Averaging principle on infinite intervals for stochastic ordinary differential equations with Lévy noise

1.   Introduction

The core idea of the averaging method is to simplify the original system so as to get an effective one which can in some sense reflect the dynamics of the original one. It was first introduced in perturbation theory by Clairaut, Laplace and Lagrange. Some rigorous results on averaging principles can date back to Krylov and Bogolyubov's work ^[1], now called Krylov-Bogolyubov-Mitropolsky method ^[2,3,4]. Driven by applications the averaging principle has been developed in mechanics, mathematics, control and other areas. There are lots of works on averaging for deterministic systems which we will not mention here. Stochastic averaging principle is proposed by Stratonovich ^[5] for nonlinear oscillation problems with random noise. For averaging principles of stochastic differential equations among others ^{[6,7,8,9,10,11,12,13,14]}.

In previous studies for stochastic averaging, researchers usually consider Gaussian noise. Although it is an ideal noise, we do agree on another fact: Gaussian noise cannot describe discontinuous situations and cannot simulate large fluctuations. It is evident that random noises in practise are more likely to be non-Gaussian. Lévy processes are essentially stochastic processes with stationary and independent increments, and they are viewed as an important and useful class of non-Gaussian processes since they are the simplest examples of random motions whose sample paths are right-continuous and have a number of (at most countable) random jump discontinuities occurring at random times, on each finite time interval. The method of averaging has been applied to stochastic differential equations with Lévy noise, but in general to the initial problem on a finite interval, such as ^[15,16]. As for the averaging method on infinite intervals to deterministic equations, the book of Burd ^[17] provides a detail description.

Motivated by the work of Cheban and Liu ^[18], we investigate the averaging principle on infinite intervals for linear and semilinear stochastic differential equations based on Lévy noise with Poisson stable (including stationary, periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent and Poisson stable) coefficients. Under some suitable conditions, the original equation has a unique bounded solution with the same recurrent property as the coefficients, see ^[19,20,21] for details. Besides we show that this recurrent solution uniformly converges to the unique stationary solution of the averaged equation on the whole real axis in distribution when the time scale goes to zero.

The paper is organized as follows. Section 2 begins with definitions of Poisson stable functions, Lévy processes and their basic properties. We simply review Lévy-Itô decomposition, B. A. Shcherbakov's comparable method by character of recurrence, and the existence of recurrent solutions for stochastic differential equations. In the third and fourth sections, we respectively investigate the averaging principles for the following equations

and

with $\varepsilon$ a small positive parameter, operator $\mathcal A$ non-stationary but bounded, coefficients $f, g, F, G$ Poisson stable in time, $t\in \mathbb R$ an infinite time interval.

2.   Preliminaries

2.1.   Poisson stable functions

Let $(\mathcal Y, \rho)$ be a complete metric space. Denote by $C(\mathbb R, \mathcal Y)$ the space of all continuous functions $\varphi :\mathbb R\to \mathcal Y$ equipped with the metric

for any $\varphi, \psi\in C(\mathbb R, \mathcal Y)$, where $d_{n}(\varphi, \psi): = \sup\limits_{|t|\le n}\rho(\varphi(t), \psi(t))$. Note that the metric $d$ generates the compact-open topology on $C(\mathbb R, \mathcal Y)$ and the space $(C(\mathbb R, \mathcal Y), d)$ is a complete metric space ^{[22,23,24,25]}.

Remark 2.1. Let $\varphi, \varphi_n\in C(\mathbb R, \mathcal Y)$ $(n\in\mathbb N)$. Then the following statements are equivalent:

(i) $d(\varphi_n, \varphi)\to 0$ as $n\to \infty$;

(ii) for each $l > 0$, $\lim\limits_{n\to \infty}\max\limits_{|t|\leq l}\rho(\varphi_n(t), \varphi(t)) = 0$;

(iii) there exists a sequence $\{l_n\}\to +\infty$ such that $\lim\limits_{n\to \infty}\max\limits_{|t|\leq l_n}\rho(\varphi_n(t), \varphi(t)) = 0$.

For given $\varphi\in C(\mathbb R, \mathcal Y)$, we use $\varphi^h$ to denote the $h$-translation of $\varphi$, where $\varphi^h(t): = \varphi(h+t)$ for $t\in\mathbb R$. The hull of $\varphi$, denoted by $H(\varphi)$, is the set of all the limits of $\varphi^{h_n}$ in $C(\mathbb R, \mathcal Y)$, i.e. $H(\varphi): = \{\psi\in C(\mathbb R, \mathcal Y):\psi = \lim\limits_{n\to\infty} \varphi^{h_n} \hbox{ for some sequence } \{h_n\} \subset \mathbb R\}$.

Remark 2.2. The mapping $\pi: \mathbb R\times C(\mathbb R, \mathcal Y)\to C(\mathbb R, \mathcal Y)$ defined by $\pi(h, \varphi) = \varphi^h$ is a dynamical system, i.e. $\pi(0, \varphi) = \varphi$, $\pi(h_1+h_2, \varphi) = \pi(h_2, \pi(h_1, \varphi))$ and the mapping $\pi$ is continuous (see ^[22,26]). In particular, the mapping $\pi$ restricted to $\mathbb R\times H(\varphi)$ is a dynamical system.

Now we recall the types of Poisson stable functions to be studied in this paper and the relations among them; see ^{[22,23,24,25]} for further details.

Definition 2.3. A function $\varphi\in C(\mathbb R, \mathcal Y)$ 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. A function $\varphi \in C(\mathbb R, \mathcal Y)$ is called quasi-periodic with the spectrum of frequencies $\nu_1, \nu_2, \ldots, \nu_m$ if the following conditions are fulfilled:

(i) the numbers $\nu_1, \nu_2, \ldots, \nu_m$ are rationally independent;

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

(iii) $\varphi(t) = \Phi(\nu_1 t, \nu_2 t, \ldots, \nu_m t)$ for all $t\in \mathbb R$.

Definition 2.5. A function $\varphi \in C(\mathbb R, \mathcal Y)$ is said to be almost periodic if for each $\varepsilon > 0$, the set $\mathcal T(\varphi, \varepsilon): = \{\tau\in\mathbb R:\sup\limits_{t\in\mathbb R}\rho(\varphi(t+\tau), \varphi(t)) < \varepsilon\}$ is relatively dense on $\mathbb R$, i.e. there exists $l = l(\varepsilon) > 0$ such that $\mathcal T(\varphi, \varepsilon)\cap [a, a+l]\not = \emptyset$ for any $a\in\mathbb R$. The set $\mathcal T(\varphi, \varepsilon)$ is called the set of $\varepsilon$-almost period of the function $\varphi$.

Definition 2.6. A function $\varphi \in C(\mathbb R, \mathcal Y)$ is said to be pseudo-periodic in the positive (respectively, negative) direction if for each $\varepsilon > 0$ and $l > 0$ there exists an $\varepsilon$-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 2.7. A function $\varphi \in C(\mathbb R, \mathcal Y)$ 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 (w.r.t.) $t\in \mathbb R$ as $n\to \infty$.

Definition 2.8. A function $\varphi \in C(\mathbb R, \mathcal Y)$ is called almost automorphic if and only if for any sequence $\{t'_{n}\} \subset \mathbb R$ there are a subsequence $\{t_n\}$ and some function $\psi: \mathbb R\to \mathcal Y$ such that

uniformly in $t$ on every compact subset from $\mathbb R$.

Definition 2.9. (i) A number $\tau\in\mathbb R$ is said to be $\varepsilon$-shift for $\varphi \in C(\mathbb R, \mathcal Y)$ if $d(\varphi^{\tau}, \varphi) < \varepsilon$; a function $\varphi \in C(\mathbb R, \mathcal Y)$ is called almost recurrent (in the sense of Bebutov) if for every $\varepsilon > 0$ the set $\{\tau : d(\varphi^{\tau}, \varphi) < \varepsilon\}$ is relatively dense.

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

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

In what follows, we denote by $(\mathcal X, \gamma)$ a complete metric space.

Definition 2.10. A function $\varphi\in C(\mathbb R, \mathcal Y)$ is called Levitan almost periodic if there exists an almost periodic function $\psi \in C(\mathbb R, \mathcal X)$ such that for any $\varepsilon > 0$ there exists $\delta = \delta (\varepsilon) > 0$ such that $d(\varphi^{\tau}, \varphi) < \varepsilon$ for all $\tau \in \mathcal T(\psi, \delta)$.

Remark 2.11. ([27,ChIV])

(i) Every almost periodic function is Levitan almost periodic but the inverse statement is not true in general.

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

Definition 2.12. A function $\varphi \in C(\mathbb R, \mathcal Y)$ is called pseudo-recurrent if for any $\varepsilon > 0$ and $l\in\mathbb R$ there exists a constant $L\ge l$ such that for any $\tau_0\in \mathbb R$ there exists a number $\tau \in [l, L]$ satisfying

Definition 2.13. A function $\varphi \in C(\mathbb R, \mathcal Y)$ 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 provided it is Poisson stable in both directions.

Remark 2.14. (^{[23,24,25,28]})

(i) Every Birkhoff recurrent function is pseudo-recurrent, but the inverse is not always true.

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

(iii) If the function $\varphi \in C(\mathbb R, \mathcal Y)$ 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.

2.2.   Shcherbakov's comparability method by character of recurrence

Let $\varphi \in C(\mathbb R, \mathcal Y)$. 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 $C(\mathbb R, \mathcal Y)$ 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 w.r.t. $t\in\mathbb R$ as $n\to \infty$.

Definition 2.15. A function $\varphi \in C(\mathbb R, \mathcal Y)$ is said to be comparable (by character of recurrence) with $\psi\in C(\mathbb R, \mathcal X)$ if $\mathfrak N_{\psi}\subseteq \mathfrak N_{\varphi}$; $\varphi$ is said to be strongly comparable (by character of recurrence) with $\psi$ if $\mathfrak M_{\psi}\subseteq \mathfrak M_{\varphi}$.

Theorem 2.16. ([23,29,Chapter II]) Let $\varphi\in C(\mathbb R, \mathcal Y)$, $\psi\in C(\mathbb R, \mathcal X)$. Then the following statements hold.

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

(ii) Let $\varphi$ be comparable with $\psi$. If the function $\psi$ is stationary (respectively, $\tau$-periodic, Levitan almost periodic, almost recurrent, Poisson stable), then so is $\varphi$.

(iii) Let $\varphi$ be strongly comparable with $\psi$. If the function $\psi$ is quasi-periodic with the spectrum of frequencies $\nu_1$, $\nu_2$, $\dots$, $\nu_m$ (respectively, almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable), then so is $\varphi$.

(iv) Let $\varphi$ be strongly comparable with $\psi$ and $\psi$ be Lagrange stable. If $\psi$ is pseudo-periodic (respectively, pseudo-recurrent), then so is $\varphi$.

Lemma 2.17. (^[19]) Let $\varphi \in C(\mathbb R, \mathcal Y)$, $\psi \in C(\mathbb R, \mathcal X)$. The following statements hold:

(i) If $\mathfrak M_{\psi}^{u} \subseteq \mathfrak M_{\varphi}^{u}$, then $\mathfrak N_{\psi}^{u}\subseteq \mathfrak N_{\varphi}^{u}$.

(ii) If $\mathfrak M_{\psi}^{u} \subseteq \mathfrak M_{\varphi}^{u}$ and the function $\psi$ is almost periodic, then so is $\varphi$.

(iii) If $\mathfrak N_{\psi}^{u} \subseteq \mathfrak N_{\varphi}^{u}$ and the function $\psi$ is pseudo periodic, then so is $\varphi$.

Denote by $BUC(\mathbb R\times \mathcal Y, \mathcal X)$ the space of all functions $f:\mathbb R\times \mathcal Y\to \mathcal X$ which are continuous in $t$ uniformly w.r.t. $y$ on every bounded subset $Q\subseteq \mathcal Y$ and bounded on every bounded subset from $\mathbb R\times \mathcal Y$. We endow the function space with the metric

where

with $Q_n\subset\mathcal Y$ being bounded, $Q_n\subset Q_{n+1}$ and $\mathcal Y = \bigcup\limits_{n\ge 1} Q_n$. Note that $(BUC(\mathbb R\times \mathcal Y, \mathcal X), d)$ is a complete metric space and $d(f_n, f)\to 0$ if and only if $f_n(t, y)\to f(t, y)$ uniformly w.r.t. $(t, y)$ on every bounded subset from $\mathbb R\times \mathcal Y$. For given $f\in BUC(\mathbb R\times \mathcal Y, \mathcal X)$ and $\tau\in\mathbb R$, denote the translation of $f$ by $f^\tau$, i.e. $f^\tau(t, y): = f(t+\tau, y)$ for $(t, y)\in \mathbb R\times \mathcal Y$, and the hull of $f$ by $H(f): = \overline{\{f^\tau:\tau\in\mathbb R\}}$ with the closure being taken under the metric $d$ given by (2.1). Note that the mapping $\pi: \mathbb R\times BUC(\mathbb R\times \mathcal Y, \mathcal X) \to BUC(\mathbb R\times \mathcal Y, \mathcal X)$ defined by $\pi(\tau, f): = f^\tau$ is a dynamical system, i.e. $\pi(0, f) = f$, $\pi(\tau_1+\tau_2, f) = \pi(\tau_2, \pi(\tau_1, f))$ and the mapping $\pi$ is continuous. See ^[22,26] or [§ 1.1] for details.

We use $BC(\mathcal Y, \mathcal X)$ to denote the space of all continuous functions $f:\mathcal Y\to \mathcal X$ which are bounded on every bounded subset $Q\subset \mathcal Y$ and equip the space with the metric

where $d_{n}(f, g): = \sup\limits_{y\in Q_n}\gamma(f(y), g(y))$ with $Q_n$ the same as (2.1). It is immediate to check that $(BC(\mathcal Y, \mathcal X), d)$ is a complete metric space.

Remark 2.18. Let $F\in BUC(\mathbb R\times \mathcal Y, \mathcal X)$ and $f :\mathbb R\to BC(\mathcal Y, \mathcal X)$ be a mapping defined by $f(t): = F(t, \cdot)$. Note that for any $F\in BUC(\mathbb R\times \mathcal Y, \mathcal X)$, we have $\mathfrak M_{F} = \mathfrak M_{f}$ and $\mathfrak M_{F}^{u} = \mathfrak M_{f}^{u}$. Here $\mathfrak M_{F}$ is the set of all sequences $\{t_n\}$ such that $\{F^{t_n}\}$ converges in the space $BUC(\mathbb R\times \mathcal Y, \mathcal X)$; $\mathfrak M_{F}^{u}$ is the set of all sequences $\{t_n\}$ such that $\{F^{t_n}\}$ converges in the space $BUC(\mathbb R\times \mathcal Y, \mathcal X)$ uniformly w.r.t. $t\in\mathbb R$.

2.3.   Semilinear stochastic differential equations with Lévy noise

Throughout the paper, we fix a complete probability space $(\Omega, \mathcal{F}, \mathbf{P})$, two real separable Hilbert spaces $(\mathbb H, |\cdot|)$ and $(U, |\cdot|_U)$, and a real separable Banach space $(\mathfrak B, |\cdot|_\mathfrak B)$. Denote by $L(\mathbb H)$ (respectively, $L(\mathfrak B)$) the Hilbert (respectively, Banach) space of all bounded linear operators on $\mathbb H$ (respectively, $\mathfrak B$) endowed with operator norm $\|\cdot\|$ (respectively, $\|\cdot\|_\mathfrak B$). Denote by $L(U, \mathbb H)$ the Banach space of all bounded linear operators from $U$ to $\mathbb H$ with the norm $\|\cdot\|_{L(U, \mathbb H)}$.

We now review the definition of Lévy processes and the important Lévy-Itô decomposition theorem (see ^[30,31]). In this paper, the Lévy processes we consider are $U$-valued.

Definition 2.19. A $U$-valued stochastic process $L = (L(t), t\ge 0)$ is called Lévy process if it has the following three properties:

(i) $L(0) = 0$ almost surely.

(ii) $L$ has stationary and independent increments, i.e. the law of $L(t+h)-L(t)$ does not depend on $t$ and for all $0\leq t_0 < t_1 < t_2 < ... < t_n < \infty$, random variables $L(t_1)-L(t_0)$, $L(t_2)-L(t_1)$, ..., $L(t_n)-L(t_{n-1})$ are independent.

(iii) $L$ is stochastically continuous, i.e. for all $\epsilon > 0$ and for all $s > 0$

Since a Lévy process $L$ is càdlàg, the associated jump process $\Delta L = (\Delta L(t), t\geq 0)$ is given by $\Delta L(t) = L(t)-L(t-)$. Let $\mathcal B(U-\{0\})$ be the Borel field of $U-\{0\}$ and $B\in \mathcal B(U-\{0\})$. Define the random counting measure

where $\chi_ B$ is the indicator function of $B$. We call $\nu(\cdot): = \mathbb{E}(N(1, \cdot))$ the intensity measure of $L$. We say that $B\in \mathcal B(U-\{0\})$ is bounded below if $0\notin \overline{B}$, the closure of $B$. Note that if $B$ is bounded below, then $N(t, B) < \infty$ holds almost surely for all $t\geq 0$ and $(N(t, B), t\geq 0)$ is a Poisson process with intensity $\nu (B)$. $N$ is called Poisson random measure. For each $t\geq 0$ and $B$ bounded below, define the compensated Poisson random measure by

Proposition 2.20 (Lévy-Itô decomposition). If $L$ is a Lévy process in $U$, then there exists $a\in U$, a $U$-valued $\mathcal Q$-Wiener process $W$ and an independent Poisson random measure $N$ on $\mathbb R^+ \times (U-\{0\})$ with intensity measure $\nu$ such that for any $t\geq 0$,

Here the intensity measure $\nu$ satisfies

and $\widetilde N$ is the compensated Poisson random measure of $N$.

As for $\mathcal Q$-Wiener processes and the stochastic integral based on them, the monograph ^[32] provides a thorough description. Assume that $L_1$ and $L_2$ are two independent, identically distributed Lévy processes with decompositions as in Proposition 2.19 and let

Then $L$ is a two-sided Lévy process. In this paper, we consider two-sided Lévy process $L$ which is defined on the filtered probability space $(\Omega, \mathcal F, \mathbf P, (\mathcal F_t)_{t\in\mathbb R})$ and suppose that the covariance operator $\mathcal Q$ of $W$ is of trace class, i.e. ${\rm Tr} \mathcal Q < \infty$.

Remark 2.21. It follows from (2.3) that $\int_{|x|_U\ge 1} \nu({\rm d } x) < \infty$. For convenience, we set hereafter

Remark 2.22. Note that the stochastic process $\widetilde L = (\widetilde L(t), t\in \mathbb R)$ given by $\widetilde L(t): = L(t+s)-L(s)$ for some $s\in \mathbb R$ is also a two-sided Lévy process which shares the same law as $L$. In particular, when $s\in\mathbb R^+$, the similar conclusion holds for one-sided Lévy processes.

Consider the linear homogeneous equation

on the space $\mathfrak B$, where $\mathcal A\in C(\mathbb R, L(\mathfrak B))$. Denote by $T_\mathcal{A}(t, \tau)$ the evolution (solving) operator of Eq (2.4), where $T_{\mathcal A}(t, \tau): = U_{\mathcal A}(t)U_{\mathcal A}^{-1}(\tau)$ with $U_{\mathcal A}(t)$ the Cauchy operator of Eq (2.4) (see ^[33]).

Definition 2.23. Eq (2.4) is said to be uniformly asymptotically stable if there are positive constants $K$ and $\omega$ such that

Lemma 24. ([34,Chapter III]) Suppose that Eq (2.4) is uniformly asymptotically stable such that Eq (2.5) holds. Then for any $t\ge \tau$ ($t, \tau \in \mathbb R$) and $\tilde{\mathcal A}\in H(\mathcal A)$

where $H(\mathcal A)$ denotes the closure in the space $C(\mathbb R, L(\mathfrak B))$ of all translations $\{\mathcal A^{h}: h\in\mathbb R\}$ with $\mathcal A^{h}(t): = \mathcal A(t+h)$ for $t\in\mathbb R$.

We now consider the stochastic differential equation driven by Lévy noise

where $\mathcal A:\mathbb R\to L(\mathbb H)$, $f:\mathbb R\times \mathbb H\to\mathbb H$, $g:\mathbb R\times \mathbb H\to L(U, \mathbb H)$; $L$ is a $U$-valued Lévy process. By Lévy-Itô decomposition (2.2), Eq (2.6) reads

where $F$ and $G$ are $\mathbb H$-valued. It allows us to study large jumps with considerable probability. We set $\mathcal F_t: = \sigma\{L(u):u\leq t\}$.

Definition 2.25. An $\mathcal{F}_t$-adapted process $\{Y(t)\}_{t\in{{\mathbb R}}}$ is called a mild solution of Eq (2.7) if it satisfies the corresponding stochastic integral equation

for all $t\geq r$ and each $r\in \mathbb R$.

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

Here $f$ varies in the space of bounded Lipschitz continuous real-valued functions on the space $\mathfrak B$ with the norm

where

A sequence $\{\mu_n\}\subset \mathcal P(\mathfrak B)$ is said to weakly converge 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 the space $\mathfrak B$. As we know 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 $n\to\infty$. See [35,§11.3] for $\beta$ metric and related properties.

Definition 2.26. A sequence of random variables $\{Y_n\}$ is said to converge in distribution to the random variable $Y$ if the corresponding laws $\{\mu_n\}$ of $\{Y_n\}$ weakly converge to the law $\mu$ of $Y$, i.e. $\beta(\mu_n, \mu)\to 0$.

Definition 2.27. Let $\{\varphi (t)\}_{t\in\mathbb R}$ be a mild solution of Eq (2.7). Then $\varphi$ is called compatible (respectively, strongly compatible) in distribution if $\mathfrak N_{(\mathcal A, f, g, F, G)} \subseteq \tilde{\mathfrak N}_{\varphi}$ (respectively, $\mathfrak M_{(\mathcal A, f, g, F, G)} \subseteq \tilde{\mathfrak M}_{\varphi}$), 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.

2.4.   Some results of linear and semilinear stochastic differential equations with Lévy noise

Define for $p\geq2$

with the norm

Then $(\mathcal L^p(\mathbf{P}; \mathbb{H}), \|\cdot\|_{\mathcal L^p(\mathbf{P}; \mathbb{H})})$ is a Banach space. Set

and define a norm by

Note that $(\mathcal L^2(\mathbf P; L(U, \mathbb H)), \|\cdot\|_{\mathcal L^2(\mathbf P; L(U, \mathbb H))})$ is a Banach space.

Remark 2.28. If the operator $\mathcal Q\in L(U)$, the space of bounded linear operators from $U$ to $U$, is nonnegative, symmetric and ${\rm Tr}\mathcal Q < \infty$, then $L\mathcal Q^\frac{1}{2}\in L_2(U, \mathbb H)$ for all $L\in L(U, \mathbb H)$, where the space $L_2(U, \mathbb H)$ is a separable Hilbert space that consists of all Hilbert-Schmidt operators from $U$ to $\mathbb H$ with inner product $\langle A, B\rangle_{L_2(U, \mathbb H)}: = \sum\limits_{k\in \mathbb N}\langle Ae_k, Be_k\rangle$ and $\{e_k\}_{k\in \mathbb N}$ an orthonormal basis of $U$. For $g\in \mathcal L^2(\mathbf P; L(U, \mathbb H))$, we have $g\mathcal Q^\frac{1}{2}\in\mathcal L^2(\mathbf P; L_2(U, \mathbb H))$ and denote $\|g\mathcal Q^\frac{1}{2}\|_{\mathcal L^2(\mathbf P; L_2(U, \mathbb H))} = \left(\mathbb E\|g\mathcal Q^\frac{1}{2}\|^2_{L_2(U, \mathbb H)}\right)^\frac{1}{2}$.

Define

where $\mathcal P_U$ is the product $\sigma$-algebra on $\Omega\times U$ and $\mathbf P_\nu = \mathbf P\otimes \nu$. For $Y\in \mathcal L^2(\mathbf P_\nu; \mathbb H)$, let

Then $\mathcal L^2(\mathbf P_\nu; \mathbb H)$ is a Hilbert space equipped with the norm $\|\cdot\|_{\mathcal L^2(\mathbf P_\nu; \mathbb H)}$.

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 $\|\varphi\|_{\infty}: = \sup\{|\varphi(t)|_{\mathfrak B}: t\in\mathbb R\}$. Note that if $f\in C_b(\mathbb R, \mathfrak B)$ and $\tilde{f}\in H(f)$, the hull of $f$, then $\|\tilde{f}\|_{\infty}\le \|f\|_{\infty}$.

Theorem 2.29. Consider the linear stochastic differential equation

Assume that $\mathcal A\in C_b(\mathbb R, L(\mathbb H))$; Eq $(2.4)$ is uniformly asymptotically stable such that $(2.5)$ holds; $f\in C_b(\mathbb{R}, \mathcal{L}^{2}(\mathbf{P}; \mathbb{H}))$, $g\in C_b(\mathbb R, \mathcal L^2(\mathbf P; L(U, \mathbb H)))$, $F, G\in C_b(\mathbb R, \mathcal L^2(\mathbf P_\nu; \mathbb H))$. Suppose that $W$ and $N$ are the same as in Section 2.3. Then Eq $(2.9)$ has a unique mild solution $\varphi\in C_b(\mathbb{R}, \mathcal{L}^{2}(\mathbf{P};\mathbb{H}))$ which satisfies

and this unique $\mathcal L^2$-bounded solution is strongly compatible in distribution (i.e. $\mathfrak M_{(\mathcal A, f, g, F, G)}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}$). Furthermore, $\mathfrak M_{(\mathcal A, f, g, F, G)}^{u}\subseteq \mathfrak {\tilde{M}}_{\varphi}^{u}$, where $\mathfrak{\tilde{M}}_{\varphi}^{u}$ is the set of all sequences $\{t_n\}$ such that the sequence $\{\varphi(t+t_n)\}$ converges in distribution uniformly in $t\in \mathbb{R}$.

Proof. The proof is analogous to Theorem 3.3 in ^[21].

Define

with $\mathcal B_U$ being the Borel $\sigma$-algebra on $U$. For $Y\in\mathcal L^2(\nu; \mathbb H)$, let

Denote by $C(\mathbb R\times\mathcal Y, \mathcal X)$ the space of all continuous functions $f:\mathbb R\times \mathcal Y\to\mathcal X$, recalling that $(\mathcal Y, \rho)$ and $(\mathcal X, \gamma)$ are complete metric spaces.

Consider Eq (2.7). Like in ^[21], assume that $f\in C(\mathbb{R}\times\mathbb H, \mathbb H)$, $g\in C(\mathbb{R}\times\mathbb H, L(U, \mathbb H))$, $F, G\in C(\mathbb{R}\times\mathbb H, \mathcal L^2(\nu; \mathbb H))$ and $f, g, F, G$ satisfy the following conditions:

(E1) There exists a number $M\geq 0$ such that for all $t\in\mathbb R$

$|f(t, 0)|\leq {M}$, $\|g(t, 0)\mathcal Q^\frac{1}{2}\|_{L_2(U, \mathbb H)}\leq {M}$,

$\int_{|x|_U < 1}|F(t, 0, x)|^2\nu({\rm d }x)\leq {M}^2$, $\int_{|x|_U\geq1}|G(t, 0, x)|^2\nu({\rm d }x)\leq {M}^2$.

(E1') There exists a number $M\geq 0$ such that for some constant $p > 2$ and all $t\in\mathbb R$

$|f(t, 0)|\leq {M}$, $\|g(t, 0)\mathcal Q^\frac{1}{2}\|_{L_2(U, \mathbb H)}\leq {M}$,

$\int_{|x|_U < 1}|F(t, 0, x)|^p\nu({\rm d }x)\leq {M}^p$, $\int_{|x|_U\geq1}|G(t, 0, x)|^p\nu({\rm d }x)\leq {M}^p$.

(E2) There exists a number $\mathcal L\geq 0$ such that for all $t\in\mathbb R$ and $y_1, y_2 \in \mathbb H$

$|f(t, y_1)-f(t, y_2)|\leq\mathcal L|y_1-y_2|$, $\|(g(t, y_1)-g(t, y_2))\mathcal Q^\frac{1}{2}\|_{L_2(U, \mathbb H)}\leq\mathcal L|y_1-y_2|$,

$\int_{|x|_U < 1}|F(t, y_1, x)-F(t, y_2, x)|^2\nu({\rm d }x)\leq\mathcal L^2|y_1-y_2|^2$,

$\int_{|x|_U\geq1}|G(t, y_1, x)-G(t, y_2, x)|^2\nu({\rm d }x)\leq\mathcal L^2|y_1-y_2|^2$.

(E2 ') There exists a number $\mathcal L\geq 0$ such that for some constant $p > 2$ and $t\in\mathbb R$, $y_1, y_2 \in \mathbb H$

$|f(t, y_1)-f(t, y_2)|\leq\mathcal L|y_1-y_2|$, $\|(g(t, y_1)-g(t, y_2))\mathcal Q^\frac{1}{2}\|_{L_2(U, \mathbb H)}\leq\mathcal L|y_1-y_2|$,

$\int_{|x|_U < 1}|F(t, y_1, x)-F(t, y_2, x)|^p\nu({\rm d }x)\leq\mathcal L^p|y_1-y_2|^p$,

$\int_{|x|_U\geq1}|G(t, y_1, x)-G(t, y_2, x)|^p\nu({\rm d }x)\leq\mathcal L^p|y_1-y_2|^p$.

(E3) $f, g, F, G$ are continuous in $t$ uniformly w.r.t. $Y$ on each bounded subset $Q\subset\mathbb H$.

Remark 2.30. (i) If $f, g, F, G$ satisfy ${\rm(E1)}-{\rm(E3)}$, then $f\in BUC(\mathbb R\times \mathbb H, \mathbb H)$, $g\in BUC(\mathbb R\times \mathbb H, L(U, \mathbb H))$, $F, G\in BUC(\mathbb{R}\times\mathbb H, \mathcal L^2(\nu; \mathbb H))$ and $H(f, g, F, G)\subset BUC (\mathbb R\times \mathbb H, \mathbb H) \times BUC (\mathbb R\times \mathbb H, L(U, \mathbb H)) \times BUC(\mathbb{R}\times\mathbb H, \mathcal L^2(\nu; \mathbb H))\times BUC(\mathbb{R}\times\mathbb H, \mathcal L^2(\nu; \mathbb H))$.

(ii) If $f, g, F, G$ satisfy ${\rm(E1)}, {\rm(E2)}$ (or ${\rm(E1 ' )}$, ${\rm(E2 ' )})$ with the constants $M$ and $\mathcal L$, then every quadruplet $(\tilde {f}, \tilde {g}, \tilde{F}, \tilde{G})\in H(f, g, F, G) : = \overline{\{(f^{\tau}, g^{\tau}, F^{\tau}, G^{\tau}): \tau\in \mathbb R\}}$, the hull of $(f, g, F, G)$, also possesses the same properties with the same constants.

Theorem 2.31. Consider Eq ${\rm(2.7)}$. Suppose that $\mathcal A\in C(\mathbb R, L(\mathbb H))$; Eq $(2.4)$ is uniformly asymptotically stable such that $(2.5)$ holds; $f\in C(\mathbb{R}\times\mathbb H, \mathbb H)$, $g\in C(\mathbb{R}\times\mathbb H, L(U, \mathbb H))$, $F, G\in C(\mathbb{R}\times\mathbb H, \mathcal L^2(\nu; \mathbb H))$. Suppose that $W$ and $N$ are the same as in Section 2.3; $f, g, F, G$ satisfy the conditions (E1) and (E2). Then the following statements hold:

(i) If $\mathcal L < \cfrac{\omega}{2K\sqrt{1+2\omega+2b}}$, then Eq ${\rm(2.7)}$ has a unique solution $\xi\in C(\mathbb R, B[0, r])$ defined by equality

where $B[0, r]: = \{Y\in\mathcal{L}^2(\mathbf {P}; \mathbb H):\|Y\|_{\mathcal L^2(\mathbf{P}; \mathbb{H})}\leq r\}$ and $r = \cfrac{2KM\sqrt{1+2\omega+2b}}{\omega-2K\mathcal L\sqrt{1+2\omega+2b}}$.

(ii) If $\mathcal{L} < \bigg\{\cfrac{\omega}{2K\sqrt{2+8\omega+4b}}\wedge \cfrac{\omega}{2K\sqrt{1+10\omega+2b}}\bigg\}$ and $f, g, F, G$ satisfy ${\rm(E1 ' )}, {\rm(E2 ' )}$ and ${\rm(E3)}$ additionally, then we have $\mathfrak M^{u}_{(\mathcal A, f, g, F, G)} \subseteq \tilde{\mathfrak M}^{u}_{\xi}$ and the solution $\xi$ is strongly compatible in distribution (i.e. $\mathfrak M_{(\mathcal A, f, g, F, G)} \subseteq \tilde{\mathfrak M}_{\xi}$).

Proof. The proof is similar to Theorem 4.6 in ^[21].

Corollary 2.32. Consider Eq $(2.7)$. Assume that the conditions of Theorem 2.31 hold.

(i) If $\mathcal A, f, g, F, G$ are jointly stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1, \nu_2, \dots, \nu_m$, almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable) in $t\in\mathbb R$ uniformly w.r.t. $Y\in \mathbb H$ on every bounded subset, then so is the unique bounded solution $\xi$ of Eq ${\rm(2.7)}$ in distribution.

(ii) If $\mathcal A, f, g, F, G$ are jointly pseudo-periodic (respectively, pseudo-recurrent) and jointly Lagrange stable in $t\in\mathbb R$ uniformly w.r.t. $Y\in \mathbb H$ on every bounded subset, then the unique bounded solution $\xi$ of Eq ${\rm(2.7)}$ is pseudo-periodic (respectively, pseudo-recurrent) in distribution.

Proof. This statement follows from Theorems 2.16, 2.31 and Remark 2.18.

3.   Averaging principle for linear equations

Let $\varepsilon_0$ be some fixed positive number. We now study an averaging principle for the following equation

where $\mathcal A\in C(\mathbb R, L(\mathbb H))$, $f\in C(\mathbb R, \mathcal L^2(\mathbf P; \mathbb H))$, $g\in C(\mathbb R, \mathcal L^2(\mathbf P; L(U, \mathbb H)))$, $F, G\in C(\mathbb R, \mathcal L^2(\mathbf P_\nu; \mathbb H))$, and $\varepsilon\in(0, \varepsilon_0]$ is a small parameter. Here $W$ and $N$ are the Lévy-Itô decomposition components of the two-sided Lévy process L as in Section 2.3.

Denote by $\Psi$ the family of all decreasing, positive bounded functions $\psi :\mathbb R_{+}\to\mathbb R_{+}$ with $\lim\limits_{t\to +\infty}\psi(t) = 0$. Let $\mathcal A\in L(\mathbb H)$. Denote by $\sigma(\mathcal A)$ the spectrum of $\mathcal A$. We respectively impose the following conditions on $\mathcal A, f, g, F, G$:

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

uniformly w.r.t. $t\in\mathbb R$;

(A2) $f\in C(\mathbb R, \mathcal L^2(\mathbf P; \mathbb H))$ and there exist functions $\omega_1 \in\Psi$, $\bar{f}\in \mathcal L^2(\mathbf P; \mathbb H)$ such that

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

(A3) $g\in C(\mathbb R, \mathcal L^2(\mathbf P; L(U, \mathbb H)))$ and there exist functions $\omega_2 \in\Psi$, $\bar{g}\in \mathcal L^2(\mathbf P; L(U, \mathbb H))$ such that

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

(A4) $F\in C(\mathbb R, \mathcal L^2(\mathbf P_\nu; \mathbb H))$ and there exist functions $\omega_3 \in\Psi$, $\bar{F}\in \mathcal L^2(\mathbf P_\nu; \mathbb H)$ such that

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

(A5) $G\in C(\mathbb R, \mathcal L^2(\mathbf P_\nu; \mathbb H))$ and there exist functions $\omega_4 \in\Psi$, $\bar{G}\in \mathcal L^2(\mathbf P_\nu; \mathbb H)$ such that

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

Theorem 3.1. ([36,Chapter IV]) Suppose that $\mathcal A\in C_{b}(\mathbb R, L(\mathfrak B))$ and

uniformly w.r.t. $t\in\mathbb R$ 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:

(i) There exists a positive constant $\alpha \leq \varepsilon_{0}$ such that the equation

where $\mathcal A_{\varepsilon }(t): = \mathcal A(\frac{t}{\varepsilon})$ for any $t\in\mathbb R$, is uniformly asymptotically stable for any $\varepsilon \in (0, \alpha]$.

(ii) There exists $\gamma_{0} > 0$ such that

Remark 3.2. (i) Under the conditions of Theorem 3.1 there are positive constants $\alpha$, $K$ and $\omega$ such that

for any $t\ge\tau$ and $\varepsilon \in (0, \alpha]$.

(ii) By Eq (3.6) there exists a function $\mathcal K:(0, \alpha)\to \mathbb R_{+}$ such that $\mathcal K(\varepsilon)\to 0$ as $\varepsilon \to 0$ and

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

Lemma 3.3. (^[18]) Let $f_{\varepsilon}\in C(\mathbb R, \mathfrak B)$ for $\varepsilon \in (0, \alpha]$ be functions satisfying the following conditions:

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

(ii) for any $l > 0$

Then for any $\omega > 0$ we have

Remark 3.4. (^[18]) If the function $f\in C_b(\mathbb R, \mathfrak B)$ and $\bar{f}\in \mathfrak B$ are such that

uniformly w.r.t. $t\in\mathbb R$, then the function $f_{\varepsilon}(\sigma): = f(\frac{\sigma}{\varepsilon})-\bar f$ satisfies the conditions of Lemma 3.3. Similarly, if the function $g$ (respectively, $F$, $G$) in (A3) (respectively, (A4), (A5)) is $\mathcal L^2$-bounded, then the function $f_{\varepsilon}(\sigma): = \mathbb E\|(g(\frac{\sigma}{\varepsilon})-\bar g)\mathcal Q^\frac{1}{2}\|_{L_2(U, \mathbb H)}^2$ (respectively, $f_{\varepsilon}(\sigma): = \int_{|x|_U < 1}\mathbb E|F(\frac{\sigma}{\varepsilon}, x)-\bar F(x)|^2\nu({{{\rm{d}}}} x)$, $f_{\varepsilon}(\sigma): = \int_{|x|_U\geq1}\mathbb E|G(\frac{\sigma}{\varepsilon}, x)-\bar G(x)|^2\nu({{{\rm{d}}}} x)$) satisfies the conditions of Lemma 3.3.

Theorem 3.5. Consider Eq (3.1). Suppose that $\mathcal A\in C_b(\mathbb R, L(\mathbb H))$, $f\in C_b(\mathbb R, \mathcal L^2(\mathbf P; \mathbb H))$, $g\in C_b(\mathbb R, \mathcal L^2(\mathbf P; L(U, \mathbb H)))$, $F, G\in C_b(\mathbb R, \mathcal L^2(\mathbf P_\nu; \mathbb H))$ and conditions $(A1)$-$(A5)$ are satisfied. Suppose further that $\bar{\mathcal A}$ in (A1) is Hurwitz such that $(3.6)$ and $(3.7)$ hold. Then for any $\varepsilon \in (0, \alpha]$, Eq $(3.1)$ has a unique solution $\varphi_{\varepsilon}\in C_{b}(\mathbb R, \mathcal L^{2}(\mathbf P; \mathbb H))$ and it is strongly compatible in distribution (i.e. $\mathfrak M_{(\mathcal A, f, g, F, G)}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}$) and $\mathfrak M_{(\mathcal A, f, g, F, G)}^{u}\subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}^{u}$. Besides we have

where $\mathcal L(X)$ denotes the law of random variable $X$ and $\bar{\phi}$ is the unique stationary solution of

Proof. Consider the following equations

and

where $\mathcal A_{\varepsilon}(t): = \mathcal A(\frac{t}{\varepsilon})$, $f_{\varepsilon}(t): = f(\frac{t}{\varepsilon})$, $g_{\varepsilon}(t): = g(\frac{t}{\varepsilon})$, $F_{\varepsilon}(t, x): = F(\frac{t}{\varepsilon}, x)$ and $G_{\varepsilon}(t, x): = G(\frac{t}{\varepsilon}, x)$ for $t\in\mathbb R$ and $\varepsilon\in(0, \varepsilon_0]$. Here $W_{\varepsilon}(t): = \sqrt{\varepsilon}W(\frac{t}{\varepsilon})$, $N_{\varepsilon}(t, x): = \sqrt{\varepsilon}N(\frac{t}{\varepsilon}, x)$ and $\widetilde N_{\varepsilon}(t, x): = \sqrt{\varepsilon}\widetilde N(\frac{t}{\varepsilon}, x)$. By Theorem 2.29, Eq (3.9) has a unique bounded solution $\psi_\varepsilon\in C_b(\mathbb R, \mathcal L^2(\mathbf P; \mathbb H))$ defined by equality

and the solution is strongly compatible in distribution (i.e. $\mathfrak M_{(\mathcal A_{\varepsilon}, f_{\varepsilon}, g_{\varepsilon}, F_{\varepsilon}, G_{\varepsilon})}\subseteq \tilde{\mathfrak M}_{\psi_{\varepsilon}}$) and $\mathfrak M^{u}_{(\mathcal A_{\varepsilon}, f_{\varepsilon}, g_{\varepsilon}, F_{\varepsilon}, G_{\varepsilon})}\subseteq \tilde{\mathfrak M}^{u}_{\psi_{\varepsilon}}$. Let $\varphi_\varepsilon (t): = \psi_\varepsilon(\varepsilon t)$, $t\in\mathbb R$. Then $\varphi_{\varepsilon}\in C_{b}(\mathbb R, \mathcal L^{2}(\mathbf P; \mathbb H))$ is the unique bounded solution of Eq (3.1). If $\{t_n\}\in \mathfrak M_{(\mathcal A, f, g, F, G)}$ (respectively, $\{t_n\}\in \mathfrak M^{u}_{(\mathcal A, f, g, F, G)}$), then by Theorem 2.29 $\{\varepsilon t_n\}\in \mathfrak M_{(\mathcal A_{\varepsilon}, f_{\varepsilon}, g_{\varepsilon}, F_{\varepsilon}, G_{\varepsilon})}\subseteq \tilde{\mathfrak M}_{\psi_{\varepsilon}}$ (respectively, $\{\varepsilon t_n\}\in \mathfrak M^{u}_{(\mathcal A_{\varepsilon}, f_{\varepsilon}, g_{\varepsilon}, F_{\varepsilon}, G_{\varepsilon})}\subseteq \tilde{\mathfrak M}^{u}_{\psi_{\varepsilon}}$). Further we have $\{t_n\}\in \tilde{\mathfrak M}_{\varphi_{\varepsilon}}$ (respectively, $\{t_n\}\in\tilde{\mathfrak M}^{u}_{\varphi_{\varepsilon}}$).

We now prove the solution $\varphi_{\varepsilon}$ of the original Eq (3.1) converges to the stationary solution $\bar{\phi}$ of the averaged Eq (3.8) uniformly in $t\in \mathbb R$ in distribution sense. By Theorem 2.29, Eq (3.10) has a unique bounded solution $\phi_{\varepsilon}\in C_{b}(\mathbb R, \mathcal L^{2}(\mathbf P; \mathbb H))$ defined by equality

By Theorem 2.29, Eq (3.8) has a unique bounded and stationary solution $\bar{\phi}$, which is given by the formula

where $T_{\bar{\mathcal A}}(t, \tau) = \exp\left\{\bar{\mathcal A}(t-\tau)\right\}$ for $t, \tau\in\mathbb R$. From (3.11) and (3.12), by the basic inequality $\Big(\sum \limits_{i = 1}^n a_i\Big)^2\leq n\Big(\sum \limits_{i = 1}^n a_i^2\Big), n\in \mathbb N$, we have

Note that

Since

we have

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

Similar to the proof of [18,Theorem 3.9], by making the change of variable $s: = \tau-t$, integrating by parts and letting $l$ be an arbitrary positive number we have

By Remark 3.4, passing to limit in (3.15) as $\varepsilon\to 0$ we obtain

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

Note that by Cauchy-Schwarz inequality, we have

as $\varepsilon \to 0$. Consequently,

As for $I_2(t, \varepsilon)$, by Itô's isometry property and Remark 3.2-(ii), we have

Since by Lemma 3.3 the integral

goes to $0$ as $\varepsilon \to 0$ uniformly w.r.t. $t\in\mathbb R$, taking to the limit in (3.16) we obtain

Note that

According to Lemma 3.3 we have

Passing to the limit in (3.17) and considering (3.18), we get

By properties of integrals for Poisson random measures, we obtain

Using the similar arguments as (3.17), we get

By Cauchy-Schwarz inequality we have

According to (3.19)-(3.21), we obtain

By Lemma 3.3 we have

so taking limits in (3.22) we obtain

Consequently we have

Since the $\mathcal L^{2}$ convergence implies convergence in probability, it follows from (3.23) that

On the other hand taking into consideration that $\mathcal L(W) = \mathcal L(W_{\varepsilon})$ and $\mathcal L(N) = \mathcal L(N_{\varepsilon})$ with compensated Poisson random measure $\widetilde N$ we have $\mathcal L(\psi_{\varepsilon}(t)) = \mathcal L(\phi_{\varepsilon}(t))$ for any $t\in \mathbb R$, and by the relationship between $\varphi_\varepsilon$ and $\psi_\varepsilon$ (i.e. $\varphi_\varepsilon (t): = \psi_\varepsilon(\varepsilon t)$, $t\in\mathbb R$) we get

The proof is complete.

Corollary 3.6. Under the assumptions of Theorem 3.5, it follows from Theorems 2.16 and 3.5 that

(i) if $\mathcal A, f, g, F, G$ are jointly stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1, \ldots, \nu_k$, almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable), then Eq (3.1) has a unique solution $\varphi_{\varepsilon} \in C_{b}(\mathbb R, \mathcal L^2(\mathbf P; \mathbb H))$ which is stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1, \ldots, \nu_k$, almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable) in distribution;

(ii) if $\mathcal A, f, g, F, G$ are Lagrange stable and jointly pseudo-periodic (respectively, pseudo-recurrent), then Eq (3.1) has a unique solution $\varphi_{\varepsilon} \in C_{b}(\mathbb R, \mathcal L^2(\mathbf P; \mathbb H))$ which is pseudo-periodic (respectively, pseudo-recurrent) in distribution;

(iii)

4.   Averaging principle for semilinear equations

Let $\varepsilon_0$ be some fixed positive number. Consider the stochastic differential equation driven by Lévy noise of type

where $\mathcal A\in C(\mathbb R, L(\mathbb H))$, $f\in C(\mathbb R\times \mathbb H, \mathbb H)$, $g\in C(\mathbb{R}\times\mathbb H, L(U, \mathbb H))$, $F, G\in C(\mathbb{R}\times\mathbb H, \mathcal L^2(\nu; \mathbb H))$; $\varepsilon\in(0, \varepsilon_0]$ is a small parameter; $W$ as well as $N$ is the Lévy-Itô decomposition components of the two-sided Lévy process $L$ with assumptions stated in Section 2.3. Assume that conditions (E1)-(E2$'$) in Section 2.4 are satisfied. We also impose the following additional conditions on $\mathcal A, f, g, F, G$:

(H1) there exists $\bar{\mathcal A}\in L(\mathbb H)$ such that

uniformly w.r.t. $t\in\mathbb R$;

(H2) there exist functions $\omega_{1}\in \Psi$ and $\bar{f}\in C(\mathbb H, \mathbb H)$ such that

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

(H3) there exist functions $\omega_{2}\in \Psi$ and $\bar{g}\in C(\mathbb H, L(U, \mathbb H))$ such that

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

(H4) there exist functions $\omega_{3}\in \Psi$ and $\bar{F}\in C(\mathbb H, \mathcal L^2(\nu; \mathbb H))$ such that

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

(H5) there exist functions $\omega_{4}\in \Psi$ and $\bar{G}\in C(\mathbb H, \mathcal L^2(\nu; \mathbb H))$ such that

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

Remark 4.1. Under the conditions (E1), (E2) and (H2)-(H5) the functions $\bar{f}, \bar{g}, \bar{F}, \bar{G}$ also possess the properties (E1), (E2) with the same constants $M$ and $\mathcal L$.

Lemma 4.2. Suppose that $f\in C(\mathbb R\times \mathbb H, \mathbb H)$, $g\in C(\mathbb{R}\times\mathbb H, L(U, \mathbb H))$, $F, G\in C(\mathbb{R}\times\mathbb H, \mathcal L^2(\nu; \mathbb H))$ and the conditions (E1) and (E2) hold. If $\varphi$ is an $\mathcal L^2$-bounded solution of the equation

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

and

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

Proof. Note that

Since $f$ satisfies ${\rm(E1)}$ and ${\rm(E2)}$, we have for any $\tau\in \mathbb R$

Using the same arguments as above we have the same estimates for functions $g, F, G$. By Cauchy-Schwartz inequality, Itô's isometry property and properties of integrals for Poisson random measures we have

Note that the BDG inequality for stochastic integrals with $\widetilde N$ is very different from the BDG inequality with Brownian stochastic integrals. Here we need the following Kunita's first inequality ([31,Theorem 2.11])

where $(Z, \mathcal Z, m)$ is a measurable space, $F:\Omega\times[t_0, t]\times Z\to \mathbb H$ is a predictable process, $p\geq 2$, and $d_p$ is continuous w.r.t. $p$.

Employing the BDG inequality (see [32,Theorem 4.36]), Kunita's first inequality, Cauchy-Schwartz inequality, Itô's isometry property and properties of integrals for Poisson random measures, we have

with $C$ denoting some positive constants and each $C$ maybe different.

Theorem 4.3. Consider Eq $(4.1)$. Suppose that the operator $\mathcal A$ is bounded; the functions $\mathcal A, f, g, F, G$ satisfy the conditions (E1), (E2), (H1)-(H5), and the operator $\bar{\mathcal A}$ in (H1) is Hurwitz, i.e. $\mathcal{R}e\ \lambda < 0$ for any $\lambda\in\sigma(\bar{\mathcal A})$. Then there exists a positive constant $\varepsilon_1\leq\alpha$ such that for any $\varepsilon\in (0, \varepsilon_{1}]$

(i) if $\mathcal L < \cfrac{\omega}{2K\sqrt{1+2\omega+2b}}$, Eq $(4.1)$ has a unique solution $\varphi_{\varepsilon}\in C_{b}(\mathbb R, \mathcal L^{2}(\mathbf P; \mathbb H))$ and $\|\varphi_{\varepsilon}\|_{\infty}\le r$, where

(ii) if $\mathcal{L} < \bigg\{\cfrac{\omega}{2K\sqrt{2+8\omega+4b}}\wedge \cfrac{\omega}{2K\sqrt{1+10\omega+2b}}\bigg\}$ and $f, g, F, G$ satisfy ${\rm(E1 ' )}$, ${\rm(E2 ' )}$ and ${\rm(E3)}$ additionally, then we have $\mathfrak M^{u}_{(\mathcal A, f, g, F, G)} \subseteq \tilde{\mathfrak M}^{u}_{\varphi_{\varepsilon}}$ and the solution $\varphi_{\varepsilon}$ is strongly compatible in distribution (i.e. $\mathfrak M_{(\mathcal A, f, g, F, G)} \subseteq \tilde{\mathfrak M}_{\varphi_{\varepsilon}}$);

(iii) if $\mathcal L < \cfrac{\omega}{2K\sqrt{3(1+2\omega+2b)}}$, we have

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

Proof. The first and second statements follow from Theorem 2.31.

We now consider the following equations

and

where $\mathcal A_{\varepsilon}(t): = \mathcal A(\frac{t}{\varepsilon})$, $f_{\varepsilon}(t, y): = f(\frac{t}{\varepsilon}, y)$, $g_{\varepsilon}(t, y): = g(\frac{t}{\varepsilon}, y)$, $F_{\varepsilon}(t, y, x): = F(\frac{t}{\varepsilon}, y, x)$ and $G_{\varepsilon}(t, y, x): = G(\frac{t}{\varepsilon}, y, x)$ for $t\in\mathbb R$, $y\in\mathbb H$ and $\varepsilon \in(0, \varepsilon_0]$. Here as before $W_{\varepsilon}(t): = \sqrt{\varepsilon}W(\frac{t}{\varepsilon})$, $\widetilde N_{\varepsilon}(t, x): = \sqrt{\varepsilon}\widetilde N(\frac{t}{\varepsilon}, x)$ and $N_{\varepsilon}(t, x): = \sqrt{\varepsilon}N(\frac{t}{\varepsilon}, x)$. Since $(f_\varepsilon, g_\varepsilon, F_\varepsilon, G_\varepsilon)$ satisfy conditions (E1) and (E2) with the same constants as $(f, g, F, G)$, according to Theorem 2.31 Eq (4.3) has a unique solution $\phi_\varepsilon$ from $C_{b}(\mathbb R, \mathcal L^{2}(\mathbf P; \mathbb H))$ with $\phi_\varepsilon \in C(\mathbb R, B[0, r])$, where

Note that

Similar to the proof of [18,Theorem 4.3] with minor modifications, we have

for any $t\in\mathbb R$ and $\varepsilon \in (0, \alpha)$. Here $A, B:(0, \alpha)\to \mathbb R_{+}$ are functions such that $A(\varepsilon), B(\varepsilon) \to 0$ as $\varepsilon \to 0$.

Next we estimate $I_3(t, \varepsilon)$ and $I_4(t, \varepsilon)$. By properties of integrals for Poisson random measures, we have

To prove

by Lemma 3.3 we only need to illustrate

Divide $[0, l]$ into intervals of size $\delta$, where $\delta > 0$ is a fixed constant depending only on $\varepsilon$. Denote 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 may assume $s > 0$ without loss of generality. Then by Lemma 4.2 we have

Denote $s(\delta): = \left[\frac{|s|}{\delta}\right]$. For $J_1$, we have

Then

and

Hence

Taking $\delta = \sqrt{\varepsilon}$ and letting $\varepsilon\rightarrow0$ in (4.7), we have

From (4.6) and (4.8) it follows that

where $C: (0, \alpha)\to \mathbb R_{+}$ is a function so that $C(\varepsilon)\to 0$ as $\varepsilon \to 0$.

Note that

The similar arguments as $I_3(t, \varepsilon)$ yield that

where $D_1(\varepsilon)$ is some constant and $D_1(\varepsilon)\to 0$ as $\varepsilon\to 0$. For $I_{42}(t, \varepsilon)$, by Cauchy-Schwartz inequality we have

We now show that

To this end, making the change of variable $s = \tau-t$ and integrating by parts, we obtain for any $l\geq0$

For $J_2$, by using Cauchy-Schwartz inequality we have

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 4.2 and Cauchy-Schwartz inequality, we obtain

For $i_1$, recalling that $s(\delta): = \left[\frac{|s|}{\delta}\right]$ by Lemma 4.2 we have

Therefore, by (4.15) and (4.16) we get

Hence, according to (4.13), (4.14) and (4.17) we have

Taking $\delta = \sqrt{\varepsilon}$ and letting $\varepsilon\rightarrow0$ in (4.11), we have

Since $l$ is arbitrary, by letting $l\to 0$ we have

From (4.12) and (4.19) it follows that

where $D_2(\varepsilon)$ is some positive constant such that $D_2(\varepsilon)\to 0$ as $\varepsilon \to 0$. So by (4.10), (4.11) and (4.20) we have

Combing (4.4), (4.5), (4.9) and (4.21), we have

By the assumption $\mathcal L < \cfrac{\omega}{2K\sqrt{3(1+2\omega+2b)}}$ the coefficient is positive, so

Since $\mathcal L^2$-convergence implies convergence in distribution,

We note that $\mathcal L(\varphi_{\varepsilon}(\frac{t}{\varepsilon})) = \mathcal L(\phi_{\varepsilon}(t))$. Then we have

The proof is complete.

Corollary 4.4. Assume that the conditions of Theorem 4.3 hold.

(i) If $\mathcal A, f, g, F, G$ are jointly stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1, \ldots, \nu_k$, almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable), then Eq (4.1) has a unique solution $\varphi_{\varepsilon} \in C_{b}(\mathbb R, \mathcal L^2(\mathbf P; \mathbb H))$ which is stationary (respectively, $\tau$-periodic, quasi-periodic with the spectrum of frequencies $\nu_1, \ldots, \nu_k$, almost periodic, almost automorphic, Birkhoff recurrent, Lagrange stable, Levitan almost periodic, almost recurrent, Poisson stable) in distribution.

(ii) If $\mathcal A, f, g, F, G$ are Lagrange stable and jointly pseudo-periodic (respectively, pseudo-recurrent), then Eq (4.1) has a unique solution $\varphi_{\varepsilon} \in C_{b}(\mathbb R, \mathcal L^2(\mathbf P; \mathbb H))$ which is pseudo-periodic (respectively, pseudo-recurrent) in distribution.

(iii)

Proof. This statement follows from Theorems 2.16, 4.3 and Remark 2.18.

5.   Applications

In this section, we illustrate our theoretical results by two examples.

Example 5.1. Consider the following stochastic ordinary differential equation driven by a two-sided Lévy noise:

where $\varepsilon$ is a small positive parameter; $W$ is a one-dimensional two-sided Brownian motion and $N$ is a Poisson random measure in $\mathbb{R}$, which is independent of $W$. Let

and define the corresponding averaged stochastic differential equation

It is clear that $\mathcal A$ generates a dissipative semigroup on $\mathbb R$ with $K = 1$, $\omega = 3$. Since the functions $\mathcal A, f, g, F, G$ are respectively stationary, periodic, stationary, stationary and quasi-periodic in $t$, uniformly with respect to $y$ on any bounded subset of $\mathbb R$, it follows that functions $\mathcal A, f, g, F, G$ are jointly quasi-periodic. Conditions ${\rm(E1)}$ and ${\rm(E1 ' )}$ always hold for any constant $M$, and the Lipschitz constants of functions $f, g, F, G$ in Conditions ${\rm(E2)}$ and ${\rm(E2 ' )}$ can be chosen as $\frac{1}{4}$ if

for $p = 2$ and some constant $p > 2$, i.e.

As for Condition ${\rm(E3)}$, it naturally holds for stochastic ordinary differential equation. Conditions (H1)-(H5) are also satisfied for functions $\mathcal A, f, g, F, G$.

Since $f, g, F, G$ satisfy ${\rm(E1)}$ and ${\rm(E2)}$, by Theorem 4.2-(i) Eq (5.1) has a unique $\mathcal L^2$-bounded solution provided $\nu(-1, 1) < \frac{25}{16}$, $b\leq 1$. The restrictions in Theorem 4.2-(ii) and (iii) respectively become

i.e. $b < \frac{5}{2}$. According to Corollary 4.3, the unique $\mathcal L^2$-bounded solution of Eq (5.1) is quasi-periodic in distribution and it uniformly converges to the unique stationary solution of the averaged Eq (5.2) on $\mathbb R$ in distribution sense.

Example 5.2. Consider the following equations:

where $0 < \omega\le a_i \le \kappa$ for $i = 1, 2, ...$, and $\kappa$ and $\omega$ are constants; $f_i, g_i\in C(\mathbb R\times l^2, l^2)$, $F_i, G_i\in C(\mathbb R\times l^2, \mathcal L^2(\nu; l^2))$, $l^2: = \{y:y = (y_1, y_2, ..., y_n, ...), \sum_{i = 1}^{\infty}y_i^2 < \infty \}$; $W$ is a one-dimensional two-sided Brownian motion and $N$ is a Poisson random measure independent of $W$. (5.3) can also be written in the following form:

where $Y = \left(\begin{array}{c} y_1 \\ \vdots \\ y_n \\ \vdots \\ \end{array} \right)$, $\mathcal A = \left(\begin{array}{cccc} -a_1 & & & \\ & \ddots & & \\ & & -a_n & \\ & & & \ddots \\ \end{array} \right)$, $f(t, Y) = \left(\begin{array}{c} f_1(t, Y) \\ \vdots \\ f_n(t, Y) \\ \vdots \\ \end{array} \right)$, $g(t, Y) = \left(\begin{array}{c} g_1(t, Y) \\ \vdots \\ g_n(t, Y) \\ \vdots \\ \end{array} \right)$,

$F(t, Y, x) = \left(\begin{array}{c} F_1(t, Y, x) \\ \vdots \\ F_n(t, Y, x) \\ \vdots \\ \end{array} \right)$, $G(t, Y, x) = \left(\begin{array}{c} G_1(t, Y, x) \\ \vdots \\ G_n(t, Y, x) \\ \vdots \\ \end{array} \right)$.

According to (H1)-(H5), we define the corresponding averaged equations:

Eq (5.5) can also be written in the following form:

where $\bar Y = \left(\begin{array}{c} \bar y_1 \\ \vdots \\ \bar y_n \\ \vdots \\ \end{array} \right)$, $\mathcal A = \left(\begin{array}{cccc} -a_1 & & & \\ & \ddots & & \\ & & -a_n & \\ & & & \ddots \\ \end{array} \right)$, $\bar f(\bar Y) = \left(\begin{array}{c} \bar f_1(\bar Y) \\ \vdots \\ \bar f_n(\bar Y) \\ \vdots \\ \end{array} \right)$, $\bar g(\bar Y) = \left(\begin{array}{c} \bar g_1(\bar Y) \\ \vdots \\ \bar g_n(\bar Y) \\ \vdots \\ \end{array} \right)$,

$\bar F(\bar Y, x) = \left(\begin{array}{c} \bar F_1(\bar Y, x) \\ \vdots \\ \bar F_n(\bar Y, x) \\ \vdots \\ \end{array} \right)$, $\bar G(\bar Y, x) = \left(\begin{array}{c} \bar G_1(\bar Y, x) \\ \vdots \\ \bar G_n(\bar Y, x) \\ \vdots \\ \end{array} \right)$.

Assume that the Lipschitz condition, growth condition and all the conditions of Theorem 4.3 are satisfied for $f_i, g_i, F_i, G_i$, (i = 1, 2, ...). Thus Theorem 4.3 and Corollary 4.4 hold. If every $f_i, g_i, F_i, G_i$, (i = 1, 2, ...) are periodic or quasi-periodic in $t$ uniformly with respect to $y$ on any bounded subset of $l^2$, then $f_i, g_i, F_i, G_i$ are jointly almost periodic. Basis on Theorem 4.3 and Corollary 4.4, Eq (5.4) has a unique almost periodic solution and it converges to the stationary solution of the averaged Eq (5.6) in distribution sense on the whole real axis.

Now we give a concrete example with infinite dimension:

and the averaged equations

It is immediate to check that the drift, diffusion, and jump coefficients of Eqs (5.7) and (5.8) satisfy all the conditions of Theorem 4.3 and Corollary 4.4. So the almost periodic solution of Eq (5.7) converges to the stationary solution of Eq (5.8) on $\mathbb R$ in distribution.

6.   Conclusions

In this paper we study an averaging principle on infinite time intervals for semilinear stochastic ordinary differential equations with Lévy noise. We establish the existence of Poisson stable (including periodic, quasi-periodic, almost periodic, almost automorphic etc) solutions of the equations by a unified framework. We show that the Poisson stable solution of the original equation converges to the stationary solution of the averaged equation uniformly on the whole real axis in distribution sense, as the time scale goes to zero. Moreover, we illustrate our theoretical results with some examples.

In this paper we only consider stochastic ordinary differential equations case. The case of the averaging on $\mathbb R$ for stochastic partial differential equations is more complicated, but it is both interesting and important. It deserves a separate paper.

Acknowledgments

This work is partially supported by NSFC Grant 11871132, and Xinghai Jieqing and DUT19TD14 funds from Dalian University of Technology. We sincerely thank Professor Zhenxin Liu for helpful discussions. We are grateful to the anonymous referees for their careful reading of our paper and valuable suggestions, which lead to significant improvement of the paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.