Averaging principle on infinite intervals for stochastic ordinary differential equations

1.   Introduction

Highly oscillating systems may be "averaged" under some suitable conditions, and the evolution of the averaged system can reflect in some sense the dynamics of the original system. This idea of averaging dates back to the perturbation theory developed by Clairaut, Laplace and Lagrange in the 18th century, and is made rigorous by Krylov, Bogolyubov, Mitropolsky [1,2,18] for nonlinear oscillations. There are vast amount of works on averaging for deterministic systems which we will not mention here. Meantime, there are also many works on averaging principle for stochastic differential equations so far, see e.g. [4,5,12,14,15,29,30,31,32] among others. But to our best knowledge, except for the centre manifold approach to averaging (see e.g. [3,22]), almost all the existing works on stochastic averaging are concerned with the so-called first Bogolyubov theorem, i.e. the convergence of the solution of the original equation to that of the averaged equation on finite intervals.

In the present paper, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for stochastic differential equations: if there exists a stationary solution for the averaged equation, then there exists in a small neighborhood (in the super-norm topology) a solution of the original equation which is defined on the whole axis and has the same recurrence property (in distribution sense) as the coefficients of the original equation. Furthermore, this recurrent solution is more general than the classical second Bogolyubov theorem, which only treats the almost periodic case. Note that the work [16] studies the averaging principle for stochastic differential equations with almost periodic coefficients, but they only show the convergence on the finite interval, not the super-norm topology on the whole axis.

To be more precise, we investigate the semilinear stochastic ordinary differential equation with Poisson stable (in particular, periodic, quasi-periodic, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, almost recurrent, pseudo periodic, pseudo recurrent) in time coefficients. Under some suitable conditions, this equation has a unique $L^2$-bounded solution which has the same recurrent properties as the coefficients, see [9,20] for details. In this paper, we show that this recurrent solution converges to the unique stationary solution of the averaged equation uniformly on the whole real axis when the time scale goes to zero.

The paper is organized as follows. In the second section we collect some known notions and facts. Namely we present the construction of shift dynamical systems, definitions and basic properties of Poisson stable functions, Shcherbakov's comparability method, and the existence of compatible solutions for stochastic differential equations. In the third and fourth sections, we investigate the averaging principle on infinite intervals for linear and semilinear stochastic differential equations respectively.

2.   Preliminaries

2.1.   Shift dynamical systems

Let $(\mathcal X,\rho)$ be a complete metric space and $(\mathcal X,\mathbb R, \pi)$ be a dynamical system (or flow) on $\mathcal X$, i.e. the mapping $\pi :\mathbb R\times \mathcal X\to \mathcal X$ is continuous, $\pi(0,x) = x$ and $\pi(t+s,x) = \pi(t,\pi(s,x))$ for any $x\in \mathcal X$ and $t,s\in\mathbb R$. The mapping $t\mapsto \pi(t,x)$ is called the motion through $x$. Denote by $C(\mathbb R,\mathcal X)$ the space of all continuous functions $\varphi :\mathbb R \to \mathcal X$ equipped with the distance

where

which generates the compact-open topology on $C(\mathbb R,\mathcal X)$. The space $(C(\mathbb R,\mathcal X),d)$ is a complete metric space (see, e.g. [25,26,27,28]).

Remark 1. (ⅰ) Let $\varphi, \varphi_{n}\in C(\mathbb R,\mathcal X)$ ($n\in\mathbb N$). Then $\lim\limits_{n\to \infty}d(\varphi_{n},\varphi) = 0$ if and only if $\lim\limits_{n\to\infty}\max\limits_{|t|\le l}\rho(\varphi_{n}(t),\varphi(t)) = 0$ for any $l>0$.

(ⅱ) If there exists a sequence $l_n\to +\infty$ such that $\lim\limits_{n\to \infty}\max\limits_{|t|\le l_n}\rho(\varphi_{n}(t),\varphi(t)) = 0$, then $\lim\limits_{n\to \infty}d(\varphi_{n},\varphi) = 0$ and vice versa. See [28] for details.

Let us now introduce two examples of shift dynamical systems which we will use later in this paper.

Example 2.1. For given $\varphi\in C(\mathbb R,\mathcal X)$, we denote by $\varphi^\tau$ the $\tau$-translation of $\varphi$, i.e. $\varphi^\tau(t) = \varphi(\tau+t)$ for $t\in\mathbb R$. Let $\sigma :\mathbb R\times C(\mathbb R,\mathcal X)\to C(\mathbb R,\mathcal X)$ be a mapping defined by equality $\sigma(\tau,\varphi): = \varphi^{\tau}$ for $(\tau,\varphi)\in \mathbb R\times C(\mathbb R,\mathcal X)$. Clearly $\sigma(0,\varphi) = \varphi$ and $\sigma(\tau_1+\tau_2,\varphi) = \sigma(\tau_2,\sigma(\tau_1,\varphi))$ for $\varphi\in C(\mathbb R,\mathcal X)$ and $\tau_1,\tau_2\in\mathbb R$. It is immediate to check (see, e.g. [7,25,26,28]) that the mapping $\sigma :\mathbb R\times C(\mathbb R,\mathcal X)\to C(\mathbb R,\mathcal X)$ is continuous, and consequently the triplet $(C(\mathbb R,\mathcal X),\mathbb R,\sigma)$ is a dynamical system which is called shift dynamical system or Bebutov dynamical system.

The hull of $\varphi$, denoted by $H(\varphi)$, is the set of all the limits of $\varphi^{\tau_n}$ in $C(\mathbb R,\mathcal X)$, i.e.

Note that the set $H(\varphi)$ is a closed and translation invariant subset of $C(\mathbb R,\mathcal X)$ and consequently it naturally defines on $H(\varphi)$ a shift dynamical system – $(H(\varphi),\mathbb R,\sigma)$.

Example 2.2. Like in [9], we denote by $BUC(\mathbb{R}\times \mathcal X,\mathcal X)$ the space of all continuous functions $f:\mathbb{R}\times \mathcal X \to \mathcal X$ which are bounded on every bounded subset from $\mathbb{R}\times \mathcal X$ and continuous in $t\in \mathbb{R}$ uniformly with respect to $x$ on each bounded subset $Q$ of $\mathcal X$. We equip this space with the topology of uniform convergence on bounded subsets of $\mathbb{R}\times \mathcal X$, which can be generated by the following metric

where

with $Q_k\subset\mathcal X$ being bounded, $Q_k\subset Q_{k+1}$ and $\cup_{k\in\mathbb N}Q_k = \mathcal X$.

For given $f\in BUC(\mathbb{R}\times \mathcal X,\mathcal X)$ and $\tau\in\mathbb{R}$, we denote by $f^{\tau}$ the $\tau$-translation of $f$, i.e. $f^{\tau}(t,x): = f(t+\tau,x)$ for $(t,x)\in\mathbb{R}\times \mathcal X$. Note that the space $BUC(\mathbb{R}\times \mathcal X, \mathcal X)$ endowed with the distance (1) is a complete metric space and invariant with respect to translations. Now we define a mapping $\sigma:\mathbb{R}\times BUC(\mathbb{R}\times \mathcal X,\mathcal X) \to BUC(\mathbb{R}\times \mathcal X,\mathcal X)$, $(\tau,f)\mapsto f^\tau$. It is clear that $\sigma(0,f) = f$ and $\sigma(\tau_2,\sigma(\tau_1,f)) = \sigma(\tau_1+\tau_2,f)$ for all $f\in BUC(\mathbb{R}\times \mathcal X,\mathcal X)$ and $\tau_1,\tau_2\in\mathbb{R}$. It is immediate to see (e.g. [7,ChI]) that the mapping $\sigma$ is continuous and consequently the triplet $(BUC(\mathbb{R}\times \mathcal X,\mathcal X),\mathbb{R} ,\sigma)$ is a dynamical system. Similar to Example 2.1, for given $f\in BUC(\mathbb{R}\times \mathcal X,\mathcal X)$, the hull $H(f)$ is a closed and translation invariant subset of $BUC(\mathbb{R}\times \mathcal X,\mathcal X)$ and consequently it naturally defines on $H(f)$ a shift dynamical system – $(H(f),\mathbb R,\sigma)$.

Denote by $BC(\mathcal X, \mathcal X)$ the space of all continuous functions $f: \mathcal X \to \mathcal X$ which are bounded on every bounded subset of $\mathcal X$ and equipped with the distance

where $Q_k$ are the same as above. Note that $(BC(\mathcal X,\mathcal X),d)$ is a complete metric space. For given $F\in BUC(\mathbb R\times \mathcal X, \mathcal X)$, define $\mathcal F: \mathbb R \to BC(\mathcal X, \mathcal X), t\mapsto \mathcal F (t)$ by letting $\mathcal F(t): = F(t,\cdot): \mathcal X\to \mathcal X$. Clearly, $\mathcal F\in C(\mathbb R, BC(\mathcal X, \mathcal X))$.

Remark 2. (ⅰ) Define the mapping $h: BUC(\mathbb R\times \mathcal X, \mathcal X)\to C(\mathbb R, BC(\mathcal X,\mathcal X))$ by equality $h(F): = \mathcal F$. It is immediate to see that the mapping $h$ is one-one and continuous, and its inverse is also continuous. So it establishes an isometry between $BUC(\mathbb R\times\mathcal X,\mathcal X)$ and $C(\mathbb R, BC(\mathcal X, \mathcal X))$.

(ⅱ) By the definition of $h$ we have $h(F^{\tau}) = \mathcal F^{\tau}$ for any $\tau\in \mathbb R$ and $F\in BUC(\mathbb R\times \mathcal X, \mathcal X)$, i.e. the shift dynamical systems $(BUC(\mathbb R\times\mathcal X,\mathcal X),\mathbb R,\sigma)$ and $\left(C(\mathbb R, BC(\mathcal X,\mathcal X)),\mathbb R,\sigma\right)$ are (dynamically) homeomorphic.

2.2.   Poisson stable functions

Let us recall the types of Poisson stable functions to be used in this paper; we refer the reader to [25,26,27,28] for further details and the relations among these types of functions.

Definition 2.3. A function $\varphi\in C(\mathbb R,\mathcal X)$ is called stationary (respectively, $\tau$-periodic) if $\varphi(t) = \varphi(0)$ (respectively, $\varphi(t+\tau) = \varphi(t)$) for all $t\in \mathbb R$.

Definition 2.4. (ⅰ) Let $\varepsilon >0$. A number $\tau \in \mathbb R$ is called $\varepsilon$-almost period of the function $\varphi:\mathbb R\rightarrow \mathcal X$ if $\rho(\varphi(t+\tau),\varphi(t))<\varepsilon$ for all $t\in\mathbb R$. Denote by $\mathcal T(\varphi,\varepsilon)$ the set of $\varepsilon$-almost periods of $\varphi$.

(ⅱ) A function $\varphi \in C(\mathbb R,\mathcal X)$ is said to be Bohr almost periodic if the set of $\varepsilon$-almost periods of $\varphi$ is relatively dense for each $\varepsilon >0$, i.e. for each $\varepsilon >0$ there exists a constant $l = l(\varepsilon)>0$ such that $\mathcal T(\varphi,\varepsilon)\cap [a,a+l]\not = \emptyset$ for all $a\in\mathbb R$.

(ⅲ) A function $\varphi \in C(\mathbb R,\mathcal X)$ is said to be pseudo-periodic in the positive (respectively, negative) direction if for each $\varepsilon >0$ and $l>0$ there exists a $\varepsilon$-almost period $\tau >l$ (respectively, $\tau <-l$) of the function $\varphi$. The function $\varphi$ is called pseudo-periodic if it is pseudo-periodic in both directions.

Remark 3. A function $\varphi \in C(\mathbb R,\mathcal X)$ is pseudo-periodic in the positive (respectively, negative) direction if and only if there is a sequence $t_n\to +\infty$ (respectively, $t_n\to -\infty$) such that $\varphi^{t_n}$ converges to $\varphi$ uniformly with respect to $t\in \mathbb R$ as $n\to \infty$.

Definition 2.5. (ⅰ) A number $\tau\in\mathbb R$ is said to be $\varepsilon$-shift of $\varphi \in C(\mathbb R,\mathcal X)$ if $d(\varphi^{\tau},\varphi)<\varepsilon$; denote $\mathfrak T(\varphi,\varepsilon): = \{\tau: d(\varphi^{\tau},\varphi)<\varepsilon\}$. A function $\varphi \in C(\mathbb R,\mathcal X)$ is called almost recurrent (in the sense of Bebutov) if for every $\varepsilon >0$ the set $\mathfrak T(\varphi,\varepsilon)$ is relatively dense.

(ⅱ) A function $\varphi\in C(\mathbb R,\mathcal X)$ is called Lagrange stable if $\{\varphi^{\tau}: \tau\in \mathbb R\}$ is a relatively compact subset of $C(\mathbb R,\mathcal X)$.

(ⅲ) A function $\varphi \in C(\mathbb R,\mathcal X)$ is called Birkhoff recurrent if it is almost recurrent and Lagrange stable.

Definition 2.6. A function $\varphi \in C(\mathbb R,\mathcal X)$ is called Poisson stable in the positive (respectively, negative) direction if for every $\varepsilon >0$ and $l>0$ there exists $\tau >l$ (respectively, $\tau <-l$) such that $d(\varphi^{\tau},\varphi)<\varepsilon$. The function $\varphi$ is called Poisson stable if it is Poisson stable in both directions.

In what follows, we denote as well $\mathcal Y$ a complete metric space.

Definition 2.7. (ⅰ) A function $\varphi\in C(\mathbb R,\mathcal X)$ is called Levitan almost periodic if there exists a Bohr almost periodic function $\psi \in C(\mathbb R,\mathcal Y)$ such that for any $\varepsilon >0$ there exists $\delta = \delta (\varepsilon)>0$ such that $\mathcal T(\psi,\delta) \subseteq \mathfrak T(\varphi,\varepsilon)$.

(ⅱ) A function $\varphi \in C(\mathbb R,\mathcal X)$ is said to be almost automorphic if it is Levitan almost periodic and Lagrange stable.

Remark 4. Note that:

1. every Bohr almost periodic function is Levitan almost periodic;

2. the function $\varphi \in C(\mathbb R,\mathbb R)$ defined by equality

is Levitan almost periodic, but it is not Bohr almost periodic [19,ChIV].

Definition 2.8. A function $\varphi \in C(\mathbb R,\mathcal X)$ is called quasi-periodic with the spectrum of frequencies $\nu_1,\nu_2,\ldots,\nu_k$ if the following conditions are fulfilled:

1. the numbers $\nu_1,\nu_2,\ldots,\nu_k$ are rationally independent;

2. there exists a continuous function $\Phi :\mathbb R^{k}\to \mathcal X$ such that $\Phi(t_1+2\pi,t_2+2\pi,\ldots,t_k+2\pi) = \Phi(t_1,t_2,\ldots,t_k)$ for all $(t_1,t_2,\ldots,t_k)\in \mathbb R^{k}$;

3. $\varphi(t) = \Phi(\nu_1 t,\nu_2 t,\ldots,\nu_k t)$ for $t\in \mathbb R$.

Let $\varphi \in C(\mathbb R,\mathcal X)$. Denote by $\mathfrak N_{\varphi}$ (respectively, $\mathfrak M_{\varphi}$) the family of all sequences $\{t_n\}\subset \mathbb R$ such that $\varphi^{t_n} \to \varphi$ (respectively, $\{\varphi^{t_n}\}$ converges) in $C(\mathbb R,\mathcal X)$ as $n\to \infty$. We denote by $\mathfrak N_{\varphi}^{u}$ (respectively, $\mathfrak M_{\varphi}^{u}$) the family of sequences $\{t_n\}\in \mathfrak N_{\varphi}$ (respectively, $\{t_n\}\in\mathfrak M_{\varphi}$) such that $\varphi^{t_n}$ converges to $\varphi$ (respectively, $\varphi^{t_n}$ converges) uniformly with respect to $t\in\mathbb R$ as $n\to \infty$.

Remark 5. (ⅰ) The function $\varphi \in C(\mathbb R,\mathcal X)$ is pseudo-periodic in the positive (respectively, negative) direction if and only if there is a sequence $\{t_n\}\in \mathfrak N_{\varphi}^{u}$ such that $t_n\to +\infty$ (respectively, $t_n\to -\infty$) as $n\to \infty$.

(ⅱ) Let $\varphi \in C(\mathbb R,\mathcal X)$, $\psi \in C(\mathbb R,\mathcal Y)$ and $\mathfrak N_{\psi}^{u} \subseteq \mathfrak N_{\varphi}^{u}$. If the function $\psi$ is pseudo-periodic in the positive (respectively, negative) direction, then so is $\varphi$.

Definition 2.9. ([23,26,27]) A function $\varphi \in C(\mathbb R,\mathcal X)$ is called pseudo-recurrent if for any $\varepsilon >0$ and $l\in\mathbb R$ there exists $L\ge l$ such that for any $\tau_0\in \mathbb R$ we can find a number $\tau \in [l,L]$ satisfying

Remark 6. ([23,26,27,28])

1. Every Birkhoff recurrent function is pseudo-recurrent, but the inverse statement is not true in general.

2. If the function $\varphi \in C(\mathbb R,\mathcal X)$ is pseudo-recurrent, then every function $\psi\in H(\varphi)$ is pseudo-recurrent.

3. If the function $\varphi \in C(\mathbb R,\mathcal X)$ is Lagrange stable and every function $\psi\in H(\varphi)$ is Poisson stable, then $\varphi$ is pseudo-recurrent.

Finally, we remark that a Lagrange stable function is not Poisson stable in general, but all other types of functions introduced above are Poisson stable.

Definition 2.10. A function $F\in BUC(\mathbb R\times \mathcal X,\mathcal X)$ is said to possess the property $A$ in $t\in\mathbb R$ uniformly with respect to $x$ on every bounded subset $Q$ of $\mathcal X$ if the motion $\sigma(\cdot,F)$ through $F$ with respect to the Bebutov dynamical system $(BUC(\mathbb R\times \mathcal X,\mathcal X),{{\mathbb R}},\sigma)$ possesses the property $A$. Here the property $A$ may be stationary, periodic, Bohr/Levitan almost periodic etc.

Remark 7. Note that a function $\varphi\in C(\mathbb{R},\mathcal X)$ possesses the property $A$ if and only if the motion $\sigma(\cdot,\varphi):\mathbb{R}\to C(\mathbb{R},\mathcal X)$ through $\varphi$ with respect to the Bebutov dynamical system $(C(\mathbb{R},\mathcal X),\mathbb{R},\sigma)$ possesses this property.

2.3.   Shcherbakov's comparability method by character of recurrence

Definition 2.11. A function $\varphi \in C(\mathbb R,\mathcal X)$ is said to be comparable (respectively, strongly comparable) by character of recurrence with $\psi\in C(\mathbb R,\mathcal Y)$ if $\mathfrak N_{\psi}\subseteq \mathfrak N_{\varphi}$ (respectively, $\mathfrak M_{\psi}\subseteq \mathfrak M_{\varphi}$).

Theorem 2.12. ([26,ChII], [24]) The following statements hold.

1. $\mathfrak M_{\psi}\subseteq \mathfrak M_{\varphi}$ implies $\mathfrak N_{\psi}\subseteq \mathfrak N_{\varphi}$, and hence strong comparability implies comparability.

2. Let $\varphi \in C(\mathbb R,\mathcal X)$ be comparable by character of recurrence with $\psi\in C(\mathbb R,\mathcal Y)$. If the function $\psi$ is stationary (respectively, $\tau$-periodic, Levitan almost periodic, almost recurrent, Poisson stable), then so is $\varphi$.

3. Let $\varphi \in C(\mathbb R,\mathcal X)$ be strongly comparable by character of recurrence with $\psi\in C(\mathbb R,\mathcal Y)$. If the function $\psi$ is quasi-periodic with the spectrum of frequencies $\nu_1,\nu_2,\dots,\nu_k$ (respectively, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable), then so is $\varphi$.

4. Let $\varphi \in C(\mathbb R,\mathcal X)$ be strongly comparable by character of recurrence with $\psi\in C(\mathbb R,\mathcal Y)$ and $\psi$ be Lagrange stable. If $\psi$ is pseudo-periodic (respectively, pseudo-recurrent), then so is $\varphi$.

Lemma 2.13. ([9]) Let $\varphi \in C(\mathbb R,\mathcal X)$, $\psi \in C(\mathbb R,\mathcal Y)$. The following statements hold.

1. If $\mathfrak M_{\psi}^{u} \subseteq \mathfrak M_{\varphi}^{u}$, then

(a) $\mathfrak N_{\psi}^{u} \subseteq \mathfrak N_{\varphi}^{u}$;

(b) if the function $\psi$ is Bohr almost periodic, then so is $\varphi$.

2. If $\mathfrak N_{\psi}^{u} \subseteq \mathfrak N_{\varphi}^{u}$ and $\psi$ is pseudo periodic, then so is $\varphi$.

2.4.   Compatible solutions of semilinear stochastic differential equations

Let $\mathfrak B$ be a real separable Banach space with the norm $|\cdot|$, and $L(\mathfrak B)$ be the Banach space of all bounded linear operators acting on the space $\mathfrak B$ equipped with operator norm $\|\cdot\|$. Consider the linear homogeneous equation

on the space $\mathfrak B$, where $\mathcal A\in C(\mathbb R,L(\mathfrak B))$. Denote by $U(t,\mathcal A)$ the Cauchy operator (see, e.g. [10]) of equation (2).

Definition 2.14. Equation (2) is said to be uniformly asymptotically stable if there are positive constants $\mathcal N$ and $\nu$ such that

where $G_{\mathcal A}(t,\tau): = U(t,\mathcal A)U^{-1}(\tau,\mathcal A)$ for any $t,\tau\in\mathbb R$.

If $\mathcal A \in C(\mathbb R,L(\mathfrak B))$, then by $H(\mathcal A)$ we denote the closure in the space $C(\mathbb R,L(\mathfrak B))$ of all translations $\{\mathcal A^{h}: h\in\mathbb R\}$, where $\mathcal A^{h}(t): = \mathcal A(t+h)$ for $t\in\mathbb R$. Denote by $C_{b}(\mathbb R,\mathfrak B)$ the Banach space of all continuous and bounded mappings $\varphi :\mathbb R\to \mathfrak B$ equipped with the norm $\|\varphi\|_{\infty}: = \sup\{|\varphi(t)|: t\in\mathbb R\}$. Note that if $f\in C_b(\mathbb R, \mathfrak B)$ and $\tilde{f}\in H(f)$, then $\|\tilde{f}\|_{\infty}\le \|f\|_{\infty}$.

Lemma 2.15. [6,ChIII] Suppose that equation (2) is uniformly asymptotically stable such that inequality (3) holds. Then

for any $t\ge \tau$ ($t,\tau \in \mathbb R$) and $\tilde{\mathcal A}\in H(\mathcal A)$.

Let $(\Omega , \mathcal F,\mathbb P)$ be a probability space, and $L^{2}(\mathbb P,\mathfrak B)$ be the space of $\mathfrak B$-valued random variables $X$ such that

Then $L^2(\mathbb P,\mathfrak B)$ is a Banach space equipped with the norm $\|X\|_{2}: = \left(\int_{\Omega}|X|^2 {{{\rm{d}}}}\mathbb P\right)^{1/2}$.

Let $\mathcal P(\mathfrak B)$ be the space of all Borel probability measures on $\mathfrak B$ endowed with the $\beta$ metric:

where $f$ are bounded Lipschitz continuous real-valued functions on $\mathfrak B$ with the norm

Recall that a sequence $\{\mu_n\}\subset \mathcal P(\mathfrak B)$ is said to weakly converges to $\mu$ if $\int f {{{\rm{d}}}}\mu_n\to \int f {{{\rm{d}}}}\mu$ for all $f\in C_b(\mathfrak B)$, where $C_b(\mathfrak B)$ is the space of all bounded continuous real-valued functions on $\mathfrak B$. It is well-known (see, e.g. [13,ChXI]) that $(\mathcal P(\mathfrak B),\beta)$ is a separable complete metric space and that a sequence $\{\mu_n\}$ weakly converges to $\mu$ if and only if $\beta(\mu_n, \mu) \to0$ as $n\to\infty$.

Definition 2.16. A sequence of random variables $\{X_n\}$ is said to converge in distribution to the random variable $X$ if the corresponding laws $\{\mu_n\}$ of $\{X_n\}$ weakly converge to the law $\mu$ of $X$, i.e. $\beta(\mu_n,\mu)\to 0$.

In the following, we assume that $\mathcal H$ is a real separable Hilbert space. We still denote the norm in $\mathcal H$ by $|\cdot|$ and the operator norm in $L(\mathcal H)$ by $\|\cdot\|$ which will not cause confusion. Let us consider the stochastic differential equation

where $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and $F,G\in C(\mathbb{R}\times \mathcal H,\mathcal H)$. Here $W$ is a two-sided standard one-dimensional Brownian motion defined on the probability space $(\Omega,\mathcal F,\mathbb P)$; that is,

where $W_1$ and $W_2$ are two independent one-sided standard one-dimensional Brownian motions. We set $\mathcal F_{t}: = \sigma\{W(u): u \le t\}$.

Definition 2.17. An $\mathcal F_{t}$-adapted process $\{X(t)\}_{t\in\mathbb R}$ is said to be a mild solution of equation (4) on ${{\mathbb R}}$ if it satisfies the following stochastic integral equation

for all $t\geq s$ and each $s\in{{\mathbb R}}$, recalling that $G_{\mathcal A}$ is defined in Definition 2.14.

Definition 2.18. We say that functions $F$ and $G$ satisfy the condition

$\rm (C1)$ if there exists a constant $M\ge 0$ such that $|F(t,0)|\vee|G(t,0)|\le M$ for $t\in \mathbb R$;

$\rm (C2)$ if there exists a constant $L\ge 0$ such that $Lip(F)\vee Lip(G)\le L$, where $Lip(F): = \sup_{t\in{{\mathbb R}}, x\neq y}\frac{|F(t,x)-F(t,y)|}{|x-y|}$ and similarly for $Lip(G)$;

$\rm (C3)$ if $F$ and $G$ are continuous in $t$ uniformly with respect to $x$ on each bounded subset $Q\subset\mathcal H$.

Remark 8. (i) If $F$ and $G$ satisfy the conditions (C1)–(C3), then $F, G\in BUC (\mathbb R\times \mathcal H, \mathcal H)$.

(ii) If $F$ and $G$ satisfy (C1)–(C2) with the constants $M$ and $L$, then every pair of functions $(\tilde{F},\tilde{G})$ in $H(F,G): = \overline{\{(F^{\tau},G^{\tau}): \tau\in \mathbb R\}}$, the hull of $(F,G)$, also possess the same property with the same constants.

Definition 2.19. Let $\{\varphi (t)\}_{t\in\mathbb R}$ be a mild solution of equation (4). Then $\varphi$ is called compatible (respectively, strongly compatible) in distribution if $\mathfrak N_{(\mathcal A,F,G)} \subseteq \tilde{\mathfrak N}_{\varphi}$ (respectively, $\mathfrak M_{(\mathcal A,F,G)} \subseteq \tilde{\mathfrak M}_{\varphi}$), where $\tilde{\mathfrak N}_{\varphi}$ (respectively, $\tilde{\mathfrak M}_{\varphi}$) means the set of all sequences $\{t_n\}\subset\mathbb R$ such that the sequence $\{\varphi(\cdot+t_n)\}$ converges to $\varphi(\cdot)$ (respectively, $\{\varphi(\cdot+t_n)\}$ converges) in distribution uniformly on any compact interval.

Theorem 2.20. Consider the equation (4). Suppose that the following conditions hold:

(a) $\sup\limits_{t\in \mathbb R}\|\mathcal{A}(t)\|<+\infty$;

(b) equation (2) is uniformly asymptotically stable such that (3) holds;

(c) the functions $F$ and $G$ satisfy the conditions (C1) and (C2).

Then the following statements hold:

1. if $L< \frac{\nu}{\mathcal N \sqrt{2+\nu}}$, then equation $(4)$ has a unique solution $\xi \in C(\mathbb{R},B[0,r])$ which satisfies

where

and

2. if additionally $F$ and $G$ satisfy (C3) and $L< \frac{\nu}{2\mathcal N \sqrt{1+\nu}}$, then

(a) $\mathfrak M^{u}_{(\mathcal A,F,G)} \subseteq\tilde{\mathfrak M}^{u}_{\xi}$, where $\tilde{\mathfrak M}^{u}_{\xi}$ means the the set of all sequences $\{t_n\}\subset\mathbb R$ such that the sequence $\{\xi(\cdot+t_n)\}$ converges in distribution uniformly on $\mathbb R$;

(b) the solution $\xi$ is strongly compatible in distribution.

Proof. The proof is analogous to Theorem 4.6 in [9].

Corollary 1. Under the conditions of Theorem 2.20 the following statements hold.

1. If the functions $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and $F,G\in$ $C(\mathbb R\times\mathcal H,\mathcal H)$ are jointly stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1,\ldots,\nu_k$, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable), then equation (4) has a unique solution $\varphi \in C_{b}(\mathbb R,L^2(\mathbb P,\mathcal H))$ which is stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1,\ldots,\nu_k$, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable) in distribution.

2. If the functions $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and $F,G\in$ $C(\mathbb R \times \mathcal H,\mathcal H)$ are Lagrange stable and jointly pseudo-periodic (respectively, pseudo-recurrent), then equation (4) has a unique solution $\varphi \in C_{b}(\mathbb R,L^2(\mathbb P,\mathcal H))$ which is pseudo-periodic (respectively, pseudo-recurrent) in distribution.

Proof. These statements follow from Theorems 2.12 and 2.20 (see also Remark 2).

3.   Averaging for linear equations

Let $\varepsilon_{0}$ be some fixed positive number. Consider the equation

where $\mathcal A\in C(\mathbb R,L(\mathcal H))$, $f,g\in C(\mathbb R,L^2(\mathbb P, \mathcal H))$, $0<\varepsilon \le \varepsilon_0$ and $W$ is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space $(\Omega,\mathcal F,\mathbb P,\mathcal F_{t})$, where $\mathcal F_{t}: = \sigma\{W(u): u\le t\}$.

Definition 3.1. Let $f:\mathbb R\times (0,\varepsilon_0]\to \mathfrak B$. Following [17] we say that $f(t;\varepsilon)$ integrally converges to $0$ if for any $L>0$ we have

If additionally there exists a constant $m>0$ such that

for any $t\in\mathbb R$ and $0<\varepsilon\le \varepsilon_{0}$, then we say that $f(t;\varepsilon)$ correctly converges to $0$ as $\varepsilon \to 0$.

Remark 9. [17,ChIV] If $f\in C(\mathbb R,\mathfrak B)$ and

uniformly with respect to $t\in\mathbb R$, then $f(t;\varepsilon): = f(\frac{t}{\varepsilon})$ integrally converges to $0$ as $\varepsilon \to 0$. If additionally the function $f$ is bounded on $\mathbb R$, then $f(t;\varepsilon)$ correctly converges to $0$ as $\varepsilon \to 0$.

Let $\mathcal A\in L(\mathcal H)$. Denote by $\sigma(\mathcal A)$ the spectrum of $\mathcal A$. Below we will use the following conditions:

$(\textbf{A1})$ $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and there exists $\bar{\mathcal A} \in L(\mathcal H)$ such that

uniformly with respect to $t\in\mathbb R$;

$(\textbf{A2})$ $f\in C(\mathbb R, L^2(\mathbb P, \mathcal H))$ and there exists $\bar{f}\in L^2(\mathbb P, \mathcal H)$ such that

uniformly with respect to $t\in\mathbb R$;

$(\textbf{A3})$ $g\in C(\mathbb R, L^2(\mathbb P, \mathcal H))$ and there exists $\bar{g}\in L^2(\mathbb P, \mathcal H)$ such that

uniformly with respect to $t\in\mathbb R$.

Denote by $\Psi$ the family of all decreasing, positive bounded functions $\psi :\mathbb R_{+}\to\mathbb R_{+}$ with $\lim\limits_{t\to +\infty}\psi(t) = 0$.

Lemma 3.2. Let $l>0$ and $\psi\in \Psi$, then

Proof. Let $\varepsilon$ and $l$ be two arbitrary positive numbers, $\nu \in(0,1)$ and $\psi\in \Psi$, then we have

Letting $\varepsilon \to 0$ we obtain the required result.

Remark 10. (ⅰ) By Lemma 2 in [8] equality (8) holds if and only if there exists a function $\omega \in\Psi$ satisfying

for any $T>0$ and $t\in\mathbb R$.

(ⅱ) Similarly equality (9) (respectively, equality (10)) holds if and only if there exists a function $\omega_{1} \in\Psi$ (respectively, $\omega_{2}\in\Psi$) satisfying

(respectively, $\frac{1}{T}\int_{t}^{t+T} \mathbb E |g(\tau)-\bar{g}|^{2}{{{\rm{d}}}}\tau \le\omega_{2}(T)$) for any $T>0$ and $t\in\mathbb R$.

Theorem 3.3. [17,ChIV] Suppose that $\mathcal A\in C_{b}(\mathbb R,L(\mathfrak B))$ and

uniformly with respect to $t\in\mathbb R$ and the operator $\bar{\mathcal A}$ is Hurwitz, i.e. $\mathcal{R}e\ \lambda <0$ for any $\lambda \in \sigma(\bar{\mathcal A})$.

Then the following statements hold:

1. there exists a positive constant $\alpha \le \varepsilon_{0}$ such that the equation

where $\mathcal A_{\varepsilon }(\tau): = \mathcal A(\frac{\tau}{\varepsilon})$ for any $\tau\in\mathbb R$, is uniformly asymptotically stable for any $0<\varepsilon\le \alpha$. Moreover there are constants $\mathcal N>0$ and $\nu>0$ such that

for any $\tau \ge\tau_{0}$ and $0<\varepsilon \le\alpha$;

2. there exists $\gamma_{0}>0$ such that

Remark 11. (ⅰ) Note that Theorem 3.3 was proved for finite-dimensional almost periodic equations (this means that the matrix-function $\mathcal A(\cdot)$ is almost periodic). For the proof for infinite-dimensional almost periodic systems see [19,ChXI].

(ⅱ) It is not difficult to show that Theorem 3.3 remains true in general case (see above) and can be proved with slight modifications of the reasoning from [17,ChIV].

(ⅲ) Under the conditions of Theorem 3.3 there are positive constants $\alpha, \mathcal N$ and $\nu$ so that

for any $0<\varepsilon \le \alpha$ and $t\ge\tau$.

Lemma 3.4. Let $f_{\varepsilon}\in C(\mathbb R, \mathfrak B)$ for $\varepsilon \in (0,\alpha]$ be functions satisfying the following conditions:

1. there exists a positive constant $A$ such that $|f_{\varepsilon}(t)|\le A$ for any $t\in\mathbb R$ and $\varepsilon\in (0,\alpha]$;

2. for any $l>0$

Then for any $\nu >0$ we have

Proof. To estimate the integral

we make the change $\tau -t = s$, then

Since

for any $s<0$, we have

Integrating by parts and taking into consideration (15) we obtain

Note that

and

By (14)–(18) we get

Then

Since

for any $l>0$, letting $\varepsilon \to 0$ in (19) we have

Since $l$ is arbitrary, it follows that

The proof is complete.

Remark 12. If the function $f\in C_b(\mathbb R,\mathfrak B)$ and $\bar{f}\in \mathfrak B$ are such that

uniformly with respect to $t\in\mathbb R$, then the function $f_{\varepsilon}(\sigma): = f(\frac{\sigma}{\varepsilon})-\bar f$ satisfies the conditions of Lemma 3.4. Indeed, note that

so the condition (13) of Lemma 3.4 holds by (20). Similarly, if the function $g$ in (A3) is $L^2$-bounded, then the function $f_{\varepsilon}(\sigma): = \mathbb E|g(\frac{\sigma}{\varepsilon})-\bar g|^2$ satisfies as well the conditions of Lemma 3.4.

Let $W_{\varepsilon}(t): = \sqrt{\varepsilon}W(\frac{t}{\varepsilon})$ for $t\in{{\mathbb R}}$. Then $W_\varepsilon$ is also a Brownian motion with the same distribution as $W$.

Theorem 3.5. Suppose that $\mathcal A\in C_{b}(\mathbb R,L(\mathcal H)), \ f,g\in C_{b}(\mathbb R,L^{2}(\mathbb P,\mathcal H))$ and conditions (A1)–(A3) are fulfilled. Suppose further that $\bar{\mathcal A}$ in (A1) is Hurwitz such that (11)–(12) holds. Then we have the following conclusions:

1. equation

has a unique bounded solution $\psi_{\varepsilon}\in C_{b}(\mathbb R,L^{2}(\mathbb P,\mathcal H))$ defined by equality

and it is strongly compatible in distribution (i.e. $\mathfrak M_{(\mathcal A_{\varepsilon},f_{\varepsilon},g_{\varepsilon})}\subseteq \tilde{\mathfrak M}_{\psi_{\varepsilon}}$) and $\mathfrak M^{u}_{(\mathcal A_{\varepsilon},f_{\varepsilon},g_{\varepsilon})}\subseteq \tilde{\mathfrak M}^{u}_{\psi_{\varepsilon}}$, where $\mathcal A_{\varepsilon}(t): = \mathcal A(\frac{t}{\varepsilon})$, $f_{\varepsilon}(t): = f(\frac{t}{\varepsilon})$ and $g_{\varepsilon}(t): = g(\frac{t}{\varepsilon})$ for $t\in\mathbb R$;

2. equation

has a unique bounded solution $\phi_{\varepsilon}\in C_{b}(\mathbb R,L^{2}(\mathbb P,\mathcal H))$ defined by equality

3.

where $\bar{\phi}$ is the unique stationary solution of equation

4. for any $\varepsilon \in (0,\alpha]$ equation (7) has a unique solution $\varphi_{\varepsilon}\in C_{b}(\mathbb R,L^{2}(\mathbb P,\mathcal H))$ and it is strongly compatible in distribution (i.e. $\mathfrak M_{(\mathcal A,f,g)}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}$) and $\mathfrak M_{(\mathcal A,f,g)}^{u}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}^{u}$;

5.

where $\mathcal L(X)$ denotes the law of random variable $X$.

Proof. The first and second statements follow directly from Theorem 2.20.

We now verify the third statement, i.e. the uniform convergence of the unique bounded solution $\phi_\varepsilon$ to the unique stationary solution $\bar\phi$ of the averaged equation. By Theorem 2.20 equation (24) has a unique bounded and stationary solution $\bar{\phi}$, which is given by the formula

where $G_{\bar{\mathcal A}}(t,\tau) = \exp\left\{\bar{\mathcal A}(t-\tau)\right\}$ for $t,\tau\in\mathbb R$. From (22) and (25) we get

By equality (11) there exists a function $\mathcal N:(0,\alpha)\to \mathbb R_{+}$ such that $\mathcal N(\varepsilon)\to 0$ as $\varepsilon \to 0$ and

for any $t\geq\tau$ ($t,\tau\in\mathbb R$).

Note that

To estimate the integral

making the change of variable $s: = \tau -t$ we obtain

Since

for any $s<0$, we have

Consequently, integrating by parts from (26) we get

Note that

so we have

for any $t\in\mathbb R$ and $s<0$.

Let now $l$ be an arbitrary positive number, then we have

and consequently

Letting $\varepsilon \to 0$ in above inequality we get

Since $l$ is arbitrary, we get by letting $l\to\infty$

Note by Theorem 3.3–(ⅱ) that

as $\varepsilon \to 0$. Consequently,

Similarly we can show that

In fact, using Itô's isometry property, the Cauchy-Schwartz inequality and reasoning as above we get

By Lemma 3.4 the integral

goes to $0$ as $\varepsilon \to 0$ uniformly with respect to $t\in\mathbb R$.

Passing to the limit in (28) and taking into account (29) we obtain (27), and consequently $\lim\limits_{\varepsilon\rightarrow0}\sup\limits_{t\in\mathbb R} \mathbb E|\phi_{\varepsilon}(t)-\bar{\phi}(t)|^2 = 0$.

To prove the fourth statement we note that the function $\varphi_{\varepsilon}(t): = \psi_{\varepsilon}(\varepsilon t)$ (for $t\in\mathbb R$) is a bounded solution of equation (7) if $\psi_{\varepsilon}$ is a bounded solution of equation (21). The uniqueness follows from the fact if $\varphi_{i}$ ($i = 1,2$) are two different bounded solutions of equation (7), then $\psi_{i}(t): = \varphi_{i}(\frac{t}{\varepsilon}), t\in{{\mathbb R}}$ ($i = 1,2$) are two different bounded solutions of equation (21), a contradiction to the first statement. It remains to show that $\mathfrak M_{(\mathcal A,f,g)}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}$ and $\mathfrak M_{(\mathcal A,f,g)}^{u}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}^{u}$. Let $\{t_n\}\in \mathfrak M_{(\mathcal A,f,g)}$ (respectively, $\{t_n\}\in \mathfrak M^{u}_{(\mathcal A,f,g)}$), then $\{\varepsilon t_n\}\in \mathfrak M_{(\mathcal A_{\varepsilon},f_{\varepsilon},g_{\varepsilon})}\subseteq \tilde{\mathfrak M}_{\psi_{\varepsilon}}$ (respectively, $\{\varepsilon t_n\}\in \mathfrak M^{u}_{(\mathcal A_{\varepsilon},f_{\varepsilon},g_{\varepsilon})}\subseteq \tilde{\mathfrak M}^{u}_{\psi_{\varepsilon}}$) by Theorem 2.20. By the relation between $\varphi_{\varepsilon}$ and $\psi_{\varepsilon}$, we have $\{t_n\}\in \tilde{\mathfrak M}_{\varphi_{\varepsilon}}$ (respectively, $\{t_n\}\in\tilde{\mathfrak M}^{u}_{\varphi_{\varepsilon}}$).

Now we are in the position to prove the last statement. Since the $L^{2}$ convergence implies convergence in probability, it follows from (23) that

On the other hand taking into consideration that $\mathcal L(W) = \mathcal L(W_{\varepsilon})$, we have $\mathcal L(\varphi_{\varepsilon}(\frac{t}{\varepsilon})) = \mathcal L(\phi_{\varepsilon}(t))$ for any $t\in \mathbb R$, and consequently

The proof is complete.

Corollary 2. Under the conditions of Theorem 3.5 the following statements hold:

1. if the functions $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and $f,g\in$ $C_{b}(\mathbb R,$ $L^2(\mathbb P,\mathcal H))$ are jointly stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1,\ldots,\nu_k$, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable), then equation (7) has a unique solution $\varphi_{\varepsilon} \in C_{b}(\mathbb R,L^2(\mathbb P,\mathcal H))$ which is stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1,\ldots,\nu_k$, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable) in distribution;

2. if the functions $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and $f,g\in$ $C_{b}(\mathbb R,$ $L^2(\mathbb P,\mathcal H))$ are Lagrange stable and jointly pseudo-periodic (respectively, pseudo-recurrent), then equation (7) has a unique solution $\varphi_{\varepsilon} \in C_{b}(\mathbb R,L^2(\mathbb P,\mathcal H))$ which is pseudo-periodic (respectively, pseudo-recurrent) in distribution;

3.

Proof. These statements follow from Theorems 2.12 and 3.5.

Remark 13. Note that the constant $\alpha>0$ in Theorem 3.3, Remark 11, Lemma 3.4 and Theorem 3.5 can be chosen the same, e.g. we may choose the minimal one of them. But there is no restriction on $\varepsilon_{0}$ which we fix at the beginning of this section.

4.   Averaging principle for semilinear stochastic differential equations

As before, let $\varepsilon_{0}$ be some fixed positive constant. Consider the following stochastic differential equation

where $\mathcal A\in C(\mathbb R,L(\mathcal H))$, $F,G\in C(\mathbb R\times \mathcal H,\mathcal H)$, $0<\varepsilon \le \varepsilon_0$ and $W$ is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space $(\Omega,\mathcal F,\mathbb P,\mathcal F_{t})$, with $\mathcal F_{t}: = \sigma\{W(u): u\le t\}$. Below we will use the following conditions:

$\textbf{(G1)}$ there exists a positive constant $M$ such that

for any $t\in{{\mathbb R}}$;

$\textbf{(G2)}$ there exists a positive constant $L$ such that

for any $x_1,x_2\in \mathcal H$ and $t\in\mathbb R$;

$\textbf{(G3)}$ there exist functions $\omega_{1}\in \Psi$ and $\bar{F}\in C(\mathcal H,\mathcal H)$ such that

for any $T>0$, $x\in \mathcal H$ and $t\in\mathbb R$;

$\textbf{(G4)}$ there exist functions $\omega_{2}\in \Psi$ and $\bar{G}\in C(\mathcal H,\mathcal H)$ such that

for any $T>0$, $x\in \mathcal H$ and $t\in\mathbb R$;

$\textbf{(G5)}$ $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and there exists $\bar{\mathcal A}\in L(\mathcal H)$ such that

uniformly with respect to $t\in\mathbb R$.

Remark 14. Under the conditions $(\textbf{G1})-(\textbf{G4})$ the functions $\bar{F}$ and $\bar{G}$ also possess the properties $(\textbf{G1})-(\textbf{G2})$ with the same constants $M$ and $L$.

We consider as well the following equations

and

where $\mathcal A_{\varepsilon}(t) : = \mathcal A(\frac{t}{\varepsilon})$, $F_{\varepsilon}(t,x): = F(\frac{t}{\varepsilon},x)$ and $G_{\varepsilon}(t,x): = G(\frac{t}{\varepsilon},x)$ for $t\in{{\mathbb R}}$, $x\in\mathcal H$ and $\varepsilon \in(0,\varepsilon_0]$. Here as before $W_{\varepsilon}(t): = \sqrt{\varepsilon}W(\frac{t}{\varepsilon})$ for $t\in{{\mathbb R}}$. Along with equations (31)–(32) we also consider the following averaged equation

Lemma 4.1. Suppose $F,G\in C(\mathbb R\times \mathcal H,\mathcal H)$ and that the conditions (G1)–(G2) hold. If $\varphi$ is an $L^2$-bounded solution (i.e. $\|\varphi\|_\infty = \sup_{t\in\mathbb R} \mathbb E|\varphi(t)|^2<+\infty$) of the equation

then there exists a constant $C>0$, depending only on $M,L,\|\varphi\|_\infty$, such that

and

for any $t\in{{\mathbb R}}$ and $h>0$.

Proof. Since

by Cauchy-Schwartz inequality and Itô's isometry property we have

Employing the BDG inequality (see, e.g. [11,Theorem 4.36] on page 114), we have

where $C$ denotes some positive constants which may change from line to line.

Theorem 4.2. Suppose that the following conditions hold:

(a) $\sup\limits_{t\in \mathbb R}\|\mathcal{A}(t)\|<+\infty$;

(b) the functions $\mathcal A$, $F$, $G$ satisfy the conditions $\rm(G1)–(G5)$, and the operator $\bar{\mathcal A}$ in $\rm(G5)$ is Hurwitz, i.e. $\mathcal{R}e\ \lambda <0$ for any $\lambda\in\sigma(\bar{\mathcal A})$;

(c)

where $\alpha,\mathcal N$ and $\nu$ are the numbers figuring in Remark 11-(ⅲ).

Then there exists a positive constant $\varepsilon_1\le \alpha$ such that for any $0<\varepsilon \le \varepsilon_{1}$

1. equation (30) has a unique solution $\varphi_{\varepsilon}\in C_{b}(\mathbb R,L^{2} (\mathbb P,\mathcal H))$ and $\|\varphi_{\varepsilon}\|_{\infty}\le r$, where $r: = \frac{\mathcal N M\sqrt{2+\nu}}{\nu-\mathcal N L\sqrt{2+\nu}}$;

2. equation (31) has a unique solution $\phi_{\varepsilon}\in C_{b}(\mathbb R,L^{2}(\mathbb P, \mathcal H))$ and $\|\phi_{\varepsilon}\|_{\infty}\le r$;

3. if additionally $F$ and $G$ satisfy (C3) and $L< \frac{\nu}{2\mathcal N \sqrt{1+\nu}}$, then the solution $\varphi_{\varepsilon}$ of equation (30) is strongly compatible in distribution (i.e. $\mathfrak M_{(\mathcal A, F,G)}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}$) and $\mathfrak M_{(\mathcal A, F, G)}^{u}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}^{u}$, recalling that $\tilde{\mathfrak M}^{u}_{\varphi_{\varepsilon}}$ means the set of all sequences $\{t_n\}$ such that $\varphi_{\varepsilon}(t+t_n)$ converges in distribution uniformly with respect to $t\in\mathbb R$;

4.

where $\bar{\phi}$ is the unique stationary solution of equation (33);

5.

Proof. By Theorem 3.3 (see also Remark 11-(ⅲ)) there exist positive constants $\mathcal N, \nu$ and $\alpha$ such that equation

is uniformly asymptotically stable for any $0<\varepsilon \le \alpha$ and

for any $t\ge\tau$. By Theorem 3.3-(ⅱ) there are $\gamma_0>0$ and $\mathcal N:(0,\alpha)\to \mathbb R_{+}$ such that $\mathcal N (\varepsilon)\to 0$ as $\varepsilon \to 0$ and

for any $t\ge\tau$.

Since $Lip(F_{\varepsilon}) = Lip(F)\le L$ and $Lip(G_{\varepsilon}) = Lip(G)\le L$, by Theorem 2.20 equation (30) (respectively, equation (31)) has a unique solution $\varphi_{\varepsilon}$ (respectively, $\phi_{\varepsilon}$) from $C_{b}(\mathbb R, L^{2}(\mathbb P,\mathfrak B))$ with $\varphi_{\varepsilon}\in C(\mathbb R, B[0,r])$ (respectively, $\phi_{\varepsilon}\in C(\mathbb R, B[0,r])$), where

and the solution $\varphi_{\varepsilon}$ is strongly compatible in distribution and $\mathfrak M_{(\mathcal A,F,G)}^{u}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}^{u}$.

Let $\bar{\phi}$ be the unique stationary solution of equation (33). We now estimate $\mathbb E|\phi_{\varepsilon}(t)-\bar{\phi}(t)|^{2}$. To this end, we note that

Since

using Cauchy-Schwartz inequality we get

We will show that

To this end, making the change of variable $s = \tau-t$ and integrating by parts, we obtain for any $l>0$

For $J_{1}$, we have

Divide $[0,l]$ into intervals of size $\delta$, where $\delta>0$ is a fixed constant depending on $\varepsilon$. Denote an adapted process $\tilde{\phi}$ such that $\tilde{\phi}(\sigma) = \bar{\phi}(t-k\delta)$ for any $\sigma\in(t-(k+1)\delta,t-k\delta]$. By Lemma 4.1, we have

For $i_{2}$, denote $s(\delta): = \left[\frac{|s|}{\delta}\right]$, we have

For $i_{2}^{1}$, by Lemma 4.1 we have

For $i_{2}^{2}$, we obtain

Therefore, (38)–(40) imply

Therefore, we have

Combing (36), (37) and (41), we have

Taking $\delta = \sqrt{\varepsilon}$ and letting $\varepsilon\rightarrow0$ in (42), we have

Since $l$ is arbitrary, by letting $l\to\infty$ we get

From (35) and (43) it follows that there exists a function $A: (0,\varepsilon_{0})\to \mathbb R_{+}$ so that $A(\varepsilon)\to 0$ as $\varepsilon \to 0$ and

for any $t\in\mathbb R$ and $\varepsilon \in (0,\varepsilon_0)$.

Now we will establish a similar estimation for $I_{2}(t,\varepsilon)$. Since

using Itô's isometry property we have

Now we prove that

By Lemma 3.4, it suffices to show that

To this end, define an adapted process $\hat{\phi}$ such that $\hat{\phi}(\sigma) = \bar{\phi}(t+k\delta)$ for any $\sigma\in[t+k\delta,t+(k+1)\delta)$. We can assume $s>0$ without loss of generality, then we have by Lemma 4.1

For $J_{3}$, we have

Then

and

Therefore we have

Taking $\delta = \sqrt{\varepsilon}$ and letting $\varepsilon\rightarrow0$ in (46), we have

From (45) and (47) it follows that

where $B(\varepsilon)$ is some positive constant such that $B(\varepsilon)\to 0$ as $\varepsilon \to 0$.

Combing (34), (44) and (48), we have

Consequently

because $1-3(\mathcal N L)^2\left(\frac{2}{\nu^2}+\frac{1}{\nu}\right)>0$.

To finish the proof of the theorem we note that $L^2$-convergence implies convergence in distribution, so

Since $\mathcal L(\varphi_{\varepsilon} (\frac{t}{\varepsilon})) = \mathcal L(\phi_{\varepsilon}(t))$, we get

The proof is complete.

Corollary 3. Under the conditions of Theorem 4.2 the following statements hold:

1. if the functions $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and $F,G\in$ $C(\mathbb R\times\mathcal H,\mathcal H)$ are jointly stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1,\ldots,\nu_k$, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable), then equation (30) has a unique solution $\varphi_{\varepsilon} \in C_{b}(\mathbb R,L^2(\mathbb P,\mathcal H))$ which is stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1,\ldots,\nu_k$, Bohr almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable) in distribution;

2. if the functions $\mathcal A\in C(\mathbb R,L(\mathcal H))$ and $F,G\in$ $C(\mathbb R \times\mathcal H,\mathcal H)$ are Lagrange stable and jointly pseudo-periodic (respectively, pseudo-recurrent), then equation (30) has a unique solution $\varphi_{\varepsilon} \in C_{b}(\mathbb R,L^2(\mathbb P,\mathcal H))$ which is pseudo-periodic (respectively, pseudo-recurrent) in distribution;

3.

with $\bar{\phi}$ being the unique stationary solution of the averaged equation (33).

Proof. This statement follows from Theorems 2.12 and 4.2 (see also Remark 2).

Remark 15. To simplify the notations and highlight the idea, we consider only the one-dimensional noise. Indeed, the main results of this paper remain hold if we replace the one-dimensional Brownian motion $W$ in Sections 3 and 4 by a $Q$-Wiener process, which brings no essential but just notational difference; see e.g. [20,21] for details.

Remark 16. In the present paper, we only consider the second Bogolyubov theorem for semilinear stochastic ordinary differential equations, i.e. the linear part $\mathcal A(\cdot)$ is bounded operator valued. We will consider the case when $\mathcal A(\cdot)$ is an unbounded operator in future work, which can be applied to related stochastic partial differential equations.

Acknowledgments

We would like to thank Professor A. J. Roberts for drawing our attention to stochastic centre manifold approach for averaging and to references [3,22]. We are grateful to the anonymous referees for their careful reading of our paper and valuable suggestions.