Recurrence of bounded solutions to a semilinear integro-differential equation perturbed by Lévy noise

Abstract

In this paper, we consider recurrence of bounded solutions to semilinear stochastic integro-differential equations via uniformly exponentially stable resolvent operators family. Concretely, we first establish the existence and uniqueness of the $L^{2}$ -bounded solution to a semilinear stochastic integro-differential equations driven by Lévy processes, and then we show this kind of $L^{2}$ -bounded solution is almost automorphic in distribution or weighted pseudo almost automorphic in distribution under suitable conditions respectively.

Keywords

Stochastic integro-differential equations almost automorphy in distribution weighted pseudo almost automorphy in distribution Lévy noise

1. Introduction

The following abstract integro-differential equation can usually be applied to describe the models arising from viscoelasticity or heat conduction in materials with memory $\begin{matrix} (1.1) & u^{'} (t) = A u (t) + \int_{- \infty}^{t} a (t - s) A u (s) d s + f (t, u (t)), t \in R, \end{matrix}$ where $A : D (A) \subseteq X \to X$ is a closed linear operator defined on a Banach space $X$ , $a (\cdot) \in L^{1} (R_{+})$ and $f (\cdot)$ is a suitable function. A typical choice of the function $a (\cdot)$ is the exponential kernel $α e^{- β t}$ , see [19,23,27] for details, thus the Eq. (1.1) is reduced to the following equation $\begin{matrix} (1.2) & u^{'} (t) = A u (t) + α \int_{- \infty}^{t} e^{- β (t - s)} A u (s) d s + f (t, u (t)), t \in R . \end{matrix}$ For this Eq. (1.2), several results on recurrence of bounded solutions have been available. For instance, Lizama and Ponce [18] studied the existence and uniqueness of bounded solutions, such as almost periodic, almost automorphic and asymptotically almost periodic solutions, and Bian et al. [6] investigated the existence of weighted pseudo almost automorphic solutions with Stepanov-like weighted pseudo almost automorphic nonlinear term. Recently, a more general choice of kernel function $a (t) : = α \frac{t^{τ - 1}}{Γ (τ)} e^{- β t}$ was investigated in [8], where $α \neq 0$ , $β > 0$ , $τ ⩾ 1$ , and $Γ (\cdot)$ stands for the Gamma function. In this case, Eq. (1.1) turns into the following integro-differential equation $\begin{matrix} (1.3) & u^{'} (t) = A u (t) + α \int_{- \infty}^{t} \frac{{(t - s)}^{τ - 1}}{Γ (τ)} e^{- β (t - s)} A u (s) d s + f (t, u (t)), t \in R . \end{matrix}$ Some sufficient conditions are established for the existence and uniqueness of bounded mild solutions to Eq. (1.3) via uniformly exponentially stable resolvent operators family.

Since noise or stochastic perturbation is unavoidable in real world, it is of great importance to consider the stochastic effects in the investigation of differential systems. Especially, Fu [13] introduced the concept of distributionally almost automorphic processes and they proved the existence and uniqueness of distributionally almost automorphic mild solutions to nonautonomous stochastic differential equations. In [17], Liu and Sun introduced the concepts of Poisson square-mean almost automorphy and almost automorphy in distribution, and established the existence of solutions which were almost automorphic in distribution for some semilinear stochastic differential equations with infinite dimensional Lévy noise. In [9], Chang and Tang investigated the notion of Poisson asymptotically almost automorphy for stochastic processes. Moreover, in [16], the author introduced the concept of weighted pseudo almost automorphy in distribution, and studied the weighted pseudo almost automorphic solutions in distribution for some semilinear nonautonomous stochastic differential equations driven by Lévy noise.

Inspired by the above mentioned work [16–18], a natural and interesting problem is to consider recurrence of bounded solutions to the Eq. (1.3) perturbed by stochastic noise. Thus the main purpose of present paper is to consider recurrence of $L^{2}$ -bounded solutions to the following semilinear stochastic integro-differential equation driven by Lévy noise $\begin{array}{rcl} d y (t) & = & [A y (t) + α \int_{- \infty}^{t} \frac{{(t - s)}^{τ - 1}}{Γ (τ)} e^{- β (t - s)} A y (s) d s] d t + f_{1} (t, y (t)) d t \\ + f_{2} (t, y (t)) d W (t) + \int_{| x |_{U} < 1} f_{3} (t, y (t -), x) \tilde{N} (d t, d x) \\ (1.4) & + \int_{| x |_{U} ⩾ 1} f_{4} (t, y (t -), x) N (d t, d x), t \in R . \end{array}$ Here $f_{1} : R \times L^{2} (P, H) \to L^{2} (P, H)$ , $f_{2} : R \times L^{2} (P, H) \to L (U, L^{2} (P, H))$ , $f_{3} : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ , $f_{4} : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ ; W and N are the Lévy–Itô decomposition components of the two-sided Lévy process L, and A is a closed linear operator defined in the Hilbert space $L^{2} (P, H)$ , where L, $P$ , $H$ , $U$ and $L (U, L^{2} (P, H))$ will be stated specifically in Section 2.

The rest of this paper is organized as follows. In Section 2, we recall some preliminary results which will be used throughout this paper. In Section 3, we establish some sufficient conditions for recurrence of bounded solutions to Eq. (1.4), and an example is also included to illustrate the obtained results.

2. Preliminaries

In this section, we introduce some basic definitions, notations, lemmas, propositions and technical results which will be used in the sequel. For more details on this section, we refer the reader to [1,2,4,5,7,12,14,16,17,20,21,24–26] and references therein.

Throughout the paper, we assume that $(H, ‖ \cdot ‖, ⟨ \cdot, \cdot ⟩)$ and $(U, | \cdot |_{U}, {⟨ \cdot, \cdot ⟩}_{U})$ are two real separable Hilbert spaces. Let $(Ω, F, P)$ be a complete probability space. The notation $L^{2} (P, H)$ stands for the space of all $H$ -valued random variables y such that $\begin{matrix} E ‖ y ‖^{2} = \int_{Ω} ‖ y ‖^{2} d P < \infty . \end{matrix}$ For $y \in L^{2} (P, H)$ , let $\begin{matrix} ‖ y ‖_{2} : = {(\int_{Ω} ‖ y ‖^{2} d P)}^{\frac{1}{2}} . \end{matrix}$ Then it is routine to check that $L^{2} (P, H)$ is a Hilbert space equipped with the norm $‖ \cdot ‖_{2}$ . The notation $L (U, H)$ stands for the collection of all bounded linear operators from $U$ into $H$ endowed with the operator topology denoted by $‖ \cdot ‖_{L (U, H)}$ .

Definition 2.1 ([3]).

A $U$ -valued stochastic process $L = (L (t), t ⩾ 0)$ is called Lévy process if:

$L (0) = 0$ almost surely;

L has independent and stationary increments;

L is stochastically continuous, i.e. for all $ε > 0$ and for all $s > 0$ $\begin{matrix} lim_{t \to s} P ({‖ L (t) - L (s) ‖}_{U} > ε) = 0 . \end{matrix}$

Definition 2.2 ([16,17]).

For a given Lévy process L, the associated jump process $Δ L = (Δ L (t), t ⩾ 0)$ is given by $Δ L (t) = L (t) - L (t -)$ for each $t ⩾ 0$ , where $L (t -) = {lim}_{s ↑ t} L (s)$ .

A Borel set $B$ in $U - {0}$ is bounded below if $0 \notin \overline{B}$ , where $\overline{B}$ is the closure of $B$ .

For any Borel set $B$ in $U - {0}$ , the counting Poisson random measure N on $U - {0}$ is defined by $\begin{matrix} N (t, B) (ω) : = ♯ {0 ⩽ s ⩽ t : Δ L (s) (ω) \in B} = \sum_{0 ⩽ s ⩽ t} χ_{B} (Δ L (s) (ω)), \end{matrix}$ where $ω \in Ω_{0}$ , $Ω_{0} \in F$ with $P (Ω_{0}) = 1$ such that $t \to L (t) (ω)$ is càdlàg for all $ω \in Ω_{0}$ . ♯ means the counting and $χ_{B}$ is the characteristic function.

N is called Poisson random measure if $B$ is bounded below, for each $t ⩾ 0$ .

$ν (\cdot) = E (N (1, \cdot))$ is called the intensity measure associated with L.

For each $t ⩾ 0$ and $B$ bounded below, the compensated Poisson random measure $\tilde{N}$ is defined by $\begin{matrix} \tilde{N} (t, B) = N (t, B) - t ν (B) . \end{matrix}$

Proposition 2.1 ([3,22]).

If L is a $U$ -valued Lévy process, then there exist $a \in U$ , a $U$ -valued Wiener process W with covariance operator $Q$ , and an independent Poisson random measure on $R^{+} \times (U - {0})$ such that for each $t ⩾ 0$ , $\begin{matrix} (2.1) & L (t) = a t + W (t) + \int_{| x |_{U} < 1} x \tilde{N} (t, d x) + \int_{| x |_{U} ⩾ 1} x N (t, d x), \end{matrix}$ where the Poisson random measure N has the intensity measure ν which satisfies $\begin{matrix} (2.2) & \int_{U} (| x |_{U}^{2} \land 1) ν (d x) < \infty \end{matrix}$ and $\tilde{N}$ is the compensated Poisson random measure of N.

Remark 2.1.
By (2.2), it follows that $\int_{| x |_{U} ⩾ 1} ν (d x) < \infty$ . For convenience, we denote $\begin{matrix} b : = \int_{| x |_{U} ⩾ 1} ν (d x) \end{matrix}$ throughout the paper.
Remark 2.2 ([17]).

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

Assume that $L_{1} (t)$ and $L_{2} (t)$ , $t ⩾ 0$ are two independent, identically distributed Lévy processes. Set $\begin{matrix} L (t) = \{\begin{matrix} L_{1} (t), & for t ⩾ 0, \\ - L_{2} (- t), & for t < 0 . \end{matrix} \end{matrix}$ Then L is a two-sided Lévy process defined on the filtered probability space $(Ω, F, P, {(F_{t})}_{t \in R})$ . We assume that $Q$ is a positive, self-adjoint and trace class operator on $U$ .

Definition 2.3.
A stochastic process $x : R \to L^{2} (P, H)$ is said to be $L^{2}$ -continuous if for any $s \in R$ , $\begin{matrix} lim_{t \to s} E {‖ x (t) - x (s) ‖}^{2} = 0 . \end{matrix}$

It is $L^{2}$ -bounded if ${sup}_{t \in R} ‖ x (t) ‖_{2} < \infty$ . Denote by $SBC (R; L^{2} (P, H))$ (resp. $SBC (R \times L^{2} (P, H); L^{2} (P, H))$ ) the collection of all the $L^{2}$ -continuous and $L^{2}$ -bounded processes.
Definition 2.4 ([16]).

A stochastic process $ϕ : R \times U \to L^{2} (P, H)$ , $(t, x) \to ϕ (t, x)$ is said to be Poisson stochastically continuous if $\begin{matrix} lim_{t \to s} \int_{U} E {‖ ϕ (t, x) - ϕ (s, x) ‖}^{2} ν (d x) = 0 . \end{matrix}$ It is called to be Poisson stochastically bounded if there exists a constant $M > 0$ such that $\int_{U} E ‖ ϕ (t, x) ‖^{2} ν (d x) ⩽ M$ . Denote by $PSBC (R \times U; L^{2} (P, H))$ (resp. $PSBC (R \times L^{2} (P, H) \times U; L^{2} (P, H))$ ) the collection of all Poisson stochastically bounded and continuous processes.

Definition 2.5 ([16]).

A stochastic process $ϕ : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ , $(t, x, z) \to ϕ (t, x, z)$ is said to be Poisson stochastically continuous if $\begin{matrix} \int_{U} E {‖ ϕ (t, x, z) - ϕ (s, y, z) ‖}^{2} ν (d x) \to 0 as (t, x) \to (s, y) . \end{matrix}$

Definition 2.6 ([16]).

An $L^{2}$ -continuous stochastic process $x : R \to L^{2} (P, H)$ is said to be square-mean almost automorphic if for every sequence of real numbers ${s_{n}^{'}}_{n \in N}$ , there exist a subsequence ${s_{n}}_{n \in N}$ and a stochastic process $y : R \to L^{2} (P, H)$ such that $\begin{matrix} lim_{n \to \infty} E {‖ x (t + s_{n}) - y (t) ‖}^{2} = 0 and lim_{n \to \infty} E {‖ y (t - s_{n}) - x (t) ‖}^{2} = 0 \end{matrix}$ hold for each $t \in R$ . The collection of all square-mean almost automorphic stochastic processes $x : R \to L^{2} (P, H)$ is denoted by $SAA (R; L^{2} (P, H))$ .

Definition 2.7 ([17]).

A function $f : R \times L^{2} (P, H) \to L (U, L^{2} (P, H))$ , $(t, x) \to f (t, x)$ , which is jointly continuous, is said to be square-mean almost automorphic in $t \in R$ uniformly for all $x \in L^{2} (P, H)$ if for every sequence of real numbers ${s_{n}^{'}}_{n \in N}$ , there exists a subsequence ${s_{n}}_{n \in N}$ and a function $\tilde{f} : R \times L^{2} (P, H) \to L (U, L^{2} (P, H))$ such that $\begin{array}{l} lim_{n \to \infty} E {‖ f (t + s_{n}, x) - \tilde{f} (t, x) ‖}_{L (U, L^{2} (P, H))}^{2} = 0 and \\ lim_{n \to \infty} E {‖ \tilde{f} (t - s_{n}, x) - f (t, x) ‖}_{L (U, L^{2} (P, H))}^{2} = 0 \end{array}$ for each $t \in R$ and each $x \in L^{2} (P, H)$ . The class of such functions will be denoted by $SAA (R \times L^{2} (P, H); L (U, L^{2} (P, H)))$ .

Definition 2.8 ([17]).

A stochastic process $f : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ , $(t, x, z) \to f (t, x, z)$ is said to be Poisson square-mean almost automorphic in $t \in R$ uniformly for all $x \in L^{2} (P, H)$ if f is Poisson stochastically continuous and for every sequence of real numbers ${s_{n}^{'}}_{n \in N}$ , there exists a subsequence ${s_{n}}_{n \in N}$ and a function $\tilde{f} : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ with $\int_{U} E ‖ \tilde{f} (t, x, z) ‖^{2} ν (d x) < \infty$ such that $\begin{matrix} lim_{n \to \infty} E \int_{U} E {‖ f (t + s_{n}, x, z) - \tilde{f} (t, x, z) ‖}^{2} ν (d x) = 0 \end{matrix}$ and $\begin{matrix} lim_{n \to \infty} E \int_{U} E {‖ \tilde{f} (t - s_{n}, x, z) - f (t, x, z) ‖}^{2} ν (d x) = 0 \end{matrix}$ for each $t \in R$ and each $x \in L^{2} (P, H)$ . The class of such functions will be denoted by $PSAA (R \times L^{2} (P, H) \times U; L^{2} (P, H))$ .

Lemma 2.1 ([16]).

$(SAA (R; L^{2} (P, H)), ‖ \cdot ‖_{\infty})$ is a Banach space when it is equipped with the norm $\begin{matrix} ‖ x ‖_{\infty} : = sup_{t \in R} {‖ x (t) ‖}_{2} = sup_{t \in R} {(E {‖ x (t) ‖}^{2})}^{\frac{1}{2}}, \end{matrix}$ for $x \in SAA (R; L^{2} (P, H))$ .

Lemma 2.2 ([17]).

If $F, F_{1}, F_{2} : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ are Poisson square-mean almost automorphic in $t \in R$ uniformly for all $x \in L^{2} (P, H)$ , then:

$F_{1} + F_{2}$ is Poisson square-mean almost automorphic.

$λ F$ is Poisson square-mean almost automorphic for every scalar λ.

For each $x \in L^{2} (P, H)$ , there exists a constant $M = M_{(x)} > 0$ such that $\begin{matrix} sup_{t \in R} \int_{U} E {‖ F (t, x, z) ‖}^{2} ν (d x) ⩽ M . \end{matrix}$

Let $U$ denote the set of all functions $ρ : R \to (0, \infty)$ , which are locally integrable over $R$ such that $ρ > 0$ almost everywhere. For a given $r > 0$ and for each $ρ \in U$ , we set $m (r, ρ) : = \int_{- r}^{r} ρ (t) d t$ and $\begin{matrix} U_{\infty} : = {ρ \in U : lim_{r \to + \infty} m (r, ρ) = \infty} . \end{matrix}$

Now for $ρ \in U_{\infty}$ , we define $\begin{array}{l} {SBC}_{0} (R, ρ) : = {φ \in SBC (R; L^{2} (P, H)) : lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ φ (t) ‖}^{2} ρ (t) d t = 0}; \\ {SBC}_{0} (R \times L^{2} (P, H), ρ) \\ : = {φ (t, y) \in SBC (R \times L^{2} (P, H); L^{2} (P, H)) : \\ lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} ‖ f (t, y) ‖ ρ (t) d t = 0 uniformly in y \in L^{2} (P, H)}; \\ {PSBC}_{0} (R \times U, ρ) : = {φ (t, x) \in PSBC (R \times U; L^{2} (P, H)) : \\ lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} \int_{U} E {‖ φ (t, x) ‖}^{2} ρ (t) ν (d x) d t = 0}; \\ {PSBC}_{0} (R \times L^{2} (P, H) \times U, ρ) \\ : = {φ (t, y, z) \in PSBC (R \times L^{2} (P, H) \times U; L^{2} (P, H)) : \\ lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} \int_{U} E {‖ φ (t, y, z) ‖}^{2} ρ (t) ν (d x) d t = 0 uniformly in y \in L^{2} (P, H)} . \end{array}$

Definition 2.9 ([16]).

An $L^{2}$ -continuous stochastic process $f : R \to L^{2} (P, H)$ is said to be square-mean weighted pseudo almost automorphic with respect to $ρ \in U_{\infty}$ if it can be decomposed as $f = g + h$ , where $g \in SAA (R; L^{2} (P, H))$ and $h \in {SBC}_{0} (R, ρ)$ . The collection of all square-mean weighted pseudo almost automorphic stochastic processes with respect to $ρ \in U_{\infty}$ is denoted by $SWPAA (R; ρ)$ .

Definition 2.10 ([16]).

An $L^{2}$ -continuous stochastic process $f : R \times L^{2} (P, H) \to L^{2} (P, H)$ is said to be square-mean weighted pseudo almost automorphic with respect to $ρ \in U_{\infty}$ in t uniformly for all $x \in L^{2} (P, H)$ if it can be decomposed as $f = g + h$ , where $g \in SAA (R \times L^{2} (P, H); L^{2} (P, H))$ and $h \in {SBC}_{0} (R \times L^{2} (P, H), ρ)$ . We denote all such stochastic process by $SWPAA (R \times L^{2} (P, H); ρ)$ .

Definition 2.11 ([16]).

A stochastic process $f : R \times U \to L^{2} (P, H)$ is said to be Poisson square-mean weighted pseudo almost automorphic with respect to $ρ \in U_{\infty}$ in $t \in R$ if f is Poisson stochastically continuous and it can be decomposed as $f = g + h$ , where $g \in PSAA (R \times U; L^{2} (P, H))$ and $h \in {PSBC}_{0} (R \times U, ρ)$ . The collection of all such stochastic processes with respect to $ρ \in U_{\infty}$ is denoted by $PSWPAA (R \times U; ρ)$ .

Definition 2.12 ([16]).

A stochastic process $f : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ , $(t, x, z) \to f (t, x, z)$ is said to be Poisson square-mean weighted pseudo almost automorphic about $ρ \in U_{\infty}$ in t uniformly for all $x \in L^{2} (P, H)$ if f is Poisson stochastically continuous and it can be decomposed as $f = g + h$ , where $g \in PSAA (R \times L^{2} (P, H) \times U; L^{2} (P, H))$ and $h \in {PSBC}_{0} (R \times L^{2} (P, H) \times U, ρ)$ . The collection of all such stochastic processes with respect to $ρ \in U_{\infty}$ is denoted by $PSWPAA (R \times L^{2} (P, H) \times U; ρ)$ .

Definition 2.13 ([16]).

A set $D$ is said to be translation invariant if for any $f (t) \in D$ , $f (t + τ) \in D$ for any $τ \in R$ .

In the following, we let $U^{inv} = {ρ \in U_{\infty} : {SBC}_{0} (R, ρ) is translation invariant}$ and $U_{P}^{inv} = {ρ \in U_{\infty} : {PSBC}_{0} (R \times U, ρ) is translation invariant}$ .

Lemma 2.3 ([16]).

For $ρ \in U^{inv}$ , $SWPAA (R; ρ)$ equipped with the norm $‖ x ‖_{\infty}$ is a Banach space.

Let $P (H)$ be the space of all Borel probability measures on $H$ endowed with the $d$ metric: $\begin{matrix} d (μ, ν) : = sup {| \int f d μ - \int f d ν | : ‖ f ‖_{BL} ⩽ 1}, μ, ν \in P (H), \end{matrix}$ where f are Lipschitz continuous real-valued functions on $H$ with the norms $\begin{matrix} ‖ f ‖_{BL} = ‖ f ‖_{L} + ‖ f ‖_{\infty}, ‖ f ‖_{L} = sup_{x \neq y} \frac{| f (x) - f (y) |}{‖ x - y ‖}, ‖ f ‖_{\infty} = sup_{x \in H} | f (x) | . \end{matrix}$ Note that the $d$ metric is a complete metric on $P (H)$ and that a sequence ${μ_{n}}$ weakly converges to μ if and only if $d (μ_{n}, μ) \to 0$ as $n \to \infty$ .

Definition 2.14 ([17]).

An $H$ -valued stochastic process $x (t)$ is said to be almost automorphic in distribution if its law $μ (t)$ is a $P (H)$ -valued almost automorphic mapping, i.e. for every sequence of real numbers ${s_{n}^{'}}_{n \in N}$ , there exists a subsequence ${s_{n}}_{n \in N}$ and a $P (H)$ -valued mapping $\tilde{μ} (t)$ such that $\begin{matrix} lim_{n \to \infty} d (μ (t + s_{n}), \tilde{μ} (t)) = 0 and lim_{n \to \infty} d (\tilde{μ} (t - s_{n}), μ (t)) = 0 \end{matrix}$ hold for each $t \in R$ .

Definition 2.15 ([16]).

An $H$ -valued stochastic process $x (t)$ is said to be weighted pseudo almost automorphic in distribution with respect to $ρ \in U_{\infty}$ , provided that it can be decomposed as $x = ξ + η$ , where ξ is almost automorphic in distribution and $η \in {SBC}_{0} (R, ρ)$ .

Lemma 2.4 ([15]).

Let $g : R \to R$ be a nonnegative continuous function such that for every $t \in R$ , $\begin{matrix} (2.3) & g (t) ⩽ ν (t) + ϱ_{1} \int_{- \infty}^{t} e^{- δ_{1} (t - s)} g (s) d s + \dots + ϱ_{n} \int_{- \infty}^{t} e^{- δ_{n} (t - s)} g (s) d s, \end{matrix}$ for some locally integrable function $ν : R \to R$ , and for some constants $ϱ_{1}, \dots, ϱ_{n} ⩾ 0$ , and some constants $δ_{1}, \dots, δ_{n} > ϱ$ with $ϱ : = \sum_{i = 1}^{n} ϱ_{i}$ . We assume that the integrals on the right-hand side of ( 2.3 ) are convergent. Let $δ : = {min}_{1 ⩽ i ⩽ n} δ_{i}$ . Then for every $γ \in (0, δ - ϱ]$ such that $\int_{- \infty}^{0} e^{γ s} ν (s) d s$ converges, we have, for every $t \in R$ , $\begin{matrix} g (t) ⩽ ν (t) + ϱ \int_{- \infty}^{t} e^{- γ (t - s)} ν (s) d s . \end{matrix}$ In particular, if ν is a constant, we have $\begin{matrix} g (t) ⩽ ν \frac{δ}{δ - ϱ} . \end{matrix}$

3. Main results

In this section, we shall consider recurrence of bounded solutions to the Eq. (1.4). Concretely, we shall investigate the existence and uniqueness of almost automorphic and weighted pseudo almost automorphic solutions in distribution for the Eq. (1.4). In order to express the bounded mild solutions of Eq. (1.4), we recall the following facts which can be found in details in the paper [8].

Lemma 3.1.

Let $a (t) : = α \frac{t^{τ - 1}}{Γ (τ)} e^{- β t}$ with $α \neq 0$ , $β > 0$ , $τ ⩾ 1$ and $Re ({(- α)}^{1 / τ} - β) < 0$ . Let $s (A) : = sup {Re λ : λ \in ρ (A)}$ denote the spectral bound of A. Assume that the following condition (HA) holds:

A generates a $C_{0}$ -semigroup on the Hilbert space $L^{2} (P, H)$ ;

$sup {Re λ, λ \in C : λ {(λ + β)}^{τ} {({(λ + β)}^{τ} + α)}^{- 1} \in σ (A)} < 0$ ;

${lim}_{μ \in R, | μ | \to \infty} ‖ {(μ_{0} + i μ - A)}^{- 1} ‖ = 0$ for some $μ_{0} > s (A)$ .

Then, there exists a uniformly exponentially stable and strongly continuous family of operators

{S (t)}_{t ⩾ 0} \subset L (L^{2} (P, H), L^{2} (P, H))

such that

S (t)

commutes with A, i.e.

S (t) D (A) \subset D (A)

A S (t) x = S (t) A x

for all

x \in D (A)

t ⩾ 0

and

\begin{matrix} S (t) x = x + \int_{0}^{t} b (t - s) A S (s) x d s, for all x \in L^{2} (P, H), t ⩾ 0, \end{matrix}

where

b (t) : = 1 + \int_{0}^{t} a (s) d s

t ⩾ 0

Definition 3.1.

Assume that the condition (HA) holds. An $F_{t}$ -progressively measurable stochastic ${y (t)}_{t \in R}$ is called a mild solution of the equation (1.4) if it verifies the following stochastic integral equation $\begin{array}{rcl} y (t) & = & \int_{- \infty}^{t} S (t - s) f_{1} (s, y (s)) d s + \int_{- \infty}^{t} S (t - s) f_{2} (s, y (s)) d W (s) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{3} (s, y (s -), x) \tilde{N} (d s, d x) \\ (3.1) & + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{4} (s, y (s -), x) N (d s, d x) . \end{array}$

The functions $f_{1} : R \times L^{2} (P, H) \to L^{2} (P, H)$ and $f_{2} : R \times L^{2} (P, H) \to L (U, L^{2} (P, H))$ are square-mean almost automorphic in t for each $y \in L^{2} (P, H)$ . Moreover, $f_{1}$ and $f_{2}$ satisfy the Lipschitz conditions in y uniformly for t, that is, there exists a constant $L_{1} > 0$ such that for all $x, y \in L^{2} (P, H)$ and $t \in R$ , $\begin{array}{l} E {‖ f_{1} (t, x) - f_{1} (t, y) ‖}^{2} ⩽ L_{1} E ‖ x - y ‖^{2}, \\ E {‖ [f_{2} (t, x) - f_{2} (t, y)] Q^{1 / 2} ‖}_{L (U, L^{2} (P, H))}^{2} ⩽ L_{1} E ‖ x - y ‖^{2} . \end{array}$

The functions $f_{3} : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ and $f_{4} : R \times L^{2} (P, H) \times U \to L^{2} (P, H)$ are Poisson square-mean almost automorphic in $t \in R$ for each $y \in L^{2} (P, H)$ . Moreover, $f_{3}$ and $f_{4}$ satisfy the Lipschitz conditions in y uniformly for t, that is, there exists a constant $L_{2} > 0$ such that for all $x, y \in L^{2} (P, H)$ and $t \in R$ , $\begin{array}{l} \int_{| x |_{U} < 1} E {‖ f_{3} (t, x, z) - f_{3} (t, y, z) ‖}^{2} ν (d x) ⩽ L_{2} E ‖ x - y ‖^{2}, \\ \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (t, x, z) - f_{4} (t, y, z) ‖}^{2} ν (d x) ⩽ L_{2} E ‖ x - y ‖^{2} . \end{array}$

Remark 3.1.

If the condition (HA) is satisfied, then by Lemma 3.1, there exist constants $M > 0$ and $ϖ > 0$ such that $‖ S (t) ‖ ⩽ M e^{- ϖ t}$ for $t ⩾ 0$ .

Theorem 3.1.

Suppose the conditions (HA), (H1) and (H2) hold, then the problem ( 1.4 ) has a unique $L^{2}$ -bounded mild solution provided that $\begin{matrix} (3.2) & \frac{L_{1} + 2 L_{2} b}{ϖ^{2}} + \frac{L_{1} + 3 L_{2}}{2 ϖ} < \frac{1}{4 M^{2}} . \end{matrix}$ In addition, this unique $L^{2}$ -bounded mild solution is almost automorphic in distribution provided that $\begin{matrix} (3.3) & \frac{L_{1} + 2 L_{2} b}{ϖ^{2}} + \frac{L_{1} + 3 L_{2}}{ϖ} < \frac{1}{8 M^{2}}, \end{matrix}$ where b is given in Remark 2.1 .

Proof.

If $y (t)$ is $L^{2}$ -bounded and the condition (HA) holds, then by Lemma 3.1, $y (t)$ is the mild solution to Eq. (1.4) if it verifies Eq. (3.1). For the sake of convenience, we break the proof into three steps.

Step 1. We show that an $L^{2}$ -bounded solution is $L^{2}$ -continuous.

If $y (t)$ is an $L^{2}$ -bounded solution of (1.4), then by the Cauchy–Schwarz inequality, Itô’s isometry and the properties of integrals for Poisson random measures, we see that for $t ⩾ a$ , $\begin{array}{l} E {‖ y (t) - y (a) ‖}^{2} \\ ⩽ 4 E {‖ \int_{a}^{t} S (t - s) f_{1} (s, y (s)) d s ‖}^{2} + 4 E {‖ \int_{a}^{t} S (t - s) f_{2} (s, y (s)) d W (s) ‖}^{2} \\ + 4 E {‖ \int_{a}^{t} \int_{| x |_{U} < 1} S (t - s) f_{3} (s, y (s -), x) \tilde{N} (d s, d x) ‖}^{2} \\ + 4 E ‖ \int_{a}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{4} (s, y (s -), x) \tilde{N} (d s, d x) \\ + \int_{a}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{4} (s, y (s -), x) ν (d x) d s ‖^{2} \\ ⩽ 4 M^{2} \int_{a}^{t} e^{- 2 ϖ (t - s)} d s \int_{a}^{t} E {‖ f_{1} (s, y (s)) ‖}^{2} d s \\ + 4 M^{2} \int_{a}^{t} e^{- 2 ϖ (t - s)} E {‖ f_{2} (s, y (s)) Q^{1 / 2} ‖}_{L (U, L^{2} (P, H))}^{2} d s \\ + 4 M^{2} \int_{a}^{t} \int_{| x |_{U} < 1} e^{- 2 ϖ (t - s)} E {‖ f_{3} (s, y (s -), x) ‖}^{2} ν (d x) d s \\ + 8 M^{2} \int_{a}^{t} \int_{| x |_{U} ⩾ 1} e^{- 2 ϖ (t - s)} E {‖ f_{4} (s, y (s -), x) ‖}^{2} ν (d x) d s \\ + 8 M^{2} \int_{a}^{t} \int_{| x |_{U} ⩾ 1} e^{- 2 ϖ (t - s)} ν (d x) d s \int_{a}^{t} \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (s, y (s -), x) ‖}^{2} ν (d x) d s \\ ⩽ 4 M^{2} e^{2 ϖ (t - a)} \int_{a}^{t} 1 d s \int_{a}^{t} E {‖ f_{1} (s, y (s)) ‖}^{2} d s \\ + 4 M^{2} e^{2 ϖ (t - a)} \int_{a}^{t} E {‖ f_{2} (s, y (s)) Q^{1 / 2} ‖}_{L (U, L^{2} (P, H))}^{2} d s \\ + 4 M^{2} e^{2 ϖ (t - a)} \int_{a}^{t} \int_{| x |_{U} < 1} E {‖ f_{3} (s, y (s -), x) ‖}^{2} ν (d x) d s \\ + 8 M^{2} e^{2 ϖ (t - a)} \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (s, y (s -), x) ‖}^{2} ν (d x) d s \\ + 8 M^{2} e^{2 ϖ (t - a)} \int_{| x |_{U} ⩾ 1} ν (d x) d s \int_{a}^{t} \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (s, y (s -), x) ‖}^{2} ν (d x) d s . \end{array}$ Now applying the same arguments of Step 1 in Proof of [17, Theorem 3.2], we obtain $\begin{array}{l} (3.4) & sup_{s \in R} E {‖ f_{1} (s, y (s)) ‖}^{2} < \infty, \\ (3.5) & sup_{s \in R} E {‖ f_{2} (s, y (s)) Q^{1 / 2} ‖}_{L (U, L^{2} (P, H))}^{2} < \infty, \\ (3.6) & sup_{s \in R} \int_{| x |_{U} < 1} E {‖ f_{3} (s, y (s -), x) ‖}^{2} ν (d x) < \infty, \\ (3.7) & sup_{s \in R} \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (s, y (s -), x) ‖}^{2} ν (d x) < \infty, \end{array}$ and hence $\begin{matrix} E {‖ y (t) - y (a) ‖}^{2} \to 0 as t \to a^{+} . \end{matrix}$ The similar argument yields that $E ‖ y (t) - y (a) ‖^{2} \to 0$ as $t \to a^{-}$ . Therefore, $y (t)$ is $L^{2}$ -continuous.

Step 2. We show that the problem (1.4) has a unique $L^{2}$ -bounded solution.

Define the nonlinear operator $Λ : SBC (R; L^{2} (P, H)) \to SBC (R; L^{2} (P, H))$ by $\begin{array}{rcl} (Λ y) (t) & = & \int_{- \infty}^{t} S (t - s) f_{1} (s, y (s)) d s + \int_{- \infty}^{t} S (t - s) f_{2} (s, y (s)) d W (s) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{3} (s, y (s -), x) \tilde{N} (d s, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{4} (s, y (s -), x) N (d s, d x) . \end{array}$ If $y (t)$ is $L^{2}$ -bounded, by (HA) and (3.4)–(3.7), it follows that $(Λ y) (t)$ is $L^{2}$ -bounded. Moreover, by the proof of Step 1, $(Λ y) (t)$ is an $L^{2}$ -continuous process if $y (t)$ is an $L^{2}$ -bounded process. Thus Λ maps $SBC (R; L^{2} (P, H))$ into itself.

Next, we show that Λ is a contraction mapping on $SBC (R; L^{2} (P, H))$ . For $y_{1}, y_{2} \in SBC (R; L^{2} (P, H))$ and $t \in R$ we have $\begin{array}{l} E {‖ (Λ y_{1}) (t) - (Λ y_{2}) (t) ‖}^{2} \\ ⩽ 4 E {‖ \int_{- \infty}^{t} S (t - s) [f_{1} (s, y_{1} (s)) - f_{1} (s, y_{2} (s))] d s ‖}^{2} \\ + 4 E {‖ \int_{- \infty}^{t} S (t - s) [f_{2} (s, y_{1} (s)) - f_{2} (s, y_{2} (s))] d W (s) ‖}^{2} \\ + 4 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) [f_{3} (s, y_{1} (s -), x) - f_{3} (s, y_{2} (s -), x)] \tilde{N} (d s, d x) ‖}^{2} \\ + 4 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) [f_{4} (s, y_{1} (s -), x) - f_{4} (s, y_{2} (s -), x)] N (d s, d x) ‖}^{2} . \end{array}$ Now we evaluate the first term of the right-hand side of the above inequality, by the Cauchy–Schwarz inequality and (H1), we have $\begin{array}{l} E {‖ \int_{- \infty}^{t} S (t - s) [f_{1} (s, y_{1} (s)) - f_{1} (s, y_{2} (s))] d s ‖}^{2} \\ ⩽ M^{2} \int_{- \infty}^{t} e^{- ϖ (t - s)} d s \int_{- \infty}^{t} e^{- ϖ (t - s)} E {‖ f_{1} (s, y_{1} (s)) - f_{1} (s, y_{2} (s)) ‖}^{2} d s \\ ⩽ \frac{M^{2} L_{1}}{ϖ^{2}} sup_{s \in R} E {‖ y_{1} (s) - y_{2} (s) ‖}^{2}, \end{array}$ and the second term satisfies $\begin{array}{l} E {‖ \int_{- \infty}^{t} S (t - s) [f_{2} (s, y_{1} (s)) - f_{2} (s, y_{2} (s))] d W (s) ‖}^{2} \\ ⩽ M^{2} \int_{- \infty}^{t} e^{- 2 ϖ (t - s)} E {‖ [f_{2} (s, y_{1} (s)) - f_{2} (s, y_{2} (s))] Q^{1 / 2} ‖}_{L (U, L^{2} (P, H))}^{2} d s \\ ⩽ \frac{M^{2} L_{1}}{2 ϖ} sup_{s \in R} E {‖ y_{1} (s) - y_{2} (s) ‖}^{2} . \end{array}$ As to the third term, using the properties of stochastic integrals for Poisson random measures and (H2), we have $\begin{array}{l} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) [f_{3} (s, y_{1} (s -), x) - f_{3} (s, y_{2} (s -), x)] \tilde{N} (d s, d x) ‖}^{2} \\ ⩽ M^{2} \int_{- \infty}^{t} \int_{| x |_{U} < 1} e^{- 2 ϖ (t - s)} E {‖ f_{3} (s, y_{1} (s -), x) - f_{3} (s, y_{2} (s -), x) ‖}^{2} ν (d x) d s \\ ⩽ \frac{M^{2} L_{2}}{2 ϖ} sup_{s \in R} E {‖ y_{1} (s) - y_{2} (s) ‖}^{2} . \end{array}$ Likewise, we have $\begin{array}{l} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) [f_{4} (s, y_{1} (s -), x) - f_{4} (s, y_{2} (s -), x)] N (d s, d x) ‖}^{2} \\ ⩽ 2 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) [f_{4} (s, y_{1} (s -), x) - f_{4} (s, y_{2} (s -), x)] \tilde{N} (d s, d x) ‖}^{2} \\ + 2 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) [f_{4} (s, y_{1} (s -), x) - f_{4} (s, y_{2} (s -), x)] ν (d x) d s ‖}^{2} \\ ⩽ \frac{M^{2} L_{2}}{ϖ} sup_{s \in R} E {‖ y_{1} (s) - y_{2} (s) ‖}^{2} \\ + 2 M^{2} \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} e^{- ϖ (t - s)} ν (d x) d s \\ \times \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} e^{- ϖ (t - s)} E {‖ f_{4} (s, y_{1} (s -), x) - f_{4} (s, y_{2} (s -), x) ‖}^{2} ν (d x) d s \\ ⩽ (\frac{M^{2} L_{2}}{ϖ} + \frac{2 M^{2} L_{2} b}{ϖ^{2}}) sup_{s \in R} E {‖ y_{1} (s) - y_{2} (s) ‖}^{2} . \end{array}$ Thus, it follows that, for each $t \in R$ , $\begin{matrix} E {‖ (Λ y_{1}) (t) - (Λ y_{2}) (t) ‖}^{2} ⩽ (\frac{4 M^{2} L_{1}}{ϖ^{2}} + \frac{2 M^{2} L_{1}}{ϖ} + \frac{6 M^{2} L_{2}}{ϖ} + \frac{8 M^{2} L_{2} b}{ϖ^{2}}) sup_{s \in R} E {‖ y_{1} (s) - y_{2} (s) ‖}^{2}, \end{matrix}$ that is, $\begin{matrix} {‖ (Λ y_{1}) (t) - (Λ y_{2}) (t) ‖}_{2}^{2} ⩽ (\frac{4 M^{2}}{ϖ^{2}} (L_{1} + 2 L_{2} b) + \frac{2 M^{2}}{ϖ} (L_{1} + 3 L_{2})) sup_{s \in R} {‖ y_{1} (s) - y_{2} (s) ‖}_{2}^{2} . \end{matrix}$ Hence $\begin{array}{rcl} ‖ Λ y_{1} - Λ y_{2} ‖_{\infty} & = & sup_{t \in R} {‖ (Λ y_{1}) (t) - (Λ y_{2}) (t) ‖}_{2} \\ ⩽ & \sqrt{(\frac{4 M^{2}}{ϖ^{2}} (L_{1} + 2 L_{2} b) + \frac{2 M^{2}}{ϖ} (L_{1} + 3 L_{2}))} ‖ y_{1} - y_{2} ‖_{\infty} . \end{array}$ Since $(\frac{4 M^{2}}{ϖ^{2}} (L_{1} + 2 L_{2} b) + \frac{2 M^{2}}{ϖ} (L_{1} + 3 L_{2})) < 1$ , by the Banach contraction mapping principle, Λ admits a unique fixed point $y^{*} \in SBC (R; L^{2} (P, H))$ such that $Λ y^{*} = y^{*}$ , which is the unique $L^{2}$ -bounded mild solution to (1.4).

Step 3. We show that the $L^{2}$ -bounded solution is almost automorphic in distribution.

Let ${s_{n}^{'}}_{n \in N}$ be an arbitrary sequence in $R$ . Owing to $f_{1} \in SAA (R; L^{2} (P, H))$ , $f_{2} \in SAA (R; L (U, L^{2} (P, H)))$ and $f_{3}, f_{4} \in PSAA (R \times L^{2} (P, H) \times U; L^{2} (P, H))$ , there exists a subsequence ${s_{n}}_{n \in N}$ of ${s_{n}^{'}}_{n \in N}$ and some stochastic process $\tilde{f_{1}}$ , $\tilde{f_{2}}$ , $\tilde{f_{3}}$ , $\tilde{f_{4}}$ such that $\begin{array}{l} (3.8) & \{\begin{matrix} {lim}_{n \to \infty} E ‖ f_{1} (t + s_{n}, y) - \tilde{f_{1}} (t, y) ‖^{2} = 0, \\ {lim}_{n \to \infty} E ‖ \tilde{f_{1}} (t - s_{n}, y) - f_{1} (t, y) ‖^{2} = 0, \end{matrix} \\ (3.9) & \{\begin{matrix} {lim}_{n \to \infty} E ‖ [f_{2} (t + s_{n}, y) - \tilde{f_{2}} (t, y)] Q^{1 / 2} ‖_{L (U, L^{2} (P, H))}^{2} = 0, \\ {lim}_{n \to \infty} E ‖ [\tilde{f_{2}} (t - s_{n}, y) - f_{2} (t, y)] Q^{1 / 2} ‖_{L (U, L^{2} (P, H))}^{2} = 0, \end{matrix} \\ (3.10) & \{\begin{matrix} {lim}_{n \to \infty} \int_{| x |_{U} < 1} E ‖ f_{3} (t + s_{n}, y, x) - \tilde{f_{3}} (t, y, x) ‖^{2} ν (d x) = 0, \\ {lim}_{n \to \infty} \int_{| x |_{U} < 1} E ‖ \tilde{f_{3}} (t - s_{n}, y, x) - f_{3} (t, y, x) ‖^{2} ν (d x) = 0, \end{matrix} \\ (3.11) & \{\begin{matrix} {lim}_{n \to \infty} \int_{| x |_{U} ⩾ 1} E ‖ f_{4} (t + s_{n}, y, x) - \tilde{f_{4}} (t, y, x) ‖^{2} ν (d x) = 0, \\ {lim}_{n \to \infty} \int_{| x |_{U} ⩾ 1} E ‖ \tilde{f_{4}} (t - s_{n}, y, x) - f_{4} (t, y, x) ‖^{2} ν (d x) = 0, \end{matrix} \end{array}$ for each $t \in R$ and $y \in L^{2} (P, H)$ .

Let $\tilde{y} (t)$ satisfy the integral equation $\begin{array}{rcl} \tilde{y} (t) & : = & \int_{- \infty}^{t} S (t - s) \tilde{f_{1}} (s, \tilde{y} (s)) d s + \int_{- \infty}^{t} S (t - s) \tilde{f_{2}} (s, \tilde{y} (s)) d W (s) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) \tilde{f_{3}} (s, \tilde{y} (s -), x) \tilde{N} (d s, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) \tilde{f_{4}} (s, \tilde{y} (s -), x) N (d s, d x) . \end{array}$ Let $W_{n} (σ) : = W (σ + s_{n}) - W (s_{n})$ , $N_{n} (σ, x) : = N (σ + s_{n}, x) - N (s_{n}, x)$ and $\tilde{N_{n}} (σ, x) : = \tilde{N} (σ + s_{n}, x) - \tilde{N} (s_{n}, x)$ for each $σ \in R$ . By Remark 2.2, $W_{n}$ is a $Q$ -Wiener process with the same distribution as W, $N_{n}$ is also a Poisson random measure having the same distribution as N with compensated Poisson random measure $\tilde{N_{n}}$ . Let $σ = s - s_{n}$ , we have $\begin{array}{rcl} y (t + s_{n}) & = & \int_{- \infty}^{t} S (t - σ) f_{1} (σ + s_{n}, y (σ + s_{n})) d σ \\ + \int_{- \infty}^{t} S (t - σ) f_{2} (σ + s_{n}, y (σ + s_{n})) d W_{n} (σ) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - σ) f_{3} (σ + s_{n}, y (σ + s_{n} -), x) \tilde{N_{n}} (d σ, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) f_{4} (σ + s_{n}, y (σ + s_{n} -), x) N_{n} (d σ, d x) . \end{array}$ We consider the process $y_{n} (t)$ which satisfies the integral equation $\begin{array}{rcl} y_{n} (t) & = & \int_{- \infty}^{t} S (t - σ) f_{1} (σ + s_{n}, y_{n} (σ)) d σ + \int_{- \infty}^{t} S (t - σ) f_{2} (σ + s_{n}, y_{n} (σ)) d W (σ) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - σ) f_{3} (σ + s_{n}, y_{n} (σ -), x) \tilde{N} (d σ, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) f_{4} (σ + s_{n}, y_{n} (σ -), x) N (d σ, d x) . \end{array}$ As before, we see that $y (t + s_{n})$ has the same distribution as $y_{n} (t)$ for each $t \in R$ and $y_{n} (t)$ is unique and $L^{2}$ -bounded.

Note that $\begin{array}{l} E {‖ y_{n} (t) - \tilde{y} (t) ‖}^{2} \\ = 4 E {‖ \int_{- \infty}^{t} S (t - σ) [f_{1} (σ + s_{n}, y_{n} (σ)) - \tilde{f_{1}} (σ, \tilde{y} (σ))] d σ ‖}^{2} \\ + 4 E {‖ \int_{- \infty}^{t} S (t - σ) [f_{2} (σ + s_{n}, y_{n} (σ)) - \tilde{f_{2}} (σ, \tilde{y} (σ))] d W (σ) ‖}^{2} \\ + 4 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - σ) [f_{3} (σ + s_{n}, y_{n} (σ -), x) - \tilde{f_{3}} (σ, \tilde{y} (σ -), x)] \tilde{N} (d σ, d x) ‖}^{2} \\ + 4 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) [f_{4} (σ + s_{n}, y_{n} (σ -), x) - \tilde{f_{4}} (σ, \tilde{y} (σ -), x)] N (d σ, d x) ‖}^{2} \\ : = I_{1} + I_{2} + I_{3} + I_{4} . \end{array}$ For $I_{1}$ , it follows from the Cauchy–Schwarz inequality that $\begin{array}{rcl} I_{1} & ⩽ & 8 E {‖ \int_{- \infty}^{t} S (t - σ) [f_{1} (σ + s_{n}, y_{n} (σ)) - f_{1} (σ + s_{n}, \tilde{y} (σ))] d σ ‖}^{2} \\ + 8 E {‖ \int_{- \infty}^{t} S (t - σ) [f_{1} (σ + s_{n}, \tilde{y} (σ)) - \tilde{f_{1}} (σ, \tilde{y} (σ))] d σ ‖}^{2} \\ ⩽ & 8 M^{2} \int_{- \infty}^{t} e^{- ϖ (t - σ)} d σ \int_{- \infty}^{t} e^{- ϖ (t - σ)} E {‖ f_{1} (σ + s_{n}, y_{n} (σ)) - f_{1} (σ + s_{n}, \tilde{y} (σ)) ‖}^{2} d σ \\ + 8 M^{2} \int_{- \infty}^{t} e^{- ϖ (t - σ)} d σ \int_{- \infty}^{t} e^{- ϖ (t - σ)} E {‖ f_{1} (σ + s_{n}, \tilde{y} (σ)) - \tilde{f_{1}} (σ, \tilde{y} (σ)) ‖}^{2} d σ \\ ⩽ & \frac{8 M^{2} L_{1}}{ϖ} \int_{- \infty}^{t} e^{- ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ + I_{1}^{n}, \end{array}$ where $\begin{matrix} I_{1}^{n} = \frac{8 M^{2}}{ϖ} \int_{- \infty}^{t} e^{- ϖ (t - σ)} E {‖ f_{1} (σ + s_{n}, \tilde{y} (σ)) - \tilde{f_{1}} (σ, \tilde{y} (σ)) ‖}^{2} d σ . \end{matrix}$ Since $\begin{matrix} {‖ f_{1} (σ + s_{n}, \tilde{y} (σ)) - f_{1} (σ + s_{n}, \tilde{y} (0)) ‖}_{2} ⩽ \sqrt{L_{1}} {‖ \tilde{y} (σ) - \tilde{y} (0) ‖}_{2}, \end{matrix}$ using the fact that f is square-mean almost automorphic in t and $\tilde{y} (t)$ is bounded in $L^{2} (P, H)$ , we obtain ${sup}_{σ \in R} ‖ f_{1} (σ + s_{n}, \tilde{y} (σ)) ‖_{2} < \infty$ , and hence ${sup}_{σ \in R} ‖ \tilde{f_{1}} (σ, \tilde{y} (σ)) ‖_{2} < \infty$ . Now by the Lebesgue dominated convergence theorem and (3.8), it follows that $I_{1}^{n} \to 0$ as $n \to \infty$ .

For $I_{2}$ , applying Itô’s isometry, we have $\begin{array}{rcl} I_{2} & ⩽ & 8 E {‖ \int_{- \infty}^{t} S (t - σ) [f_{2} (σ + s_{n}, y_{n} (σ)) - f_{2} (σ + s_{n}, \tilde{y} (σ))] d W (σ) ‖}^{2} \\ + 8 E {‖ \int_{- \infty}^{t} S (t - σ) [f_{2} (σ + s_{n}, \tilde{y} (σ)) - \tilde{f_{2}} (σ, \tilde{y} (σ))] d W (σ) ‖}^{2} \\ ⩽ & 8 M^{2} L_{1} \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ + I_{2}^{n}, \end{array}$ where $\begin{matrix} I_{2}^{n} = 8 M^{2} \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ [f_{2} (σ + s_{n}, \tilde{y} (σ)) - \tilde{f_{2}} (σ, \tilde{y} (σ))] Q^{1 / 2} ‖}_{L (U, L^{2} (P, H))}^{2} d σ . \end{matrix}$ For the same reason as for $I_{1}^{n}$ , we have ${lim}_{n \to \infty} I_{2}^{n} = 0$ .

For $I_{3}$ , by the properties of integrals for Poisson random measures, we obtain $\begin{array}{rcl} I_{3} & ⩽ & 8 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - σ) [f_{3} (σ + s_{n}, y_{n} (σ -), x) - f_{3} (σ + s_{n}, \tilde{y} (σ -), x)] \tilde{N} (d σ, d x) ‖}^{2} \\ + 8 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - σ) [f_{3} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{3}} (σ, \tilde{y} (σ -), x)] \tilde{N} (d σ, d x) ‖}^{2} \\ ⩽ & 8 M^{2} L_{2} \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ y_{n} (σ -) - \tilde{y} (σ -) ‖}^{2} d σ + I_{3}^{n}, \end{array}$ where $\begin{matrix} I_{3}^{n} = 8 M^{2} \int_{- \infty}^{t} \int_{| x |_{U} < 1} e^{- 2 ϖ (t - σ)} E {‖ f_{3} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{3}} (σ, \tilde{y} (σ -), x) ‖}^{2} ν (d x) d σ . \end{matrix}$ Now, by condition (H2) and Lemma 2.1(c), we have $\begin{matrix} \int_{| x |_{U} < 1} E {‖ f_{3} (σ + s_{n}, \tilde{y} (σ), x) - f_{3} (σ + s_{n}, \tilde{y} (0), x) ‖}^{2} ν (d x) ⩽ L_{2} E {‖ \tilde{y} (σ) - \tilde{y} (0) ‖}^{2} < \infty \end{matrix}$ and $\begin{matrix} sup_{σ \in R} \int_{| x |_{U} < 1} E {‖ f_{3} (σ + s_{n}, \tilde{y} (0), x) ‖}^{2} ν (d x) ⩽ M_{\tilde{y} (0)}, for some constant M_{\tilde{y} (0)} . \end{matrix}$ So $\begin{matrix} sup_{σ \in R} \int_{| x |_{U} < 1} E {‖ f_{3} (σ + s_{n}, \tilde{y} (σ -), x) ‖}^{2} ν (d x) < \infty \end{matrix}$ and hence $\begin{matrix} sup_{σ \in R} \int_{| x |_{U} < 1} E {‖ \tilde{f_{3}} (σ, \tilde{y} (σ -), x) ‖}^{2} ν (d x) < \infty . \end{matrix}$ Now by (3.9) and the Lebesgue dominated convergence theorem we have $I_{3}^{n} \to 0$ as $n \to \infty$ .

As for $I_{4}$ , by the properties of integrals for Poisson random measures and the Cauchy–Schwarz inequality, we have $\begin{array}{rcl} I_{4} & ⩽ & 8 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) [f_{4} (σ + s_{n}, y_{n} (σ -), x) - f_{4} (σ + s_{n}, \tilde{y} (σ -), x)] N (d σ, d x) ‖}^{2} \\ + 8 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) [f_{4} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{4}} (σ, \tilde{y} (σ -), x)] N (d σ, d x) ‖}^{2} \\ ⩽ & 16 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) [f_{4} (σ + s_{n}, y_{n} (σ -), x) - f_{4} (σ + s_{n}, \tilde{y} (σ -), x)] \tilde{N} (d σ, d x) ‖}^{2} \\ + 16 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) [f_{4} (σ + s_{n}, y_{n} (σ -), x) - f_{4} (σ + s_{n}, \tilde{y} (σ -), x)] ν (d x) d σ ‖}^{2} \\ + 16 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) [f_{4} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{4}} (σ, \tilde{y} (σ -), x)] \tilde{N} (d σ, d x) ‖}^{2} \\ + 16 E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - σ) [f_{4} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{4}} (σ, \tilde{y} (σ -), x)] ν (d x) d σ ‖}^{2} \\ ⩽ & 16 M^{2} L_{2} \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ y_{n} (σ -) - \tilde{y} (σ -) ‖}^{2} d σ \\ + 16 M^{2} b \int_{- \infty}^{t} e^{- ϖ (t - σ)} d σ \\ \times \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} e^{- ϖ (t - σ)} E {‖ f_{4} (σ + s_{n}, y_{n} (σ -), x) - f_{4} (σ + s_{n}, \tilde{y} (σ -), x) ‖}^{2} ν (d x) d σ \\ + 16 M^{2} \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} e^{- 2 ϖ (t - σ)} E {‖ f_{4} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{4}} (σ, \tilde{y} (σ -), x) ‖}^{2} ν (d x) d σ \\ + 16 M^{2} b \int_{- \infty}^{t} e^{- ϖ (t - σ)} d σ \\ \times \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} e^{- ϖ (t - σ)} E {‖ f_{4} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{4}} (σ, \tilde{y} (σ -), x) ‖}^{2} ν (d x) d σ \\ ⩽ & 16 M^{2} L_{2} \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ y_{n} (σ -) - \tilde{y} (σ -) ‖}^{2} d σ \\ + \frac{16 M^{2} L_{2} b}{ϖ} \int_{- \infty}^{t} e^{- ϖ (t - σ)} E {‖ y_{n} (σ -) - \tilde{y} (σ -) ‖}^{2} d σ + I_{4}^{n}, \end{array}$ where $\begin{array}{rcl} I_{4}^{n} & = & 16 M^{2} \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} e^{- 2 ϖ (t - σ)} E {‖ f_{4} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{4}} (σ, \tilde{y} (σ -), x) ‖}^{2} ν (d x) d σ \\ + \frac{16 M^{2} b}{ϖ} \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (σ + s_{n}, \tilde{y} (σ -), x) - \tilde{f_{4}} (σ, \tilde{y} (σ -), x) ‖}^{2} ν (d x) d σ . \end{array}$ Thus, similarly to $I_{3}^{n}$ , we have ${lim}_{n \to \infty} I_{4}^{n} = 0$ .

By the estimates for $I_{1}$ – $I_{4}$ , we have $\begin{array}{rcl} E {‖ y_{n} (t) - \tilde{y} (t) ‖}^{2} & ⩽ & I^{n} + \frac{8 M^{2} L_{1}}{ϖ} \int_{- \infty}^{t} e^{- ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ \\ + 8 M^{2} L_{1} \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ \\ + 8 M^{2} L_{2} \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ \\ + 16 M^{2} L_{2} \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ \\ + \frac{16 M^{2} L_{2} b}{ϖ} \int_{- \infty}^{t} e^{- ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ \\ ⩽ & I^{n} + \frac{8 M^{2}}{ϖ} (L_{1} + 2 L_{2} b) \int_{- \infty}^{t} e^{- ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ \\ + 8 M^{2} (L_{1} + 3 L_{2}) \int_{- \infty}^{t} e^{- 2 ϖ (t - σ)} E {‖ y_{n} (σ) - \tilde{y} (σ) ‖}^{2} d σ, \end{array}$ where $I^{n} = \sum_{i = 1}^{4} I_{i}^{n}$ . By Lemma 2.2 and (3.3), we have $\begin{matrix} E {‖ y_{n} (t) - \tilde{y} (t) ‖}^{2} \to 0 as n \to \infty for each t \in R . \end{matrix}$ Hence $y_{n} (t)$ converges in distribution to $\tilde{y} (t)$ . Moreover, because of $y (t + s_{n})$ has the same distribution as $y_{n} (t)$ , we deduce from [17, Remark 2.12] that $y (t + s_{n})$ converges in distribution to $\tilde{y} (t)$ . By analogy and using (3.8)–(3.11) we can easily prove that $\tilde{y} (t - s_{n}) \to y (t)$ in distribution as $n \to \infty$ for each $t \in R$ . The proof is complete. □

Remark 3.2.

Note that the stochastic process ${\tilde{f}}_{1}$ , ${\tilde{f}}_{2}$ and ${\tilde{f}}_{3}$ , ${\tilde{f}}_{4}$ are also Lipschitz conditions in y uniformly for t with the same Lipschitz constant $L_{1}$ and $L_{2}$ as that of $f_{1}$ , $f_{2}$ and $f_{3}$ , $f_{4}$ in (H1) and (H2). Since $\begin{array}{l} E {‖ {\tilde{f}}_{1} (t, x) - {\tilde{f}}_{1} (t, y) ‖}^{2} \\ = E {‖ {\tilde{f}}_{1} (t, x) - f_{1} (t + s_{n}, x) + f_{1} (t + s_{n}, x) + f_{1} (t + s_{n}, y) - f_{1} (t + s_{n}, y) - {\tilde{f}}_{1} (t, y) ‖}^{2} \\ ⩽ 3 E {‖ f_{1} (t + s_{n}, x) - {\tilde{f}}_{1} (t, x) ‖}^{2} + 3 E {‖ f_{1} (t + s_{n}, y) - {\tilde{f}}_{1} (t, y) ‖}^{2} \\ + 3 E {‖ f_{1} (t + s_{n}, x) - f_{1} (t + s_{n}, y) ‖}^{2}, \\ E {‖ {\tilde{f}}_{2} (t, x) - {\tilde{f}}_{2} (t, y) ‖}_{L (U, L^{2} (P, H))}^{2} \\ = E ‖ {\tilde{f}}_{2} (t, x) - f_{2} (t + s_{n}, x) + f_{2} (t + s_{n}, x) + f_{2} (t + s_{n}, y) \\ - f_{2} (t + s_{n}, y) - {\tilde{f}}_{2} (t, y) ‖_{L (U, L^{2} (P, H))}^{2} \\ ⩽ 3 E {‖ f_{2} (t + s_{n}, x) - {\tilde{f}}_{2} (t, x) ‖}_{L (U, L^{2} (P, H))}^{2} + 3 E {‖ f_{2} (t + s_{n}, y) - {\tilde{f}}_{2} (t, y) ‖}_{L (U, L^{2} (P, H))}^{2} \\ + 3 E {‖ f_{2} (t + s_{n}, x) - f_{2} (t + s_{n}, y) ‖}_{L (U, L^{2} (P, H))}^{2}, \\ \int_{| x |_{U} < 1} E {‖ {\tilde{f}}_{3} (t, x, z) - {\tilde{f}}_{3} (t, y, z) ‖}^{2} ν (d x) \\ = \int_{| x |_{U} < 1} E ‖ {\tilde{f}}_{3} (t, x, z) - f_{3} (t + s_{n}, x, z) + f_{3} (t + s_{n}, x, z) \\ + f_{3} (t + s_{n}, y, z) - f_{3} (t + s_{n}, y, z) - {\tilde{f}}_{3} (t, y, z) ‖^{2} ν (d x) \\ ⩽ 3 \int_{| x |_{U} < 1} E {‖ f_{3} (t + s_{n}, x, z) - {\tilde{f}}_{3} (t, x, z) ‖}^{2} ν (d x) \\ + 3 \int_{| x |_{U} < 1} E {‖ f_{3} (t + s_{n}, y, z) - {\tilde{f}}_{3} (t, y, z) ‖}^{2} ν (d x) \\ + 3 \int_{| x |_{U} < 1} E {‖ f_{3} (t + s_{n}, x, z) - f_{3} (t + s_{n}, y, z) ‖}^{2} ν (d x), \end{array}$ and $\begin{array}{l} \int_{| x |_{U} ⩾ 1} E {‖ {\tilde{f}}_{4} (t, x, z) - {\tilde{f}}_{4} (t, y, z) ‖}^{2} ν (d x) \\ = \int_{| x |_{U} ⩾ 1} E ‖ {\tilde{f}}_{4} (t, x, z) - f_{4} (t + s_{n}, x, z) + f_{4} (t + s_{n}, x, z) \\ + f_{4} (t + s_{n}, y, z) - f_{4} (t + s_{n}, y, z) - {\tilde{f}}_{4} (t, y, z) ‖^{2} ν (d x) \\ ⩽ 3 \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (t + s_{n}, x, z) - {\tilde{f}}_{4} (t, x, z) ‖}^{2} ν (d x) \\ + 3 \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (t + s_{n}, y, z) - {\tilde{f}}_{4} (t, y, z) ‖}^{2} ν (d x) \\ + 3 \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (t + s_{n}, x, z) - f_{4} (t + s_{n}, y, z) ‖}^{2} ν (d x) \end{array}$ for all $x, y \in L^{2} (P, H)$ and $t \in R$ . So, we can choose $ε_{i} = ε (N_{i})$ , $i = 1, 2, 3, 4$ such that the above assertion is true.

For considering the weighted pseudo almost automorphy in distribution for

L^{2}

-bounded mild solutions to Eq. (1.4), we list the following assumptions.

The functions $f_{1}, f_{2} \in SWPAA (R \times L^{2} (P, H), ρ)$ . Let $(f_{11}, f_{12})$ and $(f_{21}, f_{22})$ denote respectively the decompositions of $f_{1}$ and $f_{2}$ , namely, $\begin{array}{l} f_{1} = f_{11} + f_{12}, f_{11} \in SAA (R \times L^{2} (P, H); L^{2} (P, H)), f_{12} \in {SBC}_{0} (R \times L^{2} (P, H), ρ), \\ f_{2} = f_{21} + f_{22}, \\ f_{21} \in SAA (R \times L^{2} (P, H); L (U, L^{2} (P, H))), f_{22} \in {SBC}_{0} (R \times L^{2} (P, H), ρ) . \end{array}$ Moreover, $f_{1}$ , $f_{11}$ , $f_{2}$ and $f_{21}$ satisfy the Lipschitz conditions in y uniformly for t, that is, there exists a constant $L_{11} > 0$ such that for all $x, y \in L^{2} (P, H)$ and $t \in R$ , $\begin{array}{l} E {‖ f_{1} (t, x) - f_{1} (t, y) ‖}^{2} ⩽ L_{11} E ‖ x - y ‖^{2}, \\ E {‖ f_{11} (t, x) - f_{11} (t, y) ‖}^{2} ⩽ L_{11} E ‖ x - y ‖^{2}, \\ E {‖ [f_{2} (t, x) - f_{2} (t, y)] Q^{1 / 2} ‖}_{L (U, L^{2} (P, H))}^{2} ⩽ L_{11} E ‖ x - y ‖^{2}, \\ E {‖ [f_{21} (t, x) - f_{21} (t, y)] Q^{1 / 2} ‖}_{L (U, L^{2} (P, H))}^{2} ⩽ L_{11} E ‖ x - y ‖^{2} . \end{array}$

The functions $f_{3}, f_{4} \in PSWPAA (R \times L^{2} (P, H) \times U; ρ)$ . Let $(f_{31}, f_{32})$ and $(f_{41}, f_{42})$ denote respectively the decompositions of $f_{3}$ and $f_{4}$ , namely, $\begin{array}{l} f_{3} = f_{31} + f_{32}, \\ f_{31} \in PSAA (R \times L^{2} (P, H) \times U; L^{2} (P, H)), f_{32} \in {PSBC}_{0} (R \times L^{2} (P \times U, H), ρ), \\ f_{4} = f_{41} + f_{42}, \\ f_{41} \in PSAA (R \times L^{2} (P, H) \times U; L^{2} (P, H)), f_{42} \in {PSBC}_{0} (R \times L^{2} (P, H) \times U, ρ) . \end{array}$ In addition, $f_{3}$ , $f_{31}$ , $f_{4}$ and $f_{41}$ satisfy the Lipschitz conditions in y uniformly for t, that is, there exists a constant $L_{22} > 0$ such that for all $x, y \in L^{2} (P, H)$ and $t \in R$ , $\begin{array}{l} \int_{| x |_{U} < 1} E {‖ f_{3} (t, x, z) - f_{3} (t, y, z) ‖}^{2} ν (d x) ⩽ L_{22} E ‖ x - y ‖^{2}, \\ \int_{| x |_{U} < 1} E {‖ f_{31} (t, x, z) - f_{31} (t, y, z) ‖}^{2} ν (d x) ⩽ L_{22} E ‖ x - y ‖^{2}, \\ \int_{| x |_{U} ⩾ 1} E {‖ f_{4} (t, x, z) - f_{4} (t, y, z) ‖}^{2} ν (d x) ⩽ L_{22} E ‖ x - y ‖^{2}, \\ \int_{| x |_{U} ⩾ 1} E {‖ f_{41} (t, x, z) - f_{41} (t, y, z) ‖}^{2} ν (d x) ⩽ L_{22} E ‖ x - y ‖^{2} . \end{array}$

Theorem 3.2.

Let $ρ \in U^{inv} \cap U_{P}^{inv}$ . Suppose the conditions (HA), (H3) and (H4) hold. If $\frac{L_{11} + 2 L_{22} b}{ϖ^{2}} + \frac{L_{11} + 3 L_{22}}{ϖ} < \frac{1}{8 M^{2}}$ , then the problem ( 1.4 ) has a unique $L^{2}$ -bounded mild solution being weighted pseudo almost automorphic in distribution.

Proof.

If $y (t)$ is $L^{2}$ -bounded and the condition (HA) holds, then by Lemma 3.1, $y (t)$ is the mild solution to Eq. (1.4) if it verifies Eq. (3.1). In view of (H3)–(H4), we get $\begin{array}{rcl} y (t) & = & \int_{- \infty}^{t} S (t - s) f_{1} (s, y (s)) d s + \int_{- \infty}^{t} S (t - s) f_{2} (s, y (s)) d W (s) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{3} (s, y (s -), x) \tilde{N} (d s, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{4} (s, y (s -), x) N (d s, d x) \\ = & [\int_{- \infty}^{t} S (t - s) f_{11} (s, y (s)) d s + \int_{- \infty}^{t} S (t - s) f_{21} (s, y (s)) d W (s) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{31} (s, y (s -), x) \tilde{N} (d s, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{41} (s, y (s -), x) N (d s, d x)] \\ + [\int_{- \infty}^{t} S (t - s) f_{12} (s, y (s)) d s + \int_{- \infty}^{t} S (t - s) f_{22} (s, y (s)) d W (s) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) N (d s, d x)] \\ : = & y_{1} (t) + y_{2} (t) . \end{array}$ Similarly to the proof of Theorem 3.1 with the conditions (H3) and (H4), one can easily see that $y_{1}$ is the unique almost automorphic solution in distribution. Now let us prove that $y_{2} \in {SBC}_{0} (R, ρ)$ .

Just as the proof of [11, Theorem 4.1] and [17, Theorem 4.1] with appropriate adaptations, it follows that $y_{2} (t)$ is $L^{2}$ -continuous and $L^{2}$ -bounded. Now, we show that $\begin{matrix} lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ y_{2} (t) ‖}^{2} ρ (t) d t = 0 . \end{matrix}$ From the definition of $y_{2} (t)$ , we get $\begin{array}{l} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ y_{2} (t) ‖}^{2} ρ (t) d t \\ = \frac{1}{m (r, ρ)} \int_{- r}^{r} E ‖ \int_{- \infty}^{t} S (t - s) f_{12} (s, y (s)) d s + \int_{- \infty}^{t} S (t - s) f_{22} (s, y (s)) d W (s) \\ + \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) N (d s, d x) ‖^{2} ρ (t) d t \\ ⩽ \frac{2}{m (r, ρ)} \int_{- r}^{r} E ‖ \int_{- \infty}^{t} S (t - s) f_{12} (s, y (s)) d s \\ + \int_{- \infty}^{t} S (t - s) f_{22} (s, y (s)) d W (s) ‖^{2} ρ (t) d t \\ + \frac{2}{m (r, ρ)} \int_{- r}^{r} E ‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) \\ (3.12) & + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) N (d s, d x) ‖^{2} ρ (t) d t . \end{array}$ By the condition (H3) and [11, Theorem 4.1], we have $\begin{array}{l} lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} E ‖ \int_{- \infty}^{t} S (t - s) f_{12} (s, y (s)) d s \\ (3.13) & + \int_{- \infty}^{t} S (t - s) f_{22} (s, y (s)) d W (s) ‖^{2} ρ (t) d t = 0 . \end{array}$ On the other hand, $\begin{array}{l} \frac{1}{m (r, ρ)} \int_{- r}^{r} E ‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) \\ + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) N (d s, d x) ‖^{2} ρ (t) d t \\ ⩽ \frac{2}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) ‖}^{2} ρ (t) d t \\ + \frac{2}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) N (d s, d x) ‖}^{2} ρ (t) d t \\ ⩽ \frac{2}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) ‖}^{2} ρ (t) d t \\ + \frac{4}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) \tilde{N} (d s, d x) ‖}^{2} ρ (t) d t \\ (3.14) & + \frac{4}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) ν (d x) d s ‖}^{2} ρ (t) d t . \end{array}$ We deduce, by the properties of integrals for Poisson random measures and the Fubini theorem, $\begin{array}{l} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) ‖}^{2} ρ (t) d t \\ ⩽ \frac{M^{2}}{m (r, ρ)} \int_{- r}^{r} ρ (t) d t \int_{- \infty}^{t} \int_{| x |_{U} < 1} e^{- 2 ϖ (t - s)} E {‖ f_{32} (s, y (s -), x) ‖}^{2} ν (d x) d s \\ = \frac{M^{2}}{m (r, ρ)} \int_{- r}^{r} ρ (t) d t \int_{0}^{+ \infty} \int_{| x |_{U} < 1} e^{- 2 ϖ s} E {‖ f_{32} (t - s, y ((t - s) -), x) ‖}^{2} ν (d x) d s \\ = M^{2} \int_{0}^{+ \infty} e^{- 2 ϖ s} d s \\ (3.15) & \times \frac{1}{m (r, ρ)} \int_{- r}^{r} \int_{| x |_{U} < 1} E {‖ f_{32} (t - s, y ((t - s) -), x) ‖}^{2} ρ (t) ν (d x) d t . \end{array}$ Since $f_{32} \in {PSBC}_{0} (R \times L^{2} (P, H) \times U, ρ)$ , by (3.12) and the Lebesgue dominated convergence theorem, we infer that $\begin{matrix} (3.16) & lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) ‖}^{2} ρ (t) d t = 0 . \end{matrix}$ Using the same arguments as previous, we obtain $\begin{matrix} (3.17) & lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) \tilde{N} (d s, d x) ‖}^{2} ρ (t) d t = 0 . \end{matrix}$ We deduce, by the Cauchy–Schwarz inequality and the Fubini theorem, $\begin{array}{l} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) ν (d x) d s ‖}^{2} ρ (t) d t \\ = \frac{1}{m (r, ρ)} \int_{- r}^{r} ρ (t) d t E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) ν (d x) d s ‖}^{2} \\ ⩽ \frac{M^{2}}{m (r, ρ)} \int_{- r}^{r} ρ (t) d t \\ \times (\int_{- \infty}^{t} e^{- ϖ (t - s)} d s \int_{| x |_{U} ⩾ 1} ν (d x) \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} e^{- ϖ (t - s)} E {‖ f_{42} (s, y (s -), x) ‖}^{2} ν (d x) d s) \\ ⩽ \frac{M^{2} b}{ϖ m (r, ρ)} \int_{- r}^{r} ρ (t) d t \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} e^{- ϖ (t - s)} E {‖ f_{42} (s, y (s -), x) ‖}^{2} ν (d x) d s \\ = \frac{M^{2} b}{ϖ m (r, ρ)} \int_{- r}^{r} ρ (t) d t \int_{0}^{+ \infty} \int_{| x |_{U} ⩾ 1} e^{- ϖ s} E {‖ f_{42} (t - s, y ((t - s) -), x) ‖}^{2} ν (d x) d s \\ = \frac{M^{2} b}{ϖ} \int_{0}^{+ \infty} e^{- ϖ s} d s \\ (3.18) & \times \frac{1}{m (r, ρ)} \int_{- r}^{r} \int_{| x |_{U} ⩾ 1} E {‖ f_{42} (t - s, y ((t - s) -), x) ‖}^{2} ρ (t) ν (d x) d t . \end{array}$ Since $f_{42} \in {PSBC}_{0} (R \times L^{2} (P, H) \times U, ρ)$ , by (3.15) and the Lebesgue dominated convergence theorem, we infer that $\begin{matrix} (3.19) & lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) ν (d x) d s ‖}^{2} ρ (t) d t = 0 . \end{matrix}$

By (3.14), (3.16), (3.17) and (3.19), we obtain $\begin{array}{l} lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} E ‖ \int_{- \infty}^{t} \int_{| x |_{U} < 1} S (t - s) f_{32} (s, y (s -), x) \tilde{N} (d s, d x) \\ (3.20) & + \int_{- \infty}^{t} \int_{| x |_{U} ⩾ 1} S (t - s) f_{42} (s, y (s -), x) N (d s, d x) ‖^{2} ρ (t) d t = 0 . \end{array}$ By (3.12), (3.13) and (3.20), we have $\begin{matrix} lim_{r \to + \infty} \frac{1}{m (r, ρ)} \int_{- r}^{r} E {‖ y_{2} (t) ‖}^{2} ρ (t) d t = 0 . \end{matrix}$ Hence $y_{2} \in {SBC}_{0} (R, ρ)$ . The proof is complete. □

Finally, as a simple example we consider the following stochastic integro-differential equation with Dirichlet boundary condition $\begin{array}{rcl} (3.21) & \{\begin{matrix} \frac{\partial u}{\partial t} (t, ξ) = \frac{\partial^{2} u}{\partial ξ^{2}} (t, ξ) + \int_{- \infty}^{t} \frac{(t - s)}{2 Γ (2)} e^{- (t - s)} \frac{\partial^{2} u}{\partial ξ^{2}} (s, ξ) d s + f (t, u (t, ξ)) \\ + g (t, u (t, ξ)) \frac{\partial W}{\partial t} (t, ξ) + h (t, u (t, ξ), Z (t, ξ)) \frac{\partial Z}{\partial t} (t, ξ), t \in R, ξ \in (0, π), \\ u (t, 0) = u (t, π) = 0, t \in R, \end{matrix} \end{array}$ where f, g, h are continuous functions with appropriate properties which would be specified later, W is a $Q$ -Wiener process on $L^{2} [0, π]$ with $T r Q < \infty$ , and Z is a Lévy pure jump process on $L^{2} [0, π]$ which is independent of W. Let $H = U : = L^{2} [0, π]$ and define $A : = \frac{\partial^{2}}{\partial ξ^{2}}$ with the domain $D (A) = {x \in H : x (0) = x (π) = 0}$ . According to some deductions in [8], there exists a strongly continuous and uniformly exponentially stable resolvent operator $S (t)$ . Then Eq. (3.21) can be converted into the abstract form $\begin{array}{rcl} d y & = & [A y + α \int_{- \infty}^{t} \frac{{(t - s)}^{τ - 1}}{Γ (τ)} e^{- β (t - s)} A y d s] d t + F (t, y) d t + G (t, y) d W \\ (3.22) & + \int_{| z |_{U} < 1} H (t, y, z) \tilde{N} (d t, d z) + \int_{| z |_{U} ⩾ 1} H (t, y, z) N (d t, d z) \end{array}$ on the Hilbert space $H$ with $α = - 1 / 2$ , $β = 1$ and $τ = 2$ , where $\begin{array}{l} y : = u, F (t, y) : = f (t, u), G (t, y) d W : = g (t, u) d W, \\ h (t, u, Z) d Z : = \int_{| z |_{U} < 1} H (t, y, z) \tilde{N} (d t, d z) + \int_{| z |_{U} ⩾ 1} H (t, y, z) N (d t, d z) \end{array}$ with $\begin{matrix} Z (t, ξ) = \int_{| z |_{U} < 1} z \tilde{N} (t, d z) + \int_{| z |_{U} ⩾ 1} z N (t, d z), H (t, y, z) = h (t, u, z) z . \end{matrix}$ Here, we assume that the Lévy pure jump process Z on $L^{2} [0, π]$ is decomposed as above by the Lévy–Itô decomposition theorem. Note that, by [10, Example 3.8.], we see that the condition (HA) holds. Now we assume that the continuous functions $f (t, u)$ , $g (t, u)$ and $h (t, u, Z)$ are Lipschitz in u with Lipschitz constants independent of t, Z. Then by [22, Section 13.1] and [17, Example 5.3], the functions F and G in the associated integro-differential equation (3.22) are Lipschitz with respect to y. Assume further that f, g, h are square-mean almost automorphic in t for each $u \in L^{2} (P, H)$ , then F and G are also square-mean almost automorphic in t for each $y \in L^{2} (P, H)$ . Finally let the intensity measure ν of the Poisson process Z be such that H is Poisson square-mean almost automorphic in t and Lipschitz in y in the sense of (H2) (see [17, Example 5.3] and [22, Section 13.1] for details). Note that the Lipschitz constants of F, G, H are small if that of f, g, h are small. Therefore, by Theorem 3.1, the Eq. (3.21) has a unique square-mean almost automorphic in distribution mild solution when the Lipschitz constants of f, g, h are appropriately small such that (3.3) holds.

Footnotes

Acknowledgements

The first author was partially supported by NSF of China (11361032), and NSFRP of Shaanxi Province (2017JM1017).

References

Abbas, Pseudo almost automorphic solutions of some nonlinear integro-differential equations, Computers and Mathematics with Applications 62 (2011), 2259–2272. doi:10.1016/j.camwa.2011.07.013.

Abbas and

Xia, Existence and attractivity of k-almost automorphic solutions of a model of cellular neural network with delay, Acta Mathematica Scientia 1 (2013), 290–302. doi:10.1016/S0252-9602(12)60211-2.

Applebaum, Lévy Process and Stochastic Calculus, 2nd edn, Cambridge University Press, 2009.

Benchaabane and

Sakthivel, Sobolev-type fractional stochastic differential equations with non-Lipschitz coefficients, Journal of Computational and Applied Mathematics 312 (2017), 65–73. doi:10.1016/j.cam.2015.12.020.

Bezandry and

Diagana, Almost Periodic Stochastic Processes, Springer, New York, 2011.

Y.T.

Bian,

Y.K.

Chang and

J.J.

Nieto, Weighted asymptotic behavior of solutions to a semilinear integro-differential equation in Banach spaces, Electronic Journal of Differential Equations 2014 (2014), 91.

J.F.

Cao,

Q.G.

Yang and

Z.T.

Huang, Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations, Stochastics 83 (2011), 259–275. doi:10.1080/17442508.2010.533375.

Y.K.

Chang and

Ponce, Uniform exponential stability and its applications to bounded solutions of integro–differential equations in Banach spaces, to appear in Journal of Integral Equations and Applications.

Y.K.

Chang and

Tang, Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process, Stochastics 88 (2016), 980–1011. doi:10.1080/17442508.2016.1178748.

10.

Chen,

Xiao and

Liang, Uniform exponential stability of solutions to abstract Volterra equations, Journal of Evolution Equations 4 (2009), 661–674. doi:10.1007/s00028-009-0028-4.

11.

Chen and

Lin, Square-mean almost automorphic process and its application to stochastic evolution equations, Journal of Functional Analysis 261 (2011), 69–89. doi:10.1016/j.jfa.2011.03.005.

12.

Diagana, Weighted pseudo almost periodic functions and applications, Comptes Rendus Mathematique 343 (2006), 643–646. doi:10.1016/j.crma.2006.10.008.

13.

M.M.

Fu, Almost automorphic solutions for nonautonomous stochastic differential equations, Journal of Mathematical Analysis and Applications 393 (2012), 231–238. doi:10.1016/j.jmaa.2012.04.017.

14.

M.M.

Fu and

Z.X.

Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proceedings of the American Mathematical Society 138 (2010), 3689–3701. doi:10.1090/S0002-9939-10-10377-3.

15.

Kamenskii,

Mellah and

Raynaud de Fitte, Weak averaging of semilinear stochastic differential equations with almost periodic coefficients, Journal of Mathematical Analysis and Applications 427 (2015), 336–364. doi:10.1016/j.jmaa.2015.02.036.

16.

K.X.

Li, Weighted pseudo almost automorphic solutions for nonautonomous SPDEs driven by Lévy noise, Journal of Mathematical Analysis and Applications 427 (2015), 686–721. doi:10.1016/j.jmaa.2015.02.071.

17.

Z.X.

Liu and

Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, Journal of Functional Analysis 266 (2014), 1115–1149. doi:10.1016/j.jfa.2013.11.011.

18.

Lizama and

Ponce, Bounded solutions to a class of semilinear integro-differential equations in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications 74 (2011), 3397–3406. doi:10.1016/j.na.2011.02.018.

19.

M.A.

Meyers,

K.K.

Chawla and Chawla , Mechanical Behavior of Materials, Cambridge University Press, 2009.

20.

G.M.

N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Space, Kluwer, New York, 2001.

21.

G.M.

N’Guérékata, Topics in Almost Automorphy, Springer, New York, 2005.

22.

Peszat and

Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, 2007.

23.

Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, 1993.

24.

Ren, Reflected backward doubly stochastic differential equations driven by a Lévy process, Comptes Rendus Mathematique 348 (2010), 439–444. doi:10.1016/j.crma.2009.11.004.

25.

Ren and

Sakthivel, Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps, Journal of Mathematical Physics 53 (2012), 073517. doi:10.1063/1.4739406.

26.

Sakthivel,

Ren,

Debbouche and

N.I.

Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Applicable Analysis 95 (2016), 2361–2382. doi:10.1080/00036811.2015.1090562.

27.

Xia, Pseudo asymptotically periodic solutions for Volterra integro-differential equations, Mathematical Methods in the Applied Sciences 38 (2015), 799–810. doi:10.1002/mma.3108.

Recurrence of bounded solutions to a semilinear integro-differential equation perturbed by Lévy noise

Abstract

Keywords

1. Introduction

2. Preliminaries

Definition 2.1 ([3]).

Definition 2.2 ([16,17]).

Proposition 2.1 ([3,22]).

Remark 2.1. By (2.2), it follows that ∫ | x | U ⩾ 1 ν ( d x ) < ∞ . For convenience, we denote b : = ∫ | x | U ⩾ 1 ν ( d x ) throughout the paper. Remark 2.2 ([17]).

Definition 2.5 ([16]).

Definition 2.6 ([16]).

Definition 2.7 ([17]).

Definition 2.8 ([17]).

Lemma 2.1 ([16]).

Lemma 2.2 ([17]).

Definition 2.9 ([16]).

Definition 2.10 ([16]).

Definition 2.11 ([16]).

Definition 2.12 ([16]).

Definition 2.13 ([16]).

Lemma 2.3 ([16]).

Definition 2.14 ([17]).

Definition 2.15 ([16]).

Lemma 2.4 ([15]).

3. Main results

Footnotes

Acknowledgements

References

Remark 2.1.
By (2.2), it follows that $\int_{| x |_{U} ⩾ 1} ν (d x) < \infty$ . For convenience, we denote $\begin{matrix} b : = \int_{| x |_{U} ⩾ 1} ν (d x) \end{matrix}$ throughout the paper.
Remark 2.2 ([17]).