Electronic Journal of Qualitative Theory of Differential Equations

This paper is devoted to a general stochastic delay differential equation with infinite-dimensional diffusions in a Hilbert space. We not only investigate the existence of invariant measures with either Wiener process or Lévy jump process, but also obtain the existence of a pullback attractor under Wiener process. In particular, we prove the existence of a non-trivial stationary solution which is exponentially stable and is generated by the composition of a random variable and the Wiener shift. At last, examples of reaction-diffusion equations with delay and noise are provided to illustrate our results.


Introduction
Delay differential equations arise from evolution phenomena in physical process and biological systems (see e.g. [19,21,25]), in which time-delay is used for mathematical modelling to describe the dynamical influence from the past. Recently, the effect of noise on such functional differential equations is increasingly a focus of investigation, in particular, in the combined influence of noise and delay in dynamical systems (see e.g. [5,6,13,35,37]). In this paper, we consider the following stochastic delay differential equation in a separable Hilbert space H: } is a given real-valued stochastic process, both defined on a probability space (Ω, F , P) where filtration {F t : t ≥ 0} is the P-completion of the Borel σ-algebra on Ω.
For stochastic delay differential equations, there has been a rather comprehensive mathematical literature on both theories and applications. The existence of invariant measures is well studied in both finite and infinite dimensions by using Krylov-Bogoliubov theorem (see e.g. [7,17,18]). Scheutzow [35] formulated a sufficient condition ensuring the existence of an invariant probability measure with additive noise. For a similar approach and connections to stochastic partial differential equations, see Bakhtin and Mattingly [4]. For stochastic delay differential equations driven by Brownian motion, Mohammed [31] investigated the existence and uniqueness of strong or weak solutions under random functional Lipschitz conditions, Mao [30] discussed the method of steps, which provides a unique solution without a regular dependence of the coefficients on values in the past, Liptser and Shiryaev [26] considered weak solutions, Itô and Nisio [23] investigated the existence of weak solutions for equations with finite and infinite delay, Butkovsky and Scheutzow [8] established a general sufficient conditions ensuring the existence of an invariant measure for stochastic functional differential equations and exponential or subexponential convergence to the equilibrium. For stochastic delay differential equations driven by a Lévy process, Gushchin and Küchler [20] established some necessary and sufficient conditions ensuring the existence and uniqueness of stationary solutions, Reiß, Riedle, and van Gaans [32] proved that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space, and also investigated the existence of an invariant measure by proving the tightness of the segments using semimartingale characteristics and the Krylov-Bogoliubov method. Existence and uniqueness of global solutions have been established under local Lipschitz and linear growth conditions (see e.g. [30,40]) or weak one-sided local Lipschitz (or monotonicity) conditions. Recently, Liu [28] considered stationary distributions of a class of second-order stochastic delay evolution equations driven by Wiener process or Lévy jump process in Hilbert space. In this paper we shall prove the existence of an invariant measure for (1.1) without boundedness conditions on the diffusion coefficient. Note that the segment process takes values in the infinite dimensional space L, boundedness in probability does not generally imply tightness. In this case, one usually uses compactness of the orbits of the underlying deterministic equation to obtain tightness. However, such a compactness property does not hold for functional differential equation (1.1). For more details, see [7]. In this work, we will study the existence of invariant measures of (1.1) by applying the Krylov-Bogoliubov method.
A criterion for the existence of random attractors for random dynamical systems is established by Crauel and Flandoli [14], who also obtained the invariant Markov measures supported by the random attractor. Caraballo, Kloeden and Keal [10] proved the existence of random attractors of an ordinary differential equation with a random stationary delay. Kloeden and Lorenz [24] pointed out that the classical theory of pathwise random dynamical systems with a skew product (see e.g. [3]) does not apply to nonlocal dynamics such as when the dynamics of a sample path depends on other sample paths through an expectation or when the evolution of random sets depends on nonlocal properties such as the diameter of the sets. In [24], Kloeden and Lorenz showed that such nonlocal random dynamics can be characterized by a deterministic two-parameter process from the theory of nonautonomous dynamical systems acting on a state space of random variables or random sets with the meansquare topology and provided a definition of mean-square random dynamical systems and their attractors. Wu and Kloeden [39] investigated the existence of a random attractor for a mean-square random dynamical system (MS-RDS) generated by a stochastic delay differential equation with random delay for which the drift term is dominated by a nondelay component satisfying a one-sided dissipative Lipschitz condition. The exponential stability of trivial stationary solutions for stochastic partial differential equations has been extensively analyzed (see e.g. [12,22,29]). Caraballo, Kloeden and Schmalfuß [11] obtained the existence of a nontrivial stationary solution and a random fixed point which is exponentially stable. In this paper, we shall generalize the relevant results of Caraballo, Kloeden and Schmalfuß [11] to such a stochastic evolution equation with delay as (1.1). In particular, we shall prove the existence of a random fixed point, which generates the exponentially stable stationary solution of (1.1). Moreover, this stationary solution attracts bounded sets of initial conditions.
In this paper, we first establish a non-autonomous random dynamical system generated by equation (1.1). Then we show the existence of an invariant measure of (1.1) driven by Wiener process. In particular, we obtain a random pullback attractor consisting of a single point which is exponentially stable. Next, the existence of invariant measures of (1.1) driven by Lévy jump process is obtained by using Lévy-Itô decomposition formula. Finally, we apply our results to reaction-diffusion equations with noise and delay.

Preliminaries
Throughout this paper, we always assume that H is a separable Hilbert space, and there exists a Gelfand triplet V ⊂ H ⊂ V ′ of separable Hilbert spaces, where V ′ denotes the dual of V and V = Dom(A 1 2 ) (see page 55 of [38] for more details). The inner product in H is denoted by ⟨·, ·⟩, and the duality mapping between V ′ and V by ⟨·, ·⟩ V . We denote by a 1 > 0 the constant of the injection V ⊂ H, i.e., a 1 ∥u∥ 2 ≤ ∥u∥ 2 V for u ∈ V, and let −A : V → V ′ be a positive, linear and continuous operator for which there exists an a 2 > 0 such that ⟨−Au, u⟩ V ≥ a 2 ∥u∥ 2 V for all u ∈ V. It is well known (see, for instance, [6,9,15]) that A is the generator of a strongly continuous semigroup Φ(t) = e tA on H satisfying that where λ = a 1 a 2 > 0 and L (H) is a space of bounded linear operators on H.
where X t (φ) represents X t (φ)(s) = X(t + s, φ) for s ∈ [−τ, 0] and t ≥ 0. where where P t is called the transition operator of (1.1) and C b (L) denotes the set of all bounded and continuous real-valued functions on L. Let µ X t (ϕ) be the distribution of X t (ϕ), t ≥ 0. If an F 0 -measurable ϕ ∈ L 2 (Ω, L) is such that µ X t (ϕ) = µ ϕ for all t ≥ 0, then µ ϕ is called a stationary distribution of (1.1) and X(t, ϕ) is then called a stationary solution.
It follows from the above definition that an invariant measure µ is a stationary distribution of (1.1) if and only if when F 0 is assumed to be rich enough to allow the existence of an F 0 -measurable random variable with distribution µ.
Definition 2.2. Denote by P(L) the set of Borel probability measures on L endowed with the topology of weak convergence of measures. For µ 1 , µ 2 ∈ P(L) define a metric on P(L) by It is well known that P(L) is complete under the metric d(·, ·) (see [16,Theorem 2.4.9]). In order to show the existence of an invariant measure, we consider the segments of a solution. In contrast to the scalar solution process, the process of segment {X t (ϕ) : t ≥ 0} is a Markov process [17,18]. It is shown that the segment process is also Feller and there exists a solution of which the segments are tight (see, for example, [17] for more details). Then we apply the Krylov-Bogoliubov method. In fact, we have the following result. Lemma 2.3. Suppose that for any bounded subset U of L, (ii) sup t≥0 sup ϕ∈U E∥X t (ϕ)∥ 2 L < ∞. Then, for any initial condition ϕ ∈ L, the solution of equation (1.1) converges to an invariant measure.
Proof. It suffices to show that for any initial condition ϕ ∈ L, {P(ϕ, t, ·) : t ≥ 0} is Cauchy in the space P(L) with the metric d(·, ·) in Definition 2.2. For this purpose, we only need to show that for any initial data ϕ ∈ L and ε > 0, there exists a time T > 0 such that The proof is referred to Lemma 5.1 in [28]. Here we shall provide the details for the sake of completeness. For any f ∈ M and t, s > 0, note that where L R = {ϕ ∈ L : ∥ϕ∥ L ≤ R} and L c R = L − L R . By virtue of condition (i), there exists a time T 2 > 0 such that On the other hand, condition (ii) implies that there exists a positive sufficiently large constant R such that Hence (2.3) holds and the transition probability P(X s (ϕ), ·) of X t (ϕ) converges weakly to some µ ∈ P(L). For every f ∈ C b (L) the Markovian property of X t (ϕ), t ≥ 0 gives that That is, µ is an invariant measure for X t (ϕ), t ≥ 0. The proof is completed.

Stochastic systems driven by Wiener process
In this section we consider equation where Q is a linear, symmetric and nonnegative bounded operator on U. In particular, we shall call {W(t) : t ≥ 0}, a U -valued Q-Wiener process with respect to {F t : t ≥ 0}. First, we shall show the solution process is tight. Let L Q 2 (U , H) is the space of all Hilbert-Schmidt operators from U to H with ∥G∥ 2 where G * (s) and Φ * (s) are the adjoint operators of G(s) and Φ(s), respectively. We suppose that Throughout this section, the operator F: L → H is supposed to be Lipschitz continuous while the operator G: L → L Q 2 (U , H) is supposed to be Lipschitz continuous with respect to the Hilbert-Schmidt norm L Q 2 (U , H) of linear operators from U to H: for all x, y ∈ L, where K, K 1 , K 2 are nonpositive constants. Note that under hypotheses (2.1) and (3.2), (1.1) has a unique mild solution of which the segment is a Markov and Feller process (see [33,34,39] for more details). In the subsequent two subsections, we investigate the existence of invariant measure and random attractor as well as the exponential stability of stationary solutions.
Then we will prove the segment process of solution to (1.1) is bounded with Wiener process.
Then the solution process (2.2) is ultimately bounded in the mean-square sense, i.e., for any bounded set U of L, Proof. It follows from 2K 2 Note that Following (2.1), (3.2) and Hölder's inequality we have for t > τ. It follows from (2.1), (3.2) and the Burkholder-Davis-Gundy inequality that for t > τ. Thus, (3.9), (3.10) and (3.11) together imply that for t > τ, . Then Gronwall's inequality gives that This completes the proof. By Lemmas 2.3, 3.1 and 3.2, we can have the following result about the existence of invariant measures of equation (1.1) driven by Wiener process. Now we show the uniqueness of invariant measures. If µ, µ ′ ∈ P(L) are two different invariant measures for X t of (1.1), for any f ∈ M, by virtue of (3.7), Hölder's inequality and the invariance of µ(·), µ ′ (·), it follows that

Random attractor
We consider the canonical probability space (Ω, F , P), where and F is the Borel σ-algebra induced by the compact open topology of Ω (see [3]), while P is the corresponding Wiener measure on (Ω, F , P). Define a shift operators by flow θ = {θ t } t∈R on Ω: Then, (Ω, θ 0 is the identity on Ω, θ s+t = θ t θ s for all s, t ∈ R, and θ t (P) = P for all t ∈ R.
More precisely, P is ergodic with respect to θ. In addition, with respect to the filtration we have that for any t, s ∈ R, whereF is the completion of F , see [3, Definition 2.3.4] for more details. For the sake of convenience, from now on, we will abuse the notation slightly and write the space Ω as Ω.
The Wiener process with covariance Q is adapted to the filtration {F s+t } t≥0 . First we define a mean-square random dynamical system referring to [24,39]. Let for each t ∈ R.

Definition 3.5 ([39, Definition 11]). A family
and Ψ-positively invariant if Let X(·, t 0 , ϕ 0 ) be the solution of the following equation with initial value ϕ 0 ∈ Π t 0 , For each (t, t 0 , ϕ 0 ) ∈ R 2 ≥ × Π t 0 , define solution mapping of (3.13): It is easy to see that Ψ satisfies the initial value property, Existence and uniqueness of solution of (3.13) show that Ψ satisfies the two-parameter semigroup evolution property. Moreover, Ψ is continuous for all (t, t 0 , ϕ 0 ) ∈ R 2 ≥ × Π t 0 since solution X t (·, t 0 , ϕ 0 ) is continuous with respect to t, ϕ 0 . Thus, (3.13) generates a continuous MS-RDS Ψ = {Ψ(t, t 0 , ·), (t, t 0 ) ∈ R 2 ≥ } with state space L. It follows from Lemma 3.2 that for any bounded set U of Π there exist constants B > 0 and T U ≥ 0 such that for all t ≥ t 0 + T U and ϕ 0 ∈ U ∩ Π t 0 , which can be represented in the pullback sense that E∥Ψ(t, t n , ϕ n )∥ 2 < B for all t n ≤ t − T B and ϕ n ∈ U ∩ Π t n . Lemma 3.1 shows that any two solutions converge together in the mean-square sense uniformly for different initial conditions at the same starting time. Namely, for any ϕ 0 , ψ 0 ∈ Π t 0 , with the convergence being uniform for initial values in a common bounded subset as well as in the initial time t 0 . Let U B be a bounded ball about the origin of radius B in L. Consider a sequence t n → −∞ as n → ∞ with t n < −T U − τ and t n+1 ≤ t n − T B U and define a sequence for an arbitrary ϕ n ∈ U B ∩ Π t n . Namely, for all s ∈ [−τ, 0]. Then {χ n } ∞ n=1 are obviously mean-square bounded by B for all ϕ n taking values in U B ∩ Π t n .
Lemma 3.7. {χ n } n∈N is a Cauchy sequence with values in U B ∩ Π 0 and there exists a unique limit Proof. It suffices to prove that for every ε > 0 there exists N ε > 0 such that Let t m < t n < 0. Then we have where ϕ n,m := Ψ(t n , t m , ϕ m ) ∈ U B ∩ Π t n . Indeed, Thus, it follows from Lemma 3.1 that (3.15) holds and all solutions starting in the common bounded subset U B converge in Π 0 . Since Π 0 is complete, the Cauchy sequence has a unique limit χ * 0 ∈ U B ∩ Π 0 . The proof is completed.
From the above process, we can repeat with 0 in (3.14) replaced by −1 to obtain a limit χ * −1 ∈ Π −1 . It is easy to see from the construction that χ * 0 = Ψ(0, −1, χ * −1 ). Follow this way, we can construct a sequence {χ * −n } n∈N and hence obtain an entire MS-RDS χ * t for all t ∈ R. Moreover, all other MS-RDS trajectories converge to χ * t in the mean-square sense. Theorem 3.8. Under the assumptions of Lemmas 3.1 and 3.2, there exists a pullback random attractor for the random dynamical system generated by (1.1) which consists of singleton sets. Furthermore, the random attractor pullback attracts all other solution processes in mean-square sense.
Proof. The above arguments shows the existence of random attractor consisting of singleton sets A t = {χ * t } and attracts all other solution processes in the mean-square sense. Next we show the random attractor is unique. Suppose there is another entire trajectoryχ * t ∈ A t for all t ∈ R and there exists a constant ε 0 > 0 such that On the other hand, it follows from the convergence in Lemma 3.1 that there exists T ≥ 0 such that ε 0 2 for t ≥ T, which is a contradiction. This completes the proof. Remark 3.9. If the random attractor A(ω), ω ∈ Ω consists of a single point, then A defines a random fixed point which attracts tempered random sets.
Now we can show the existence of the fixed point.
Indeed, for any fixed t, from (3.16), Lemma 3.10 and semigroup property we have This completes the proof.

Systems driven by Lévy jump process
In this section, we will give the existence of invariant measures of (1.1) with Lévy jump process in separable Hilbert space U . To this end, it suffices to verify the assertions in Lemma 2.3 hold. Let Z = {Z(t) : t ≥ 0} be a stochastic process defined on a probability space (Ω, F , P). We say that Z is a Lévy process if: We have the following property of Lévy measures on separable Hilbert spaces (see [36]). ν(·) is also called a Lévy measure.
Let Γ ∈ B(U − {0}) with 0 / ∈Γ and f : Γ → U measurable. Define the following integral This is a finite sum P-a.s. since the number of summands is finite P-a.s. For f ∈ L 2 ν ≜ L 2 (U − {0}, ν| U −{0} ; U ), the next proposition defines the integral with respect to the compensated Poisson random measure (see [36] for more details).
Let Z be a U -valued Lévy process with its Lévy triplet (0, Q, ν) below. By Lemma 4.1, ν(Γ) is a Lévy measure with Γ ∈ B(Γ − {0}). Note that an adapted Lévy process with zero mean is martingale, and that a Lévy process is martingale if and only if it is integrable and b + ∥z∥ U ≥1 zν(dz) = 0.
It follows from Lévy-Itô decomposition that the Lévy process can be written as In view of Proposition 4.4, we have Throughout this section, we always assume that the operators F and G in (1.1) satisfy for all x, y ∈ L, where K, K 1 , K 2 are nonnegative constants, L (U , H) is the space of bounded linear operators from U to H.
The boundedness of solution with Lévy jump process is given as follows.
Now we only need to show the tightness of solution (2.2) with Lévy jump process.
In view of Lemmas 4.6 and 4.7, it suffices to show the uniqueness of invariant measures of (1.1) driven by Lévy jump process. If µ,μ ∈ P(L) are two different invariant measures, then for any f ∈ M, it follows from (4.12) and the invariance of µ,μ ∈ P(L) that Thus, we obtain the following main result immediately.

Application
Let T := R/(2πZ) be equipped with the usual Riemannian metric, and let dξ denote the Lebesgue measure on T. For any p ≥ 1, let It is easy to see that H is a real separable Hilbert space with the inner product ⟨x, y⟩ = T x(ξ)y(ξ)dξ, x, y ∈ H, and the norm ∥x∥ = ⟨x, x⟩. In the following two subsections, we consider two stochastic reaction-diffusion equations on torus T.

A Brownian motion case
Consider a stochastic reaction-diffusion equation driven by a Brownian motion {W(t)} t≥0 on torus T as follows: where ϕ ∈ C := C([−1, 0], W 1,2 (T)), f : R → R and g: R → R are Lipschitz continuous and satisfy the linear growth, i.e., there exist positive constants K, K 1 , and K 2 such that for all u, v ∈ R. Obviously, A = ∂ 2 ∂ξ 2 is a self-adjoint operator on H with the discrete spectral. More precisely, there exist an orthogonal basis {e k = exp{ik(·)} : k ∈ Z * } with Z * = Z \ {0}, and a sequence of real numbers {λ k = k 2 : k ∈ Z * } such that −Ae k = λ k e k . Let V be the domain of the fractional operator (−A) 1/2 , that is, and with the norm ∥u∥ V = ⟨u, u⟩ V = ∥(−A) 1/2 u∥. Clearly, V is densely and compactly embedded in H. For every u ∈ H, there exists {a k } k∈Z * ⊂ R such that u = ∑ x∈Z * a k e k . Thus, we have Thus, we obtain (2.1) with λ = λ 2 1 . We consider a symmetric positive linear operator Q in H such that Qe k = q k e k for k ∈ Z * , where {q k } k∈Z * is a bounded sequence of nonnegative real numbers. Thus, Tr(Q) ≜ ∑ k∈Z * ⟨Qe k , e k ⟩ = ∑ k∈Z * q k < ∞, and Q is also called a trace class operator. Let {W(t)} t≥0 be a H-valued Q-Wiener process given by where {W k (t) : t ≥ 0} k∈Z * be a sequence of independent standard one-dimensional Brownian motions on some filtered probability space (Ω, F , {F t } t≥0 , P), that is, W k (t) ∼ N (0, t), EW k (t) = 0, E[W k (t)] 2 = t, and E[W k (t)W k (s)] = min{t, s}. It is easy to see that the infinite series of W(t) converges in L 2 (Ω), and satisfies E⟨W(t), W(t)⟩ = tTr(Q), E (⟨W(t), a⟩⟨W(s), b⟩) = (t ∧ s)⟨a, b⟩.
Then we can rewrite system (5.1) into the abstract form (1.1) with τ = 1, F(u t ) = f (u(t − 1, ·)) and G(u t ) = g(u(t − 1, ·)). Note that the segment process u t = u(t + s, ξ), s ∈ [−1, 0] is equipped with norm in C, i.e., In what follows, we shall verify that F and G satisfy hypothesis (3.2). In fact, it follows from (5.2) and Minkowski inequality that Then we can set the parameter values in (3.2) as follows K = L f + L g ∑ k∈Z * q k 1 2 , Thus, from Theorems 3.3 and 3.8 we have the following result.
Then equation (5.1) has a unique invariant measure and a pullback attractor.

S. Li and S. Guo
It follows from Section 5.1 that It is easy to check that Then we set the parameter values in (4.3) as follows K = L f + L g , Thus the result of existence of invariant measure of (5.3) follows from Theorems 4.6 and 4.7.
Then equation (5.3) has a unique invariant measure.