Wasserstein convergence rates in the invariance principle for deterministic dynamical systems

Abstract

In this paper, we consider the convergence rate with respect to Wasserstein distance in the invariance principle for deterministic non-uniformly hyperbolic systems. Our results apply to uniformly hyperbolic systems and large classes of non-uniformly hyperbolic systems including intermittent maps, Viana maps, unimodal maps and others. Furthermore, as a non-trivial application to the homogenization problem, we investigate the Wasserstein convergence rate of a fast–slow discrete deterministic system to a stochastic differential equation.

1 Introduction

It is well known that deterministic dynamical systems can exhibit some statistical limit properties if the system is chaotic enough and the observable satisfies some regularity conditions. In recent years, we have seen growing research interests in statistical limit properties for deterministic systems such as the law of large numbers (or Birkhoff’s ergodic theorem), central limit theorem (CLT), weak invariance principle (WIP), almost sure invariance principle (ASIP), large deviations and so on.

The WIP (also known as the functional CLT) states that a stochastic process constructed by the sums of random variables with suitable scale converges weakly to Brownian motion, which is a far-reaching generalization of the CLT. Donsker’s theorem [13] is the prototypical invariance principle, which deals with independent and identically distributed random variables. Later, different versions were extensively studied. In particular, many authors studied the WIP and ASIP for dynamical systems with some hyperbolicity. Denker and Philipp [11] proved the ASIP for uniformly hyperbolic diffeomorphisms and flows. The results are stronger, as the ASIP implies the WIP and CLT. Melbourne and Nicol [24] investigated the ASIP for non-uniformly expanding maps and non-uniformly hyperbolic diffeomorphisms that can be modelled by a Young tower [33, 34], and they also obtained corresponding results for flows. After that, there are many works on the WIP for non-uniformly hyperbolic systems, which we will not mention here.

To the best of our knowledge, there are only two works on rates of convergence in the WIP for deterministic dynamical systems in spite of the fact that there are many results on the convergence itself. In his PhD thesis [1], Antoniou obtained the rate of convergence in the Lévy–Prokhorov distance for uniformly expanding maps using the martingale approximation method and applying an estimate for martingale difference arrays [22] by Kubilyus. Then, following the same method, together with Melbourne, he [2] further generalized the convergence rates to non-uniformly expanding/hyperbolic systems. Specifically, they apply a new version of the martingale-coboundary decomposition [21] by Korepanov et al.

The Wasserstein distance has been used extensively in recent years to metrize weak convergence. It is stronger and contains more information than the Lévy–Prokhorov distance since it involves the metric of the underlying space. This distance finds important applications in the fields of optimal transport, geometry, partial differential equations and so on; see, e.g., Villani [32] for details. There are some results on Wasserstein convergence rates for the CLT in the community of probability and statistics; see, e.g., [10, 26, 29]. However, to our knowledge, there are no related results on the invariance principle for dynamical systems. Motivated by [1, 2], we aim to estimate the Wasserstein convergence rate in the WIP for non-uniformly hyperbolic systems.

Following the procedure of [2], we first consider a martingale as an intermediary process. In [2], the authors apply a result of Kubilyus [22] and the key is to estimate the distance between $W_n$ , defined in (3.1) below, and the intermediary process. In the present paper, we use the ideas in [2] to estimate the distance between $W_n$ and the intermediary process. Hence, most of our efforts are to deal with the Wasserstain distance between the intermediary process and Brownian motion, which is handled by a martingale version of the Skorokhod embedding theorem. In this way, we obtain the rate of convergence $O(n^{-1/4+\delta })$ in the Wasserstein distance, where $\delta $ depends on the degree of non-uniformity. When the system that can be modelled by a Young tower has a superpolynomial tail, $\delta $ can be arbitrarily small.

Our results are applicable to uniformly hyperbolic systems and large classes of non-uniformly hyperbolic systems modelled by a Young tower with superpolynomial and polynomial tails. In comparison with [2], when the dynamical system has a superpolynominal tail, we can obtain the same convergence rate $O(n^{-1/4+\delta })$ for $\delta $ arbitrarily small. However, in our case, the price to pay is that the dynamical system needs to have stronger mixing properties. For example, we consider the Pomeau–Manneville intermittent map (3.2), which has a polynomial tail. By [2], the convergence rate is $O(n^{-1/4+{\gamma }/{2}+\delta })$ in the Lévy–Prokhorov distance for $\gamma \in (0,\tfrac 12)$ , but we obtain the Wasserstein convergence rate $O(n^{-1/4+{\gamma }/(4(1-\gamma ))+\delta })$ only for $\gamma \in (0,\tfrac 14)$ . See Example 3.6 for details.

As a non-trivial application, we consider the deterministic homogenization in fast–slow dynamical systems. In [15], Gottwald and Melbourne proved that the slow variable with suitable scales converges weakly to the solution of a stochastic differential equation. Then Antoniou and Melbourne [2] studied the weak convergence rate of the above problem based on the convergence rate in the WIP with respect to the Lévy–Prokhorov distance. In this paper, we obtain the Wasserstein convergence rate for the homogenization problem based on our results. In comparison with [2], for uniformly hyperbolic fast systems, we obtain the same convergence rate $O(\epsilon ^{1/3-\delta })$ , where $\delta $ can be arbitrarily small and $\epsilon $ is identified with $n^{-1/2}$ . However, for non-uniformly hyperbolic fast systems, we need to request stronger mixing properties than in [2]. See Remark 5.2 for details.

The remainder of this paper is organized as follows. In §2, we give the definition and basic properties of Wasserstein distances. In §3, we review the definitions of non-uniformly expanding maps and non-uniformly hyperbolic diffeomorphisms and we state the main results in this paper. In §4, we first introduce the method of martingale approximation and summarize some required properties and then we prove the main results. In the last section, we give an application to fast–slow systems.

Throughout the paper, we use $1_A$ to denote the indicator function of measurable set A. As usual, $a_n=o(b_n)$ means that $\lim _{n\to \infty } a_n/b_n=0$ , $a_n=O(b_n)$ means that there exists a constant $C>0$ such that $|a_n|\le C |b_n|$ for all $n\ge 1$ and $\|\cdot \|_{L^p}$ means the $L^p$ -norm. For simplicity, we write C to denote constants independent of n and C may change from line to line. We use $\rightarrow _{w}$ to denote the weak convergence in the sense of probability measures [5]. We denote by $C[0,1]$ the space of all continuous functions on $[0,1]$ equipped with the supremum distance $d_C$ , that is,

$$ \begin{align*} d_C(x,y):=\sup_{t\in [0,1]}|x(t)-y(t)|, \quad x,y\in C[0,1]. \end{align*} $$

We use ${\mathbb P}_X$ to denote the law/distribution of random variable X and use $X=_d Y$ to mean $X, Y$ sharing the same distribution.

2 Preliminaries

In this section, we review the definition of Wasserstein distances and some important properties about the distance. See, e.g., [8, 28, 32] for details.

Let $(\mathcal {X}, d)$ be a Polish space, that is, a complete separable metric space, equipped with the Borel $\sigma $ -algebra $\mathcal {B}$ . Given two probability measures $\mu $ and $\nu $ on $\mathcal {X}$ , take two random variables X and Y such that $\mathrm {law} (X)=\mu , \mathrm {law} (Y)=\nu $ . Then the pair $(X,Y)$ is called a coupling of $\mu $ and $\nu $ ; the joint distribution of $(X,Y)$ is also called a coupling of $\mu $ and $\nu $ .

Definition 2.1. Let $q\in [1,\infty )$ . Then, for any two probability measures $\mu $ and $\nu $ on $\mathcal {X}$ , the Wasserstein distance of order q between them is defined by

$$ \begin{align*} \mathcal{W}_{q}(\mu,\nu) & :={\bigg( \inf_{\pi\in\Pi(\mu,\nu)}\int_{\mathcal{X}} {d(x,y)}^q\,\mathrm{d}\pi(x,y)\bigg)}^{1/q}\\ &= \inf \{ [\mathbf{E} {d(X,Y)}^q]^{1/q}; \mathrm{law} (X)=\mu, \mathrm{law} (Y)=\nu \},\nonumber \end{align*} $$

where $\Pi (\mu ,\nu )$ is the set of all couplings of $\mu $ and $\nu $ .

Proposition 2.2. (See [8, Lemma 5.2])

Given two probability measures $\mu $ and $\nu $ on $\mathcal {X}$ , the infimum in Definition 2.1 can be attained for some coupling $(X, Y)$ of $\mu $ and $\nu $ .

Those couplings achieving the infimum in Proposition 2.2 are called optimal couplings of $\mu $ and $\nu $ . Note also that the distance $\mathcal {W}_{q}(\mu ,\nu )$ can be bounded above by the $L^q$ distance of any coupling $(X,Y)$ of $\mu $ and $\nu $ .

Proposition 2.3. (See [8, Theorem 5.6] or [32, Definition 6.8])

$\mathcal {W}_{q}(\mu _n, \mu )\to 0$ if and only if the following two conditions hold:

1. (1) $\mu _{n}\rightarrow _{w}\mu ;$ and

2. (2) $\int _{\mathcal X} d(x,x_{0})^{q}\,\mathrm {d}\mu _{n}(x)\rightarrow \int _{\mathcal X} d(x,x_{0})^{q}\,\mathrm {d}\mu (x)$ for some (thus any) $x_{0}\in \mathcal {X}$ .

In particular, if d is bounded, then the convergence with respect to $\mathcal {W}_{q}$ is equivalent to the weak convergence.

Proposition 2.4. Suppose that $\mathcal {G}:\mathcal {X}\rightarrow \mathcal {X}$ is Lipschitz continuous with constant K. Then, for any two probability measures $\mu $ and $\nu $ on $\mathcal {X}$ and $q\in [1,\infty )$ ,

$$ \begin{align*}\mathcal{W}_q(\mu\circ \mathcal{G}^{-1},\nu\circ \mathcal{G}^{-1})\le K \mathcal{W}_q(\mu,\nu).\end{align*} $$

Proof. By Proposition 2.2, we can choose an optimal coupling $(X, Y)$ of $\mu $ and $\nu $ such that

$$ \begin{align*} [{\mathbf E} d(X, Y)^{q}]^{1/q}= \mathcal{W}_q(\mu,\nu). \end{align*} $$

Then

$$ \begin{align*} \mathcal{W}_q(\mu\circ \mathcal{G}^{-1},\nu\circ \mathcal{G}^{-1})&\le [{\mathbf E} d(\mathcal{G}(X),\mathcal{G}(Y))^{q}]^{1/q}\\ &\le K[{\mathbf E} d(X,Y)^{q}]^{1/q}= K \mathcal{W}_q(\mu,\nu).\\[-3pc] \end{align*} $$

Remark 2.5. In the following, for simplicity, we use the notation $\mathcal {W}_p(X,Y)$ to mean $\mathcal {W}_p({\mathbb P}_X, {\mathbb P}_Y)$ . However, we should keep in mind that $(X,Y)$ need not be an optimal coupling of $({\mathbb P}_X, {\mathbb P}_Y)$ .

The following result is known; see, e.g., [8, Lemma 5.3] or [28, Corollary 8.3.1] for details. However, the forms or proofs in these references are different from the following, which is more appropriate for our purpose. For the convenience of the reader, we also give a proof.

Proposition 2.6. For any given probability measures $\mu $ and $\nu $ on $\mathcal {X}$ and $p\in [1,\infty )$ ,

$$ \begin{align*} \pi(\mu,\nu)\le \mathcal{W}_{p}(\mu,\nu)^{{p}/({p+1})}, \end{align*} $$

where $\pi $ is the Lévy–Prokhorov distance defined by

$$ \begin{align*} \pi(\mu,\nu):=\inf\{\epsilon> 0: \mu(A)\le \nu(A^{\epsilon})+\epsilon \mathrm{ for~all~closed~sets~} A\in \mathcal{B}\}. \end{align*} $$

Here $A^{\epsilon }$ denotes the $\epsilon $ -neighbourhood of A.

Proof. Let A be a closed set. Then, for any coupling $(X,Y)$ of $\mu $ and $\nu $ ,

$$ \begin{align*} {\mathbb P}(X\in A)&\le {\mathbb P}(Y\in A^{\epsilon})+{\mathbb P}(d(X,Y)\ge \epsilon)\\ &\le {\mathbb P}(Y\in A^{\epsilon})+\frac{{\mathbf E} d(X,Y)^p}{\epsilon^p}. \end{align*} $$

Note that ${\mathbb P}(X\in A)$ and ${\mathbb P}(Y\in A^{\epsilon })$ depend on X and Y only through their distributions. So by the arbitrariness of the coupling $(X,Y)$ of $\mu $ and $\nu $ ,

$$ \begin{align*} {\mathbb P}(X\in A)\le {\mathbb P}(Y\in A^{\epsilon})+ \epsilon^{-p}\mathcal{W}_{p}(\mu,\nu)^{p}. \end{align*} $$

Choosing $\epsilon =\mathcal {W}_{p}(\mu ,\nu )^{{p}/({p+1})}$ , we deduce that ${\mathbb P}(X\in A)\le {\mathbb P}(Y\in A^{\epsilon }) + \epsilon $ . Hence,

$$ \begin{align*} \pi(\mu,\nu)\le \mathcal{W}_{p}(\mu,\nu)^{{p}/({p+1})}.\\[-37pt] \end{align*} $$

3 Non-uniformly expanding/hyperbolic maps

3.1 Non-uniformly expanding map

Let $(M,d)$ be a bounded metric space with Borel probability measure $\rho $ . Let $T:M\rightarrow M$ be a non-singular (that is, $\rho (T^{-1}E)=0$ if and only if $\rho (E)=0$ for all Borel measurable sets E) ergodic transformation. Suppose that Y is a subset of M with positive measure and that $\{Y_j\}$ is an at most countable measurable partition of Y with $\rho (Y_j)>0$ . Let $R:Y\rightarrow \mathbb {Z}^{+}$ be an integrable function that is constant on each $Y_j$ and let $T^{R(y)}(y)\in Y$ for all $y\in Y$ . We call R the return time and ${F=T^{R}:Y\rightarrow Y}$ is the corresponding induced map. We do not require that R is the first return time to Y.

Let $\nu =({\mathrm {d}\rho |_Y})/({\mathrm {d}\rho |_Y\circ F})$ be the inverse Jacobian of F with respect to $\rho $ . We assume that there are constants $\unicode{x3bb}>1$ , $K, C>0$ and $\eta \in (0,1]$ such that, for any $x,y$ in a same partition element $Y_j$ :

1. (1) $F|_{Y_j}=T^{R(Y_j)}:Y_j\rightarrow Y$ is a (measure-theoretic) bijection for each j;

2. (2) $d(Fx,Fy)\geq \unicode{x3bb} d(x,y)$ ;

3. (3) $d(T^{l}x,T^{l}y)\leq C d(Fx,Fy)$ for all $0\leq l < R(Y_j)$ ; and

4. (4) $|\log \nu (x)-\log \nu (y)|\leq Kd(Fx,Fy)^{\eta }$ .

Then, such a dynamical system $T:M\rightarrow M$ is a non-uniformly expanding map. If ${R\in L^p(Y)}$ for some $p\ge 1$ , then we call $T:M\rightarrow M$ a non-uniformly expanding map of order p. It is standard that there is a unique absolutely continuous F-invariant probability measure $\mu _Y$ on Y with respect to the measure $\rho $ .

We define the Young tower as in [33, 34]. Let $\Delta :=\{(x, l):x\in Y, l=0,1,\ldots , R(x)-1\}$ , and define an extension map $f:\Delta \rightarrow \Delta $ by

$$ \begin{align*} f(x,l):= \begin{cases} (x,l+1) & \text{if } l+1<R(x),\\ (Fx,0) & \text{if } l+1=R(x). \end{cases} \end{align*} $$

We have a projection map $\pi _\Delta :\Delta \rightarrow M$ given by $\pi _\Delta (x,l):=T^lx$ and it is a semiconjugacy satisfying $T\circ \pi _\Delta =\pi _\Delta \circ f$ . Then we obtain an ergodic f-invariant probability measure $\mu _\Delta $ on $\Delta $ given by $\mu _\Delta :=\mu _Y\times m/\int _Y R\, \mathrm {d} \mu _Y$ , where m denotes the counting measure on ${\mathbb N}$ . Hence, there exists an extension space $(\Delta , \mathcal {M}, \mu _\Delta )$ , where $\mathcal {M}$ is the underlying $\sigma $ -algebra on $(\Delta , \mu _\Delta )$ . Further, the push-forward measure $\mu =(\pi _\Delta )_\ast \mu _\Delta $ is an absolutely continuous T-invariant probability measure.

Given a Hölder observable $v:M\rightarrow {\mathbb R}$ with exponent $\eta \in (0,1]$ , define

$$ \begin{align*} |v|_\infty:=\sup_{x\in M}|v(x)|, \quad |v|_\eta:=\sup_{x\neq y}\frac{|v(x)-v(y)|}{d(x,y)^{\eta}}. \end{align*} $$

Let $C^\eta (M)$ denote the Banach space of Hölder observables with norm $\|v\|_\eta =|v|_\infty +|v|_\eta <\infty .$ Consider the continuous processes $W_{n}$ defined by

(3.1) $$ \begin{align} W_{n}(t):=\frac{1}{\sqrt{n}}\bigg[\sum_{j=0}^{[nt]-1}v\circ T^j+(nt-[nt])v\circ T^{[nt]}\bigg] \quad\mathrm{for~all~} t\in[0,1], \end{align} $$

where $v\in C^\eta (M)$ with $\int _M v\, \mathrm {d} \mu =0$ . Let $v_n:=\sum _{i=0}^{n-1}v\circ T^i$ denote the Birkhoff sum.

The following lemma is a summary of known results; see [16, 21, 24, 25] for details.

Lemma 3.1. Suppose that $T:M\rightarrow M$ is a non-uniformly expanding map of order $p\ge 2$ . Let $v:M\rightarrow \mathbb {R}$ be a Hölder observable with $\int _M v \,\mathrm {d}\mu =0$ . Then the following statements hold.

1. (a) The limit $\sigma ^2=\lim _{n\rightarrow \infty }\int _M(n^{-1/2}v_n)^2\,\mathrm {d} \mu $ exists.

2. (b) $n^{-1/2}v_{n}\rightarrow _{w} G$ as $n\rightarrow \infty $ , where G is normal with mean zero and variance $\sigma ^2$ .

3. (c) $W_n\rightarrow _{w} W$ in $C[0,1]$ as $n\rightarrow \infty $ , where W is a Brownian motion with mean zero and variance $\sigma ^2$ .

4. (d) If $\mu _{Y}(R>n)=O(n^{-(\beta +1)}), \beta >1$ , then

   $$ \begin{align*} \lim_{n\rightarrow\infty}\int_{M}|n^{-{1}/{2}}v_{n}|^{q}\,\mathrm{d}\mu= {\mathbf E}|G|^{q} \quad\mathrm{for~all~} q\in[0,2\beta). \end{align*} $$

5. (e) $\|\!\max _{k\leq n}|\!\sum _{i=0}^{k-1}v\circ T^{i}|\|_{L^{2(p-1)}}\leq C\|v\|_{\eta }n^{1/2}$ for all $n\geq 1$ .

Proof. Items (a)–(c) are well known; see, e.g., [16, 21, 24]. Item (d) can be found in [25, Theorem 3.5]. For item (e), see [21, Corollary 2.10] for details.

Remark 3.2. In the case of (d), Melbourne and Török [25] gave examples to illustrate that the qth moments diverge for $q> 2\beta $ . Hence, the result on the order of convergent moments is essentially optimal.

Theorem 3.3. Let $T:M\rightarrow M$ be a non-uniformly expanding map of order $p> 2$ . Suppose that $v:M\rightarrow \mathbb {R}$ is a Hölder observable with $\int _M v\, \mathrm {d}\mu =0$ . Then ${\mathcal {W}_{q}(W_{n},W)\to 0}$ in $C[0,1]$ for all $1\le q< 2(p-1)$ .

Proof. It follows from Lemma 3.1(e) that $W_n$ has a finite moment of order $2(p-1)$ . This, together with the fact that $W_n\to _{w} W$ as $n\to \infty $ in Lemma 3.1(c), implies that, for each $q<2(p-1)$ ,

$$ \begin{align*} \lim_{n \to \infty}\mathbf{E}\sup_{t\in [0,1]}|W_n(t)|^q=\mathbf{E}\sup_{t\in [0,1]}|W(t)|^q \end{align*} $$

by [9, Theorem 4.5.2]. On the other hand, by the fact that $W_n: M\to C[0,1]$ and the definition of push-forward measures,

$$ \begin{align*} \int_{C[0,1]} d_C(x,0)^q \, \mathrm{d} \mu\circ W_n^{-1}(x)=\int_M \sup_{t\in [0,1]}|W_n(t,\omega)|^q\, \mathrm{d} \mu(\omega)=\mathbf{E}\sup_{t\in [0,1]}|W_n(t)|^q. \end{align*} $$

Similarly,

$$ \begin{align*} \int_{C[0,1]} d_C(x,0)^q \, \mathrm{d} \mu\circ W^{-1}(x)=\mathbf{E}\sup_{t\in [0,1]}|W(t)|^q. \end{align*} $$

Hence,

$$ \begin{align*} \lim_{n \to \infty}\int_{C[0,1]}d_C(x,0)^q \, \mathrm{d} \mu\circ W_n^{-1}(x)=\int_{C[0,1]}d_C(x,0)^q \, \mathrm{d} \mu\circ W^{-1}(x). \end{align*} $$

By taking $\mu _n=\mu \circ W_n^{-1}, \mu =\mu \circ W^{-1}$ and $x_0=0$ in Proposition 2.3 and the fact that $W_n\to _{w} W$ in Lemma 3.1(c), the result follows.

Theorem 3.4. Let $T:M\rightarrow M$ be a non-uniformly expanding map of order $p\ge 4$ and suppose that $v:M\rightarrow \mathbb {R}$ is a Hölder observable with $\int _M v \,\mathrm {d}\mu =0$ . Then there exists a constant $C>0$ such that $\mathcal {W}_{{p}/{2}}(W_{n},W)\leq Cn^{-1/4+1/(4(p-1))}$ for all $n\geq 1$ .

We postpone the proof of Theorem 3.4 to §4.

Remark 3.5.

1. (1) Since $\mathcal {W}_{q}\ \le \mathcal {W}_{p}$ for $q\le p$ , Theorem 3.4 provides an estimate for $\mathcal {W}_{q}(W_{n},W)$ for all $1\le q\le p/2$ , $p\ge 4$ .

2. (2) Our result implies a convergence rate $O(n^{-1/4+\delta '})$ with respect to the Lévy–Prokhorov distance, where $\delta '$ depends only on p and $\delta '$ can be arbitrarily small as $p\to ~\infty $ . Indeed, for two given probability measures $\mu $ and $\nu $ , we have $\pi (\mu ,\nu )\le \mathcal {W}_{p}(\mu ,\nu )^{{p}/({p+1})}$ ; see Proposition 2.6.

3. (3) The convergence rate in Theorem 3.4 may not be optimal. However, it is well known that one cannot get a better result than $O(n^{-1/4})$ by means of the Skorokhod embedding theorem; see [6, 30] for details.

Example 3.6. (Pomeau–Manneville intermittent maps)

A typical example of non-uniformly expanding systems with polynomial tails is the Pomeau–Manneville intermittent map [23, 27]. Consider the map $T:[0,1]\rightarrow [0,1]$ given by

(3.2) $$ \begin{align} T(x)= \begin{cases} x(1+2^\gamma x^\gamma) & \text{if } x\in\big[0,\frac{1}{2}\big),\\ 2x-1 & \text{if } x\in\big[\frac{1}{2},1\big], \end{cases} \end{align} $$

where $\gamma \ge 0$ is a parameter. When $\gamma =0$ , this is $Tx=2x $ mod $1$ , which is a uniformly expanding system. It is well known that, for each $0\le \gamma <1$ , there is a unique absolutely continuous invariant probability measure $\mu $ . By [34], for $0<\gamma <1$ , the map can be modelled by a Young tower with tails $O(n^{-{1}/{\gamma }})$ . Further for $\gamma \in [0,\tfrac 12)$ , the CLT and WIP hold for Hölder continuous observables. We restrict the parameter $\gamma \in (0,\tfrac 12)$ ; then the map is a non-uniformly expanding system of order p for any $p<{1}/{\gamma }$ . By Theorem 3.4, we obtain $\mathcal {W}_{{p}/{2}}(W_{n},W)\leq Cn^{-1/4+\gamma/(4(1-\gamma))+\delta}$ for all $\gamma \in (0,\tfrac 14)$ .

Example 3.7. (Viana maps)

Consider the Viana maps [31] $T_\alpha :S^1\times {\mathbb R}\rightarrow S^1\times {\mathbb R}$

$$ \begin{align*} T_\alpha(\omega, x)=(l\omega \text{~mod~} 1, a_0+\alpha\sin 2\pi \omega-x^2). \end{align*} $$

Here, $a_0\in (1,2)$ is chosen in such a way that $x=0$ is a preperiodic point for the map $g(x)=a_0-x^2$ , $\alpha $ is fixed to be sufficiently small and $l\in {\mathbb N}$ with $l\ge 16$ . The results in [17] show that any T close to the map $T_\alpha $ in the $C^3$ topology can be modelled by a Young tower with stretched exponential tails, which is a non-uniformly expanding map of order p for all $p\ge 1$ . Hence by Theorem 3.4, for all $p\geq 4$ , $\mathcal {W}_{{p}/{2}}(W_{n},W)\leq Cn^{-1/4+1/(4(p-1))}$ .

3.2 Non-uniformly hyperbolic diffeomorphism

In this subsection, we introduce the main results for non-uniformly hyperbolic systems in the sense of Young [33, 34]. In this case, we follow the argument in [21, 24].

Let $T:M \rightarrow M$ be a diffeomorphism (possibly with singularities†) defined on a Riemannian manifold $(M,d)$ . As in [24], consider a subset $Y\subset M$ which has a hyperbolic product structure: that is, there exist a continuous family of unstable disks $\{W^u\}$ and a continuous family of stable disks $\{W^s\}$ such that:

1. (1) $\mathrm {dim} W^s+ \mathrm {dim} W^u=\mathrm {dim} M$ ;

2. (2) each $W^u$ -disk is transversal to each $W^s$ -disk in a single point; and

3. (3) $Y=(\cup W^u)\cap (\cup W^s)$ .

For $x\in Y$ , $W^s(x)$ denotes the element in $\{W^s\}$ containing x.

Furthermore, there is a measurable partition $\{Y_j\}$ of Y such that each $Y_j$ is a union of elements in $\{W^s\}$ and a $W^u$ such that each element of $\{W^s\}$ intersects $W^u$ in one point. Defining an integrable return time $R:Y\rightarrow \mathbb {Z}^{+}$ that is constant on each partition $Y_j$ , we can get the corresponding induced map $F=T^{R}:Y\rightarrow Y$ . The separation time $s(x,y)$ is the greatest integer $n\ge 0$ such that $F^nx,F^ny$ lie in the same partition element of Y.

We assume that there exist $C>0$ and $\gamma \in (0,1)$ such that:

1. (1) $F(W^s(x))\subset W^s(Fx)$ for all $x\in Y$ ;

2. (2) $d(T^n(x),T^n(y))\le C \gamma ^n$ for all $x\in Y$ , $y\in W^s(x)$ and $n\ge 0$ ; and

3. (3) $d(T^n(x),T^n(y))\le C \gamma ^{s(x,y)}$ for $x,y\in W^u$ and $0\le n<R$ .

As for the non-uniformly expanding map, we can define a Young tower. Let ${\Delta :=\{(x,l):x\in Y, l=0,1,\ldots ,R(x)-1\}}$ and define an extension map $f:\Delta \rightarrow \Delta $ ,

$$ \begin{align*} f(x,l):= \begin{cases} (x,l+1) & \text{if } l+1<R(x),\\ (Fx,0) & \text{if } l+1=R(x). \end{cases} \end{align*} $$

We have a projection map $\pi _\Delta :\Delta \rightarrow M$ given by $\pi _\Delta (x,l):=T^lx$ and it is a semiconjugacy satisfying $T\circ \pi _\Delta =\pi _\Delta \circ f$ .

Let $\bar {Y}=Y/\thicksim $ , where $y\thicksim y'$ if $y'\in W^s(y)$ ; denote by $\bar {\pi }:Y\rightarrow \bar {Y}$ the natural projection. We can also obtain a partition $\{\bar {Y}_j\}$ of $\bar {Y}$ , a well-defined return time $\bar {R}:\bar {Y}\rightarrow \mathbb {Z}^{+}$ and a corresponding induced map $\bar {F}:\bar {Y}\rightarrow \bar {Y}$ , as in the case of Y. In addition, we assume that:

1. (1) $\bar {F}|_{\bar {Y}_j}=\bar {T}^{\bar {R}(\bar {Y}_j)}:\bar {Y}_j\rightarrow \bar {Y}$ is a bijection for each j; and

2. (2) $\nu _0={d\bar {\rho }}/({d\bar {\rho }\circ \bar {F}})$ satisfies $|\log \nu _0(y)-\log \nu _0(y')|\le K\gamma ^{s(y,y')} $ , for all $y,y'\in \bar {Y}_j$ , where $\bar {\rho }=\bar {\pi }_\ast \rho $ with $\rho $ being the Riemannian measure.

Let $\bar {f}:\bar {\Delta }\rightarrow \bar {\Delta }$ denote the corresponding extension map. The projection $\bar {\pi }:Y\rightarrow \bar {Y}$ extends to the projection $\bar {\pi }:\Delta \rightarrow \bar {\Delta }$ ; here we use the same notation $\bar {\pi }$ , which should not cause confusion. There exist an $\bar {f}$ -invariant probability measure $\bar {\mu }$ on $\bar {\Delta }$ and an f-invariant probability measure $\mu _{\Delta }$ on $\Delta $ , such that $\bar {\pi }:\Delta \rightarrow \bar {\Delta }$ and $\pi _\Delta :\Delta \rightarrow M$ are measure preserving.

Theorem 3.8. Let $T:M\rightarrow M$ be a non-uniformly hyperbolic transformation of order $p> 2$ . Suppose that $v:M\rightarrow \mathbb {R}$ is a Hölder observable with $\int _M v \,\mathrm {d}\mu =0$ . Then $\mathcal {W}_{q}(W_{n},W)\to 0$ in $C[0,1]$ for all $1\le q< 2(p-1)$ .

Proof. By [21, Corollary 5.5], $W_n$ has a finite moment of order $2(p-1)$ . The remaining proof is similar to that of Theorem 3.3.

Theorem 3.9. Let $T:M\rightarrow M$ be a non-uniformly hyperbolic transformation of order $p\ge 4$ and suppose that $v:M\rightarrow \mathbb {R}$ is a Hölder observable with $\int _M v \,\mathrm {d}\mu =0$ . Then there exists a constant $C>0$ such that $\mathcal {W}_{{p}/{2}}(W_{n},W)\leq Cn^{-1/4+{1}/{(4(p-1))}}$ for all $n\geq 1$ .

We postpone the proof of Theorem 3.9 to the next section.

Example 3.10. (Non-uniformly expanding/hyperbolic systems with exponential tails)

In this case, the return time $R\in L^p$ for all p. Hence, for all $p\geq 4$ , $\mathcal {W}_{{p}/{2}}(W_{n},W)\leq Cn^{-1/4+1/(4(p-1))}$ . Specific examples are:

• • some partially hyperbolic systems with a mostly contracting direction [7, 12];

• • unimodal maps and multimodal maps as in [20] for a fixed system; and

• • Hénon-type attractors [19]. Let $T_{a,b}:{\mathbb R}^2\rightarrow {\mathbb R}^2$ be defined by $T_{a,b}(x,y)= (1-ax^2+y, bx)$ for $a<2$ , $b>0$ , where b is small enough depending on a. It follows from [3, 4] that T admits an Sinai–Ruelle–Bowen (SRB) measure and T can be modelled by a Young tower with exponential tails.

4 Proof of Theorems 3.4 and 3.9

4.1 Martingale approximation

The martingale approximation method [14] is one of the main methods for studying statistical limit properties. In [21], Korepanov et al obtained a new version of martingale-coboundary decomposition, which is applicable to non-uniformly hyperbolic systems. In this subsection, we recall some required properties in [21].

Proposition 4.1. Let $T:M\rightarrow M$ be a non-uniformly expanding map of order $p\ge 1$ and suppose that $v:M\rightarrow \mathbb {R}$ is a Hölder observable with $\int _M v\,\mathrm{d}\mu =0$ . Then there is an extension $f:\Delta \rightarrow \Delta $ of T such that, for any $v\in C^\eta (M)$ , there exist $m\in L^{p}(\Delta )$ and $\chi \in L^{p-1}(\Delta )$ with

$$ \begin{align*} v\circ\pi_\Delta=m+\chi\circ f-\chi,\quad \mathbf{E}(m|f^{-1}\mathcal{M})=0. \end{align*} $$

Moreover, there is a constant $C>0$ such that, for all $v\in C^\eta (M)$ ,

$$ \begin{align*} \|m\|_{L^p}\le C\|v\|_{\eta},\quad \|\chi\|_{L^{p-1}}\le C\|v\|_{\eta} \end{align*} $$

and, for $n\geq 1$ ,

$$ \begin{align*} \Big\|\!\max_{0\le j\le n}|\chi\circ f^j-\chi|\Big\|_{L^p}\le C\|v\|_{\eta}n^{1/p}. \end{align*} $$

Proof. The proposition is a summary of Propositions 2.4, 2.5 and 2.7 in [21].

Proposition 4.2. Fix $n\ge 1$ . Then $\{m\circ f^{n-i},f^{-(n-i)}\mathcal {M};1\le i\le n\}$ is a martingale difference sequence.

Proof. See, for example [21, Proposition 2.9].

Proposition 4.3. If $p\geq 2$ , then $\|\!\max _{k\leq n}|\!\sum _{i=1}^{k}m\circ f^{n-i}|\|_{L^{p}}\leq C\|m\|_{L^p}n^{1/2}$ for all $n\ge 1$ .

Proof. See the proof in [21, Corollary 2.10].

4.2 Proof of Theorem 3.4

Define

$$ \begin{align*} \zeta_{n,j}:=\frac{1}{\sqrt{n}\sigma}m\circ f^{n-j},\quad \mathcal{F}_{n,j}:=f^{-(n-j)}\mathcal{M} \quad\mathrm{for}\ 1\le j\le n. \end{align*} $$

For $1\le l\le n$ , define the conditional variance

$$ \begin{align*} V_{n,l}:=\sum_{j=1}^{l}\mathbf{E}(\zeta_{n,j}^2|\mathcal{F}_{n,j-1}). \end{align*} $$

We set $V_{n,0}=0$ .

Define the stochastic process $X_n$ with sample paths in $C[0,1]$ by

(4.1) $$ \begin{align} X_{n}(t):=\sum_{j=1}^{k}\zeta_{n,j}+\frac{tV_{n,n}-V_{n,k}}{V_{n,k+1}-V_{n,k}}\zeta_{n,k+1} \quad \textrm{if } V_{n,k}\leq tV_{n,n}<V_{n,k+1}. \end{align} $$

Step 1. Estimate of the Wasserstein distance between ${X}_{{n}}$ and $B$ . Let B be a standard Brownian motion, that is, $B=_d1/\sigma W$ .

Lemma 4.4. Let $p\ge 4$ . Then, for any $\delta>0$ , there exists a constant $C>0$ such that $\mathcal {W}_{{p}/{2}}(X_{n},B)\leq C n^{-(1/4-\delta )}$ for all $n\geq 1$ .

Proof. (1) Fix $n>0$ . It suffices to deal with a single row of the array $\{\zeta _{n,j},\mathcal {F}_{n,j}, 1\le j\le n\}$ . By the Skorokhod embedding theorem (see Theorem A.1), there exists a probability space (depending on n) supporting a standard Brownian motion, still denoted by B, which should not cause confusion, and a sequence of non-negative random variables $\tau _1,\ldots , \tau _n$ such that, for $T_i=\sum _{j=1}^{i}\tau _j$ , we have $\sum _{j=1}^{i}\zeta _{n,j}=B(T_i)$ with $1\le i\le n$ . In particular, we set $T_{0}=0$ . Then, on this probability space and for this Brownian motion, we aim to show that, for any $\delta>0$ , there exists a constant $C>0$ such that

$$ \begin{align*} \Big\|\!\sup_{t\in[0,1]}|X_{n}(t)-B(t)|\Big\|_{L^{{p}/{2}}}\leq Cn^{-({1}/{4}-\delta)} \quad \mathrm{for~all~}n\ge1. \end{align*} $$

Thus, the result follows from Definition 2.1.

For ease of exposition when there is no ambiguity, we will write $\zeta _j$ and $V_k$ instead of $\zeta _{n,j}$ and $V_{n,k}$ , respectively. Then, by (4.1),

(4.2) $$ \begin{align} X_{n}(t)=B(T_{k})+\bigg(\frac{tV_{n}-V_{k}}{V_{k+1}-V_{k}}\bigg)(B(T_{k+1})-B(T_{k}))\quad \mathrm{if}\ V_{k}\leq tV_{n}<V_{k+1}. \end{align} $$

(2) Note that Theorem A.1(3) implies that

$$ \begin{align*} T_k-V_k=\sum_{i=1}^{k}(\tau_{i}-\mathbf{E}(\tau_{i}|\mathcal{B}_{i-1})) \quad \mathrm{if}\ 1\le k\le n, \end{align*} $$

where $\mathcal {B}_{i}$ is the $\sigma $ -field generated by all events up to $T_i$ for $1\le i\le n$ . Therefore, $\{T_k-V_k, \mathcal {B}_{k}, 1\le k\le n\}$ is a martingale. By the Burkholder inequality and the conditional Jensen inequality, for all $p\ge 4$ ,

$$ \begin{align*} \Big\|\!\max_{1\le k\le n}|T_k-V_{k}|\Big\|_{L^{{p}/{2}}} &\le Cn^{{1}/{2}} \max_{1\le k\le n}\|\tau_{k}-\mathbf{E}(\tau_{k}|\mathcal{B}_{k-1})\|_{L^{{p}/{2}}}\\ &\le Cn^{{1}/{2}} \max_{1\le k\le n}\|\tau_{k}\|_{L^{{p}/{2}}}. \end{align*} $$

It follows from Theorem A.1(4) that $\mathbf {E}(\tau _{k}^{p/2})\le 2\Gamma ( {p}/{2}+1)\mathbf {E}(\zeta _{k}^{p})$ for each k. So

(4.3) $$ \begin{align} \Big\|\!\max_{1\le k\le n}|T_k-V_{k}|\Big\|_{L^{{p}/{2}}} \le Cn^{{1}/{2}} \max_{1\le k\le n}\|\zeta_{k}\|_{L^p}^2= Cn^{-{1}/{2}}\|m\|_{L^p}^{2}. \end{align} $$

On the other hand, it follows from [2, Proposition 4.1] that

(4.4) $$ \begin{align} \|V_{n}-1\|_{L^{{p}/{2}}}\le C n^{-{1}/{2}}\|v\|_{\eta}^{2}. \end{align} $$

(3) Based on the above estimates, by Chebyshev’s inequality,

(4.5) $$ \begin{align} \begin{aligned} \mu(|T_{n}-1|>1)&\le \mathbf{E}|T_{n}-1|^{{p}/{2}} \le 2^{p/2-1}\{\mathbf{E}|T_{n}-V_{n}|^{{p}/{2}}+\mathbf{E}|V_{n}-1|^{{p}/{2}}\}\\ &\le Cn^{-{p}/{4}}(\|m\|_{L^p}^{p}+\|v\|_{\eta}^{p}).\\ \end{aligned} \end{align} $$

According to the Hölder inequality, (4.5) and Proposition 4.3, we deduce that

$$ \begin{align*} I:&=\Big\| 1_{\{|T_{n}-1|>1\}}\sup_{t\in[0,1]}|X_{n}(t)-B(t)|\Big\|_{L^{{p}/{2}}}\\ &\le (\mu(|T_{n}-1|>1))^{1/p}\Big\| \sup_{t\in[0,1]}|X_{n}(t)-B(t)|\Big\|_{L^{p}}\\ &\le (\mu(|T_{n}-1|>1))^{1/p}\Big(\Big\|\!\sup_{t\in[0,1]}|X_{n}(t)|\Big\|_{L^{p}}+\Big\|\!\sup_{t\in[0,1]}|B(t)|\Big\|_{L^{p}}\Big)\\ &\le Cn^{-{1}/{4}}. \end{align*} $$

(4) We now estimate $|X_{n}-B|$ on the set $\{|T_{n}-1|\le 1\}$ : that is,

$$ \begin{align*} &\Big\| 1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|X_{n}(t)-B(t)|\Big\|_{L^{{p}/{2}}}\\ &\quad\le \Big\| 1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|X_{n}(t)-B(T_{k})|\Big\|_{L^{p/2}}+\Big\|1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|B(T_{k})-B(t)|\Big\|_{L^{{p}/{2}}}\\ &\quad =: I_1 + I_2. \end{align*} $$

For $I_1$ , it follows from (4.2) that

$$ \begin{align*} \sup_{t\in[0,1]}|X_{n}(t)-B(T_{k})|\le \max_{0\le k\le n-1}|B(T_{k+1})-B(T_{k})|=\max_{0\le k\le n-1}|\zeta_{k+1}|. \end{align*} $$

By Proposition A.2,

$$ \begin{align*} I_1&=\Big\| 1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|X_{n}(t)-B(T_{k})|\Big\|_{L^{p}}\nonumber\\ &\le \Big\| 1_{\{|T_{n}-1|\le 1\}}\max_{0\le k\le n-1}|\zeta_{k+1}|\Big\|_{L^{p}}\nonumber\\ &\le \Big\| \max_{0\le k\le n-1}|\zeta_{k+1}|\Big\|_{L^{p}}\nonumber\\ &\le Cn^{-({1}/{2}-{1}/{p})}. \end{align*} $$

(5) We now consider $I_{2}$ on the set $\{|T_{n}-1|\le 1\}$ . Take $p_1>p$ . Then it is well known that

(4.6) $$ \begin{align} \mathbf{E}|B(t)-B(s)|^{p_1}\le c|t-s|^{{p_1}/{2}} \quad \text{for all~} s,t\in [0,2]. \end{align} $$

So, it follows from Kolmogorov’s continuity theorem that, for each $0<\gamma <1/2-{1}/ ({p_1})$ , the process $B(\cdot )$ admits a version, still denoted by B, such that, for almost all $\omega $ , the sample path $t\mapsto B(t,\omega )$ is Hölder continuous with exponent $\gamma $ and

$$ \begin{align*} \Big\|\!\sup_{s,t\in[0,2]\atop s\neq t}\frac{|B(s)-B(t)|}{|s-t|^{\gamma}}\Big\|_{L^{p_1}}< \infty. \end{align*} $$

In particular,

(4.7) $$ \begin{align} \Big\|\!\sup_{s,t\in[0,2]\atop s\neq t}\frac{|B(s)-B(t)|}{|s-t|^{\gamma}}\Big\|_{L^{p}}< \infty. \end{align} $$

As for $|T_{k}-t|$ ,

$$ \begin{align*} \sup_{t\in[0,1]}|T_{k}-t|&\le \max_{0\le k\le n-1}\sup_{t\in[{V_{k}}/{V_{n}},{V_{k+1}}/{V_{n}})}|T_{k}-t|\\ &\le \max_{0\le k\le n-1}\bigg|T_{k}-\frac{V_{k}}{V_{n}}\bigg|+\max_{0\le k\le n-1}\sup_{t\in[{V_{k}}/{V_{n}},{V_{k+1}}/{V_{n}})}\bigg|\frac{V_{k}}{V_{n}}-t\bigg|\\ &\le \max_{0\le k\le n}\bigg|T_{k}-\frac{V_{k}}{V_{n}}\bigg|+\max_{0\le k\le n-1}\bigg|\frac{V_{k+1}}{V_{n}}-\frac{V_{k}}{V_{n}}\bigg|\\ &\le \max_{0\le k\le n}|T_{k}-V_{k}|+\max_{0\le k\le n}\bigg|V_{k}-\frac{V_{k}}{V_{n}}\bigg|+\max_{0\le k\le n-1}\bigg|\frac{V_{k+1}}{V_{n}}-V_{k+1}\bigg|\\ &\quad +\max_{0\le k\le n-1}|V_{k+1}-V_{k}|+\max_{0\le k\le n-1}\bigg|V_{k}-\frac{V_{k}}{V_{n}}\bigg|\\ &\le \max_{0\le k\le n}|T_{k}-V_{k}| + 3\max_{0\le k\le n}\bigg|V_{k}-\frac{V_{k}}{V_{n}}\bigg| +\max_{0\le k\le n-1}|V_{k+1}-V_{k}|. \end{align*} $$

Note that $T_{0}=V_{0}=0$ and $\gamma \le 1$ , so

$$ \begin{align*} \sup_{t\in[0,1]}|T_{k}-t|^{\gamma}\le \max_{1\le k\le n}|T_{k}-V_{k}|^{\gamma} + 3^\gamma\max_{1\le k\le n}\bigg|V_{k}-\frac{V_{k}}{V_{n}}\bigg|^{\gamma} +\max_{0\le k\le n-1}|V_{k+1}-V_{k}|^{\gamma}. \end{align*} $$

Hence,

(4.8) $$ \begin{align} &\Big\|\!\sup_{t\in[0,1]}|T_{k}-t|^{\gamma}\Big\|_{L^{p}}\nonumber\\ &\quad\le \Big\|\!\max_{1\le k\le n}|T_{k}-V_{k}|\Big\|_{L^{\gamma p}}^{\gamma} +3^\gamma \bigg\|\!\max_{1\le k\le n}\bigg|V_{k}-\frac{V_{k}}{V_{n}}\bigg|\bigg\|_ {L^{\gamma p}}^{\gamma} +\Big\|\!\max_{0\le k\le n-1}|V_{k+1}-V_{k}|\Big\|_ {L^{\gamma p}}^{\gamma}. \end{align} $$

For the first term, since $\gamma < \tfrac 12$ , it follows from (4.3) that

(4.9) $$ \begin{align} \Big\|\!\max_{1\le k\le n}|T_{k}-V_{k}|\Big\|_{L^{\gamma p}}^{\gamma}\le Cn^{-{\gamma}/{2}}. \end{align} $$

For the second term, since $|V_{k}-{V_{k}}/{V_{n}}|=V_k|1-1/{V_n}|$ ,

$$ \begin{align*}\max_{1\le k\le n}\bigg|V_{k}-\frac{V_{k}}{V_{n}}\bigg|=V_n \bigg|1-\frac1{V_n}\bigg|=|V_{n}-1|.\end{align*} $$

Hence, by (4.4),

(4.10) $$ \begin{align} \bigg\|\!\max_{1\le k\le n}\bigg|V_{k}-\frac{V_{k}}{V_{n}}\bigg|\bigg\|_ {L^{\gamma p}}^{\gamma} =\|V_{n}-1\|_{L^{\gamma p}}^{\gamma}\le Cn^{-{\gamma}/{2}}. \end{align} $$

As for the last term, note that $|V_{k}-V_{k-1}|=\mathbf {E}(\zeta _{k}^2|\mathcal {F}_{k-1}) =\mathbf {E}(({1}/{n\sigma ^2})m^2|f^{-1}\mathcal {M})\circ f^{n-k}$ for all $1\le k\le n$ . So,

(4.11) $$ \begin{align} \Big\|\!\max_{0\le k\le n-1}|V_{k+1}-V_{k}|\Big\|_ {L^{\gamma p}}^{\gamma}=\bigg\|\!\max_{1\le k\le n}\bigg|\mathbf{E}\bigg(\frac{m^2}{n\sigma^2}|f^{-1}\mathcal{M}\bigg)\circ f^{n-k}\bigg|\bigg\|_{L^{\gamma p}}^{\gamma} \le Cn^{-(\gamma-{2\gamma}/{p})}, \end{align} $$

where the inequality follows from Proposition A.2.

Based on the above estimates (4.9)–(4.11),

(4.12) $$ \begin{align} \Big\|\!\sup_{t\in[0,1]}|T_{k}-t|^{\gamma}\Big\|_{L^{p}} \le C(n^{-{\gamma}/{2}}+n^{-(\gamma-{2\gamma}/{p})}) \le C n^{-{\gamma}/{2}}, \end{align} $$

where the last inequality holds since $\gamma < \tfrac 12$ , $1-{2}/{p}\ge \tfrac 12$ .

On the set $\{|T_{n}-1|\le 1\}$ , note that

$$ \begin{align*} \sup_{t\in[0,1]}|B(T_{k})-B(t)|\le \bigg[\,\sup_{s,t\in[0,2]\atop s\neq t}\frac{|B(s)-B(t)|}{|s-t|^{\gamma}}\bigg]\Big[\sup_{t\in[0,1]}|T_{k}-t|^{\gamma}\Big]. \end{align*} $$

Since $0<\gamma <\tfrac 12-{1}/({p_1})$ , by the Hölder inequality, (4.7) and (4.12),

$$ \begin{align*} I_2&=\Big\|1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|B(T_{k})-B(t)|\Big\|_{L^{{p}/{2}}}\\ &\le \bigg\|\bigg[\!\sup_{s,t\in[0,2]\atop s\neq t}\frac{|B(s)-B(t)|}{|s-t|^{\gamma}}\bigg]\Big[\!\sup_{t\in[0,1]}|T_{k}-t|^{\gamma}\Big]\bigg\|_{L^{{p}/{2}}}\\ &\le \bigg\|\!\sup_{s,t\in[0,2]\atop s\neq t}\frac{|B(s)-B(t)|}{|s-t|^{\gamma}}\bigg\|_{L^{p}}\Big\|\!\sup_{t\in[0,1]}|T_{k}-t|^{\gamma}\Big\|_{L^{p}}\\ &\le C n^{-{\gamma}/{2}}. \end{align*} $$

Note that $p_1$ can be taken arbitrarily large in (4.6), which implies that $\gamma $ can be chosen sufficiently close to $\tfrac 12$ . So, for any $\delta>0$ , we can choose $p_1$ large enough such that $I_2\le Cn^{-1/4+\delta }$ . The result now follows from the above estimates for $I,I_1$ and $I_2$ .

Step 2. Estimate of the convergence rate between ${W}_{{n}}$ and ${X}_{{n}}$ . The proof is almost identical to that in [2, §4.1], so we only sketch it here.

Proposition 4.5. [2, Proposition 4.6]

For $n\ge 1$ , define

$$ \begin{align*} Z_n:=\max_{0\le i,l\le\sqrt{n}}\bigg|\!\sum_{j=i\sqrt{n}}^{i\sqrt{n}+l-1} v\circ T^j\bigg|. \end{align*} $$

Then:

1. (a) $|\!\sum _{j=a}^{b-1}v\circ T^j|\le Z_n((b-a)(n^{1/2}-1)^{-1}+3)$ for all $0\le a<b \le n$ ; and

2. (b) $\|Z_n\|_{L^{2(p-1)}}\le C\|v\|_{\eta }n^{1/4+1/(4(p-1))}$ for all $n\ge 1$ .

Define a continuous transformation $g:C[0,1]\rightarrow C[0,1]$ by $g(u)(t):= u(1)- u (1-t)$ .

Lemma 4.6. Let $p>2$ . Then there exists a constant $C>0$ such that $\mathcal {W}_{p-1}(g\circ W_{n}\circ \pi _\Delta ,\sigma X_{n})\leq Cn^{-1/4+{1}/({4(p-1)})}$ for all $n\geq 1$ , recalling that $\pi _\Delta :\Delta \to M$ is the projection map.

Proof. Since $\mathcal {W}_{p-1}(g\circ W_{n}\circ \pi _\Delta ,\sigma X_{n})\le \|\!\sup _{t\in [0,1]}|g\circ W_{n}(t)\circ \pi _\Delta -\sigma X_n(t)|\|_{L^{p-1}}$ , following the proof of [2, Lemma 4.7], we can obtain the conclusion.

Proof of Theorem 3.4

Note that $g\circ g= \mathrm {Id}$ and g is Lipschitz with $\mathrm {Lip}g \le 2$ . It follows from Proposition 2.4 that

$$ \begin{align*} \mathcal{W}_{{p}/{2}}(W_{n},W)= \mathcal{W}_{{p}/{2}}(g(g\circ W_{n}),g(g\circ W))\le 2\mathcal{W}_{{p}/{2}}(g\circ W_{n},g\circ W). \end{align*} $$

Since $\pi _\Delta $ is a semiconjugacy, $W_n \circ \pi _\Delta =_d W_n$ . Also, $g(W)=_d W=_d\sigma B$ . By Lemmas 4.4 and 4.6, for $p\ge 4$ ,

$$ \begin{align*} \mathcal{W}_{{p}/{2}}(g\circ W_{n},g\circ W)&=\mathcal{W}_{{p}/{2}}(g\circ W_{n}\circ \pi_\Delta ,W)\\&\le \mathcal{W}_{{p}/{2}}(g\circ W_{n}\circ \pi_\Delta ,\sigma X_n)+\mathcal{W}_{{p}/{2}}(\sigma X_n ,\sigma B)\\&\le Cn^{-1/4+1/(4(p-1))}+Cn^{-1/4+\delta}\le Cn^{-1/4+1/(4(p-1))}, \end{align*} $$

where the last inequality holds because $\delta>0$ can be taken arbitrarily small.

4.3 Proof of Theorem 3.9

The proof is based on the following Lemma 4.7 which is presented in detail in [21, §5].

Lemma 4.7. Let $p\ge 1$ , $\eta \in (0,1]$ . Suppose that $T:M\rightarrow M$ is a non-uniformly hyperbolic transformation with the return time $R\in L^p$ and that $v:M\rightarrow \mathbb {R}$ is a Hölder observable. Then:

1. (1) $\bar {f}:\bar {\Delta }\rightarrow \bar {\Delta }$ is a non-uniformly expanding map of order p; and

2. (2) there exists $\theta \in (0,1)$ such that, for all $v\in C^\eta (M)$ , there exist $\phi \in C^\theta (\bar {\Delta })$ and ${\psi \in L^\infty (\Delta )}$ such that $v\circ \pi _\Delta =\phi \circ \bar {\pi }+\psi -\psi \circ f$ . Moreover, $|\psi |_\infty \le C\|v\|_\eta $ , $\|\phi \|_\theta \le C\|v\|_\eta $ .

Proof of Theorem 3.9

As the definition of $W_n$ in (3.1), define $\overline {W}_n(t)\kern1.2pt{:=}\kern1.2pt({1}/{\sqrt {n}})\sum _{j=0}^{nt-1}\phi \kern1.2pt{\circ} \bar {f}^j$ for $t={j}/{n}, 1\le j\le n$ , and linearly interpolate to obtain the process ${\overline {W}_n\in C[0,1]}$ , where $\phi $ is from Lemma 4.7. By Lemma 4.7, we have $|\overline {W}_n(t)\circ \bar {\pi }-W_n\circ \pi _\Delta |_\infty \le Cn^{-1/2}|\psi |_\infty $ by simple computations. Since $\bar {\pi }, \pi _\Delta $ are semiconjugacies, $\mathcal {W}_{{p}/{2}}(\overline {W}_n,\kern-0.5pt W_{n})\kern1pt{=}\mathcal {W}_{{p}/{2}}(\overline {W}_n\circ \bar {\pi },W_n\circ \pi _\Delta )\le Cn^{-1/2}$ . It follows from Lemma 4.7 that $\bar {f}$ is a non-uniformly expanding map of order p and that $\phi $ is a Hölder continuous observable with $\int _{\bar {\Delta }} \phi \, \mathrm {d}\bar {\mu }=0$ . By Theorem 3.4, for $p\ge 4$ , $\mathcal {W}_{{p}/{2}}(\overline {W}_{n},W)\le Cn^{-1/4+1/(4(p-1))}$ . Hence, $\mathcal {W}_{{p}/{2}}(W_{n},W)\le \mathcal {W}_{{p}/{2}}(W_{n},\overline {W}_n)+ \mathcal {W}_{{p}/{2}}(\overline {W}_{n},W)\le Cn^{-1/4+1/(4(p-1))}$ .

5 Application to homogenization problem

We consider fast–slow systems of the discrete form

(5.1) $$ \begin{align} x_\epsilon(n+1)=x_\epsilon(n)+\epsilon^2g(x_\epsilon(n),y(n),\epsilon)+\epsilon h(x_\epsilon(n))v(y(n)),\quad ~x_\epsilon(0)=\xi, \end{align} $$

where $g:\mathbb {R}\times M\times {\mathbb R}^+\rightarrow \mathbb {R}$ , $h:\mathbb {R}\rightarrow \mathbb {R}$ satisfy some regularity conditions and ${v\in C^\eta (M)}$ with $\int _M v \,\mathrm {d}\mu =0$ . The fast variables $y(n)\in M$ are generated by iterating a non-uniformly expanding map: that is $y(n+1)=Ty(n)$ , $y(0)=y_0$ . Here, $T:M\to M$ satisfies the setting in §3.1 The initial condition $\xi \in {\mathbb R}$ is fixed and $y_0\in M$ is chosen randomly, which is the reason for the emergence of randomness from deterministic dynamical systems.

We have the following regularity conditions.

1. (1) $ g:\mathbb {R}\times M\times {\mathbb R}^+\rightarrow \mathbb {R}$ is bounded.

2. (2) $ g(x,y,0)$ is Lipschitz in x uniformly in y with Lipchitz constant L; that is, $|g(x_1,y,0)-g(x_2,y,0)|\le L|x_1-x_2|$ for all $x_1,x_2\in \mathbb {R}, y\in M$ .

3. (3) $\sup _{x\in {\mathbb R}}\sup _{y\in M}|g(x,y,\epsilon )-g(x,y,0)|\le C\epsilon ^{1/3}$ .

4. (4) $ g(x,y,0)$ is Hölder continuous in y uniformly in x: that is, $\sup _{x\in {\mathbb R}}|g(x,\cdot ,0)|_\eta <\infty $ .

5. (5) h is exact; that is, $h={1}/{\psi '}$ , where $\psi $ is a monotone differentiable function and $\psi '$ denotes the derived function. Moreover, $h, h',h", {1}/{h}$ are bounded.

Let ${\hat x}_\epsilon (t)=x_\epsilon (t\epsilon ^{-2})$ for $t=0, \epsilon ^2, 2\epsilon ^2,\ldots, $ and linearly interpolate to obtain ${\hat x}_\epsilon \in C[0,1]$ . Then it follows from [15, Theorem 1.3] that ${\hat x}_\epsilon \rightarrow _{w} X$ in $C[0,1]$ , where X is the solution to the Stratonovich stochastic differential equation (SDE)

(5.2) $$ \begin{align} \mathrm{d}X=\bigg\{ \bar{g}(X)-\frac{1}{2}h(X)h'(X)\int_{M}v^{2}\,\mathrm{d}\mu\bigg\}\mathrm{d}t+h(X)\circ \mathrm{d}W,\quad ~X(0)=\xi. \end{align} $$

Here, W is a Brownian motion with mean zero and variance $\sigma ^2$ and $\bar {g}(x)=\int _{M}g(x,y,0)\,\mathrm {d}\mu (y)$ .

Define $W_\epsilon (t)=\epsilon \sum _{j=0}^{t\epsilon ^{-2}-1}v(y(j))$ for $t=0, \epsilon ^2, 2\epsilon ^2,\ldots ,$ and linearly interpolate to obtain $W_\epsilon \in C[0,1]$ . Comparing $W_\epsilon $ with $W_n$ , we can see that $\epsilon $ is identified with $n^{-1/2}$ . Hence, it follows from Theorem 3.4 that $\mathcal {W}_{{p}/{2}}(W_\epsilon ,W)=O(\epsilon ^{({p-2})/{2(p-1)}})$ .

Theorem 5.1. Let $T:M\rightarrow M$ be a non-uniformly expanding map of order $p\ge 4$ . Suppose that the regularity conditions hold. Then there exists a constant $C>0$ such that

$$ \begin{align*} \mathcal{W}_{{p}/{2}}({\hat x}_\epsilon, X)\le \begin{cases} C\epsilon^{({p-2})/{2p}}\quad & \text{if }4\le p\le 6,\\ C\epsilon^{{1}/{3}}(-\log\epsilon)^{{1}/{4}}\quad & \text{if }p>6. \end{cases} \end{align*} $$

Proof. The proof follows from the argument in [2]. First, suppose that $h(x)\equiv 1$ and let $N=[\epsilon ^{-4/3}]$ . By [2, Proposition 5.4], we can write

$$ \begin{align*} {\hat x}_\epsilon(t)=\xi+W_\epsilon(t)+D_\epsilon(t)+E_\epsilon(t)+\int_{0}^{t}\bar{g}({\hat x}_\epsilon(s))\,\mathrm{d} s, \end{align*} $$

where

$$ \begin{align*} &D_\epsilon(t)=\epsilon^{{2}/{3}}\sum_{n=0}^{[t\epsilon^{-{2}/{3}}]-1}J_{\epsilon}(n), \quad J_\epsilon(n)=\epsilon^{{4}/{3}}\sum_{j=nN}^{(n+1)N-1}\tilde {g}(x_\epsilon(nN),y(j)),\\ &\qquad\qquad\quad\qquad\qquad \tilde {g}(x,y)=g(x,y,0)-\bar g(x) \end{align*} $$

and

(5.3) $$ \begin{align} \Big\|\!\sup_{t\in[0,1]}|E_\epsilon(t)|\Big\|_{L^{2(p-1)}}\le C\epsilon^{{1}/{3}}. \end{align} $$

Let $B_\epsilon (R_\epsilon )=\{\sup _{t\in [0,1]}|\hat {x}_{\epsilon }(t)|\le R_\epsilon \}$ , where $R_\epsilon =(-32\sigma ^2\log \epsilon )^{1/2}$ . By [2, Lemma 5.5],

$$ \begin{align*} \mu\Big(\sup_{t\in[0,1]}|\hat{x}_{\epsilon}(t)|\ge R_\epsilon\Big)\le C\epsilon^{({p-2})/{2p}}. \end{align*} $$

Since

$$ \begin{align*} \mu\Big(\sup_{t\in[0,1]}|D_\epsilon 1_{B^c_\epsilon(R_\epsilon)}(t)|>0\Big)\le \mu\Big(\sup_{t\in[0,1]}|\hat{x}_{\epsilon}(t)|\ge R_\epsilon\Big)\le C\epsilon^{({p-2})/{2p}} \end{align*} $$

and $D_\epsilon $ is bounded, we get

$$ \begin{align*} \Big\|\!\sup_{t\in[0,1]}|D_\epsilon 1_{B^c_\epsilon(R_\epsilon)}(t)|\Big\|_{L^\infty}\le C\epsilon^{({p-2})/{2p}}. \end{align*} $$

Moreover, it follows from [2, Lemma 5.6] that $\|\!\sup _{t\in [0,1]}|D_\epsilon 1_{B_\epsilon (R_\epsilon )}(t)|\|_{L^{2(p-1)}}\le C\epsilon ^{1/3}(-\log \epsilon )^{1/4}$ . Hence,

(5.4) $$ \begin{align} \Big\|\!\sup_{t\in[0,1]}|D_\epsilon(t)|\Big\|_{L^{2(p-1)}}= O(\epsilon^{({p-2})/{2p}}+\epsilon^{{1}/{3}}(-\log\epsilon)^{{1}/{4}}). \end{align} $$

Next, define a continuous map $G:C[0,1]\rightarrow C[0,1]$ as $G(u)=v$ , where v is the unique solution to $v(t)=\xi +u(t)+\int _{0}^{t}\bar {g}(v(s))\,\mathrm {d} s$ . Since $\bar {g}$ is Lipschitz, according to the existence and uniqueness of solutions to ordinary differential equations, G is well defined. By Gronwall’s inequality, G is Lipschitz with $\mathrm {Lip}G\le e^{\mathrm {Lip} \bar g}$ .

Since $X=G(W)$ and ${\hat x}_\epsilon =G(W_\epsilon +D_\epsilon +E_\epsilon )$ ,

$$ \begin{align*} \mathcal{W}_{{p}/{2}}({\hat x}_\epsilon, X)=\mathcal{W}_{{p}/{2}}(G(W_\epsilon+D_\epsilon+E_\epsilon), G(W))\le e^{\mathrm{Lip} \bar g}\mathcal{W}_{{p}/{2}}(W_\epsilon +D_\epsilon+E_\epsilon, W). \end{align*} $$

Following $\mathcal {W}_{{p}/{2}}(W_\epsilon ,W)=O(\epsilon ^{({p-2})/({2(p-1)})})$ and the above estimates (5.3)–(5.4), we obtain that $\mathcal {W}_{{p}/{2}}({\hat x}_\epsilon , X)\le C(\epsilon ^{({p-2})/{2p}}+\epsilon ^{1/3}(-\log \epsilon )^{1/4})$ .

When $h\not \equiv 1$ , by a change of variables, $z_\epsilon (n)=\psi (x_\epsilon (n))$ , ${\hat z}_\epsilon (t)=\psi ({\hat x}_\epsilon (t))$ , we can reduce the case of multiplicative noise to the case of additive noise: that is,

$$ \begin{align*} z_\epsilon(n+1)-z_\epsilon(n)=\epsilon v(y(n))+\epsilon^2G(z_\epsilon(n),y(n),\epsilon),\quad z_\epsilon(0)=\psi(\xi), \end{align*} $$

where $G(z,y,\epsilon ):=\psi '(\psi ^{-1}z)g(\psi ^{-1}z,y,\epsilon )+\tfrac 12\psi "(\psi ^{-1}z) (\psi '(\psi ^{-1}z))^{-2}v^2(y)+O(\epsilon )$ ; see [15, 21] for the calculations. Moreover, we can verify that $G(z,y,\epsilon )$ satisfies the regularity conditions (1)–(4).

Let

$$ \begin{align*} \bar{G}(z):=\psi'(\psi^{-1}z)\bar{g}(\psi^{-1}(z))+\frac{1}{2}\psi"(\psi^{-1}(z)) (\psi'(\psi^{-1}(z)))^{-2}\int_{M}v^2\,\mathrm{d}\mu. \end{align*} $$

Consider the SDE

(5.5) $$ \begin{align} \mathrm{d} Z=\mathrm{d} W+\bar{G}(Z)\mathrm{d} t, \quad Z(0)=\psi(\xi). \end{align} $$

Then ${\hat z}_\epsilon \rightarrow _{w}Z$ , where Z is the solution to (5.5) and $\mathcal {W}_{{p}/{2}}({\hat z}_\epsilon ,Z)=O(\epsilon ^{({p-2})/{2p}}+\epsilon ^{1/3}(-\log \epsilon )^{1/4})$ . Because the Stratonovich integral satisfies the usual chain rule, we can see that $Z=\psi (X)$ satisfies the SDE (5.5) as in [15]. Hence,

$$ \begin{align*} \mathcal{W}_{{p}/{2}}({\hat x}_\epsilon, X)&=\mathcal{W}_{{p}/{2}}(\psi^{-1}({\hat z}_\epsilon),\psi^{-1}(Z))\le \mathrm{Lip}(\psi^{-1})\mathcal{W}_{{p}/{2}}({\hat z}_\epsilon,Z)\\ &=O(\epsilon^{({p-2})/{2p}}+\epsilon^{{1}/{3}}(-\log\epsilon)^{{1}/{4}}). \end{align*} $$

The proof is complete.

Remark 5.2.

1. (1) Our result is also applicable to the case where the fast variables are generated by iterating a non-uniformly hyperbolic transformation.

2. (2) In [2], the authors obtained the convergence rate $O(\epsilon ^{1/3-\delta })$ with respect to the Lévy–Prokhorov distance, where $\delta $ depends only on p and $\delta $ can be arbitrarily small as $p\to \infty $ . Compared with [2], our result implies the convergence rate in [2] by Proposition 2.6. However, the price to pay is that the non-uniformly hyperbolic fast systems need to have stronger mixing properties than in [2]. To be more specific, in [2], the fast systems are non-uniformly hyperbolic with return time $R\in L^p (p>2)$ , while, in our case, the return time $R\in L^p (p\ge 4)$ .