array}$$

Observe the relation

$$S^n(\underline{\omega },x)=({\theta }^n\underline{\omega },T_{\underline{\omega }}^{n}x),\mathrm{for} \mathrm{any} n\ge 0.$$

A probability measure $\mu$ on $[0,1]$ is stationary if and only if the measure ${\mathbb{P}}^{\otimes \mathbb{N}}\otimes \mu$ is invariant under S.

The total variation of $f:[0,1]\to \mathbb{C}$ on an interval $[c,b]\subset [0,1]$ is given by

$$\mathit{TV}\left(f\right)=sup\{\sum _{i=0}^n|f\left(a_{i+1}\right)-f\left(a_i\right)|:n\ge 1,c=a_0\le \dots \le a_n=b\}.$$

The usual $L^1\left(m\right)$ norm of g with respect to the Lebesgue measure m is

$${\parallel g\parallel }_1={\int }_0^1\left|g\left(x\right)\right|dm\left(x\right).$$

The function space with bounded variation is

$${BV:=\{f:[0,1]\to R:\parallel g\parallel }_{BV}<+{\infty \},\parallel g\parallel }_{BV}=\mathit{TV}\left(g\right)+{\parallel g\parallel }_1$$

In the sequel, we may assume one of the following hypotheses:

Hypothesis 1.

For any non-trivial subinterval $I\subset [0,1]$, there exists $n\ge 1$ such that $T_{\underline{\omega }}^n\left(I\right)=[0,1]$ for almost every $\underline{\omega }\in {\Omega }^n$.

Hypothesis 2.

$T_{{\omega }_i}\in \mathcal{E}$ for all ${\omega }_i\in \Omega$.

3. Main Results

In this section, we state our main results. They result from two theorems. Theorem 1 presents two very important results. In the first part, the ${\parallel .\parallel }_{L^1\left(m\right)}$-norm of the difference between ${\mathcal{L}}_{\omega }^n{\varphi }_1$ and ${\mathcal{L}}_{\omega }^n{\varphi }_2$ converges to zero exponentially. In the second part of Theorem 1, the quantity defined in Equation (2) is bounded. Theorem 2 shows that the decay of correlations of quantity in Equation (3) is exponential correlation with respect an arbitrary invariant measure $\mu$.

Theorem 1.

There exists $\alpha \in (0,1)$, which the two following properties hold.

(1) 

For almost every ω, there exists $C\left(\omega \right)>0$, and one has

$$\forall n\ge 0,\parallel {\mathcal{L}}_{{\omega }_n}\dots {\mathcal{L}}_{{\omega }_1}({\varphi }_1-{\varphi }_2){\parallel }_{L^1\left(m\right)}\le C\left(\omega \right)max\left(\parallel {\varphi }_1{\parallel }_{BV},{\parallel {\varphi }_2\parallel }_{BV}\right){\alpha }^n,$$

(4)

where ${\varphi }_1,{\varphi }_2\in BV$ are two probability density functions.

(2) 

For a probability measure $\nu =\varphi m$, where $\varphi \in BV$, we have

$$\begin{array}{c}\hfill \forall n\ge 0,|\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}d\nu |\le {C\left(\omega \right)\parallel \varphi \parallel }_{BV}\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }{\alpha }^n,\end{array}$$

(5)

for all complex-valued functions $g_1\in BV$ and $g_2\in L^{\infty }\left(m\right)$.

The next theorem presents the upper bound for quantity Equation (2) when we consider an arbitrary invariant measure $\mu$ with density function $\varphi \in BV$.

Theorem 2.

There exist two constants $\alpha \in (0,1)$ and $C>0$ such that

$$\forall n\ge 0,|\int g_1·{\mathcal{Q}}^n{g}_{2}d\mu -\int g_{1}d\mu \int g_{2}d\mu |\le C\left(\omega \right)\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }{\alpha }^n,$$

(6)

for all $g_2\in L^{\infty }\left(m\right)$ and complex-valued functions $g_1\in BV$.

Remark 1.

We have the following remarks:

1. 

The bound in Equation (4) states that the $L^1$-norm (${\parallel .\parallel }_{L^1\left(m\right)}$) between, for special sequences ω, the push-forwards (i.e., ${\mathcal{L}}_{\omega }^n{\varphi }_1$ and ${\mathcal{L}}_{\omega }^n{\varphi }_2$) of two densities with bounded variation ${\varphi }_1$ and ${\varphi }_2$ converges to zero exponentially.

2. 

When an arbitrary probability measure $\nu =\varphi m$, where ϕ belongs to the BV space, $g_1$ and $g_2\circ T_{\omega }^n$ become asymptotically uncorrelated at an exponential rate.

3. 

From Equation (6), when μ is an invariant measure, then $g_1$ and ${\mathcal{Q}}^n{g}_2$ are exponentially correlated and

$$\int g_1·{\mathcal{Q}}^n{g}_{2}d\mu converges to\int g_{1}d\mu \int g_{2}d\mu .$$

We use the coupling method to give the proof of Theorems 1 and 2. This method is presented in Section 5.

4. Regularity of Densities

In this section, we state very important results. The first result is concerning some properties of the density function under the covering property of our random system. Furthermore, we use the Lasota–Yorke inequality to study the total variation in the push-forward density. For $({\omega }_0,\dots ,{\omega }_{n-1})\in {\Omega }^n$, we denote by $[{\omega }_0,\dots ,{\omega }_{n-1}]$ the cylinder

$$\{{\underline{\omega }}^{\prime }\in {\Omega }^{\mathbb{N}}:{\omega }_i^{\prime }={\omega }_i,i=0,\dots ,n-1\}.$$

Given that $K>0$ will be chosen later, we consider a finite partition $\mathcal{U}=\{I_1,\dots ,I_k\}$ of $[0,1]$ in intervals of lengths less than $1/\left(2K\right)$. Point (1) in the next result is Lemma 1.

Since the system has a random covering property, i.e., it satisfies $\left(H_1\right)$, then

$$\forall i\in \{1,\dots ,k\},\exists n_i\ge 1,\mathrm{and}{\omega }^{\left(i\right)}\in {\Omega }^{n_i},\mathrm{such}\mathrm{that}{T}_{{\omega }^{\left(i\right)}}^{n_i}{I}_i=[0,1].$$

In the next lemma, we study the properties of a probability density function $\varphi$ that belongs to the cone ${\mathcal{H}}_k$ under the property of covering systems.

Lemma 1.

For $K>0$, let ${\mathcal{H}}_K$ be the cone

$${\mathcal{H}}_K=\left\{\varphi :[0,1]\to {\mathbb{R}}_{+}\in \mathrm{BV},{\int }_0^1\varphi \left(x\right)dx=1,\mathit{TV}\left(\varphi \right)\le K\right\}.$$

For $\varphi \in {\mathcal{H}}_K$, there exists $\kappa >0$ and $i\in \{1,\dots ,k\},I_i\subset [0,1],$ such that

1. 

${inf}_{x\in I_i}\varphi \left(x\right)\ge 1/2$;

2. 

$inf{\mathcal{L}}_{\omega }^n\varphi \ge \kappa$, for any $n\ge n_i$ and any $\omega \in \left[{\omega }^{\left(i\right)}\right]$.

Proof. 

Suppose that $T_{\omega }$ verifies the covering property, i.e., $T_{\omega }^{n_0\left(\omega \right)}{I}_{\varphi }=[0,1]$. Then, there exists $y\in I_{\varphi }$ such that $T_{\omega }^{n_0\left(\omega \right)}\left(y\right)=x$. Thus,

$${\mathcal{L}}_{\omega }^{n_0\left(\omega \right)}\left(1_{I_{\varphi }}\right)\left(x\right)\ge 1_{I_{\varphi }}\left(y\right)·\frac{1}{|{\left(T_{\omega }^{n_0\left(\omega \right)}\right)}^{\prime }\left(y\right)|}\ge \frac{1}{|{\left(T_{\omega }^{n_0\left(\omega \right)}\right)}^{\prime }\left(y\right)|}.$$

It is enough to take $\kappa ={inf}_{y\in I_{\varphi }}{\left|{\left(T^{n_0\left(\omega \right)}\right)}^{\prime }\left(y\right)\right|}^{-1}$. Using Lemma 1, we see that since $\varphi \ge 0$, then for all $I_{\varphi }\subset I$,

$${\mathcal{L}}_{\omega }^{n_0\left(\omega \right)}\left(\varphi \right)\ge {\mathcal{L}}_{\omega }^{n_0\left(\omega \right)}\left(1_{I_{\varphi }}\varphi \right)\ge {\mathcal{L}}_{\omega }^{n_0\left(\omega \right)}\left(\frac{1}{2}{1}_{I_{\varphi }}\right)=\frac{1}{2}{\mathcal{L}}_{\omega }^{n_0\left(\omega \right)}\left(1_{I_{\varphi }}\right)\ge \frac{\kappa }{2}.$$

□

In the next Lemma we use the Lasota–Yorke inequality to bound the total variation of the push-forward density ${\mathcal{L}}_{\omega }^n\varphi$.

Lemma 2.

For all $n\ge 1$, there exists $\eta \in (0,1)$, such that the following holds: for every function ϕ with bounded variation, where $B>0$,

$$\mathit{TV}\left({\mathcal{L}}_{\omega }^n\varphi \right)\le {\eta }^n\mathit{TV}\left(\varphi \right)+B.$$

(7)

Proof. 

Through the Lasota–Yorke inequality [6], there exists $A\left(T_{\omega }\right)>0$, and one has

$$\mathit{TV}\left({\mathcal{L}}_{\omega }\varphi \right)\le \frac{2}{\lambda \left(T_{\omega }\right)}\mathit{TV}\left(\varphi \right)+A\left(T_{\omega }\right){\parallel \varphi \parallel }_{L_m^1}.$$

Thus,

$$\begin{array}{ccc}\hfill \mathit{TV}\left({\mathcal{L}}_{\omega }^n\varphi \right)& \le & \frac{2}{\lambda \left(T_{{\omega }_n}\right)}\mathit{TV}\left({\mathcal{L}}_{\omega }^{n-1}\varphi \right)+A\left(T_{{\omega }_n}\right){\parallel {\mathcal{L}}_{\omega }^{n-1}\varphi \parallel }_{L_m^1}.\hfill \end{array}$$

For $p\in \{1,\dots ,n\}$, we have

$$\parallel {\mathcal{L}}_{\omega }^p{\varphi \parallel }_{L_{\mathbf{m}}^1}\le {\parallel \varphi \parallel }_{L_m^1},$$

and by induction, we have

$$\begin{array}{ccc}\hfill \mathit{TV}\left({\mathcal{L}}_{\omega }^n\varphi \right)& \le & \frac{2}{\lambda \left(T_{{\omega }_n}\right)}\mathit{TV}\left({\mathcal{L}}_{\omega }^{n-1}\varphi \right)+A\left(T_{{\omega }_n}\right){\parallel \varphi \parallel }_{L_m^1}\hfill \\ \hfill ⋮\\ & \le & \frac{2^n}{{\prod }_{i=1}^n\lambda \left(T_{{\omega }_i}\right)}\mathit{TV}\left(\varphi \right)+\left(1+\frac{2}{\lambda \left(T_{{\omega }_n}\right)}+\dots +\frac{2^{n-1}}{{\prod }_{i=2}^n\lambda \left(T_{{\omega }_i}\right)}\right)A{\parallel \varphi \parallel }_{L_m^1}\hfill \\ \\ & \le & {\eta }^n\mathit{TV}\left(\varphi \right)+\left(1+\eta +{\eta }^2+\dots +{\eta }^{n-1}\right)A{\parallel \varphi \parallel }_{L_m^1}\hfill \\ \\ & \le & {\eta }^n\mathit{TV}\left(\varphi \right)+B,\hfill \end{array}$$

with $n_0\in \mathbb{N}$ such that $\lambda \left(T_{{\omega }_{n_0}}\right)=inf\left(\lambda \left(T_{{\omega }_1}\right),\dots ,\lambda \left(T_{{\omega }_n}\right)\right)$ and $\eta =2/\lambda \left(T_{{\omega }_{n_0}}\right)$. □

Remark 2.

Equation (7) plays an important role in the regularity of the push-forward density.

In the next corollary, we present a useful result concerning the regularity of densities that belong to the cone ${\mathcal{H}}_{K^{\prime }}$. Precisely, if we take $\varphi$, an arbitrary density function belonging to the cone ${\mathcal{H}}_{K^{\prime }}$, and apply ${\mathcal{L}}_{\omega }^n$, then the density function ${\mathcal{L}}_{\omega }^n\varphi$ belongs to ${\mathcal{H}}_K$.

Corollary 1.

Assume the conditions of Lemma 2. Let $\kappa \in (0,1)$, $K>B/(1-\kappa )$ and $K^{\prime }:=K/(1-\kappa )$. Then, there exists $n_{\kappa }>0$ such that

$${\mathcal{L}}_{\omega }^n\left({\mathcal{H}}_{K^{\prime }}\right)\subset {\mathcal{H}}_K,\mathit{for}\mathit{all}\mathit{n}\ge {\mathit{n}}_{\kappa }\mathit{and}\mathit{all}\omega \in {\Omega }^{\mathbb{N}}.$$

Proof. 

The set ${\mathcal{H}}_K$ enjoys the following simple property. If $\varphi \in {\mathcal{H}}_K$, and if ${\displaystyle \varphi \ge \kappa }$ and $\kappa \in (0,1)$, then

$$\mathit{TV}\left(\frac{\varphi -\kappa }{1-\kappa }\right):=sup\{\sum _{i=0}^n|\tilde{\varphi }\left(a_{i+1}\right)-\tilde{\varphi }\left(a_i\right)\left|\right\}=\frac{1}{1-\kappa }\mathit{TV}\left(\varphi \right)\le \frac{K}{1-\kappa },$$

(8)

which shows that

$$\tilde{\varphi }:=\frac{\varphi -\kappa }{1-\kappa }\in {\mathcal{H}}_{K^{\prime }}.$$

Using (8), then there exists $n_{\kappa }>0$ such that

$${\mathcal{L}}_{\omega }^{n_{\kappa }}\tilde{\varphi }:=\frac{1}{1-\kappa }{\mathcal{L}}_{\omega }^{n_{\kappa }}(\varphi -\kappa )\in {\mathcal{L}}_{\omega }^{n_{\kappa }}\left({\mathcal{H}}_{K^{\prime }}\right),$$

Using Lemma 2, we obtain

$$\begin{array}{ccc}\hfill \mathit{TV}\left({\mathcal{L}}_{\omega }^{n_{\kappa }}\tilde{\varphi }\right)& =& \mathit{TV}(\frac{1}{1-\kappa }{\mathcal{L}}_{\omega }^{n_{\kappa }}(\varphi -\kappa ))\hfill \\ & =& \frac{1}{1-\kappa }\mathit{TV}\left({\mathcal{L}}_{\omega }^{n_{\kappa }}\varphi \right)\le \frac{1}{1-\kappa }\left({\eta }^{n_{\kappa }}\mathit{TV}\left(\varphi \right)+B\right)\le {\eta }^{n_{\kappa }}{K}^{\prime }+\frac{B}{1-\kappa }.\hfill \end{array}$$

If $K>\frac{B}{1-\kappa }$, then there exists a large enough constant $n>n_{\kappa }>0$ such that

$$K>\frac{B}{1-\kappa }\ge \mathit{TV}\left({\mathcal{L}}_{\omega }^n\varphi \right)\ge \mathit{TV}\left({\mathcal{L}}_{\omega }^{n_{\kappa }}\varphi \right)$$

Finally, this concludes that there exists a large enough constant $n>n_{\kappa }>0$ such that

$${\mathcal{L}}_{\omega }^n\varphi \in {\mathcal{H}}_K,ifK>\frac{B}{1-\kappa }.$$

□

5. Coupling Method

In this section, we introduce the coupling method (for more details, see [7,8,9]). It plays an important role to find the statistical properties for random dynamical systems.

We start with two arbitrary probability density functions, ${\varphi }_1,{\varphi }_2\in {\mathcal{H}}_{K^{″}}$, for some $K^{″}>0$, and we couple a $\kappa$-fraction with its n-step push-forwards ${\varphi }_{j,n}={\mathcal{L}}_{\omega }^n{\varphi }_j$, for a large enough n, so that these push-forwards belong to ${\mathcal{H}}_K$ and remain bounded from below by $2\kappa$. For such n, we can write

$${\varphi }_{j,n}=\kappa +(1-\kappa ){\tilde{\varphi }}_n^j,\mathrm{where}{\tilde{\varphi }}_n^j=\frac{{\varphi }_{j,n}-\kappa }{1-\kappa }\in {\mathcal{H}}_{K^{\prime }}.$$

By (8), it follows that

$$\parallel {\varphi }_{1,n}-{\varphi }_{2,n}{\parallel }_{L^1\left(m\right)}\le (1-\kappa ){\parallel {\tilde{\varphi }}_n^1-{\tilde{\varphi }}_n^2\parallel }_{L^1\left(m\right)}.$$

(9)

We can then continue the procedure inductively, treating ${\tilde{\varphi }}_n^j$ as the initial densities, both belonging to ${\mathcal{H}}_{K^{\prime }}$. In view of the above, to obtain densities in ${\mathcal{H}}_K$ that are bounded from below by $2\kappa$, we first have to wait $n_{K^{″}}$ steps in order to enter the cone ${\mathcal{H}}_K$, and then, according to point (2) in Lemma 1, wait a sufficient number of steps to experience all the finite words ${\omega }^{\left(i\right)}$, $i=1,\dots ,k$ in the sequence ${\theta }^{n_{K^{″}}}\omega$. This leads us to define, for $i=1,\dots ,k$, the random times

$$R_i\left(\omega \right)=inf\{n\ge 0:{\theta }^n\omega \in \left[{\omega }^{\left(i\right)}\right]\},R=\underset{i=1,\dots ,k}{max}{R}_i.$$

We then define

$${\tau }_0\left(\omega \right)=0,n_0\left(\omega \right)=0,{\tau }_1\left(\omega \right)=n_{K^{″}}+R\left({\theta }^{n_{K^{″}}}\omega \right)+\ell ,n_1\left(\omega \right)={\tau }_1\left(\omega \right),$$

and, for $k\ge 2$,

$${\tau }_k\left(\omega \right)=n_{\kappa }+R\left({\theta }^{n_{k-1}\left(\omega \right)}\omega \right)+\ell ,n_k\left(\omega \right)=n_{k-1}\left(\omega \right)+{\tau }_k\left(\omega \right),$$

where $\ell ={max}_{i=1,\dots ,k}\left|{\omega }^{\left(i\right)}\right|$ is the maximal length of the words ${\omega }^{\left(i\right)}$, $i=1,\dots ,k$. Thus, $n_k$ is the time at which the k-th coupling will occur, and ${\tau }_k$ is the k-th inter-coupling time. Both depend on $K^{″}$ and $\omega$, but we suppress this from the notation for readability. Via the coupling argument, we obtain that for all $n\ge 0$ and all pairs ${\varphi }^1,{\varphi }^2\in {\mathcal{H}}_{K^{″}}$, it holds that

$$\begin{array}{c}\hfill \parallel {\varphi }_{1,n}-{\varphi }_{2,n}{\parallel }_{L_m^1}\le 2{(1-\kappa )}^{N_n},\end{array}$$

(10)

where $N_n$ is the coupling number that has occurred at time n, defined by

$$N_n\left(\omega \right)=max\{k\ge 0:n_k\left(\omega \right)\le n\}.$$

(11)

To prove the exponential decay of correlations, we now need to show that $N_n$ grows at least linearly with n, for almost every $\omega$. The following lemma gives us some needed properties of the random time R:

Lemma 3.

There exists a constant $0<q<1$ such that $\tilde{\mathbb{P}}(R>n)=\mathcal{O}\left(q^n\right)$.

Proof. 

It is enough to prove that $\tilde{\mathbb{P}}(R_i>n)$ decays exponentially fast for each i separately. In other words, we want to prove that, for the full-shift $\theta$ on ${\Omega }^{\mathbb{N}}$ on a finite alphabet $\Omega$, endowed with a Bernoulli product measure $\mu$ associated to a probability vector $p={\left\{p_{\omega }\right\}}_{\omega \in \Omega }$, the hitting time T to any cylinder $\left[\omega \right]=[{\omega }_0,\dots ,{\omega }_{j-1}]$ has exponential tails: $\mu (T>n)=\mathcal{O}\left(q^n\right)$ for some $q<1$.

(1) We first reduce this problem in order to consider only cylinders of length 1. For this, we consider the new alphabet $\widehat{\Omega }={\Omega }^j$, and identify the full shift on ${\Omega }^{\mathbb{N}}$ to a subshift on ${\left({\Omega }^j\right)}^{\mathbb{N}}$, via the map ${\left({\omega }_n\right)}_{n\ge 0}\mapsto {\left(({\omega }_n,\dots ,{\omega }_{n+j-1})\right)}_{n\ge 0}$. The image of this map is the subshift of finite type on ${\widehat{\Omega }}^{\mathbb{N}}$, whose transition matrix A satisfies $A_{({\omega }_0\dots {\omega }_{j-1}),({\omega }_0^{\prime }\dots {\omega }_{j-1}^{\prime })}=1$ if and only if ${\omega }_1={\omega }_0^{\prime },\dots ,{\omega }_{j-1}={\omega }_{j-2}^{\prime }$.

The Bernoulli measure $\mu$ is then identified with the Markov measure $\nu$ on ${\widehat{\Omega }}^{\mathbb{N}}$ associated to the stochastic matrix P defined by $P_{({\omega }_0\dots {\omega }_{j-1}),({\omega }_1\dots {\omega }_j)}=p_{{\omega }_j}$ and the invariant probability vector $\widehat{p}$ defined by ${\widehat{p}}_{({\omega }_0\dots {\omega }_{j-1})}=p_{{\omega }_0}\dots p_{{\omega }_{j-1}}$. It is important to check that A and P are irreducible and aperiodic. The hitting time $\widehat{T}$ on ${\widehat{\Omega }}^{\mathbb{N}}$ to the cylinder $[{\omega }_0^{\prime }\dots {\omega }_{j-1}]$ of length 1 satisfies $\nu (\widehat{T}>n)=\mu (T>n)$.

(2) Then, the problem is reduced to prove that, for a subshift of finite type with finite alphabet $\widehat{\Omega }$ with transition matrix A, endowed with a Markov measure $\nu$ associated to a primitive stochastic matrix P and an invariant probability vector $\widehat{p}$, the hitting time $\widehat{T}$ to any cylinder $\left[{\omega }^{★}\right]$ of length 1 has an exponential tail: $\nu (\widehat{T}>n)=\mathcal{O}\left(q^n\right)$ for some $q<1$. The rest of the proof is now devoted to proving this fact, and will not mention of the particular nature of the new alphabet $\widehat{\Omega }$.

(3) $\forall n\ge 0$, one has

$$\nu (\widehat{T}=n)=\nu \left(\bigcup _{{\omega }_0,\dots ,{\omega }_{n-1}\in \widehat{\Omega }\setminus \left\{{\omega }^{★}\right\}}[{\omega }_0,\dots ,{\omega }_{n-1},{\omega }^{★}]\right)=\sum _{{\omega }_0,\dots ,{\omega }_{n-1}\ne {\omega }^{★}}{\widehat{p}}_{{\omega }_0}{P}_{{\omega }_0,{\omega }_1}\dots P_{{\omega }_{n-1},{\omega }^{★}}.$$

Introducing the new matrix Q defined by

$$Q_{\omega ,{\omega }^{\prime }}=\left\{\begin{array}{cc}{P}_{\omega ,{\omega }^{\prime }}\hfill & \mathrm{if}\omega \ne {\omega }^{★},\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

we can write

$$\nu (\widehat{T}=n)=\sum _{{\omega }_0,\dots ,{\omega }_{n-1}\in \widehat{\Omega }}{\widehat{p}}_{{\omega }_0}{Q}_{{\omega }_0,{\omega }_1}\dots Q_{{\omega }_{n-1},{\omega }^{★}}=\sum _{{\omega }_0}{\widehat{p}}_{{\omega }_0}{Q}_{{\omega }_0,{\omega }^{★}}^n.$$

Since P is stochastic and primitive, $0\le Q\le P$ and $Q\ne P$, via the Perron–Frobenius theory for positive matrices (see item (e) of Theorem 1.1 in [21]), we have that the spectral radius of Q is strictly less than 1. Consequently, all the entries $Q_{\omega ,{\omega }^{\prime }}^n$ decay exponentially fast with n, from which it follows that $\nu (\widehat{T}>n)={\sum }_{m>n}\nu (\widehat{T}=m)$ converges to 0 exponentially fast. □

Note that we can prove the exponential decay of quenched and annealed correlations exactly as in [8].

6. Discussion

In this section, we discuss our main results. We use the coupling method to find statistical properties of our RDS that consists of coupling the push-forwards densities. For this, we start with two probability density functions ${\varphi }_1,{\varphi }_2$ belonging to the cone ${\mathcal{H}}_{K^{″}}$ defined by

$${\mathcal{H}}_{K^{″}}=\left\{\varphi :[0,1]\to {\mathbb{R}}_{+}\in \mathrm{BV},{\int }_0^1\varphi \left(x\right)dx=1,\mathit{TV}\left(\varphi \right)\le K^{″}\right\}.$$

for some $K^{″}>0$, we couple a $\kappa$-fraction ${\varphi }_{j,n}={\mathcal{L}}_{\omega }^n{\varphi }_j=\kappa +(1-\kappa ){\tilde{\varphi }}_n^j$, where

$${\tilde{\varphi }}_n^j=\frac{{\varphi }_{j,n}-\kappa }{1-\kappa }\in {\mathcal{H}}_{K^{\prime }}.$$

For a large enough n, using Lemma 2, we prove that ${\mathcal{L}}_{\omega }^n{\varphi }_j$ belongs to ${\mathcal{H}}_K$ and remains bounded from below by $2\kappa$. That means the the cone ${\mathcal{H}}_{K^{\prime }}$ is included in the cone ${\mathcal{H}}_K$. We continue the coupling procedure and we analyze the random time defined by

$$R_i\left(\omega \right)=inf\{n\ge 0:{\theta }^n\omega \in \left[{\omega }^{\left(i\right)}\right]\},R=\underset{i=1,\dots ,k}{max}{R}_i.$$

Then, there exists a constant $0<q<1$ such that $\tilde{\mathbb{P}}(R>n)=\mathcal{O}\left(q^n\right)$. The last probability leads us to obtain an exponential decay of quenched and annealed correlations exactly as in [8].

Furthermore, Inequality (4) in Theorem 1 shows that the difference between two push-forward densities ${\mathcal{L}}_{\omega }^n{\varphi }_1$ and ${\mathcal{L}}_{\omega }^n{\varphi }_2$ converges to 0 exponentially. Moreover, Inequality (5) in Theorem 1 shows that when $\nu =\varphi m$, a measure of probability where $\varphi$ is a probability density function belonging to the BV space, then the bound of the quantity in (2) is bounded by ${C\left(\omega \right)\parallel \varphi \parallel }_{BV}\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }{\alpha }^n$ and $g_1$ and $g_2\circ T_{\omega }^n$ are asymptotically uncorrelated. Finally, Inequality (6) shows that $g_1$ and ${\mathcal{Q}}^n{g}_2$ are exponentially correlated and

$$\int g_1·{\mathcal{Q}}^n{g}_{2}d\mu \mathrm{converges}\mathrm{to}\int g_{1}d\mu \int g_{2}d\mu ,$$

when $\mu$ is an invariant measure.

7. Proofs of Theorems

In this section, we prove our main results presented in Section 3. For this, we need to introduce the following technical lemma. It leads to a bound limit of the total variation between two BV observables.

Lemma 4.

For all observable ϕ and $\psi \in BV$, we have

$$\mathit{TV}\left(\varphi \psi \right)\le {\parallel \varphi \parallel }_{BV}{\parallel \psi \parallel }_{BV}.$$

Proof. 

For all observable $\varphi$ and $\psi \in BV$, we have

$$\begin{array}{ccc}\hfill \mathit{TV}\left(\varphi \psi \right)& =& sup\{\sum _{i=0}^n|\left(\varphi \psi \right)\left(a_{i+1}\right)-\left(\varphi \psi \right)\left(a_i\right)|,0=a_0<\cdots <a_n=1\}\hfill \\ & =& sup\{\sum _{i=0}^n|\varphi \left(a_{i+1}\right)\psi \left(a_{i+1}\right)-\varphi \left(a_i\right)\psi \left(a_i\right)|,0=a_0<\cdots <a_n=1\}\hfill \\ & =& sup\{\sum _{i=0}^n|\varphi \left(a_{i+1}\right)\psi \left(a_{i+1}\right)-\varphi \left(a_{i+1}\right)\psi \left(a_i\right)+\varphi \left(a_{i+1}\right)\psi \left(a_i\right)\hfill \\ & & -\varphi \left(a_i\right)\psi \left(a_i\right)|,0=a_0<\cdots <a_n=1\}\hfill \\ & \le & sup\{\sum _{i=0}^n|\varphi \left(a_{i+1}\right)||\psi \left(a_{i+1}\right)-\psi \left(a_i\right)|,0=a_0<\cdots <a_n=1\}\hfill \\ & & +sup\{\sum _{i=0}^n|\psi \left(a_{i+1}\right)||\varphi \left(a_{i+1}\right)-\varphi \left(a_i\right)|,0=a_0<\cdots <a_n=1\}\hfill \\ & \le & {\parallel \varphi \parallel }_1\mathit{TV}\left(\psi \right)+{\parallel \psi \parallel }_{1}Var\left(\varphi \right)\hfill \\ & \le & {\parallel \varphi \parallel }_1\mathit{TV}\left(\psi \right)+{\parallel \psi \parallel }_1\mathit{TV}\left(\varphi \right)+Var\left(\varphi \right)\mathit{TV}\left(\psi \right)+{\parallel \varphi \parallel }_1{\parallel \psi \parallel }_1={\parallel \varphi \parallel }_{BV}{\parallel \psi \parallel }_{BV}.\hfill \end{array}$$

□

Proof of Theorem 1.

Given two sequences of p.d.f.s ${\varphi }_{1,n}$, ${\varphi }_{2,n}\in {\mathcal{H}}_K$, we have

$$\parallel {\varphi }_{1,n}-{\varphi }_{2,n}{\parallel }_{L^1\left(m\right)}\le \underset{A}{\underbrace{{\int }_{\{\tilde{n}>n\}}|{\varphi }_{1,n}-{\varphi }_{2,n}|dm}}+\underset{B}{\underbrace{{\int }_{\{\tilde{n}\le n\}}|{\varphi }_{1,n}-{\varphi }_{2,n}|dm}}.$$

Using Inequality (10) and

$$\parallel {\mathcal{L}}_{\omega }^{n}1\parallel =\int 1.1\circ T_{{\omega }_n}\circ \cdots \circ T_{{\omega }_1}\le 1⟹\parallel {\mathcal{L}}^n{\varphi \parallel }_{\infty }\le {\parallel \varphi \parallel }_{\infty },$$

we obtain

$$A\le \left(\parallel {\varphi }_{1,n}{\parallel }_{\infty }+{\parallel {\varphi }_{2,n}\parallel }_{\infty }\right)\mathbb{P}\left(\{\tilde{n}>n\}\right)\le Cmax(\parallel {\varphi }^1{\parallel }_{BV},\parallel {\varphi }^2{\parallel }_{BV})q^n,$$

and

$$B\le 2{(1-\kappa )}^{tn-1}.$$

Finally, the first result of Theorem 1 is verified with $\alpha =max({(1-\kappa )}^t,q)$. Let us denote

$$\chi \left(n\right)=1_{\{n\ge \tilde{n}\}}+{(1-\kappa )}^{tn-1}.$$

Define a perturbed probability density function ${\psi }_r$ by

$${\varphi }_g=\frac{\varphi +r}{1+r},r>0.$$

Since $\varphi +r\ge r$, we obtain

$$\mathit{TV}\left({\varphi }_r\right)=\frac{1}{1+r}\mathit{TV}\left(\varphi \right)<\frac{1}{r}\mathit{TV}\left(\varphi \right).$$

Thus,

$${\varphi }_r\in {\mathcal{H}}_1⟹\frac{1}{r}\mathit{TV}\left(\varphi \right)<1⟹\mathit{TV}\left(\varphi \right)<r.$$

Using (9), we have

$$\parallel {\varphi }_{1,n}-{\varphi }_{2,n}{\parallel }_{L^1\left(m\right)}\le 2{(1-k)}^{\left[tn\right]}+2q^{N_n}\le 2{(1-k)}^{tn-1}.$$

Consider a complex function $g_2\in L^{\infty }\left(m\right)$ and a real function $g_1\in BV$ with $\int g_{1}dm=0$. Define

$$\widetilde{g_1}=\frac{g_1+2\mathit{TV}\left(g_1\right)}{2\mathit{TV}\left(g_1\right)}.$$

It is clear that $\widetilde{g_1}\in {\mathcal{H}}_1$. Therefore,

$$\begin{array}{ccc}\hfill |\int g_1·g_2\circ T_{\omega }^{n}dm|=|\int {\mathcal{L}}_{\omega }^n{g}_1·g_{2}dm|& \le & \parallel g_2{\parallel }_{\infty }{\parallel {\mathcal{L}}_{\omega }^n{g}_1\parallel }_{L^1\left(m\right)}\hfill \\ & \le & 2\mathit{TV}\left(g_1\right)\parallel g_2{\parallel }_{\infty }{\parallel {\mathcal{L}}_{\omega }^n(\widetilde{g_1}-1)\parallel }_{L^1\left(m\right)}\hfill \\ & \le & 4\mathit{TV}\left(g_1\right){\parallel g_2\parallel }_{\infty }\chi \left(n\right).\hfill \end{array}$$

In general, since $\int g_{1}dm=0$, we obtain the following inequality:

$$\begin{array}{c}\hfill |\int g_1·g_2\circ T_{\omega }^{n}dm-\int g_{1}dm\int g_2\circ T_{\omega }^{n}dm|\le 4\mathit{TV}\left(g_1\right){\parallel g_2\parallel }_{\infty }\chi \left(n\right).\end{array}$$

Define

$$H=\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}dm,$$

and

$$G=\int g_2\circ T_{\omega }^{n}d\nu -\int g_2\circ T_{\omega }^{n}dm.$$

Now, if we replace $dm$ by $d\nu =\varphi dm$, where $\varphi$ is a density probability in BV, and use Lemma 4, we obtain

$$\begin{array}{ccc}\hfill \left|H\right|& =& |\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}dm|\hfill \\ \hfill \\ & =& |\int g_1·g_2\circ T_{\omega }^n\varphi dm-\int g_1\varphi dm\int g_2\circ T_{\omega }^{n}dm|\hfill \\ \hfill \\ & \le & 4\parallel g_1{\varphi \parallel }_{BV}\parallel g_2{\parallel }_{\infty }\chi \left(n\right)\le {4\parallel \varphi \parallel }_{BV}\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }\chi \left(n\right).\hfill \end{array}$$

(12)

and

$$\begin{array}{ccc}\hfill \left|G\right|& =& |\int g_2\circ T_{\omega }^{n}d\nu -\int g_2\circ T_{\omega }^{n}dm|\hfill \\ & =& |\int \varphi g_2\circ T_{\omega }dm-\int \varphi dm\int \varphi g_2\circ T_{\omega }^{n}dm|\hfill \\ & \le & 4\mathit{TV}\left(\varphi \right)\parallel g_2{\parallel }_{\infty }\chi \left(n\right)\le {4\parallel \varphi \parallel }_{BV}{\parallel g_2\parallel }_{\infty }\chi \left(n\right).\hfill \end{array}$$

(13)

Using Inequalities (12) and (13), we obtain

$$\begin{array}{ccc}\hfill \left|V\right|& =& |\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}d\nu |\hfill \\ & =& |\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}dm\hfill \\ & \le & |\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}dm|\hfill \\ & & +|\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}dm|\hfill \\ & \le & \left|H\right|+|\int g_{1}d\nu |\left|G\right|\hfill \\ & \le & \left|H\right|+\parallel g_1{\parallel }_{BV}\left|G\right|\le {8\parallel \varphi \parallel }_{BV}\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }\chi \left(n\right).\hfill \end{array}$$

Then, the proof of Inequality (5) is finished for real-valued $g_1$. Through similarity, we prove the same bound for the $g_1$ complex function. □

Proof of Theorem 2.

We take $\mu =\nu$. Since $g_1$ and $g_2$ are bounded, we obtain

$$\begin{array}{ccc}\hfill Z& =& |\int g_1·{\mathcal{Q}}^n{g}_{2}d\mu -\int g_{1}d\mu \int g_{2}d\mu |\hfill \\ & =& |\int g_1·{\mathcal{Q}}^n{g}_{2}d\mu -\int g_{1}d\mu \int {\mathcal{Q}}^n{g}_{2}d\mu |\hfill \\ & =& |\int g_1·\mathbb{E}[g_2\circ T_{\omega }^n]d\mu -\int g_{1}d\mu \int \mathbb{E}[g_2\circ T_{\omega }^n]d\mu |\hfill \\ & \le & \mathbb{E}\left[|\int g_1·g_2\circ T_{\omega }^{n}d\mu -\int g_{1}d\mu \int g_2\circ T_{\omega }^{n}d\mu |\right]\hfill \\ & \le & \mathbb{E}\left[{8\parallel \psi \parallel }_{BV}\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }\chi \left(n\right)\right]\hfill \\ & \le & {C\parallel \psi \parallel }_{BV}\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }\mathbb{E}\left[\chi \left(n\right)\right]\hfill \\ & \le & {C\parallel \psi \parallel }_{BV}\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }\left(D^{\prime }{q}^n+{(1-k)}^{tn-1}\right).\hfill \end{array}$$

□

8. Conclusions and Perspectives

In the present paper, we prove that the decay of correlations for ${\mathcal{C}}^2$ piecewise expanding maps from $[0,1]$ into itself is an exponential decay of correlations. For this, we apply the Lasota–Yorke inequality (see Lemma 2) for our random dynamical systems. This inequality leads to the existence of an absolutely continuous invariant measure $\nu =\psi m$, where $\psi \in BV$. Then, we use a very useful method called the coupling method. Firstly, this method consists of starting with two arbitrary probability density functions ${\psi }_1$ and ${\psi }_2$ belonging to the cone ${\mathcal{H}}_{K^{″}}$, where $K^{″}$ is a positive constant. Secondly, we couple the $\kappa$-fraction with its n-steps ${\mathcal{L}}_{\omega }^n{\varphi }_1$ and ${\mathcal{L}}_{\omega }^n{\varphi }_2$ belonging to the cone ${\mathcal{H}}_K$, and this remains bounded by $2\kappa$. Finally, we continue the procedure of coupling and treat the initial densities ${\tilde{\varphi }}_n^1$ and ${\tilde{\varphi }}_n^2$, both belonging to ${\mathcal{H}}_{K^{\prime }}$, to obtain densities in ${\mathcal{H}}_K$ that are bounded from below by $2\kappa$. For this step, we wait $n_{K^{″}}$ steps in order to enter the cone ${\mathcal{H}}_K$, and then, according to point (2) in Lemma 1, we wait a sufficient number of steps to experience all the finite words ${\omega }^{\left(i\right)}$, $i=1,\dots ,k$, in the sequence ${\theta }^{n_{K^{″}}}\omega$. In addition, using the coupling argument, we obtain that for all $n\ge 0$ and all pairs ${\varphi }^1,{\varphi }^2\in {\mathcal{H}}_{K^{″}}$, it holds that

$$\begin{array}{c}\hfill \parallel {\varphi }_{1,n}-{\varphi }_{2,n}{\parallel }_{L_m^1}\le 2{(1-\kappa )}^{N_n},\end{array}$$

where $N_n$ is defined in Equation (11). Furthermore, the analysis of $N_n$ leads us to prove the decay of correlations for our random dynamical systems. The first result about the decay of correlations is given by the following inequality: there exists $0<\alpha <1$ such that for almost every $\omega$, there exists $C\left(\omega \right)>0$

$$\forall n\ge 0,\parallel {\mathcal{L}}_{{\omega }_n}\dots {\mathcal{L}}_{{\omega }_1}({\varphi }_1-{\varphi }_2){\parallel }_{L^1\left(m\right)}\le C\left(\omega \right)max\left(\parallel {\varphi }_1{\parallel }_{BV},{\parallel {\varphi }_2\parallel }_{BV}\right){\alpha }^n,$$

where ${\varphi }_1$ and ${\varphi }_2$ two probability density functions belonging to the $BV$ space. For the second result, given a probability distribution $d\nu =\varphi dm,\varphi \in BV$, there exists $0<\alpha <1$ such that for almost every $\omega$, there exists $C\left(\omega \right)>0$

$$\begin{array}{c}\hfill \forall n\ge 0,|\int g_1·g_2\circ T_{\omega }^{n}d\nu -\int g_{1}d\nu \int g_2\circ T_{\omega }^{n}d\nu |\le {C\left(\omega \right)\parallel \psi \parallel }_{BV}{\parallel f\parallel }_{BV}{\parallel g\parallel }_{\infty }{\alpha }^n,\end{array}$$

for all complex-valued functions $g_1\in BV$ et $g\in L^{\infty }\left(m\right)$. The third result is given by the following: there exist two constants, $\alpha \in (0,1)$ and $C>0$, such that

$$\forall n\ge 0,|\int g_1·{\mathcal{Q}}^n{g}_{2}d\mu -\int g_{1}d\mu \int g_{2}d\mu |\le C\parallel g_1{\parallel }_{BV}{\parallel g_2\parallel }_{\infty }{\alpha }^n,$$

where $g_2\in L^{\infty }\left(m\right)$ and $g_1\in BV$ are complex-valued functions.

In the future, we will work to prove other statistical properties for our random dynamical systems like the central limit theorem, the loss memory and the almost-sure invariance principle using the coupling method.