Averaging Principles for Nonautonomous Two-Time-Scale Stochastic Reaction-Diffusion Equations with Jump

In this paper, we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven by Poisson randommeasures. /e coefficients of the equation are assumed to be functions of time, and some of them are periodic or almost periodic. /erefore, the Poisson term needs to be processed, and a new averaged equation needs to be given. For this reason, the existence of time-dependent evolution family of measures associated with the fast equation is studied and proved that it is almost periodic. Next, according to the characteristics of almost periodic functions, the averaged coefficient is defined by the evolution family of measures, and the averaged equation is given. Finally, the validity of the averaging principle is verified by using the Khasminskii method.


Introduction
e slow-fast systems are widely encountered in biology, ecology, and other application areas. In this paper, we are concerned with the following nonautonomous slow-fast systems of stochastic partial differential equations (SPDEs) on a bounded domain O of R d (d ≥ 1): where ϵ ≪ 1 is a positive parameter and α is a sufficiently large fixed constant. e operators N 1 and N 2 are boundary operators. e stochastic perturbations ω Q 1 , ω Q 2 , and N 1 , N ϵ 2 are mutually independent Wiener processes and Poisson random measures on the same complete stochastic basis (Ω, F, F t t≥0 , P), which will be described in Section 2. For i � 1, 2, the operator A i (t) and the functions b i , f i , and g i depend on time, and we assume that the operator A 2 (t) is periodic and the functions b 1 , b 2 , f 2 , and g 2 are almost periodic. e goal of this paper is to establish an effective approximation for the slow equation of original system (1) by using the averaging principle. e averaged equation is obtained as follows: where B 1 is the averaged coefficient, which will be given in equation (6). To demonstrate the validity of the averaging principle, we prove that, for any T > 0 and η > 0, it yields lim ϵ⟶0 P sup where u is the solution of the averaged equation (2). e theory of the averaging principle, originated by Laplace and Lagrange, has been applied in celestial mechanics, oscillation theory, radiophysics, and other fields. e firstly rigorous results for the deterministic case were given by Bogolyubov and Mitropolskii [1]. Moreover, Volosov [2] and Besjes [3] also promoted the development of the averaging principle. en, great interest has appeared in its application to dynamical systems under random perturbations. An important contribution was that, in 1968, Khasminskii [4] originally proposed the averaging principle for stochastic differential equations (SDEs) driven by Brownian motion. Since then, the averaging principle has been an active area of research. Many studies on the averaging principle of SDEs have been presented, e.g, Givon [5], Freidlin and Wentzell [6], Duan [7], ompson [8], and Xu and his coworkers [9][10][11]. Recently, effective approximations for slow-fast SPDEs have been received extensive attention. Cerrai [12,13] investigated the validity of the averaging principle for a class of stochastic reaction-diffusion equations with multiplicative noise. In addition, Wang and Roberts [14], Pei and Xu [15][16][17], and Li [18] and Gao [19][20][21] also studied the averaging principles for the slow-fast SPDEs. e abovementioned works mainly considered autonomous systems. For the autonomous systems, as long as the initial value is given, the solution of which only depends on the duration of time, not on the selection of the initial time. However, if the initial time is different, the solution of nonautonomous equations with the same initial data will also be different. erefore, compared with the autonomous systems, the dynamic behaviors of the nonautonomous systems are more complex, which can portray more actual models. Chepyzhov and Vishik [22] studied long-time dynamic behaviors of nonautonomous dissipative system. Carvalho [23] dealt with the theory of attractors for the nonautonomous dynamical systems. Bunder and Roberts [24] considered the discrete modelling of nonautonomous PDEs.
In 2017, the averaging principle has been presented for nonautonomous slow-fast system of stochastic reactiondiffusion equations by Cerrai [25]. But, the system of this paper was driven by Gaussian noises, which is considered as an ideal noise source and only can simulate fluctuations near the mean value. Actually, due to the complexity of the external environment, random noise sources encountered in practical fields usually exhibit non-Gaussian properties, which may cause sharp fluctuations. It should be pointed out that Poisson noise, one of the most ubiquitous noise sources in many fields [26][27][28], can provide an accurate mathematical model to describe discontinuous random processes, some large moves, and unpredictable events [29][30][31]. So, in this paper, we are devoted to developing the averaging principle for nonautonomous systems of reaction-diffusion equations driven by Wiener processes and Poisson random measures.
e key to using the averaging principle to analyze system (1) is the fast equation with a frozen slow component

Complexity
By dealing with the Poisson terms, we prove that an evolution family of measures (μ x t ; t ∈ R) on L 2 (O) for fast equation (4) exists. en, assuming that A 2 (t) is periodic and b 2 , f 2 , and g 2 are almost periodic, we prove that the evolution family of measures is almost periodic. Moreover, with the aid of eorem 2.10 in [32], we prove that the family of functions is uniformly almost periodic for any x in a compact set According to the characteristics of almost periodic function ( [25], eorem 3.4), we define the averaged coefficient B 1 as follows: Finally, the averaged equation is obtained through the averaged coefficient B 1 . Using the classical Khasminskii method to the present situation, the averaging principle is effective.
e abovementioned notations will be given in Section 2, and in this paper, c > 0 below with or without subscripts will represent a universal constant whose value may vary in different occasions.

Notations, Assumptions, and Preliminaries
Let O be a bounded domain of R d (d ≥ 1) having a smooth boundary. In this paper, we denote H the separable Hilbert space L 2 (O), endowed with the usual scalar product, and with the corresponding norm ‖·‖ H . e norm in L ∞ (O) will be denoted by ‖·‖ ∞ . Furthermore, the subspace D((− A) θ ) [33][34][35] of the generator A is dense in H, and endowed with the norm, for 0 ≤ θ < 1, 0 < t ≤ T. According to eorem 6.13 in [36], there exists c θ > 0, such that We denote by B b (H) the Banach space of the bounded Borel functions φ: H ⟶ R, endowed with the sup-norm In slow-fast system (1), the Gaussian noises zω Q 1 /zt(t, ξ) and zω Q 2 /zt(t, ξ) are assumed to be white in time and colored in space in the case of space dimension d > 1, for t ≥ 0 and ξ ∈ O. And ω Q i (t, ξ)(i � 1, 2) is the cylindrical Wiener processes defined as where e k k∈N is a complete orthonormal basis in H, β k (t) k∈N is a sequence of mutually independent standard Brownian motion defined on the same complete stochastic basis (Ω, F, F t t≥0 , P), and Q i is a bounded linear operator on H. Next, we give the definitions of Poisson random measures N 1 (dt, dz) and N ϵ 2 (dt, dz). Let (Z, B(Z)) be a given measurable space and v(dz) be a σ-finite measure on it. D p i t , i � 1, 2 represents two countable subsets of R + . Moreover, let p 1 t , t ∈ D p 1 t be a stationary F t -adapted Poisson point process on Z with the characteristic v, and p 2 t , t ∈ D p 2 t be the other stationary F t -adapted Poisson point process on Z with the characteristic v/ϵ. Denote by N i (dt, dz), i � 1, 2 the Poisson counting measure associated with p i t , i.e., Let us denote the two independent compensated Poisson measures where v 1 (dz)dt and 1/ϵv 2 (dz)dt are the compensators. Refer to [28,37] for a more detailed description of the stochastic integral with respect to a cylindrical Wiener process and Poisson random measure.
For any t ∈ R, the operators A 1 (t) and A 2 (t) are second-order uniformly elliptic operators, having continuous coefficients on O. e operators N 1 and N 2 are the boundary operators, which can be either the identity operator (Dirichlet boundary condition) or a first-order operator (coefficients satisfying a uniform nontangentiality condition). We shall assume that the operator A i (t) has the following form: where A i is a second order uniformly elliptic operator [38,39] with continuous coefficients on O, which is independent of t. In addition, L i (t) is a first-order differential operator, which has the form: e realizations of the differential operators A i and L i in H are A i and L i . Moreover, A 1 and A 2 generate two analytic semigroups e tA 1 and e tA 2 , respectively.

Complexity 3
Now, we give the following assumptions: (A1) (a) For i � 1, 2, the function c i : R ⟶ R is continuous, and there exist c 0 , c > 0 such that (A2) For i � 1, 2, there exists a complete orthonormal system e i,k k∈N in H and two sequences of nonnegative real numbers α i,k k∈N and λ i,k k∈N such that for some constants ρ i ∈ (2, +∞] and β i ∈ (0, +∞) such that Remark 1. For more comments and examples about the assumption (A2) of the operators A i and Q i , the reader can read [12].
Lipschitz continuous and linearly growing, uniformly with respect to (t, ξ, z) ∈ R × O × Z. Moreover, for all p ≥ 1, there exist positive constants c 1 and c 2 , such that, for all Lipschitz continuous and linearly growing, uni- Moreover, for all q ≥ 1, there exist positive constants c 3 and c 4 , such that, for all Due to (A3) and (A4), for any fixed (t, z) ∈ (R, Z), the mappings are Lipschitz continuous and have linear growth conditions. Now, for i � 1, 2, we define and for any ϵ > 0 and β ≥ 0, set For ϵ � 1, we write U β,i (t, s), and for ϵ � 1 and β � 0, we write U i (t, s).
Remark 3. By using the same argument as Chapter 5 of the book [40], we can get that there exists a unique evolution Complexity

A Priori Bounds for the Solution
With all notations introduced above, system (1) can be rewritten in the following abstract form: According to Remark 2, we know that the coefficients of system (28) satisfy global Lipschitz and linear growth conditions, and the assumptions (A1)-(A4) are uniform with respect to t ∈ R. So, using the same argument as [28,41,42], it is easy to prove that, for any ϵ > 0, T > 0 and x, y ∈ H, there exists a unique adapted mild solution (u ϵ , v ϵ ) to system (28) in L p (Ω; D([0, T]; H × H)). is means that there exist two unique adapted processes u ϵ and v ϵ in Proof. For fixed ϵ ∈ (0, 1] and x, y ∈ H, for any t ∈ [0, T], we denote For any p ≥ 2, because B 1 (·) is Lipschitz continuous, using Young's inequality, we have E sup Substituting (38) and (39) into (37), we obtain Now, we have to estimate For any t ∈ [0, T], we set For any p ≥ 2, because α > 0 is large enough, by proceeding as in (34), we can get According to the Gronwall inequality, we have 6 Complexity According to the definition of Λ 2,ϵ (t), for any t ∈ [0, T], we have erefore, by integrating with respect to t, using Young's inequality, we obtain According to the Burkholder-Davis-Gundy inequality, by proceeding as Proposition 4.2 in [12], we can easily get Concerning the stochastic term Ψ 2,ϵ (t), using Kunita's first inequality, we have By integrating with respect to t both sides and using Young's inequality, we have Substituting (49) and (51) into (48), we get As c p (0) � 0 and c p (t) is a continuous increasing function, we can fix t 0 > 0, such that for any t ≤ t 0 , we have c p (t) ≤ 1/2; so, Due to (41) and (53), for any t ∈ [0, t 0 ], we can get Similarly, we also can fix 0 < t 1 ≤ t 0 , such that for any t ≤ t 1 , we have c p,T (t) ≤ 1/2; so, According to the Gronwall inequality, we get (56) For any p ≥ 2, by repeating this in the intervals [t 1 , 2t 1 ], [2t 1 , 3t 1 ] etc., we can easily get (31). Substituting (31) into (41) and using the Gronwall inequality again, we get (30). Using the Hölder inequality, we can estimate (30) and (31) for p � 1.

Lemma 3. Under (A1)-(A4), for any θ
(69) By proceeding as the proof of Proposition 4.4 in [12] and (60), fix θ ∈ [0, θ), for any p ≥ 1, and it is possible to show that According to the proof of (39), using the Hölder inequality and (30), we have en, if we take p > 1, such that We can get As we are assuming |h| ≤ 1, (68) follows for any p ≥ p by taking From the Hölder inequality, we can estimate (68) for p < p, So, we have (68). According to the above lemma, we get that, for every ϵ ∈ (0, 1], the function u ϵ (t) is uniformly bounded about t ∈ [0, T], and it is also equicontinuous at every point of t ∈ [0, T]. In view of eorem 12.3 in [43], we can infer that the set u ϵ ϵ∈(0,1] is relatively compact in D([0, T]; H). In addition, according to eorem 13.2 in [43] and the above lemma, by using Chebyshev's inequality, we also can get that the family of probability measures L(u ϵ ) ϵ∈(0,1] is tight in P (D([0, T]; H)).

An Evolution Family of Measures for the Fast Equation
For any frozen slow component x ∈ H, any initial condition y ∈ H, and any s ∈ R, we introduce the following problem: where for two independent Q 2 -Wiener processes w Q 2 1 (t) and w Q 2 2 (t) and two independent compensated Poisson measures N 1′ (dt, dz) and N 3′ (dt, dz) with the same Lévy measure are both defined as in Section 2.
where t ∈ R. en, for every In the following, for any x ∈ H and any adapted process v, we set For any 0 < δ < α and any v 1 and v 2 with s < t, by proceeding as in the proof of Lemma 7.1 in [44], it is possible to show that there exists p > 1, such that, for any p ≥ p, we have sup r∈ [s,t] where L f 2 is the Lipschitz constant of f 2 and c p,1 and c p,2 are two suitable positive constants independent of α > 0 and s < t.
For the stochastic term Ψ α (v; s)(t), using Kunita's first inequality ( [26], eorem 4.4.23), we get 10 Complexity where L g 2 is the Lipschitz constant of g 2 and c p,1 and c p,2 are two suitable positive constants independent of α > 0 and s < t. Moreover, using (A4), we can show that sup r∈ [s,t] sup r∈ [s,t] where M f 2 and M g 2 are the linear growth constants of f 2 and g 2 and c p,1 and c p,2 are two suitable positive constants independent of α > 0 and s < t.
For any fixed adapted process v, let us introduce the problem: We denote that its unique mild solution is ρ α (v; s). is means that ρ α (v; s) solves the equation: Due to (82) and (84), using the same argument as (5.8) in [25], it is easy to prove that for any process v 1 , v 2 , and 0 < δ < α, we have sup r∈ [s,t] where, Similarly, thanks to (85) and (86), for any process v and 0 < δ < α, we can prove that sup r∈ [s,t] e δp(r− s) E ρ α (v; s)(r)
rough the above proof, eorem 1 is established. □

Data Availability
No data were used to support this study. 20 Complexity

Conflicts of Interest
e authors declare that they have no conflicts of interest.