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

Authors: Yong Xu; Ruifang Wang
Journal: Complexity
 (2020)

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## Abstract

In this paper, we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven by Poisson random measures. The coefficients of the equation are assumed to be functions of time, and some of them are periodic or almost periodic. Therefore, 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.


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## Body

## 1. Introduction

The 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 domainO of ℝdd≥1:(1)∂uϵ∂tt,ξ=A1tuϵt,ξ+b1t,ξ,uϵt,ξ,vϵt,ξ+f1t,ξ,uϵt,ξ∂ωQ1∂tt,ξ+∫ℤg1t,ξ,uϵt,ξ,z∂N˜1∂tt,ξ,dz,∂vϵ∂tt,ξ=1ϵA2t−αvϵt,ξ+b2t,ξ,uϵt,ξ,vϵt,ξ+1ϵf2t,ξ,uϵt,ξ,vϵt,ξ∂ωQ2∂tt,ξ+∫ℤg2t,ξ,uϵt,ξ,vϵt,ξ,z∂N˜2ϵ∂tt,ξ,dz,uϵ0,ξ=xξ,vϵ0,ξ=yξ,ξ∈O,N1uϵt,ξ=N2vϵt,ξ=0,t≥0,ξ∈∂O,where ϵ≪1 is a positive parameter and α is a sufficiently large fixed constant. The operators N1 and N2 are boundary operators. The stochastic perturbations ωQ1,ωQ2, and N˜1,N˜2ϵ are mutually independent Wiener processes and Poisson random measures on the same complete stochastic basis Ω,ℱ,ℱtt≥0,ℙ, which will be described in Section 2. For i=1,2, the operator Ait and the functions bi,fi,andgi depend on time, and we assume that the operator A2t is periodic and the functions b1,b2,f2,andg2 are almost periodic.The goal of this paper is to establish an effective approximation for the slow equation of original system (1) by using the averaging principle. The averaged equation is obtained as follows:(2)∂u¯∂tt,ξ=A1tu¯t,ξ+B¯1u¯tξ+f1t,ξ,u¯t,ξ∂ωQ1∂tt,ξ+∫ℤg1t,ξ,u¯t,ξ,z∂N˜1∂tt,ξ,dz,u¯0,ξ=xξ,ξ∈O,N1u¯t,ξ=0,t≥0,ξ∈∂O,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(3)limϵ⟶0ℙsupt∈0,Tuϵt−u¯tL2O>η=0,where u¯ is the solution of the averaged equation (2).The theory of the averaging principle, originated by Laplace and Lagrange, has been applied in celestial mechanics, oscillation theory, radiophysics, and other fields. The 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. Then, 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], Thompson [8], and Xu and his coworkers [9–11]. 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 [15–17], and Li [18] and Gao [19–21] also studied the averaging principles for the slow-fast SPDEs.The 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. Therefore, 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 reaction-diffusion 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 [26–28], can provide an accurate mathematical model to describe discontinuous random processes, some large moves, and unpredictable events [29–31]. 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.The key to using the averaging principle to analyze system (1) is the fast equation with a frozen slow component x∈L2O:(4)∂vx,y∂tt,ξ=A2t−αvx,yt,ξ+b2t,ξ,xξ,vx,yt,ξ+f2t,ξ,xξ,vx,yt,ξ∂ωQ2∂tt,ξ+∫ℤg2t,ξ,xξ,vx,yt,ξ,z∂N˜2ϵ∂tt,ξ,dz,vx,ys,ξ=yξ,ξ∈O,N2vx,yt,ξ=0,t≥s,ξ∈∂O.By dealing with the Poisson terms, we prove that an evolution family of measuresμtx;t∈ℝ on L2O for fast equation (4) exists. Then, assuming that A2t is periodic and b2,f2,andg2 are almost periodic, we prove that the evolution family of measures is almost periodic. Moreover, with the aid of Theorem 2.10 in [32], we prove that the family of functions(5)t∈ℝ⟼∫L2OB1t,x,yμtxdy,is uniformly almost periodic for any x in a compact set K⊂L2O, where B1t,x,yξ=b1t,ξ,xξ,yξ for any x,y∈L2O and ξ∈O.According to the characteristics of almost periodic function ([25], Theorem 3.4), we define the averaged coefficient B¯1 as follows:(6)B¯1x≔limT⟶∞1T∫0T∫L2OB1t,x,yμtxdydt,x∈L2O.Finally, the averaged equation is obtained through the averaged coefficientB¯1. Using the classical Khasminskii method to the present situation, the averaging principle is effective.The abovementioned notations will be given in Section2, and in this paper, c>0 below with or without subscripts will represent a universal constant whose value may vary in different occasions.


## 2. Notations, Assumptions, and Preliminaries

LetO be a bounded domain of ℝdd≥1 having a smooth boundary. In this paper, we denote ℍ the separable Hilbert space L2O, endowed with the usual scalar product,(7)x,yℍ=∫Oxξyξdξ,and with the corresponding norm ⋅ℍ. The norm in L∞O will be denoted by ⋅∞.Furthermore, the subspaceD−Aθ [33–35] of the generator A is dense in ℍ, and endowed with the norm,(8)Λθ=−AθΛℍ,Λ∈D−Aθ,for 0≤θ<1,0<t≤T. According to Theorem 6.13 in [36], there exists cθ>0, such that(9)−AθeAtℍ≤cθt−θ.We denote byBbℍ the Banach space of the bounded Borel functions φ:ℍ⟶ℝ, endowed with the sup-norm(10)φ0≔supx∈ℍ|φx,and Cbℍ is the subspace of the uniformly continuous mappings. Moreover, Ds,T;ℍ denotes the space of all càdlàg path from s,T into ℍ.We shall denote thatℒℍ is the space of the bounded linear operators in ℍ and denote ℒ2ℍ the subspace of Hilbert–Schmidt operators, endowed with the norm(11)Q2=TrQ∗Q.In slow-fast system (1), the Gaussian noises ∂ωQ1/∂tt,ξ and ∂ωQ2/∂tt,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 ωQit,ξi=1,2 is the cylindrical Wiener processes defined as(12)ωQit,ξ=∑k=1∞Qiekξβkt,i=1,2,where ekk∈ℕ is a complete orthonormal basis in ℍ, βktk∈ℕ is a sequence of mutually independent standard Brownian motion defined on the same complete stochastic basis Ω,ℱ,ℱtt≥0,ℙ, and Qi is a bounded linear operator on ℍ.Next, we give the definitions of Poisson random measuresN˜1dt,dz and N˜2ϵdt,dz. Let ℤ,ℬℤ be a given measurable space and vdz be a σ-finite measure on it. Dpti,i=1,2 represents two countable subsets of ℝ+. Moreover, let pt1,t∈Dpt1 be a stationary ℱt-adapted Poisson point process on ℤ with the characteristic v, and pt2,t∈Dpt2 be the other stationary ℱt-adapted Poisson point process on ℤ with the characteristic v/ϵ. Denote by Nidt,dz,i=1,2 the Poisson counting measure associated with pti, i.e.,(13)Nit,Λ≔∑s∈Dpti,s≤tIΛpti,i=1,2.Let us denote the two independent compensated Poisson measures(14)N˜1dt,dz≔N1dt,dz−v1dzdt,N˜2ϵdt,dz≔N2dt,dz−1ϵv2dzdt,where v1dzdt and 1/ϵv2dzdt 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 anyt∈ℝ, the operators A1t and A2t are second-order uniformly elliptic operators, having continuous coefficients on O. The operators N1 and N2 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 Ait has the following form:(15)Ait=γitAi+ℒit,t∈ℝ,i=1,2,where Ai is a second order uniformly elliptic operator [38, 39] with continuous coefficients on O, which is independent of t. In addition, ℒit is a first-order differential operator, which has the form:(16)ℒit,ξuξ=lit,ξ,∇uξℝd,t∈ℝ,ξ∈O.The realizations of the differential operatorsAi and ℒi in ℍ are Ai and Li. Moreover, A1 and A2 generate two analytic semigroups etA1 and etA2, respectively.Now, we give the following assumptions:(A1)
(a)
Fori=1,2, the function γi:ℝ⟶ℝ is continuous, and there exist γ0,γ>0 such that(17)γ0≤γit≤γ,t∈ℝ.(b)
Fori=1,2, the function li:ℝ×O⟶ℝd is continuous and bounded.(A2)
Fori=1,2, there exists a complete orthonormal system ei,kk∈ℕ in ℍ and two sequences of nonnegative real numbers αi,kk∈ℕ and λi,kk∈ℕ such that(18)Aiei,k=−αi,kei,k,Qiei,k=λi,kei,k,k≥1,κi≔∑k=1∞λi,kρiei,k∞2<∞,ζi≔∑k=1∞αi,k−βiei,k∞2<∞,for some constantsρi∈2,+∞ and βi∈0,+∞ such that(19)βiρi−2ρi<1.Remark 1.
For more comments and examples about the assumption (A2) of the operatorsAi and Qi, the reader can read [12].(A3)
The mappingsb1:ℝ×O×ℝ2⟶ℝ,f1:ℝ×O×ℝ⟶ℝ,g1:ℝ×O×ℝ×ℤ⟶ℝ are measurable, and the mappings b1t,ξ,⋅:ℝ2⟶ℝ,f1t,ξ,⋅:ℝ⟶ℝ, g1t,ξ,⋅,z:ℝ⟶ℝ are Lipschitz continuous and linearly growing, uniformly with respect to t,ξ,z∈ℝ×O×ℤ. Moreover, for all p≥1, there exist positive constants c1andc2, such that, for all x1,x2∈ℝ, we have(20)supt,ξ∈ℝ×O∫ℤg1t,ξ,x1,zpυ1dz≤c11+x1p,supt,ξ∈ℝ×O∫ℤg1t,ξ,x1,z−g1t,ξ,x2,zpυ1dz≤c2x1−x2p.(A4)
The mappingsb2:ℝ×O×ℝ2⟶ℝ,f2:ℝ×O×ℝ2⟶ℝ,g2:ℝ×O×ℝ2×ℤ⟶ℝ are measurable, and the mappings b2t,ξ,⋅:ℝ2⟶ℝ,f2t,ξ,⋅:ℝ2⟶ℝ,g2t,ξ,⋅,z:ℝ2⟶ℝ are Lipschitz continuous and linearly growing, uniformly with respect to t,ξ,z∈ℝ×O×ℤ. Moreover, for all q≥1, there exist positive constants c3andc4, such that, for all xi,yi∈ℝ2,i=1,2, we have(21)supt,ξ∈ℝ×O∫ℤg2t,ξ,x1,y1,zqυ2dz≤c31+x1q+y1q,supt,ξ∈ℝ×O∫ℤg2t,ξ,x1,y1,z−g2t,ξ,x2,y2,zqυ2dz≤c4x1−x2q+y1−y2q.Remark 2.
For anyt,ξ∈ℝ×O and x,y,h∈ℍ,z∈ℤ, we shall set(22)B1t,x,yξ≔b1t,ξ,xξ,yξ,B2t,x,yξ≔b2t,ξ,xξ,yξ,F1t,xhξ≔f1t,ξ,xξhξ,F2t,x,yhξ≔f2t,ξ,xξ,yξhξ,G1t,x,zhξ≔g1t,ξ,xξ,zhξ,G2t,x,y,zhξ≔g2t,ξ,xξ,yξ,zhξ.
Due to (A3) and (A4), for any fixedt,z∈ℝ,ℤ, the mappings(23)B1t,⋅:ℍ×ℍ⟶ℍ,B2t,⋅:ℍ×ℍ⟶ℍ,F1t,⋅:ℍ⟶ℒℍ,F2t,⋅:ℍ×ℍ⟶ℒℍ,G1t,⋅,z:ℍ⟶ℒℍ,G2t,⋅,z:ℍ×ℍ⟶ℒℍ,are Lipschitz continuous and have linear growth conditions.
Now, fori=1,2, we define(24)γit,s≔∫stγirdr,s<t,and for any ϵ>0 and β≥0, set(25)Uβ,ϵ,it,s=e1/ϵγit,sAi−β/ϵt−s,s<t.
Forϵ=1, we write Uβ,it,s, and for ϵ=1 and β=0, we write Uit,s.
Next, for anyϵ>0,β≥0 and for any u∈Cs,t;W01,pO, we define(26)ψβ,ϵ,iu;sr=1ϵ∫srUβ,ϵ,ir,ρLiρuρdρ,s<r<t.
In the caseϵ=1, we write ψβ,iu;sr, and in the case ϵ=1 and β=0, we write ψiu;sr.Remark 3.
By using the same argument as Chapter 5 of the book [40], we can get that there exists a unique evolution system Ut,s=eA∫stγrdr0≤s≤t≤T for γtA. Moreover, we also can get that ψβ,ϵ,iu;st is the solution of(27)dut=1ϵAit−βutdt,t>s,us=0.


## 3. A Priori Bounds for the Solution

With all notations introduced above, system (1) can be rewritten in the following abstract form:(28)duϵt=A1tuϵt+B1t,uϵt,vϵtdt+F1t,uϵtdωQ1t+∫ℤG1t,uϵt,zN˜1dt,dz,dvϵt=1ϵA2t−αvϵt+B2t,uϵt,vϵtdt+1ϵF2t,uϵt,vϵtdωQ2t+∫ℤG2t,uϵt,vϵt,zN˜2ϵdt,dz,uϵ0=x,vϵ0=y.According to Remark2, 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 uϵ,vϵ to system (28) in LpΩ;D0,T;ℍ×ℍ. This means that there exist two unique adapted processes uϵ and vϵ in LpΩ;D0,T;ℍ such that(29)uϵt=U1t,0x+ψ1uϵ;0t+∫0tU1t,rB1r,uϵr,vϵrdr+∫0tU1t,rF1r,uϵrdwQ1r+∫0t∫ℤU1t,rG1r,uϵr,zN˜1dr,dz,vϵt=Uα,ϵ,2t,0y+ψα,ϵ,2vϵ;0t+1ϵ∫0tUα,ϵ,2t,rB2r,uϵr,vϵrdr+1ϵ∫0tUα,ϵ,2t,rF2r,uϵr,vϵrdwQ2r+∫0t∫ℤUα,ϵ,2t,rG2r,uϵr,vϵr,zN˜2ϵdr,dz.Lemma 1.
Under (A1)–(A4), for anyp≥1 and T>0, there exists a positive constant cp,T, such that for any x,y∈ℍ and ϵ∈0,1, we have(30)Esupt∈0,Tuϵtℍp≤cp,T1+xℍp+yℍp,(31)∫0TEvϵtℍpdt≤cp,T1+xℍp+yℍp.Proof.
For fixedϵ∈0,1 and x,y∈ℍ, for any t∈0,T, we denote(32)Γ1,ϵt≔∫0tU1t,rF1r,uϵrdwQ1r,Ψ1,ϵt≔∫0t∫ℤU1t,rG1r,uϵr,zN˜1dr,dz.
SettingΛ1,ϵt≔uϵt−Γ1,ϵt−Ψ1,ϵt, we have(33)ddtΛ1,ϵt=γ1tA1Λ1,ϵt+L1tΛ1,ϵt+Γ1,ϵt+Ψ1,ϵtB1t,Λ1,ϵt+Γ1,ϵt+Ψ1,ϵt,vϵt,Λ1,ϵ0=x.
For anyp≥2, because B1⋅ is Lipschitz continuous, using Young’s inequality, we have(34)1pddtΛ1,ϵtℍp=γ1tA1Λ1,ϵt,Λ1,ϵtℍΛ1,ϵtℍp−2+L1tΛ1,ϵt+Γ1,ϵt+Ψ1,ϵt,Λ1,ϵtℍΛ1,ϵtℍp−2+B1t,Λ1,ϵt+Γ1,ϵt+Ψ1,ϵt,vϵt−B1t,Γ1,ϵt+Ψ1,ϵt,vϵt,Λ1,ϵtℍΛ1,ϵtℍp−2+B1t,Γ1,ϵt+Ψ1,ϵt,vϵt−B1t,Ψ1,ϵt,vϵt,Λ1,ϵtℍΛ1,ϵtℍp−2+B1t,Ψ1,ϵt,vϵt,Λ1,ϵtℍΛ1,ϵtℍp−2≤cΛ1,ϵtℍp+cΓ1,ϵtℍΛ1,ϵtℍp−1+cΨ1,ϵtℍΛ1,ϵtℍp−1+cB1t,Ψ1,ϵt,vϵtℍΛ1,ϵtℍp−1≤cpΛ1,ϵtℍp+cp1+Γ1,ϵtℍp+Ψ1,ϵtℍp+vϵtℍp.
This implies that(35)Λ1,ϵtℍp≤ecptxℍp+cp∫0tecpt−r1+Γ1,ϵrℍp+Ψ1,ϵrℍp+vϵrℍpdr.
According to the definition ofΛ1,ϵt, for any t∈0,T, we have(36)uϵtℍp≤cp,T1+xℍp+supr∈0,TΓ1,ϵrℍp+supr∈0,TΨ1,ϵrℍp+cp,T∫0Tvϵrℍpdr.
So,(37)Esupt∈0,Tuϵtℍp≤cp,T1+xℍp+Esupt∈0,TΓ1,ϵtℍp+Esupt∈0,TΨ1,ϵtℍp+cp,T∫0TEvϵrℍpdr.
According to Lemma 4.1 in [12] with θ=0, it is easy to prove that(38)Esupt∈0,T‖Γ1,ϵt‖ℍp≤cp,T∫0T1+E‖uϵr‖ℍpdr.
Due to (A3), using Kunita’s first inequality, we get(39)Esupt∈0,TΨ1,ϵtℍp≤cpE∫0T∫ℤeγ1t,rA1G1r,uϵr,zℍ2v1dzdrp/2+cpE∫0T∫ℤeγ1t,rA1G1r,uϵr,zℍpv1dzdr≤cpE∫0T1+uϵrℍ2drp/2+cpE∫0T1+uϵrℍpdr≤cp,T∫0T1+Euϵrℍpdr,(40)Esupt∈0,TΨ1,ϵtℍp≤cp,T∫0T1+Euϵrℍpdr.
Substituting (38) and (39) into (37), we obtain(41)Esupt∈0,Tuϵtℍp≤cp,T1+xℍp+∫0TEvϵrℍpdr+cp,T∫0TEsupσ∈0,ruϵσℍpdr.
Now, we have to estimate(42)∫0TEvϵrℍpdr.
For anyt∈0,T, we set(43)Γ2,ϵt≔1ϵ∫0tUα,ϵ,2t,rF2r,uϵr,vϵrdwQ2r,Ψ2,ϵt≔∫0t∫ℤUα,ϵ,2t,rG2r,uϵr,vϵr,zN˜2ϵdr,dz.
LetΛ2,ϵt≔vϵt−Γ2,ϵt−Ψ2,ϵt, we have(44)ddtΛ2,ϵt=1ϵγ2tA2−αΛ2,ϵt+1ϵL2tΛ2,ϵt+Γ2,ϵt+Ψ2,ϵt+1ϵB2t,uϵt,Λ2,ϵt+Γ2,ϵt+Ψ2,ϵt,Λ2,ϵ0=y.
For anyp≥2, because α>0 is large enough, by proceeding as in (34), we can get(45)1pddtΛ2,ϵtℍp≤−α2ϵΛ2,ϵtℍp+cpϵ1+uϵtℍp+Γ2,ϵtℍp+Ψ2,ϵtℍp.
According to the Gronwall inequality, we have(46)Λ2,ϵtℍp≤cpϵ∫0te−αp/2ϵt−r1+uϵrℍp+Γ2,ϵrℍp+Ψ2,ϵrℍpdr+e−αp/2ϵtyℍp.
According to the definition ofΛ2,ϵt, for any t∈0,T, we have(47)Evϵtℍp≤cpEΓ2,ϵtℍp+cpEΨ2,ϵtℍp+cpe−αp/2ϵtyℍp+cpϵ∫0te−αp/2ϵt−r1+Euϵrℍp+EΓ2,ϵrℍp+EΨ2,ϵrℍpdr.
Therefore, by integrating with respect tot, using Young’s inequality, we obtain(48)∫0tEvϵrℍpdr≤cptyℍp+cp∫0tEΓ2,ϵrℍpdr+∫0tEΨ2,ϵrℍpdr+cpϵ∫0t1+Euϵrℍp+EΓ2,ϵrℍp+EΨ2,ϵrℍpdr×∫0te−αp/2ϵrdr≤cpt1+yℍp+cp∫0tEuϵrℍpdr+∫0tEΓ2,ϵrℍpdr+∫0tEΨ2,ϵrℍpdr.
According to the Burkholder–Davis–Gundy inequality, by proceeding as Proposition 4.2 in [12], we can easily get(49)∫0tEΓ2,ϵrℍpdr≤cpt∫0t1+Euϵrℍp+Evϵrℍpdr.
Concerning the stochastic termΨ2,ϵt, using Kunita’s first inequality, we have(50)EΨ2,ϵtℍp≤cpE1ϵ∫0t∫ℤe−α/ϵt−reγ2t,r/ϵA2G2r,uϵr,vϵr,zℍ2v2dzdrp/2+cpϵE∫0t∫ℤe−α/ϵt−reγ2t,r/ϵA2G2r,uϵr,vϵr,zℍpv2dzdr≤cpϵp/2E∫0te−αp−2ϵt−r1+uϵrℍp+vϵrℍpdr∫0te−αp/ϵp−2t−rdrp−2/2+cpϵE∫0te−αp/ϵt−r1+uϵrℍp+vϵrℍpdr≤cpϵ∫0te−αp/2ϵt−r1+Euϵrℍp+Evϵrℍpdr.
By integrating with respect tot both sides and using Young’s inequality, we have(51)∫0tEΨ2,ϵrℍpdr≤cpϵ∫0te−αp/2ϵrdr⋅∫0t1+Euϵrℍp+Evϵrℍpdr≤cpt∫0t1+Euϵrℍp+Evϵrℍpdr.
Substituting (49) and (51) into (48), we get(52)∫0tEvϵrℍpdr≤cpt1+yℍp+∫0tEuϵrℍpdr+cp,Tt∫0tEvϵrℍpdr.
Ascp0=0 and cpt is a continuous increasing function, we can fix t0>0, such that for any t≤t0, we have cpt≤1/2; so,(53)∫0tEvϵrℍpdr≤cpt1+yℍp+Esupr∈0,tuϵrℍp,t∈0,t0.
Due to (41) and (53), for any t∈0,t0, we can get(54)Esupr∈0,tuϵrℍp≤cp,T1+xℍp+yℍp+cp,TtEsupr∈0,tuϵrℍp+cp,T∫0tEsupσ∈0,ruϵσℍpdr.
Similarly, we also can fix0<t1≤t0, such that for any t≤t1, we have cp,Tt≤1/2; so,(55)Esupr∈0,tuϵrℍp≤cp,T1+xℍp+yℍp+cp,T∫0tEsupσ∈0,ruϵσℍpdr,t∈0,t1.
According to the Gronwall inequality, we get(56)Esupr∈0,tuϵrℍp≤cp,T1+xℍp+yℍp,t∈0,t1.
Substituting (56) into (53), it yields(57)∫0tEvϵrℍpdr≤cp,T1+xℍp+yℍp,t∈0,t1.
For anyp≥2, by repeating this in the intervals t1,2t1,2t1,3t1 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 2.
Under (A1)–(A4), there existsθ¯>0, such that for any T>0,p≥1,x∈D−A1θ with θ∈0,θ¯ and y∈ℍ, there exist a positive constant cp,θ,T>0 such that(58)supϵ∈0,1Esupt∈0,Tuϵtθp≤cp,θ,T1+xθp+yℍp.Proof.
Assuming thatx∈D−A1θθ≥0, for any t∈0,T, we have(59)uϵt=U1t,0x+ψ1uϵ;0t+∫0tU1t,rB1r,uϵr,vϵrdr+∫0tU1t,rF1r,uϵrdwQ1r+∫0t∫ℤU1t,rG1r,uϵr,zN˜1dr,dz.
Concerning the second termψ1uϵ;0t, we get(60)ψ1uϵ;0tθp≤cp∫0t−A1θeγ1t,rA1L1ruϵrdrℍp≤cp,θ∫0tt−r−θL1ruϵrℍdrp≤cp,θsupr∈0,Tuϵrℍp∫0tt−r−θdrp≤cp,θ,T1+xℍp+yℍp.
For anyp≥2, according to the proof of Proposition 4.3 in [12], and thanks to (30) and (31), it is possible to show that there exists a θ˜≥0, such that for any θ≤θ˜∧1/2, we have(61)Esupt∈0,T∫0tU1t,rB1r,uϵr,vϵrdrθp≤cp,θ,T1+xℍp+yℍp.(62)Esupt∈0,TΓ1,ϵtθp≤cp,θ,T1+xℍp+yℍp.
Concerning the stochastic termΨ1,ϵt, using the factorization argument, we have(63)Ψ1,ϵt=cθ∫0tt−rθ−1eγ1t,rA1ϕϵ,θrdr,where(64)ϕϵ,θr=∫0r∫ℤr−σ−θeγ1r,σA1G1σ,uϵσ,zN˜1dσ,dz.
Next, for anyp≥2, let θ^=1/4−1/2p∧1/2p, for any θ≤θ^, according to (A3) and Lemma 1, using Kunita’s first inequality and the Hölder inequality, we get(65)Ψ1,ϵtθp≤cθ∫0tt−rθ−1ϕϵ,θrθdrp≤cθsupr∈0,tϕϵ,θrθp∫0tt−rθ−1drp≤cp,θ,Tsupr∈0,t∫0r∫ℤr−σ−θ−A1θeγ1r,σA1G1σ,uϵσ,zN˜1dσ,dzℍp≤cp,θ,Tsupr∈0,t∫0r∫ℤr−σ−2θr−σ−2θG1σ,uϵσ,zℍ2v1dzdσp/2+cp,θ,Tsupr∈0,t∫0r∫ℤr−σ−pθr−σ−pθG1σ,uϵσ,zℍpv1dzdσ≤cp,θ,Tsupr∈0,t∫0r∫ℤG1σ,uϵσ,zℍ2v1dzp/2dσ×∫0rr−σ−4pθ/p−2dσp−2/2+cp,θ,Tsupr∈0,t∫0rr−σ−2pθdσ⋅supσ∈0,r∫ℤG1σ,uϵσ,zℍpv1dz≤cp,θ,T1+supσ∈0,Tuϵσℍp.
So, due to (30), we get(66)Esupt∈0,TΨ1,ϵtθp≤cp,θ,T1+xℍp+yℍp.
Hence, if we chooseθ¯≔1/8∧θ˜∧θ^, thanks to (60)–(62) and (66), for any p≥2 and θ<θ¯, we get(67)Esupt∈0,Tuϵtθp≤cp,θ,T1+xθp+yℍp.
Using the Hölder inequality, we can estimate (58) for p=1.Lemma 3.
Under (A1)–(A4), for anyθ∈0,θ¯ and 0≤h≤1, there exists βθ>0, such that, for any T>0,p≥1,x∈D−A1θ,y∈ℍ, and t∈0,T, it holds(68)supϵ∈0,1Euϵt−uϵt+hℍp≤cp,θ,Thβθp+h1+xθp+yℍp.Proof.
For anyt≥0 with t,t+h∈0,T, we have(69)uϵt+h−uϵt=U1t+h,t−Iuϵt+ψ1uϵ;tt+h+∫tt+hU1t+h,rB1r,uϵr,vϵrdr+∫tt+hU1t+h,rF1r,uϵrdwQ1r+∫tt+h∫ℤU1t+h,rG1r,uϵr,zN˜1dr,dz≔∑i=15ℐti.
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(70)Eℐt1ℍp≤cp,θ,Thθp1+xθp+yℍp,Eℐt2ℍp≤cp,Thp−11+xθp+yℍp,Eℐt3ℍp≤cp,Thp1+xθp+yℍp,Eℐt4ℍp≤cp,Thp−2/2−β1ρ1−2/ρ1p/21+xθp+yℍp.
According to the proof of (39), using the Hölder inequality and (30), we have(71)Eℐt5ℍp≤cpE∫tt+h1+uϵrℍ2drp/2+cpE∫tt+h1+uϵrℍpdr,≤cphp−2/2+1∫tt+h1+Euϵrℍpdr≤cp,Thp2+h1+xℍp+yℍp.
Then, if we takep¯>1, such that(72)β1ρ1−2ρ1p¯p¯−2<1.
We can get(73)Euϵt+h−uϵtℍp≤cp,Th1−1/p¯p+hp+h1/2−1/p¯−β1ρ1−2/2ρ1p+hp/2+h×1+xℍp+yℍp+cp,θ,Thθp1+xθp+yℍp.
As we are assumingh≤1, (68) follows for any p≥p¯ by taking(74)βθ≔minθ,1−1p¯,1,12−1p¯−β1ρ1−22ρ1,12.
From the Hölder inequality, we can estimate (68) for p<p¯,(75)Euϵt+h−uϵtℍp≤Euϵt+h−uϵtℍp¯p/p¯.
So, we have (68).
According to the above lemma, we get that, for everyϵ∈0,1, the function uϵt is uniformly bounded about t∈0,T, and it is also equicontinuous at every point of t∈0,T. In view of Theorem 12.3 in [43], we can infer that the set uϵϵ∈0,1 is relatively compact in D0,T;ℍ. In addition, according to Theorem 13.2 in [43] and the above lemma, by using Chebyshev’s inequality, we also can get that the family of probability measures ℒuϵϵ∈0,1 is tight in PD0,T;ℍ.


## 4. An Evolution Family of Measures for the Fast Equation

For any frozen slow componentx∈ℍ, any initial condition y∈ℍ, and any s∈ℝ, we introduce the following problem:(76)dvt=A2t−αvt+B2t,x,vtdt+F2t,x,vtdω¯Q2t+∫ℤG2t,x,vt,zN˜2′dt,dz,vs=y,where(77)w¯Q2t=w1Q2t,w2Q2−t,ift≥0,ift<0,N˜2′t,z=N˜1′t,z,N˜3′−t,z,ift≥0,ift<0,for two independent Q2-Wiener processes w1Q2t and w2Q2t 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.According to the definition of the operatorψα,2⋅;s, we know that the mapping ψα,2⋅;s:Cs,T;ℍ⟶Cs,T;ℍ is a linearly bounded operator, and it is Lipschitz continuous. Hence, we have that, for any x,y∈ℍ,p≥1 and s<T, there exists a unique mild solution vx⋅;s,y∈LpΩ;D0,T;ℍ (see [28]) in the following form:(78)vxt;s,y=Uα,2t,sy+ψα,2vx⋅;s,y;st+∫stUα,2t,rB2r,x,vxr;s,ydr+∫stUα,2t,rF2r,x,vxr;s,ydw¯Q2r+∫st∫ℤUα,2t,rG2r,x,vxr;s,y,zN˜2′dr,dz.Moreover, if the spaceDℝ;ℍ endowed with the topology of uniform convergence on bounded intervals, an ℱtt∈ℝ-adapted process vx∈LpΩ;D0,T;ℍ is a mild solution of the equation:(79)dvt=A2t−αvt+B2t,x,vtdt+F2t,x,vtdω¯Q2t+∫ℤG2t,x,vt,zN˜2′dt,dz,where t∈ℝ. Then, for every s<t, we have(80)vxt=Uα,2t,svxs+ψα,2vx;st+∫stUα,2t,rB2r,x,vxrdr+∫stUα,2t,rF2r,x,vxrdw¯Q2r+∫st∫ℤUα,2t,rG2r,x,vxr,zN˜2′dr,dz.In the following, for anyx∈ℍ and any adapted process v, we set(81)Γαv;st≔∫stUα,2t,rF2r,x,vrdw¯Q2r,t>s,Ψαv;st≔∫st∫ℤUα,2t,rG2r,x,vr,zN˜2′dr,dz,t>s.For any0<δ<α and any v1andv2 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(82)supr∈s,teδpr−sEΓαv1;sr−Γαv2;srℍp≤cp,1Lf2pα−δcp,2×supr∈s,teδpr−sEv1r−v2rℍp,where Lf2 is the Lipschitz constant of f2 and cp,1andcp,2 are two suitable positive constants independent of α>0 and s<t.For the stochastic termΨαv;st, using Kunita’s first inequality ([26], Theorem 4.4.23), we get(83)EΨαv1;st−Ψαv2;stℍp≤cpE∫st∫ℤe−αt−reγ2t,rA2G2r,x,v1r,z−G2r,x,v2r,zℍ2v2′dzdrp/2+cpE∫st∫ℤe−αt−reγ2t,rA2G2r,x,v1r,z−G2r,x,v2r,zℍpv2′dzdr≤cpE∫st∫ℤe−2αt−rG2r,x,v1r,z−G2r,x,v2r,zℍ2v2′dzdrp/2+cpE∫st∫ℤe−αpt−rG2r,x,v1r,z−G2r,x,v2r,zℍpv2′dzdr≤cpLg2p∫ste−2α−δt−re−2δt−se2δr−sEv1r−v2rℍ2drp/2+cpLg2p∫ste−pα−δt−re−δpt−seδpr−sEv1r−v2rℍpdr≤cpLg2p∫ste−2α−δt−rdrp/2+∫ste−pα−δt−rdr×e−δpt−ssupr∈s,teδpr−sEv1r−v2rℍp≤cp,1Lg2pα−δcp,2e−δpt−ssupr∈s,teδpr−sEv1r−v2rℍp.so(84)supr∈s,teδpr−sEΨαv1;sr−Ψαv2;srℍp≤cp,1Lg2pα−δcp,2×supr∈s,teδpr−sEv1r−v2rℍp,where Lg2 is the Lipschitz constant of g2 and cp,1andcp,2 are two suitable positive constants independent of α>0 and s<t.Moreover, using (A4), we can show that(85)supr∈s,teδpr−sEΓαv;srℍp≤cp,1Mf2pα−δcp,2supr∈s,teδpr−s1+Evrℍp,(86)supr∈s,teδpr−sEΨαv;srℍp≤cp,1Mg2pα−δcp,2supr∈s,teδpr−s1+Evrℍp,where Mf2andMg2 are the linear growth constants of f2andg2 and cp,1andcp,2 are two suitable positive constants independent of α>0 and s<t.For any fixed adapted processv, let us introduce the problem:(87)dρt=A2t−αρtdt+F2t,x,vtdω¯Q2t+∫ℤG2t,x,vt,zN˜2′dt,dz,ρs=0.We denote that its unique mild solution isραv;s. This means that ραv;s solves the equation:(88)ραv;st=ψα,2ραv;s;st+Γαv;st+Ψαv;st,s<t<T.Due to (82) and (84), using the same argument as (5.8) in [25], it is easy to prove that for any process v1,v2, and 0<δ<α, we have(89)supr∈s,teδpr−sEραv1;sr−ραv2;srℍp≤cp,1Lpα−δcp,2×supr∈s,teδpr−sEv1r−v2rℍp,where, L=maxLb2,Lf2,Lg2.Similarly, thanks to (85) and (86), for any process v and 0<δ<α, we can prove that(90)supr∈s,teδpr−sEραv;srℍp≤cp,1Mpα−δcp,2supr∈s,teδpr−sE1+vrℍp,where M=maxMb2,Mf2,Mg2.Lemma 4.
Under (A1)–(A4), there existsδ>0, such that, for any x,y∈ℍ and p≥1,(91)Evxt;s,yℍp≤cp1+xℍp+e−δpt−syℍp,s<t.Proof.
We setΛαt≔vxt;s,y−ραt, where ραt=ραvx⋅;s,y;st is the solution of problem (87) with v=vx⋅;s,y. Using Young’s inequality, we have(92)1pddtΛαtℍp≤A2t−αΛαt,ΛαtℍΛαtℍp−2+B2t,x,Λαt+ραt−B2t,x,ραt,ΛαtℍΛαtℍp−2+B2t,x,ραt,ΛαtℍΛαtℍp−2≤−αΛαtℍp+cΛαtℍp+c1+xℍ+ραtℍΛαtℍp−1≤−αΛαtℍp+cpΛαtℍp+cp1+xℍp+ραtℍp.
Becauseα is large enough, we can find η=α−cp>0, such that(93)ddtΛαtℍp≤−ηpΛαtℍp+cp1+xℍp+ραtℍp.
According to the Gronwall inequality, we have(94)Λαtℍp≤e−ηpt−syℍp+cp1+xℍp+cp∫ste−ηpt−rραrℍpdr.
So, for anyp≥1,(95)vxt;s,yℍp≤cpραtℍp+cpe−ηpt−syℍp+cp1+xℍp+cp∫ste−ηpt−rραrℍpdr.
Fixing0<δ<η, according to (90), we get(96)eδpt−sEvxt;s,yℍp≤cpeδpt−sEραtℍp+cpepδ−ηt−syℍp+cpeδpt−s1+xℍp+cp∫steδpr−sEραrℍpdr≤cp,1Mpα−δcp,2supr∈s,teδpr−s1+Evxr;s,yℍp+cpepδ−ηt−syℍp+cpeδpt−s1+xℍp+cp,1Mpα−δcp,2∫stsupr∈s,teδpr−s1+Evxr;s,yℍpdr.
Takingα1=2cp,1Mp1/cp,2+δ, when α≥α1, we have(97)supr∈s,teδpr−sEvxr;s,yℍp≤cpyℍp+cpeδpt−s1+xℍp+∫stsupr∈s,teδpr−sEvxr;s,yℍpdr.
Due to the Gronwall lemma, we have(98)supr∈s,teδpr−sEvxr;s,yℍp≤cpyℍp+cpeδpt−s1+xℍp.
Hence, we get (91).Lemma 5.
Under (A1)–(A4), for anyt∈ℝ and x,y∈ℍ, for all p≥1, there exists ηxt∈LpΩ;ℍ such that(99)lims⟶−∞Evxt;s,y−ηxtℍp=0.Moreover, for anyp≥1, there exists some δp>0, such that(100)Evxt;s,y−ηxtℍp≤cpe−δpt−s1+xℍp+yℍp.Finally,ηx is a mild solution in ℝ of (79).Proof.
Fixh>0 and define(101)ρt=vxt;s,y−vxt;s−h,y,s<t.
We know thatρt is the unique mild solution of the problem:(102)dρt=A2t−αρt+B2t,x,vxt;s,y−B2t,x,vxt;s−h,ydt+F2t,x,vxt;s,y−F2t,x,vxt;s−h,ydw¯Q2t+∫ℤG2t,x,vxt;s,y−G2t,x,vxt;s−h,yN˜2′dt,dz,ρs=y−vxs;s−h,y,ρt=Uα,2t,sy−vxs;s−h,y+ψα,2ρ;st+∫stUα,2t,rB2r,x,vxr;s,y−B2r,x,vxr;s−h,ydr+∫stUα,2t,rF2r,x,vxr;s,y−F2r,x,vxr;s−h,ydw¯Q2r+∫st∫ℤUα,2t,rG2r,x,vxr;s,y−G2r,x,vxr;s−h,yN˜2′dr,dz.
Multiply both sides of the above equation byeδpt−s. Because α large enough, according to Lemma 2.4 in [25], we have(103)eδpt−sEρtℍp≤cpeδ−αpt−sEeγ2t,sA2y−vxs;s−h,yℍp+cpE∫steγ2t,rA2eδ−αt−reδr−sB2r,x,vxr;s,y−B2r,x,vxr;s−h,ydrℍp+cpE∫steγ2t,rA2eδ−αt−reδr−sF2r,x,vxr;s,y−F2r,x,vxr;s−h,ydw¯Q2rℍp+cpE∫st∫ℤeγ2t,rA2eδ−αt−reδr−sG2r,x,vxr;s,y−G2r,x,vxr;s−h,yN˜2′dr,dzℍp≔∑i=14ℐti.
According to Lemma 3.1 in [12], we know that, for any J∈ℒL∞D,ℍ∩ℒℍ,L1D with J=J∗ and s≥0, we have(104)esAiJQi22≤Kis−βiρi−2/ρie−αρi+2/ρisJℒL∞D,ℍ2,where(105)Ki=βieβiρi−2/ρiζiρi−2/ρiκi2/ρi.
Takingp¯>1, such that p¯β2ρ2−2/ρ2p¯−2<1. Then, using the Burkholder–Davis–Gundy inequality and Kunita’s first inequality, we can get that, for any p≥p¯ and 0<δ<α, it yields(106)ℐt2≤cpLb2psupr∈s,teδpr−sEρrℍp⋅∫steδ−αt−rdrp≤cpLb2pα−δpsupr∈s,teδpr−sEρrℍp,(107)ℐt3≤cpE∫steγ2t,rA2eδ−αt−reδr−sF2r,x,vxr;s,y−F2r,x,vxr;s−h,yQ222drp/2≤cpLf2pK2p/2∫ste2δ−αt−rγ2t,r−β2ρ2−2/ρ2e−αρ2+2/ρ2γ2t,rdrp/2×supr∈s,teδpr−sEρrℍp≤cpLf2psupr∈s,teδpr−sEρrℍp⋅∫ste2δ−αt−rγ0t−r−β2ρ2−2/ρ2drp/2≤cpLf2psupr∈s,teδpr−sEρrℍp⋅∫0t−sr−β2ρ2−2/ρ2p/p−2drp−2/2∫0t−se−pα−δrdr≤cpLf2pα−δsupr∈s,teδpr−sEρrℍp,ℐt4≤cpE∫st∫ℤeγ2t,rA2eδ−αt−reδr−sG2r,x,vxr;s,y−G2r,x,vxr;s−h,yℍ2v2′dzdrp/2.+cpE∫st∫ℤeγ2t,rA2eδ−αt−reδr−sG2r,x,vxr;s,y−G2r,x,vxr;s−h,yℍpv2′dzdr≤cpLg2psupr∈s,teδpr−sEρrℍp⋅∫0t−se−2α−δrdrp/2+∫0t−se−pα−δrdr≤cpLg2pα−δcpsupr∈s,teδpr−sEρrℍp.
Hence, we have(108)supr∈s,teδpr−sEρrℍp≤cpy−vxs;s−h,yℍp+cp,1Lpα−δcp,2supr∈s,teδpr−sEρrℍp.
Therefore, forα>0 large enough, we can find 0<δ¯p<α, such that(109)cp,1Lα−δ¯pcp,2<1.
This implies that(110)supr∈s,tepδ¯pr−sEρrℍp≤cpy−vxs;s−h,yℍp.
Letδp=pδ¯p, thanks to Lemma 4, we have(111)Evxt;s,y−vxt;s−h,yℍp≤cpe−δpt−sy−vxs;s−h,yℍp≤cpe−δpt−s1+xℍp+yℍp+e−δphyℍp.
BecauseLpΩ;ℍ has completeness, for any p≥p¯, let s⟶−∞ in (111), there exists ηxt∈LpΩ;ℍ such that (99) holds. Then, if we let h⟶∞ in (111), we obtain (100). Using the Hölder inequality, we can get (99), and (100) holds for any p<p¯.
If we takey1,y2∈ℍ, use the same arguments for vxt;s,y1−vxt;s,y2,s<t, we have(112)Evxt;s,y1−vxt;s,y2ℍp≤cpe−δpt−sy1−y2ℍp,s<t.
Lets⟶−∞, this means that the limit ηxt does not depend on the initial condition y∈ℍ.
Finally, we prove thatηxt is a mild solution of (79). Due to the limit ηxt does not depend on the initial condition, we can let initial condition y=0. For any s<t and h>0, we have(113)vxt;s−h,0=Uα,2t,svxs;s−h,0+ψα,2vx⋅;s−h,0;st+∫stUα,2t,rB2r,x,vxr;s−h,0dr+∫stUα,2t,rF2r,x,vxr;s−h,0dw¯Q2r+∫st∫ℤUα,2t,rG2r,x,vxr;s−h,0,zN˜2′dr,dz.
Leth⟶∞ on both sides, due to (99), we can get, for any s<t, have(114)ηxt=Uα,2t,sηxs+ψα,2ηx;st+∫stUα,2t,rB2r,x,ηxrdr+∫stUα,2t,rF2r,x,ηxrdw¯Q2r+∫st∫ℤUα,2t,rG2r,x,ηxr,zN˜2′dr,dz.
This means thatηxt is a mild solution of (79).
For anyt∈ℝ and x∈ℍ, we denote that the law of the random variable ηxt is μtx, and we introduce the transition evolution operator:(115)Ps,txφy=Eφvxt;s,y,s<t,y∈ℍ,where φ∈ℬbℍ.
Due to (91) and (99), for any p≥1, we have(116)supt∈ℝEηxtℍp≤cp1+xℍp,x∈ℍ,so that(117)supt∈ℝ∫ℍyℍpμtxdy≤cp1+xℍp,x∈ℍ.
According to the above conclusion, by using the same arguments as ([25], Proposition 5.3), we know that the family μtxt∈ℝ defines an evolution system of probability measures on ℍ for equation (76). This means that, for any t∈ℝ, μtx is a probability measure on ℍ, and it holds that(118)∫ℍPs,txφyμsxdy=∫ℍφyμtxdy,s<t,for every φ∈Cbℍ. Moreover, we also have(119)Ps,txφy−∫ℍφyμtxdy≤ce−δ1t−s1+xℍ.Lemma 6.
Under (A1)–(A4), the family of measures(120)Λ≔μtx:t∈ℝ,x∈ℍis tight in PD0,T;ℍ.Proof.
According to the definition ofvx⋅;s,0, for any t>s and h>0, we have(121)vxt+h;s,0−vxt;s,0=Uα,2t+h,t−Ivxt;s,0+ψα,2vx⋅;s,0;tt+h+∫tt+hUα,2t+h,rB2r,x,vxr;s,0dr+∫tt+hUα,2t+h,rF2r,x,vxr;s,0dw¯Q2r+∫tt+h∫ℤUα,2t+h,rG2r,x,vxr;s,0,zN˜2′dr,dz.
Due to the assumption (A4), we know that the mappingsB2,F2,andG2 are linearly growing. For any t>s,0<h<1, and x∈ℍ, analogous to the proof of Lemma 3 and using the estimate (91), we can get(122)Evxt+h;s,0−vxt;s,0ℍp≤cphκp+h1+xℍp,where κp is a function of p and satisfies κp>0.
From Lemma5, we know that the limit ηxt does not depend on the initial condition, and we can get(123)Eηxt+h−ηxtℍp≤Eηxt+h−vxt+h;s,0ℍp+Evxt+h;s,0−vxt;s,0ℍp+Evxt;s,0−ηxtℍp.
Then, thanks to (100) and (122), let s⟶∞, we have(124)Eηxt+h−ηxtℍp≤cphκp+h1+xℍp.
Moreover, we have proved that the familyμtxt∈ℝ defines an evolution system of probability measures for (76). By using Chebyshev’s inequality and (116), it yields(125)limn⟶∞μtxηxt:ηxtℍ≥n≤limn⟶∞Eηxtℍ2n2≤limn⟶∞c21+xℍ2n2=0.
In addition, for anyt>s,h>0, and fixed ϵ>0, according to the (124), we have(126)limh⟶0μtxηxt:ηxt+h−ηxtℍ≥ϵ≤limh⟶0Eηxt+h−ηxtℍ2ϵ2≤limh⟶0c2hκ2+h1+xℍ2ϵ2=0.
According to Theorem 13.2 in [43], (125) and (126) imply that the family of measures(127)Λ≔μtx:t∈ℝ,x∈ℍ,is tight in PD0,T;ℍ.
In order to get the averaged equation, we must ensure the existence of the averaged coefficientB¯1. So, we need the evolution family of measures satisfying some nice properties. We give the following assumption:(A5)
(a)
The functionsγ2:ℝ⟶0,∞ and l2:ℝ×O⟶ℝd are periodic, with the same period.(b)
The families of functions(128)B1,R≔b1⋅,ξ,σ:ξ∈O,σ∈Bℝ2R,B2,R≔b2⋅,ξ,σ:ξ∈O,σ∈Bℝ2R,FR≔f2⋅,ξ,σ:ξ∈O,σ∈Bℝ2R,GR≔g2⋅,ξ,σ,z:ξ∈O,σ∈Bℝ2R,z∈ℤare uniformly almost periodic for anyR>0.Remark 4.
Similar with the proof of Lemma 6.2 in [25], we get that, under (A5), for any R>0, the families of functions(129)B1⋅,x,y:x,y∈Bℍ×ℍR,B2⋅,x,y:x,y∈Bℍ×ℍR,F2⋅,x,y:x,y∈Bℍ×ℍR,G2⋅,x,y,z:x,y,z∈Bℍ×ℍR×ℤ,are uniformly almost periodic.
As we know above,A2⋅ is periodic, Remark 4 holds, and the family of measures Λ is tight. By proceeding as [45], we can prove that the mapping(130)t⟼μtx,t∈ℝ,x∈ℍ,is almost periodic.


## 5. The Averaged Equation

Lemma 7.
Under (A1)–(A5), for any compact setK⊂ℍ, the family of functions(131)t∈ℝ⟼∫ℍB1t,x,yμtxdy,x∈K,is uniformly almost periodic.Proof.
AsK is a compact set in ℍ, so it is bounded and there exist some R>0 such that K⊂BℍR. That is, for any x∈K, we have x≤R.
Now, let us define(132)Φt,x=∫ℍB1t,x,yμtxdy,t,x∈ℝ×ℍ.
Then, for anyt,τ∈ℝ,n∈ℕ and x∈K, according to assumption (A3) and equation (117), we can get(133)Φt+τ,x−Φt,x≤∫yℍ≤nB1t+τ,x,yμt+τxdy−∫yℍ≤nB1t+τ,x,yμtxdy+∫yℍ>nB1t+τ,x,yμt+τxdy+∫‖y‖ℍ>nB1t+τ,x,yμtxdy+∫ℍB1t+τ,x,yμtxdy−∫ℍB1t,x,yμtxdy≤supx∈K,yℍ≤nB1t+τ,x,yℍ∫ℍμt+τx−μtxdy+c1+xn+∫ℍB1t+τ,x,y−B1t,x,yℍμtxdy.
Fixed somen¯ such that c1+x/n¯≤ϵ/3. Then, for any t∈ℝ, due to the mapping t⟼μtx and the families of functions B1⋅,x,y are almost periodic, we can find some τ∈ℝ such that(134)Φt+τ,x−Φt,x<ϵ.
So, the functionΦ⋅,x is almost periodic for any x∈K.
By preceding as in Lemma5, we can get that, under (A1)–(A4), for any fixed x1,x2∈K and p≥1, there exists cp>0 such that(135)sups<tEvx1t;s,0−vx2t;s,0ℍp≤cpx1−x2ℍp.
According to (99) and (100), let s⟶0, it yields(136)supt∈ℝEηx1t−ηx2tℍp≤cpx1−x2ℍp.
Hence, thanks to (A3) and (136), we can conclude that, for any x1,x2∈K, we have(137)Φt,x1−Φt,x2ℍ≤EB1t,x1,ηx1t−B1t,x2,ηx2tℍ≤cx1−x2ℍ+Eηx1t−ηx2tℍ21/2≤cx1−x2ℍ.
This means that the family of functionsΦt,⋅:t∈ℝ is equicontinuous about x.
In view of Theorem 2.10 in [32], the above conclusions indicate that Φ⋅,x:x∈K is uniformly almost periodic. Hence, the proof is complete.
According to Theorem 3.4 in [25], we define(138)B¯1x≔limT⟶∞1T∫0T∫ℍB1t,x,yμtxdydt,x∈ℍ.
And, thanks to (A3) and (117), we have that(139)B¯1xℍ≤c1+xℍ.Lemma 8.
Under (A1)–(A5), for anyT>0,s,σ∈ℝ, and x,y∈ℍ,(140)E1T∫ss+TB1t,x,vxt;s,ydt−B¯1xℍ≤cT1+xℍ+yℍ+αT,x,for some mapping α:0,∞×ℍ⟶0,∞ such that(141)supT>0αT,x≤c1+xℍ,x∈ℍ.And, for any compact setK⊂ℍ, we have(142)limT⟶∞supx∈KαT,x=0.Proof.
We denote(143)ψxB1t,y≔B1t,x,y−∫ℍB1t,x,wμtxdw.
So,(144)E1T∫ss+TB1t,x,vxt;s,y−∫ℍB1t,x,wμtxdwdt2=2T2∫ss+T∫rs+TEψxB1r,vxr;s,yψxB1t,vxt;s,ydtdr=2T2∫ss+T∫rs+TEψxB1r,vxr;s,yPr,txψxB1r,vxr;s,ydtdr≤2T2∫ss+T∫rs+TEψxB1r,vxr;s,y21/2EPr,txψxB1r,vxr;s,y21/2dtdr.
Due to (A3), (91), and (117), we have(145)EψxB1r,vxr;s,y2≤cEB1r,x,vxr;s,yℍ2+cE∫ℍB1r,x,wℍμrxdw2≤c1+xℍ2+Evxr;s,yℍ2≤c1+xℍ2+e−2δr−syℍ2,and according to (100), we get(146)EPr,txψxB1r,vxr;s,y2=EPr,txB1r,x,vxr;s,y−∫ℍB1r,x,wμrxdwℍ2=EB1t,x,vxt;s,y−B1t,x,ηxtℍ2≤cEvxt;s,y−ηxtℍ2≤ce−δ2t−s1+xℍ2+yℍ2.
Letδ=δ2/2, it follows(147)E1T∫ss+TB1t,x,vxt;s,y−∫ℍB1t,x,wμtxdwdtℍ≤cT1+xℍ+yℍ∫ss+T∫rs+Te−δt−sdtdr1/2≤cT1+xℍ+yℍ.
Since the family of functions (131) is uniformly almost periodic, according to Theorem 3.4 in [25], we can get that the limit(148)limT⟶∞1T∫ss+T∫ℍB1t,x,wμtxdwdt,converges to B¯1x uniformly with respect to s∈ℝ and x∈K (K is a compact set in ℍ). Therefore, if we define(149)αT,x=1T∫ss+T∫ℍB1t,x,wμtxdwdt−B¯1xℍ,we get the conclusion (142). Moreover, thanks to the assumption (A3), equations (117), and (139), we also can get(150)αT,x≤1T∫ss+T∫ℍB1t,x,wℍμtxdwdt+B¯1xℍ≤c1+xℍ.
Hence, the proof is complete.
Now, we introduce the averaged equation:(151)dut=A1tut+B¯1utdt+F1t,utdwQ1t+∫ℤG1t,ut,zN˜1dt,dz,u0=x∈ℍ.
Due to assumption (A3), we can easily get that the mappingB¯1 is Lipschitz continuous. So, for any x∈ℍ,T>0 and p≥1, equation (151) admits a unique mild solution u¯∈LpΩ;D0,T;ℍ.


## 6. Averaging Principles

In this section, we prove that the slow motionuϵ converges to the averaged motion u¯, as ϵ⟶0.Theorem 1.
Under (A1)–(A5), fixingx∈D−A1θθ∈0,θ¯ and y∈ℍ, for any T>0 and η>0, we have(152)limϵ⟶0ℙsupt∈0,Tuϵt−u¯tℍ>η=0,where u¯ is the solution of the averaged (151).Proof.
For anyh∈DA1∩L∞O, we have(153)uϵt,hℍ=x,hℍ+∫0tA1ruϵr,hℍdr+∫0tB¯1uϵr,hℍdr+∫0tF1r,uϵrdwQ1r,hℍ+∫0t∫ℤG1r,uϵr,zN˜1dr,dz,hℍ+Rϵt,where(154)Rϵt≔∫0tB1r,uϵr,vϵr−B¯1uϵr,hℍdr.
According to the proof in Section3, we know that the family ℒuϵϵ∈0,1 is tight in PD0,T;ℍ. Hence, in order to prove Theorem 1, it is sufficient to prove limϵ⟶0Esupt∈0,TRϵt=0.
For anyϵ>0 and some deterministic constant δϵ>0, we divide the interval 0,T in subintervals of the size δϵ. In each time interval kδϵ,k+1δϵ,k=0,1,…,T/δϵ, we define the following auxiliary fast motion v^ϵ:(155)dv^ϵt=1ϵA2t−αv^ϵt+B2t,uϵkδϵ,v^ϵtdt+1ϵF2t,uϵkδϵ,v^ϵtdωQ2t+∫ℤG2t,uϵkδϵ,v^ϵtN˜2ϵdt,dz.
According to the definition ofv^ϵ, we know that an analogous estimate to (31) holds. So, for any p≥1, we have(156)∫0TEv^ϵtℍpdt≤cp,T1+xℍp+yℍp.Lemma 9.
Under (A1)–(A5), fixingx∈D−A1θθ∈[0,θ¯ and y∈ℍ, there exists a constant κ>0, such that if(157)δϵ=ϵlnϵ−κ,we have(158)limϵ⟶0supt∈0,TEv^ϵt−vϵtℍp=0.Proof.
Letϵ>0 be fixed. For k=0,1,…,T/δϵ and t∈kδϵ,k+1δϵ, let ρϵt be the solution of the problem(159)dρϵt=1ϵA2t−αρϵtdt+1ϵKϵtdωQ2t+∫ℤHϵt,zN˜2ϵdt,dz,ρϵkδϵ=0,where(160)Kϵt≔F2t,uϵkδϵ,v^ϵt−F2t,uϵt,vϵt,Hϵt,z≔G2t,uϵkδϵ,v^ϵt,z−G2t,uϵt,vϵt,z.
We get(161)ρϵt=ψα,ϵ,2ρϵ;kδϵt+Γϵt+Ψϵt,t∈kδϵ,k+1δϵ,where(162)Γϵt=1ϵ∫kδϵtUα,ϵ,2t,rKϵrdwQ2r,Ψϵt=∫kδϵt∫ℤUα,ϵ,2t,rHϵr,zN˜2ϵdr,dz.
If we denoteΛϵt≔v^ϵt−vϵt and ϑϵt≔Λϵt−ρϵt, we have(163)dϑϵt=1ϵA2t−αϑϵt+B2t,uϵkδϵ,v^ϵt−B2t,uϵt,vϵtdt.
Due to Lemma3, for α>0 large enough, using Young’s inequality, we have(164)1pddtϑϵtℍp≤1ϵγ2tA2+L2t−αϑϵt,ϑϵtℍϑϵtℍp−2+1ϵB2t,uϵkδϵ,v^ϵt−B2t,uϵt,vϵt,ϑϵtℍϑϵtℍp−2≤cϵϑϵtℍp−αϵϑϵtℍp+cϵuϵkδϵ−uϵtℍϑϵtℍp−1+cϵv^ϵt−vϵtℍϑϵtℍp−1≤−α2ϵϑϵtℍp+cpϵuϵkδϵ−uϵtℍp+cpϵv^ϵt−vϵtℍp≤−α2ϵϑϵtℍp+cpϵ1+xθp+yℍpδϵβθp+δϵ+cpϵv^ϵt−vϵtℍp.
Using the Gronwall inequality, we get(165)ϑϵtℍp≤cpϵ1+xθp+yℍpδϵβθp+1+δϵ2+cpϵ∫kδϵtv^ϵr−vϵrℍpdr.
By proceeding as Lemma 6.3 in [13], we prove that(166)EΓϵtℍp≤cpϵ1+xθp+yℍpδϵβθp+1+δϵ2+cpϵ∫kδϵtEv^ϵr−vϵrℍpdr.
For the other stochastic termΨϵt, using Kunita’s first inequality and the Hölder inequality, we get(167)EΨϵtℍp≤cpE1ϵ∫kδϵt∫ℤe−α/ϵt−reγ2t,r/ϵA2Hϵr,zℍ2v2dzdrp/2+cpϵE∫kδϵt∫ℤe−α/ϵt−reA2γ2t,r/ϵHϵr,zℍpv2dzdr≤cpϵp/2E∫kδϵte−2α/ϵt−ruϵkδϵ−uϵrℍ2+v^ϵr−vϵrℍ2drp/2+cpϵ∫kδϵtEuϵkδϵ−uϵrℍp+Ev^ϵr−vϵrℍpdr≤cpϵp/2∫kδϵtEuϵkδϵ−uϵrℍp+Ev^ϵr−vϵrℍpdr∫kδϵte−2α/ϵp/p−2t−rdrp−2/2+cpϵ∫kδϵtEuϵkδϵ−uϵrℍp+Ev^ϵr−vϵrℍpdr≤cpϵ∫kδϵtEuϵkδϵ−uϵrℍp+Ev^ϵr−vϵrℍpdr≤cpϵ1+xθp+yℍpδϵβθp+1+δϵ2+cpϵ∫kδϵtEv^ϵr−vϵrℍpdr.
According to Lemma 2.4 in [25] and equations (161)–(167), we obtain(168)Ev^ϵt−vϵtℍp≤cpϵ1+xθp+yℍpδϵβθp+1+δϵ2+cpϵ∫kδϵtEv^ϵr−vϵrℍpdr.
From the Gronwall lemma, this means(169)Ev^ϵt−vϵtℍp≤cpϵδϵβθp+1+δϵ2ecp/ϵδϵ1+xθp+yℍp.
Fort∈0,T, selecting δϵ=ϵlnϵ−κ, then if we take κ<βθp/βθp+1+cp∧1/2+cp, we have (158).Lemma 10.
Under the same assumptions as in Theorem1, for any T>0, we have(170)limϵ⟶0Esupt∈0,TRϵt=0.Proof.
According to the definition ofB¯1, we get that the mapping B¯1:ℍ⟶ℍ is Lipschitz continuous. Using assumption (A3) and Lemmas 3 and 9, we have(171)limϵ⟶0Esupt∈0,TRϵt≤limϵ⟶0E∫0TB1r,uϵr,vϵr−B1r,uϵr/δϵδϵ,v^ϵr,hℍdr+limϵ⟶0Esupt∈0,T∫0tB1r,uϵr/δϵδϵ,v^ϵr−B¯1uϵr,hℍdr≤limϵ⟶0cThℍsupr∈0,TEuϵr−uϵr/δϵδϵℍ+supr∈0,TEv^ϵr−vϵrℍ+limϵ⟶0∑k=0T/δϵE∫kδϵk+1δϵB1r,uϵr/δϵδϵ,v^ϵr−B¯1uϵkδϵ,hℍdr+limϵ⟶0hℍ∑k=0T/δϵ∫kδϵk+1δϵEB¯1uϵkδϵ−B¯1uϵrℍdr≤limϵ⟶0cThℍ1+xθ+yℍδϵβθ+δϵ+supr∈0,TEv^ϵr−vϵrℍ+limϵ⟶0∑k=0T/δϵE∫kδϵk+1δϵB1r,uϵkδϵ,v^ϵr−B¯1uϵkδϵ,hℍdr+limϵ⟶0cThℍ1+xθ+yℍδϵβθ+1+δϵ2T/δϵ+1.
So, we have to show that(172)limϵ⟶0∑k=0T/δϵE∫kδϵk+1δϵB1r,uϵkδϵ,v^ϵr−B¯1uϵkδϵ,hℍdr=0.
If we setζϵ=δϵ/ϵ, we have(173)E∫kδϵk+1δϵB1r,uϵkδϵ,v^ϵr−B¯1uϵkδϵ,hℍdr=E∫0δϵB1kδϵ+r,uϵkδϵ,v^ϵkδϵ+r−B¯1uϵkδϵ,hℍdr=E∫0δϵB1kδϵ+r,uϵkδϵ,v˜uϵkδϵ,vϵkδϵr/ϵ−B¯1uϵkδϵ,hℍdr=δϵE1ζϵ∫0ζϵB1kδϵ+ϵr,uϵkδϵ,v˜uϵkδϵ,vϵkδϵr−B¯1uϵkδϵ,hℍdr,where v˜uϵkδϵ,vϵkδϵr is the solution of the fast motion (76) with the initial datum given by vϵkδϵ and the frozen slow component given by uϵkδϵ. In addition, the noises in (76) are independent of uϵkδϵ and vϵkδϵ. According to the proof of Lemma 8, we get(174)E∫kδϵk+1δϵB1r,uϵkδϵ,v^ϵr−B¯1uϵkδϵ,hℍdr≤δϵcζϵ1+Euϵkδϵℍ+Evϵkδϵℍhℍ+δϵhℍEαζϵ,uϵkδϵ.
In Section3, we proved that the family(175)uϵkδϵ:ϵ>0,k=0,…,T/δϵ,is tight. Then, by proceeding as the proof of equation (8.21) in [25], we also can get that, for any η>0, there exists a compact set Kη⊂ℍ, such that(176)Eαζϵ,uϵkδϵ≤supx∈Kηαζϵ,x+ηc1+xℍ+yℍ.
Due to the arbitrariness ofη and according to Lemma 8, we can get (172). Furthermore, (170) holds.
Through the above proof, Theorem1 is established.



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