Dynamical stability of random delayed FitzHugh-Nagumo lattice systems driven by nonlinear Wong-Zakai noise

Dynamical stability of random delayed FitzHugh-Nagumo lattice systems Dynamical stability of random delayed FitzHugh-Nagumo lattice systems driven by nonlinear Wong-Zakai noise Shuang Yang,1, a) Yangrong Li,2, b) and Tom´ as Caraballo3, c) 1)School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China 2)School of Mathematics and statistics, Southwest University, Chongqing 400715, People’s Republic of China 3)Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, C/Tarfia s/n, 41012-Sevilla, Spain In this paper, two problems related to FitzHugh-Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh-Nagumo lattice systems driven by nonlinear Wong-Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation time of Wong-Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delayed lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli-Arzel`a theorem and uniform tailestimates. We then show that the upper semi-continuous of attractors as the correlation time tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model, and study the convergence of attractors when the delay approaches zero. Namely, the upper semicontinuity of attractors for the delayed system to the non-delayed one is proved. Keywords: Random delay lattice system; FitzHugh-Nagumo system; Nonlinear Wong-Zakai noise; Pullback random attractor; Upper semicontinuity I. INTRODUCTION Some lattice dynamical systems can be derived from spatial discretization of continuum systems. They have wide applications in our daily life, including physics, chemistry, biology, engineering and other fields of science (see, e.g.,5,11,15–17). As far as the authors are aware, one of the most interesting lattice systems is FitzHugh-Nagumo model, which simulates the process of signal transmission across axons. It is known that, lattice dynamical systems with delay have been receiving much attention for many years (9,44,46). The motivation of this paper is to study the long-time dynamics of pullback random attractors for the following delayed FitzHugh-Nagumo lattice system driven by a nonlinear Wong-Zakai noise:            dui dt −(ui−1−2ui+ui+1) + λui+αvi=Fi(ui(t)) + fi(ui(t−(ρ)(t))) + gi(t) + Gi(t, ui)Gδ(t, ω), dvi dt +ςvi−βui=hi(t) + fi(vi(t−(ρ)(t))), ui(τ+s) = φi(s), vi(τ+s) = υi(s), i ∈Z, t > τ, τ ∈R, s ∈[−ρ, 0], (1) where λ,α,ς,β,γand ρare positive constants, (ρ)is a variable delayed function with maximum delay ρ,Fiis a nonlinear drift function with polynomial growth of arbitrary order, fiis an external force affected by memory during the interval of delay time [−ρ, 0], the deterministic time-dependent forcings gi, hi∈L2 loc(R, L2(Rn)), φi, ψiare the initial data on the internal [−ρ, 0],Giis a nonlinear diffusion, Gδis the Wong-Zakai process with a)Electronic mail: [email protected] b)Corresponding author; Electronic mail: [email protected]u.cn c)Electronic mail: [email protected] 2 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems correlation time δ > 0, which is δ-difference of a two-side scalar Wiener process Won a probability space (Ω,F,P), given by Gδ(t, ω) := 1 δ(W(t+δ, ω)−W(t, ω)),∀δ > 0, t ∈R, ω ∈Ω.(2) This type of Wong-Zakai noise was first introduced by40,41 in which the authors used deterministic differential equations to approximate stochastic ones for one-dimensional Brownian motions. Later, a growing number of authors extended the idea of Wong-Zakai approximations to higher-dimensional Brownian motions, martingales and semimartingales (see18,19,29–35 ). Recently, the convergence of solutions and attractors for random equations with such a noise when δ→0has been extensively studied (see1,4,6,12,13,48). Note that Wong-Zakai approximation systems in these references were only considered in the case of linear noise, that is, the sequence of diffusion functions G= (Gi)i∈Zin (1) is u= (ui)i∈Zor independent of u. Moreover, so were all the results on the semicontinuity of random attractors, see3,22,37,43 for autonomous stochastic equations, and4,7,20,21,23,26,39,42,45,47 for non-autonomous stochastic equations. However, there exist very few works on studying the attractors of random delayed equations driven by the nonlinear Wong-Zakai noise, even in autonomous version. To our knowledge, the only two papers for such autonomous equations were published by Li et al. in24,25. In this paper, we investigate the dynamics of nonautonomous random delayed FitzHugh-Nagumo lattice system with a nonlinear Wong-Zakai noise (1). The present article is divided into two parts. In the first part, we prove the existence of a pullback random attractor Aδ={Aδ(t, ω)}for the random delayed system (1) with a nonlinear noise and its upper semicontinuity when δ→+∞. This is different from the general situation δ→0. To prove the existence of a pullback random attractor Aδin Xρ σ=C([−ρ, 0],Xσ)with Xσ=ℓ2 σ×ℓ2 σ, where ℓ2 σis weighted space for each δ > 0, we must verify the random dynamical system (or cocycle) Ψδ, induced by Eq. (1) driven by the Wong-Zakai nonlinear noise, is pullback asymptotically compact in Xρ σ. The ideas of uniform estimates and the Ascoli-Arzel`a theorem are the crucial tools to prove it. As for the upper semi-convergence of Aδas δ→+∞, we need the help of the logarithm law of the Wiener process, which establishes W(t, ω)/log(log |t|)→0as t→ ±∞, as well as the result (11) in Lemma II.1 which ensures lim δ→+∞sup t∈[a,b]Gδ(t, ω) = 0,P-a.s. ω∈Ω, a ≤b. Based on the previous arguments, we consider the limiting system of random delayed lattice model (1) when δ→+∞as the deterministic delayed lattice system:            dˆui dt −(ˆui−1−2ˆui+ ˆui+1) + λˆui+αˆvi=Fi(ˆui(t)) + fi(ˆui(t−(ρ)(t))) + gi(t), dˆvi dt +ςˆvi−βˆui=hi(t) + fi(ˆvi(t−(ρ)(t))), ˆui(τ+s) = ˆ φi,ˆvi(τ+s) = ˆυi, i ∈Z, t > τ, τ ∈R, s ∈[−ρ, 0]. (3) Under some appropriate conditions (see Hypotheses E, F1, F2, G1-G3 later), we find out that the system (3) generates a pullback attractor denoted by A∞(t)whose existence has been established, see2,8,36. Then we need to check that the random pullback attractor Aδ(t, ω)semi-converges to A∞(t)as δ→+∞(see Theorem IV.2), that is, lim δ→+∞dXρ σ(Aδ(t, ω),A∞(t)) = 0,∀t∈R, ω ∈Ω,(4) where the distance dXρ σis defined for all subsets Aand Bof Xρ σby dXρ σ(A, B) := sup a∈A inf b∈Bsup ν∈[−ρ,0] ka(ν)−b(ν)kXσ. Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 3 For this end, we prove the solutions to random system (1) converge to that of the corresponding deterministic system (3) when δ→+∞. Note that the pullback attractor A∞(t)in (4) is written as Aρ(t)to indicate its dependence on the delay ρfor later purpose. In the second part of this article, our goal is to further establish the upper semicontinuity of the pullback attractor Aρ(t)as ρ→0(see Theorem V.4), that is, lim ρ→0d∗ Xρ σ(Aρ(t),A0(t)) = 0,∀t∈R,(5) where A0(t)is a pullback attractor for the non-delayed version of Eq. (3), and the distance d∗ Xρ σis defined for all subset Aof Xρ σand Bof Xσby d∗ Xρ σ(A, B) := sup a∈A inf b∈Bsup ν∈[−ρ,0] ka(ν)−bkXσ. Due to the validity of all the estimates we obtained in Section III of the first part, especially in two cases of the non-delayed case (ρ= 0) and the deterministic case (δ→+∞) for (1). Therefore, we immediately deduce the existence of the pullback attractor A0(t). As for (5), the main task is to prove the convergence of solutions to system (3) as ρ→0. The article is organized as follows. In the next section, we introduce Wong-Zai process, weighted spaces and some notations, impose some suitable assumptions, and define a family of continuous cocycles. In Section III, we prove the existence of pullback random attractors for problem (1). In Section IV, we further establish its upper semicontinuity as δ→+∞. The last section is devoted to the upper semicontinuity of pullback attractors for problem (3) as ρ→0. II. RANDOM DELAYED FITZHUGH-NAGUMO LATTICE SYSTEM DRIVEN BY WONG-ZAKAI NOISE In this section, we first prove some useful results on Wong-Zakai processes and weighted spaces. We then define a continuous cocycle (non-autonomous random dynamical system) Ψδassociated with the random delayed FitzHugh-Nagumo lattice system (1) for all δ > 0, and establish some suitable assumptions. A. Wong-Zakai process As usual, we identify the Wiener process W(t, ω)with the path ω(t)on the metric dynamical system (Ω,F,P, θ}, i.e., W(t, ω) = ω(t), where Ω = {ω∈C(R,R) : ω(0) = 0}with the compact-open topology, Fis the Borel σ-algebra, Pis the Wiener measure on (Ω,F),θ={θt:t∈R}is a group on (Ω,F,P) denoted by θtω(·) = ω(·+t)−ω(t), and there is a θ-invariant full-measure set Ω0⊂Ωsatisfying lim t→± ω(t) t= 0,∀ω∈Ω0.(6) For convenience, we write Ω0as Ω. For each δ > 0, define a random variable Gδby Gδ(ω) := Gδ(0, ω) = ω(δ) δ,∀δ > 0, ω ∈Ω,(7) which implies that the Wong-Zakai process has another form: Gδ(t, ω) = 1 δ(W(t+δ, ω)−W(t, ω)) = Gδ(θtω),∀δ > 0, t ∈R, ω ∈Ω.(8) The following Lemma gives several conclusions on Gδ. 4 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems Lemma II.1. For each (δ, ω)∈R+×Ω, we obtain the following results (i) The mapping t→ Gδ(θtω)is continuous such that lim δ→0sup t∈[a,b]Zt 0Gδ(θsω)ds −ω(t)= 0; (9) (ii) The mapping t→ Gδ(θtω)is of sublinear growth, i.e., lim t→±∞ Gδ(θtω) t= 0; (10) (iii) The mapping δ→ Gδ(θtω)is continuous on (0,+∞)and uniformly continuous on [δ0,+∞)for all δ0>0such that lim δ→+∞sup t∈[a,b]|Gδ(θtω)|= 0; (11) (iv) For any ς1, ς2>0and ω∈Ωsuch that for all δ > 0, Z0 −∞ eς1t|Gδ(θtω)|ς2dt < +∞,and lim δ→+∞Z0 −∞ eς1t|Gδ(θtω)|ς2dt = 0.(12) Proof. (i) It follows from28 (lemma 2.1) that (9) holds true. (ii) According to (7), we obtain lim t→±∞ Gδ(θtω) t= lim t→±∞ ω(t+δ)−ω(t) δt = lim t→±∞ ω(t+δ) t+δ·t+δ δt −1 δlim t→±∞ ω(t) t= 0.(13) (iii) Since t→ω(t)is continuous, one can imply that δ→ Gδ(θtω)is continuous on (0,+∞). We now prove that it is uniformly continuous on [δ0,+∞)for all δ0>0. And thus we need to imply that lim δ→+∞sup t∈[a,b]Gδ(θtω) = lim δ→+∞sup t∈[a,b] ω(t+δ)−ω(t) δ = lim δ→+∞sup t∈[a,b] ω(t+δ) δ−lim δ→+∞inf t∈[a,b] ω(t) δ= 0.(14) On the one hand, for given ǫ > 0and ω∈Ω, note that ω(t) t→0as t→+∞, so there exists T1:= T1(ǫ, ω)>0 such that |ω(t)| ≤ ǫt for all t≥T1. For each a, b ∈Rand a≤b, then [a, b]is compact. Then, for all δ≥T1−a, and so t+δ≥T1>0whenever t∈[a, b], sup t∈[a,b] ω(t+δ) t+δ≤sup t∈[a,b] |ω(t+δ)| t+δ≤sup t∈[a,b] ǫ(t+δ) t+δ=ǫ. (15) We then easily check that 0≤t+δ δ≤2for all δ≥max{|a|,|b|}. Let δ0= max{T1−a, |a|,|b|}, then for all δ≥δ0and t∈[a, b]such that sup t∈[a,b] ω(t+δ) δ= sup t∈[a,b] ω(t+δ) t+δ·t+δ δ≤2ǫ, which implies lim δ→+∞sup t∈[a,b] ω(t+δ) δ= 0.(16) Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 5 On the other hand, since the minimum of ω(·)over [a, b]exists and is finite, we deduce lim δ→+∞inf t∈[a,b] ω(t) δ= 0.(17) Combining (16) and (17), we obtain we obtain (14). This implies (11), which together with the continuity of δ→ Gδ(θtω), yields the uniform continuity on [δ0,+∞). (iv) By (ii), t→ Gδ(θtω)is of sublinear growth as t→ −∞, which along with the continuity of δ→ Gδ(θtω) shows that eς1t|Gδ(θtω)|ς2is integrable with respect to t∈(−∞,0] for any ς1, ς2>0. By ω(t) t→0as t→ ±∞, there is a T:= T(ω)>0such that ω(t) t≤1for all |t| ≥ T, |ω(t)| ≤ |t|+C(ω),∀t∈R,(18) where C(ω) = sup t∈[−T,T ]|ω(t)|<+∞. For all δ≥1, we then proves the following inequality holds true. |Gδ(θtω)| ≤ 2C(ω)−2t+ 1,∀t≤0, ω ∈Ω.(19) Case A: If t∈[−δ, 0], then for all δ≥1, |Gδ(θtω)|=1 δ|ω(t+δ)−ω(t)| ≤ 1 δ(|ω(t+δ)|+|ω(t)|) ≤1 δ(t+δ+C(ω)) + (−t+C(ω))= 1 + 2 δC(ω)≤2C(ω)−2t+ 1.(20) Case B: If t∈(−∞,−δ], then for all δ≥1, |Gδ(θtω)|=1 δ|ω(t+δ)−ω(t)| ≤ 1 δ(|ω(t+δ)|+|ω(t)|) ≤1 δ(−t−δ+C(ω)) + (−t+C(ω))≤2C(ω)−2t+ 1.(21) Combining two cases Aand B, we have (19) as desired. Thus, we easily show Z0 −∞ eς1t|Gδ(θtω)|ς2dt ≤Z0 −∞ eς1t(2C(ω)−2t+ 1)ς2dt < +∞,∀ς1, ς2>0. According to the Lebesgue control convergence theorem and (11), we deduce lim δ→+∞Z0 −∞ eς1t|Gδ(θtω)|ς2dt =Z0 −∞ eς1tlim δ→+∞|Gδ(θtω)|ς2dt = 0, which proves (12) as desired. All proofs are complete. B. Weighted spaces and continuous cocycles Given p≥1and σ > 1 2, we define the weighted p-times summation space by ℓp σ=(u={ui}i∈Z:kukσ,p = (X i∈Z ξi|ui|p)1 p),(22) 6 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems where ξi= (1 + i2)−σfor i∈Z, and so ξ= (ξi)i∈Z∈ℓpfor any p≥1. Thanks to4,15,(ℓp σ,k · kσ,p)is a separable Banach space. In particular, ℓ2 σis a Hilbert space with inner product and norm, respectively: (u, v)σ=X i∈Z ξiuivi,kukσ= (u, u)1 2 σ,∀u, v ∈ℓ2 σ.(23) By the H¨older inequality, for p > q ≥1, we have kkq σ,q ≤ kξk p−q p ℓ1kkq σ,p,∀∈ℓp σ. More precisely, kkq σ,q =X i∈Z ξi||q=X i∈Z ξ p−q p iξ q p i||q ≤X i∈Z (ξ p−q p i)p p−qp−q pX i∈Zξ q p i|i|qp qq p=kξk p−q p ℓ1kkq σ,p.(24) Taking into account the delay, let Xρ σ=C([−ρ, 0], ℓ2 σ), which is the space of all continuous functions from [−ρ, 0] to ℓ2 σwith the following norm kυkXρ σ= sup s∈[−ρ,0] kυ(s)kσ= sup s∈[−ρ,0] X i∈Z ξi|υi(s)|21 2,∀υ∈Xρ σ.(25) For convenience, the delay shift of ϕ= (u, v) : R×R→R2is defined by ϕt= (ut, vt) : [−ρ, 0] ×[−ρ, 0] →R2, ϕt(s, x) = (u(t+s), v(t+s)) (26) for all s∈[−ρ, 0]. Let Xσ=ℓ2 σ×ℓ2 σand Xρ σ=C([−ρ, 0],Xσ)be equipped by the norms kϕk2 Xσ=βkuk2 σ+αkvk2 σ,∀ϕ= (u, v),(27) and kϕtk2 Xρ σ=βkutk2 Xρ σ+αkvtk2 Xρ σ=βsup s∈[−ρ,0] kut(s)k2 σ+αkvt(s)k2 σ,∀ϕ= (u, v),(28) where αand βare as in (1). Then, we introduce the discrete Laplace and gradient operators by (Au)i=−ui−1+ 2ui−ui+1,(Bu)i=ui+1 −ui,(B∗u)i=ui−1−ui,(29) which shows that A=BB∗=B∗B, see14. Note that for all u, v ∈l2such that (Bu, v) = (u, B∗v),(Au, v) = (Bu, Bv). It is simple to obtain that for all i∈Z, 0.4σ≤ξi+1 ξi =1 + i2 1 + (i+ 1)2σ ≤2.5σ,and 0.4σ≤ξi−1 ξi =1 + (i−1)2 1 + i2σ ≤2.5σ,(30) which implies that ξi±1≤2.5σξi,|(Bξ)i|=|ξi+1 −ξi| ≤ 2.5σξi,|(B∗ξ)i| ≤ 2.5σξi.(31) Let F(x, u(t)) = (Fi(ui(t)))i∈Z, f(u(t−(ρ)(t))) = (fi(ui(t−(ρ)(t))))i∈Z, f(v(t−(ρ)(t))) = (fi(vi(t− (ρ)(t))))i∈Z, g(x, t) = (gi(t))i∈Z, G(t, u) = (Gi(t, ui))i∈Z,and h(x, t) = (hi(t))i∈Z. Then system (1) can be rewritten as            du dt +Au +λu +αv =F(x, u(t)) + f(u(t−(ρ)(t))) + g(x, t) + G(t, u)Gδ(θtω), dv dt +ςv −βu =h(x, t) + f(v(t−(ρ)(t))), u(τ+s) = φ(s), v(τ+s) = υ(s), t > τ, τ ∈R, s ∈[−ρ, 0], ρ > 0. (32) Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 7 Hypothesis E. The delay function (ρ)(·)is a positive continuously differentiable function satisfying ρ:= sup t∈R (ρ)(t)<+∞, ρ∗:= sup ρ∈(0,ρ0] sup t∈R d dt(ρ)(t)<1.(33) Therefore, the memory time ρ∈(0, ρ0]for some ρ0>0. Hypothesis F1. For the nonlinear drift function Fi∈C1(R,R), we assume that for all s∈Rand i∈Z, Fi(s)s≤ −α1|s|p+µ1,i, µ1= (µ1,i)i∈Z∈ℓ 2p−2 p σ,(34) Fi(s)≤α2|s|p−1+µ2,i, µ2= (µ2,i)i∈Z∈ℓ2 σ,(35) ∂Fi ∂s (s)≤ −α3|s|p−2+µ3,i, µ3= (µ3,i)i∈Z∈ℓ∞,(36) where p≥2, α1, α2and α3are positive constants. Hypothesis F2. The nonlinear delayed term fiis continuous such that for all s1, s2∈R, fi(0) = 0,∀i∈Z, sup i∈Z sup s1,s2∈R|fi(s1)−fi(s2)| ≤ Lf|s1−s2|,(37) where Lf>0is constant. From now on, let κ= min{λ, ς}and σ0:= 4 ×2.52σ+4 3(2.53σ+ 2kµ3kℓ∞). Besides, we assume σ0+4L2 f κ(1−ρ∗)< κ. In this case, there exists m0>0small enough such that for all m∈(0, m0), m+σ0−κ+4L2 femρ0 κ(1 −ρ∗)<0.(38) In particular, m−κ+4L2 femρ0 κ(1−ρ∗)<0. Hypothesis G1. Let Gi(·,·)be continuous from R2to Rsatisfying |Gi(t, s)| ≤ α4|s|q−1+µ4,i(t), µ4= (µ4,i)i∈Z∈L∞(R, ℓp σ),(39) where 2≤q < p, α4>0. We further impose the following assumptions. Hypothesis G2. The forces gand hare backward tempered: Υ(τ) := sup r≤τZ0 −∞ emν(kg(ν+r)k2 σ+kh(ν+r)k2 σ)dν < +∞,∀τ∈R.(40) Hypothesis G3. The forces gand hare backward tail-small: lim k→∞ sup r≤τZ0 −∞ emν X |i|≥k ξi(|gi(ν+r)|2+|hi(ν+r)|2)dν = 0,∀τ∈R.(41) Under the assumptions (33)-(41), similarly to the Galerkin method, we can show that for each δ > 0,τ∈ R, ω ∈Ωand ψτ= (uτ, vτ)∈ Xρ σ=C([−ρ, 0], ℓ2 σ×ℓ2 σ), the random delayed FitzHugh-Nagumo lattice system (32) possesses a unique solution ϕδ(·, τ, ω, ψτ) = (uδ(·, τ, ω, uτ), vδ(·, τ, ω, vτ)) such that ϕδ∈C([τ−ρ, +∞), ℓ2 σ×ℓ2 σ).(42) 8 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems Besides, the solution ϕδis continuous with respect to the initial data ψτin Xρ σ. By the same method as in10, one can prove that ϕδ(t, τ, ω, ψτ)is (F,B(Xρ σ))-measurable in ω∈Ω. Then, for each δ > 0, we can define a family of continuous cocycles Ψδ:R+×R×Ω×Xρ σ7→ Xρ σgiven by Ψδ(t, τ, ω)ψτ=ϕδ t+τ(·, τ, θ−τω, ψτ).(43) Let Dbe the universe of all backward tempered bi-parametric sets in Xρ σ, where a bi-parametric set D:= {D(τ, ω) : (τ, ω)∈R×Ω}in Xρ σis called backward tempered, that is, D ∈ Dif and only if lim t→+∞e−γt sup r≤τkD(r−t, θ−tω)k2 Xρ σ= 0,∀(γ, τ, ω)∈R+×R×Ω.(44) We easily check that Dis backward-union closed in the sense of ˆ D ∈ Dwhenever D ∈ D, where ˆ D(τ, ω) = [ r≤τD(r, ω),∀(τ, ω)∈R×Ω.(45) However, the usual universe e Dof all tempered bi-parametric sets is not backward-union closed, where e D ∈ e D if and only if lim t→+∞e−γtke D(τ−t, θ−tω)k2 Xρ σ= 0,∀(γ, τ, ω)∈R+×R×Ω.(46) III. EXISTENCE OF PULLBACK RANDOM ATTRACTORS This subsection is devoted to the existence of pullback random attractors for the random delayed FitzHughNagumo system (32). We first derive a variety of backward uniform estimates of solutions to Eq. (32), including the backward uniform absorption and tail-estimates. We then prove the pullback asymptotic compactness of the solutions via the Ascoli-Arzel`a theorem in Xρ σ=C([−ρ, 0],Xσ), where Xσ=ℓ2 σ×ℓ2 σ. Finally, we prove the existence of tempered random attractors for Eq. (32). A. Backward uniform absorption Lemma III.1. Let the hypotheses E, F1, F2, G1, G2 and (38) be satisfied. Then, for each (τ, ω, D)∈R×Ω×D and ψr−t= (φr−t, υr−t)∈ D(r−t, θ−tω), there exists a T:= T(τ, ω, D)≥3ρ+ 1 such that for all t≥T, the solution ϕδ= (uδ, vδ)to (32) satisfies sup r≤τ sup s∈[−2ρ−1,0] kϕδ(r+s, r −t, θ−rω, ψr−t)k2 Xσ≤cRδ(τ, ω),(47) sup r≤τZr r−t em(ν−r)kϕδ(ν)k2 Xσdν + sup r≤τZr r−t em(ν−r)kϕδ(ν)k2 Xσ+kuδ(ν)kp σ,pdν ≤cRδ(τ, ω),(48) where Rδ(τ, ω) = 1 + Υ(τ) + ηδ(ω)with Υ(τ) = sup r≤τZ0 −∞ emν (kg(ν+r)k2 σ+kh(ν+r)k2 σ)dν, ηδ(ω) = Z0 −∞ emν |Gδ(θνω)|p p−qdν. (49) Proof. Taking the inner product of (32) with (2βuδ,2αvδ) := (2βuδ(ν, r −t, θ−rω, φr−t),2αvδ(ν, r − t, θ−rω, υr−t)) in Xσ=ℓ2 σ×ℓ2 σ(when no ambiguity is possible, we delete the superscript δbelow), we obtain d dν kϕk2 Xσ+ 2κkϕk2 Xσ=−2βX i∈Z ξi(Au)iui+ 2β(F(x, u), u)σ(50) Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 9 + 2β(f(u(ν−(ρ)(ν))), u)σ+ 2α(f(v(ν−(ρ)(ν))), v)σ + 2β(g(x, ν), u)σ+ 2α(h(x, ν), v)σ+ 2βGδ(θν−rω)(G, u)σ, where we recall that kϕk2 Xσ=βkuk2 σ+αkvk2 σ,κ= min{λ, ς}. By (31) and (B(ξu))i= (Bξ)iui+1+ξi(Bu)i, we have −2βX i∈Z ξi(Au)iui=−2βX i∈Z (Bu)i(B(ξu))i=−2βX i∈Z (Bξ)i(Bu)iui+1 −2βX i∈Z ξi|(Bu)i|2 ≤2βX i∈Z 2.5σξi|(Bu)i||ui+1|−2βX i∈Z ξi|(Bu)i|2≤βX i∈Z ξi(2.52σ|ui+1|2+|(Bu)i|2)−2βX i∈Z ξi|(Bu)i|2 ≤2.52σβX i∈Z ξi|ui+1|2≤2.53σβX i∈Z ξi+1|ui+1|2= 2.53σβkuk2 σ.(51) By (34) in the hypothesis F1, we imply 2β(F(x, u), u)σ= 2βX i∈Z ξiFi(ui)ui ≤ −2α1βX i∈Z ξi|ui|p+ 2βX i∈Z ξiµ1,i ≤ −2α1βkukp σ,p + 2βkµ1kσ,1.(52) According to the Young inequality and (37) in the hypothesis F2, we deduce 2β(f(u(ν−(ρ)(ν))), u)σ+ 2α(f(v(ν−(ρ)(ν))), v)σ = 2βX i∈Z ξifi(ui(ν−(ρ)(ν)))ui+ 2αX i∈Z ξifi(vi(ν−(ρ)(ν)))vi ≤4L2 f κX i∈Zβ|ui(ν−(ρ)(ν))|2+α|vi(ν−(ρ)(ν))|2+κ 4kϕk2 Xσ ≤4L2 f κkϕ(ν−(ρ)(ν))k2 Xσ+κ 4kϕk2 Xσ.(53) The Young inequality gives 2β(g(x, ν), u)σ+ 2α(h(x, ν), v)σ≤κ 4kϕk2 Xσ+c1(kg(ν)k2 σ+kh(ν)k2 σ),(54) where c1=c1(β, α, κ)>0. By (39) in the hypothesis G1, we have 2βGδ(θν−rω)(G, u)σ≤2β|Gδ(θν−rω)|X i∈Z ξi|Gi(ν, ui)||ui| ≤2βα4|Gδ(θν−rω)|X i∈Z ξi|ui|q+ 2β|Gδ(θν−rω)|X i∈Z ξi|µ4,i(ν)||ui| ≤c2|Gδ(θν−rω)|kukq σ,q +2 ˆqβ|Gδ(θν−rω)|kµ4(ν)kˆq σ,ˆq,(55) where c2= 2βα4+2 qβand 1 ˆq+1 q= 1. Now, we estimate the last two terms in (55), respectively. On the one hand, by (24), we obtain c2|Gδ(θν−rω)|kukq σ,q ≤c2|Gδ(θν−rω)|kξk p−q p ℓ1kukq σ,p ≤1 2α1βkukp σ,p +c3|Gδ(θν−rω)|p p−q,(56) 16 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems Let ιk,i := ι(|i| k)for each k≥1and i∈Z. It is not hard to check that ιk= (ιk,i)i∈Z∈ℓ∞and for all k≥1, i ∈Z, |ιk,i+1 −ιk,i| ≤ c∗ k.(93) Lemma III.4. Let the hypotheses E, F1, F2, G1-G3 and (38) be satisfied. Then, for each (τ, ω, D)∈R×Ω× D, then the solution ϕδ= (uδ, vδ)to (32) satisfies lim k,t→+∞sup r≤τ sup s∈[−ρ,0] kϕδ(r+s, r −t, θ−rω, ψr−t)k2 Xσ(|x|≥k)= 0 (94) uniformly in ψr−t= (φr−t, υr−t)∈ D(r−t, θ−tω). Moreover, the convergence in (94) is uniform with respect to large δ, that is, there exists a δ0:= δ0(ω)which is independent of τ, Dsuch that lim k,t→+∞sup δ≥δ0 sup r≤τ sup s∈[−ρ,0] kϕδ(r+s, r −t, θ−rω, ψr−t)k2 Xσ(|x|≥k)= 0.(95) Proof. Taking the inner product of (32) with (2βιk,iξiui(ν),2αιk,iξivi(ν)) := (2βιk,iξiui(ν, r−t, θ−rω, ur−t), 2αιk,iξivi(ν, r −t, θ−rω, vr−t)) and summing up the product over i∈Z, it follows d dν X i∈Z ιk,iξi(β|ui|2+α|vi|2) + 2κX i∈Z ιk,iξi(β|ui|2+α|vi|2) + 2βX i∈Z ιk,iξi(Au)iui = 2βX i∈Z ιk,iξiFi(ui)ui+ 2βX i∈Z ιk,iξif(ui(ν−(ρ)(ν)))ui + 2αX i∈Z ιk,iξif(vi(ν−(ρ)(ν)))vi+ 2βX i∈Z ιk,iξigi(ν)ui + 2αX i∈Z ιk,iξihi(ν)vi+ 2βGδ(θν−rω)X i∈Z ιk,iξiGi(ν, ui)ui,(96) where we recall that κ= min{λ, ς}. By (ιk,iξiui,(Au)i) = ((Bιkξu)i,(Bu)i) = (ιk,i+1ξi+1ui+1 − ιk,iξiui,(Bu)i)and (B(ξu))i= (Bξ)iui+1 +ξi(Bu)i, we obtain −2βX i∈Z ιk,iξi(Au)iui= 2βX i∈Z (ιk,iξiui−ιk,i+1ξi+1ui+1)(Bu)i = 2βX i∈Z (ιk,i −ιk,i+1)ξi+1ui+1(Bu)i−2βX i∈Z ιk,i(ξi+1ui+1 −ξiui)(Bu)i = 2βX i∈Z (ιk,i −ιk,i+1)ξi+1ui+1(Bu)i−2βX i∈Z ιk,i(Bξ)iui+1(Bu)i−2βX i∈Z ιk,iξi|(Bu)i|2.(97) By (93) and ξi+1 ≤2.5σξias in (31), we deduce 2βX i∈Z (ιk,i −ιk,i+1)ξi+1ui+1(Bu)i≤2βc∗ kX i∈Z ξi+1(|ui+1|2+|ui+1||ui|) ≤2βc∗ kX i∈Z ξi+13 2|ui+1|2+1 2|ui|2 ≤3βc∗ kkuk2 σ+ 2.5σβc∗ kkuk2 σ= (3 + 2.5σ)βc∗ kkuk2 σ.(98) By (93) and (31) again, we have −2βX i∈Z ιk,i(Bξ)iui+1(Bu)i−2βX i∈Z ιk,iξi|(Bu)i|2 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 17 ≤2βX i∈Z ιk,i2.5σξi|ui+1||(Bu)i|−2βX i∈Z ιk,iξi|(Bu)i|2 ≤βX i∈Z ιk,iξi(2.52σ|ui+1|2+|(Bu)i|2)−2βX i∈Z ιk,iξi|(Bu)i|2 ≤2.52σβX i∈Z ιk,iξi|ui+1|2≤2.53σβX i∈Z ιk,iξi+1|ui+1|2 = 2.53σβX i∈Z ιk,i+1ξi+1|ui+1|2+ 2.53σβX i∈Z (ιk,i −ιk,i+1)ξi+1|ui+1|2 ≤2.53σβX i∈Z ιk,iξi|ui|2+ 2.53σβc∗ kkuk2 σ.(99) Using (98) and (99) in (97), we deduce −2βX i∈Z ιk,iξi(Au)iui≤2.53σβX i∈Z ιk,iξi|ui|2+c1 kkuk2 σ,(100) where c1=β(3 + 2.5σ+ 2.53σ)c∗.By (34) in the hypothesis F1, we imply 2βX i∈Z ιk,iξiFi(ui)ui≤ −2α1βX i∈Z ιk,iξi|ui|p+ 2βX i∈Z ιk,iξi|µ1,i|.(101) Applying the Young inequality and using (37) in the hypothesis F2, we yield 2βX i∈Z ιk,iξif(ui(ν−(ρ)(ν)))ui+ 2αX i∈Z ιk,iξif(vi(ν−(ρ)(ν)))vi ≤4L2 f κX i∈Z ιk,iξiβ|ui(ν−(ρ)(ν))|2+α|vi(ν−(ρ)(ν))|2+κ 4X i∈Z ιk,iξi(β|ui|2+α|vi|2).(102) The Young inequality gives 2βX i∈Z ιk,iξigi(ν)ui+ 2αX i∈Z ιk,iξihi(ν)vi ≤c2X i∈Z ιk,iξi(|gi(ν)|2+|hi(ν)|2) + κ 4X i∈Z ιk,iξi(β|ui|2+α|vi|2).(103) According to (39) in the hypothesis G1, the last term of (96) is bounded by 2βGδ(θν−rω)X i∈Z ιk,iξiGi(ν, ui)ui ≤2β|Gδ(θν−rω)|X i∈Z ιk,iξi|Gi(ν, ui)||ui| ≤2βα4|Gδ(θν−rω)|X i∈Z ιk,iξi|ui|q+ 2β|Gδ(θν−rω)|X i∈Z ιk,iξi|µ4,i(ν)||ui| ≤c3|Gδ(θν−rω)|X i∈Z ιk,iξi|ui|q+2 ˆqβ|Gδ(θν−rω)|X i∈Z ιk,iξi|µ4,i(ν)|ˆq,(104) where c3= 2βα4+2 qβ, we recall that 1 ˆq+1 q= 1. Now, we estimate the last two terms in (104), respectively. On the one hand, by the Young inequality and the same method as in the proof of (24), we imply c3|Gδ(θν−rω)|X i∈Z ιk,iξi|ui|q=c3X i∈Zι q p k,iξ q p i|ui|qι p−q p k,i ξ p−q p i|Gδ(θν−rω)| 18 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems ≤1 2α1βX i∈Zι q p k,iξ q p i|ui|qp q+c4X i∈Zι p−q p k,i ξ p−q p i|Gδ(θν−rω)|p p−q =1 2α1βX i∈Z ιk,iξi|ui|p+c4|Gδ(θν−rω)|p p−qX i∈Z ιk,iξi,(105) where c4=c4(p, q, c3, β, α1). On the other hand, note that q≥2and so ˆq≤2≤q < p, we have 2 ˆqβ|Gδ(θν−rω)|X i∈Z ιk,iξi|µ4,i(ν)|ˆq≤2 ˆqβ|Gδ(θν−rω)|X i∈Zι ˆq p k,iξ ˆq p i|µ4,i(ν)|ˆqι p−ˆq p k,i ξ p−ˆq p i ≤2 ˆqβ|Gδ(θν−rω)|X i∈Zιk,iξi|µ4,i(ν)|pˆq pX i∈Z ιk,iξip−ˆq p ≤2 ˆqβ|Gδ(θν−rω)|kµ4(ν)kˆq σ,pX i∈Z ιk,iξip−ˆq p≤c5|Gδ(θν−rω)|X i∈Z ιk,iξip−ˆq p,(106) where c5=2 ˆqβkµ4kˆq L∞(R,ℓp σ)<+∞. Using (105) and (106) in (104), we obtain 2βGδ(θν−rω)X i∈Z ιk,iξiGi(ν, ui)ui≤1 2α1βX i∈Z ιk,iξi|ui|p+c4|Gδ(θν−rω)|p p−qX i∈Z ιk,iξi +c5|Gδ(θν−rω)|X i∈Z ιk,iξip−ˆq p.(107) From the above estimates, (96) can be rewritten: d dν X i∈Z ιk,iξi(β|ui|2+α|vi|2) + κX i∈Z ιk,iξi(β|ui|2+α|vi|2) +κ 2X i∈Z ιk,iξi(β|ui|2+α|vi|2) + 3 2α1βX i∈Z ιk,iξi|ui|p ≤2.53σβX i∈Z ιk,iξi|ui|2+c1 kkuk2 σ+4L2 f κX i∈Z ιk,iξiβ|ui(ν−(ρ)(ν))|2+α|vi(ν−(ρ)(ν))|2 +c6X i∈Z ιk,iξi(|µ1,i(ν)|+|gi(ν)|2+|hi(ν)|2) +c4|Gδ(θν−rω)|p p−qX i∈Z ιk,iξi+c5|Gδ(θν−rω)|X i∈Z ιk,iξip−ˆq p,(108) where c6= 2β+c2. Note that 2.53σβX i∈Z ιk,iξi|ui|2= 2.53σβX i∈Z (ι p−2 p k,i ξ p−2 p i)(ι 2 p k,iξ 2 p i|ui|2) ≤1 2α1βX i∈Z ιk,iξi|ui|p+c7X i∈Z ιk,iξi.(109) Thus, d dν X i∈Z ιk,iξi(β|ui|2+α|vi|2) + κX i∈Z ιk,iξi(β|ui|2+α|vi|2) Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 19 +κ 2X i∈Z ιk,iξi(β|ui|2+α|vi|2) + α1βX i∈Z ιk,iξi|ui|p ≤c1 kkuk2 σ+4L2 f κX i∈Z ιk,iξiβ|ui(ν−(ρ)(ν))|2+α|vi(ν−(ρ)(ν))|2 +c8X i∈Z ιk,iξi(1 + |µ1,i(ν)|+|gi(ν)|2+|hi(ν)|2) +c4|Gδ(θν−rω)|p p−qX i∈Z ιk,iξi+c5|Gδ(θν−rω)|X i∈Z ιk,iξip−ˆq p.(110) Multiplying (110) by emν and integrating it about ν∈[r−t, r +s], where r≤τand s∈[−ρ, 0], we deduce em(r+s)X i∈Z ιk,iξi(β|ui(r+s)|2+α|vi(r+s)|2) + κ 2Zr+s r−t emν X i∈Z ιk,iξi(β|ui(ν)|2+α|vi(ν)|2)dν +α1βZr+s r−t emν X i∈Z ιk,iξi|ui(ν)|pdν ≤em(r−t)βX i∈Z ιk,iξi|ur−t,i|2+αX i∈Z ιk,iξi|vr−t,i|2 + (m−κ)Zr+s r−t emνX i∈Z ιk,iξi(β|ui(ν)|2+α|vi(ν)|2)dν +4L2 f κZr+s r−t emν X i∈Z ιk,iξiβ|ui(ν−(ρ)(ν))|2+α|vi(ν−(ρ)(ν))|2dν +c1 kZr+s r−t emνku(ν)k2 σdν +c8Zr+s r−t emν X i∈Z ιk,iξi(1 + |µ1,i(ν)|+|gi(ν)|2+|hi(ν)|2)dν +c4Zr+s r−t emν|Gδ(θν−rω)|p p−qdν X i∈Z ιk,iξi+c5Zr+s r−t emν|Gδ(θν−rω)|dνX i∈Z ιk,iξip−ˆq p.(111) For the third line of (111), we easily deduce em(r−t)βX i∈Z ιk,iξi|ur−t,i|2+αX i∈Z ιk,iξi|vr−t,i|2≤em(r−t)kψr−t(0)k2 Xσ.(112) The fifth line of (111) is bounded by 4L2 f κZr+s r−t emν X i∈Z ιk,iξiβ|ui(ν−(ρ)(ν))|2+α|vi(ν−(ρ)(ν))|2dν ≤4L2 f κ(1 −ρ∗)Zr+s r−t−ρ em(µ+(ρ)(ν)) X i∈Z ιk,iξiβ|ui(µ)|2+α|vi(µ)|2dµ ≤4L2 femρ0 κ(1 −ρ∗)Zr−t r−t−ρ emµ X i∈Z ιk,iξiβ|ui(µ)|2+α|vi(µ)|2dµ +4L2 femρ0 κ(1 −ρ∗)Zr+s r−t emµ X i∈Z ιk,iξiβ|ui(µ)|2+α|vi(µ)|2dµ 20 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems ≤4L2 femρ0 mκ(1 −ρ∗)em(r−t)kψr−tk2 Xρ σ +4L2 femρ0 κ(1 −ρ∗)Zr+s r−t emµ X i∈Z ιk,iξiβ|ui(µ)|2+α|vi(µ)|2dµ. (113) By (38) and (113), we can rewrite (111) by X i∈Z ιk,iξi(β|ui(r+s)|2+α|vi(r+s)|2) + κ 2Zr+s r−t em(ν−r−s)X i∈Z ιk,iξi(β|ui(ν)|2+α|vi(ν)|2)dν +α1βZr+s r−t em(ν−r−s)X i∈Z ιk,iξi|ui(ν)|pdν ≤c9em(−t−s)kψr−tk2 Xρ σ+c1 kZr+s r−t em(ν−r−s)ku(ν)k2 σdν +c8Zr+s r−t em(ν−r−s)X i∈Z ιk,iξi(1 + |µ1,i(ν)|+|gi(ν)|2+|hi(ν)|2)dν (114) +c4Zr+s r−t em(ν−r−s)|Gδ(θν−rω)|p p−qdν X i∈Z ιk,iξi +c5Zr+s r−t em(ν−r−s)|Gδ(θν−rω)|dνX i∈Z ιk,iξip−ˆq p. By s∈[−ρ, 0] and ρ∈(0, ρ0], we have X i∈Z ιk,iξi(β|ui(r+s)|2+α|vi(r+s)|2) + κ 2Zr+s r−t em(ν−r)X i∈Z ιk,iξi(β|ui(ν)|2+α|vi(ν)|2)dν +α1βZr+s r−t em(ν−r)X i∈Z ιk,iξi|ui(ν)|pdν ≤c9emρ0e−mtkψr−tk2 Xρ σ+c1emρ01 kZr r−t em(ν−r)ku(ν)k2 σdν +c8emρ0Zr r−t em(ν−r)X i∈Z ιk,iξi(1 + |µ1,i(ν)|+|gi(ν)|2+|hi(ν)|2)dν +c4emρ0Zr r−t em(ν−r)|Gδ(θν−rω)|p p−qdν X i∈Z ιk,iξi +c5emρ0Zr r−t em(ν−r)|Gδ(θν−rω)|dνX i∈Z ιk,iξip−ˆq p.(115) Since ψr−t= (φr−t, υr−t)∈ D(r−t, θ−tω), we imply e−mtkψr−tk2 Xρ σ≤e−mt sup r≤τkD(r−t, θ−tω)k2 Xρ σ→0,as t→ ∞.(116) By (47) in Lemma III.1, since Υ(τ), ηδ(ω)<+∞such that for each δ > 0, 1 ksup r≤τZr r−t em(ν−r)ku(ν)k2 σdν ≤c kRδ(τ, ω)→0,as k, t →+∞.(117) Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 21 According to (41) in the hypothesis G3, we obtain sup r≤τZr r−t em(ν−r)X i∈Z ιk,iξi(1 + |µ1,i(ν)|+|gi(ν)|2+|hi(ν)|2)dν (118) ≤sup r≤τZ0 −∞ emν X |i|≥k ξi(1 + |µ1,i(ν+r)|+|gi(ν+r)|2+|hi(ν+r)|2)dν →0, as k, t →+∞. It follows from (12) in Lemma II.1 and ξ∈ℓ1that for all δ > 0, sup r≤τZr r−t em(ν−r)|Gδ(θν−rω)|p p−qdν X i∈Z ιk,iξi≤Z0 −∞ emν|Gδ(θνω)|p p−qdν X |i|≥k ξi→0,(119) as k, t →+∞. In fact, the convergence of (119) is uniform convergence for large δ. By (12) in Lemma II.1, there exists a δ1=δ1(ω)>0such that ηδ(ω) = Z0 −∞ emν|Gδ(θνω)|p p−qdν ≤1,∀δ≥δ1,(120) which implies that sup δ≥δ1 sup r≤τZr r−t em(ν−r)|Gδ(θν−rω)|p p−qdν X i∈Z ιk,iξi≤X |i|≥k ξi→0,as k, t →+∞.(121) Using the same method, we obtain for all δ > 0, sup r≤τZr r−t em(ν−r)|Gδ(θν−rω)|dνX i∈Z ιk,iξip−ˆq p≤sup r≤τZ0 −∞ emν|Gδ(θνω)|dνX |i|≥k ξip−ˆq p→0,(122) as k, t →+∞. And the above convergence is also uniform convergence for large δ. More precisely, by (12) in Lemma II.1 again, R0 −∞ emr|Gδ(θrω)|dr →0as δ→+∞, hence, there exists a δ2=δ2(ω)>0such that sup δ≥δ2 sup r≤τZr r−t em(ν−r)|Gδ(θν−rω)|dνX i∈Z ιk,iξip−ˆq p≤X |i|≥k ξip−ˆq p→0,(123) as k, t →+∞. It follows from (115)-(123) that sup r≤τ sup s∈[−ρ,0] X |i|≥2k ξi(β|uδ i(r+s)|2+α|vδ i(r+s)|2)(124) ≤sup r≤τ sup s∈[−ρ,0] X i∈Z ιk,iξi(β|uδ i(r+s)|2+α|vδ i(r+s)|2)→0,as k, t →+∞, for all δ > 0and uniformly in large δ. This completes the proof. C. Backward asymptotic compactness of solutions and existence of pullback random attractors The following lemma is useful for verifying the asymptotic compactness of solutions. 22 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems Lemma III.5. Let the hypotheses E, F1, F2, G1, G2 and (38) be satisfied. Then, for each (τ, ω, D)∈R×Ω×D and ψr−t= (φr−t, υr−t)∈ D(r−t, θ−tω), there exists a T:= T(τ, ω, D)≥3ρ+ 1 such that for all t≥T, the solution ϕδ= (uδ, vδ)to (32) satisfies sup r≤τZr r−ρ  d dν uδ(ν, r −t, θ−rω, ψr−t)   2 σ dν + sup r≤τZr r−ρ  d dν vδ(ν, r −t, θ−rω, ψr−t)   2 σ dν ≤ce Rδ(τ, ω),(125) where e Rδ(τ, ω)is given by Lemma III.3. Proof. Multiplying the first equation in (32) with du/dν, where u(ν) := u(ν, r −t, θ−rω, ψr−t), we obtain   du dν    2 σ≤ckAuk2 σ+ckuk2 σ+ckvk2 σ+ckF(u(ν))k2 σ+ckf(u(ν−(ρ)(ν)))k2 σ +ckg(ν)k2 σ+c|Gδ(θν−rω)|2kG(ν, u)k2 σ ≤ckuk2 σ+ckvk2 σ+ckF(u(ν))k2 σ+ckf(u(ν−(ρ)(ν)))k2 σ +ckg(ν)k2 σ+c|Gδ(θν−rω)|2kG(ν, u)k2 σ.(126) Integrating from r−ρto rand taking the supremum over r∈(−∞, τ], we deduce sup r≤τZr r−ρ  d dν u(ν, r −t, θ−rω, ψr−t)   2 σdν ≤csup r≤τZr r−ρ (ku(ν)k2 σ+kv(ν)k2 σ)dν +csup r≤τZr r−ρkF(u(ν))k2 σdν +csup r≤τZr r−ρkf(u(ν−(ρ)(ν)))k2 σdν +csup r≤τZr r−ρkg(ν)k2 σdν +csup r≤τZr r−ρ|Gδ(θν−rω)|2kG(ν, u)k2 σdν. (127) By (48) in Lemma III.1, there exists T:= T(τ, ω, D)≥3ρ+ 1 such that for all t≥T, the first term on the right-hand side of (127) is bounded by sup r≤τZr r−ρ (ku(ν)k2 σ+kv(ν)k2 σ)dν ≤cem(3ρ+1) sup r≤τZr r−3ρ−1 em(ν−r)kϕ(ν)k2 σdν ≤cem(3ρ0+1)Rδ(τ, ω).(128) By (35) in the hypothesis F1 and (72) in Lemma III.3, we have sup r≤τZr r−ρkF(u(ν))k2 σdν ≤sup r≤τZr r−ρ (2α2 2kuk2p−2 σ,2p−2+ 2kµ2k2 σ)dν ≤2α2 2sup r≤τZr r−ρkuk2p−2 σ,2p−2dν +c≤2α2 2ce Rδ(τ, ω) + c, (129) where we used µ2∈ℓ2 σ. According to (37) in the hypothesis F2 and (87), we obtain sup r≤τZr r−ρkf(u(ν−(ρ)(ν)))k2 σdν ≤L2 fsup r≤τZr r−ρku(ν−(ρ)(ν))k2 σdν ≤cL2 f(ρ+ 1)Rδ(τ, ω).(130) As done in (88), we have sup r≤τZr r−ρkg(ν)k2 σdν ≤em(ρ+1) sup r≤τZ0 −∞ emνkg(ν+r)k2 σdν ≤em(ρ0+1)Υ(τ)<+∞.(131) Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 23 It follows from Lemma II.1 (i) that t→ Gδ(θtω)is continuous. Thus, there exists an L0>0such that sup ν∈[−ρ,0] |Gδ(θνω)|2≤L0.(132) According to (39) in the hypothesis G1, we obtain sup r≤τZr r−ρ|Gδ(θν−rω)|2kG(ν, u)k2 σdν ≤L0sup r≤τZr r−ρX i∈Z ξi|Gi(ν, ui)|2dν ≤L0sup r≤τZr r−ρX i∈Z ξi(2α2 4|ui(ν)|2q−2+ 2|µ4,i(ν)|2)dν =2α2 4L0sup r≤τZr r−ρX i∈Z ξi|ui(ν)|2q−2dν + 2L0sup r≤τZr r−ρkµ4(ν)k2 σdν. (133) Now, we estimate the last two lines of (133) separately. On the one hand, by 2≤q < p, and so 2q−2> 0,2p−2 2q−2>1, and by (72) in Lemma III.3, we deduce sup r≤τZr r−ρX i∈Z ξi|ui(ν)|2q−2dν ≤csup r≤τZr r−ρX i∈Z ξi(|ui(ν)|2p−2+ 1)dν =csup r≤τZr r−ρku(ν)k2p−2 σ,2p−2dν +csup r≤τZr r−ρkξkℓ1dν ≤ce Rδ(τ, ω).(134) On the other hand, by (24), ξ∈ℓ1and µ4∈L∞(R, ℓp σ), sup r≤τZr r−ρkµ4(ν)k2 σdν ≤sup r≤τZr r−ρkξk p−2 p ℓ1kµ4(ν)k2 σ,pdν ≤ρkξk p−2 p ℓ1kµ4k2 L∞(R,ℓp σ)<+∞.(135) Using (134) and (135) in (133), we imply sup r≤τZr r−ρ|Gδ(θν−rω)|2kG(ν, u)k2 σdν ≤ce Rδ(τ, ω).(136) By (127)-(136), we deduce sup r≤τZr r−ρ  d dν uδ(ν, r −t, θ−rω, ψr−t)   2 σdν ≤ce Rδ(τ, ω).(137) One can similarly prove that sup r≤τZr r−ρ  d dν vδ(ν, r −t, θ−rω, ψr−t)   2 σdν ≤ce Rδ(τ, ω).(138) This together with (137) yields (125) as desired. Proposition III.6. Let the hypotheses E, F1, F2, G1-G3 and (38) be satisfied. For each δ > 0, the cocycle Ψδassociated with the random delayed FitzHugh-Nagumo lattice system (1) is D-backward asymptotically compact in Xσ=ℓ2 σ×ℓ2 σ. More precisely, for all s∈[−ρ, 0], (Ψδ(tn, rn−tn, θ−tnω)ψn)(s) = ϕδ(rn+s, rn−tn, θ−rnω, ψn) has a convergent subsequence in Xσwhenever rn≤τ, tn↑+∞and ψn= (φn, υn)∈ D(rn−tn, θ−tnω). 24 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems Proof. Let (τ, ω, D)∈R×Ω×Dbe fixed and suppose that rn≤τ, tn↑+∞and ψn∈ D(rn−tn, θ−tnω). For each s∈[−ρ, 0], we define Yn(s) := Ψδ(tn, rn−tn, θ−tnω)ψn(s). It suffices to prove that the sequence {Yn(s)}∞ n=1 has a convergent subsequence in Xσ. Besides, we write Yn(s) = (Yn i(s))i∈Zfor each n∈N. Given ǫ > 0, by Lemma III.4, there exist k1, n1∈Nsuch that sup r≤τ sup s∈[−ρ,0] kYn(s)k2 Xσ(|x|≥k1)≤ǫ2,∀n≥n1, k ≥k1.(139) According to (47) in Lemma III.1, there exists n2≥n1such that for all n≥n2, kYn(s)k2 Xσ≤cRδ(τ, ω)<+∞, which implies that the sequence {Yn(s)}∞ n=1 is bounded in Xσ. In particular, the sequence {(Yn i(s))|i|<k1}∞ n=1 is bounded and pre-compact in the finite-dimensional space R2k1−1. In this case, there is a subsequence {(Yn∗ i(s))|i|<k1}∞ n=1 such that it is a Cauchy sequence in R2k1−1. Hence, there exists n3≥n2such that for all n∗, m∗≥n3, X |i|<k1 ξi|Yn∗ i(s)−Ym∗ i(s)|2≤X |i|<k1 |Yn∗ i(s)−Ym∗ i(s)|2≤ǫ2,(140) where we recall that ξi= (1 + i2)−σ≤1for all i∈Zand σ > 1 2. By (139) and (140), we show that for all n∗, m∗≥n3, kYn∗(s)−Ym∗(s)k2 Xσ=X |i|<k1 ξi|Yn∗ i(s)−Ym∗ i(s)|2+X |i|≥k1 ξi|Yn∗ i(s)−Ym∗ i(s)|2 ≤ǫ2+ 2 X |i|≥k1 ξi|Yn∗ i(s)|2+ 2 X |i|≥k1 ξi|Ym∗ i(s)|2≤5ǫ2, which shows kYn∗(s)−Ym∗(s)kXσ≤√5ǫ. Therefore, {Yn∗(s)}is a Cauchy subsequence of {Yn(s)}and convergent in Xσ. We are now in a position to show the existence of D-pullback random attractors for the cocycle Ψδ. Theorem III.7. Suppose all hypotheses E, F1, F2, G1-G3 and (38) are satisfied. For each δ > 0and s∈ [−ρ, 0], the cocycle Ψδassociated with the random delayed FitzHugh-Nagumo lattice system (1) has a Dpullback random attractor Aδ∈Dand a e D-pullback random attractor e Aδ∈e Din Xρ σ=C([−ρ, 0],Xσ), respectively. Moreover, Aδ=e Aδ. Proof. We mainly proof that Ψδis D-backward asymptotically compact in Xρ σ. That is, for any sequences rn≤τ, tn↑+∞and ψn= (φn, υn)∈ D(rn−tn, θ−tnω), the sequence Ψδ(tn, rn−tn, θ−tnω)ψn=ϕδ rn(·, rn−tn, θ−rnω, ψn) has a convergent subsequence in Xρ σ. For this end, we need to check the following three steps. Step 1. For each s∈[−ρ, 0], we prove {(Ψδ(tn, rn−tn, θ−tnω)ψn)(s)}n∈Nis pre-compact in Xσ=ℓ2 σ×ℓ2 σ. The conclusion holds true on account of Proposition III.6. Step 2. We show the sequence {Ψδ(tn, rn−tn, θ−tnω)ψn}n∈Nin Xρ σis equi-continuous from [−ρ, 0] to Xσ. Let s1, s2∈[−ρ, 0] with s2> s1. By Lemma III.5, there exists an N∈Nsuch that tN≥Tand thus, for all n≥N, k(Ψδ(tn, rn−tn, θ−tnω)ψn)(s1)−(Ψδ(tn, rn−tn, θ−tnω)ψn)(s2)kXσ =kϕδ(rn+s1, rn−tn, θ−rnω, ψn)−ϕδ(rn+s2, rn−tn, θ−rnω, ψn)kXσ Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 25 ≤cZrn+s2 rn+s1  d dν uδ(ν, rn−tn, θ−rnω, φn)  σdν +cZrn+s2 rn+s1  d dν vδ(ν, rn−tn, θ−rnω, υn)  σdν ≤cZrn rn−ρ  d dν uδ(ν, rn−tn, θ−rnω, φn)   2 σdν1 2|s2−s1|1 2 +cZrn rn−ρ  d dν vδ(ν, rn−tn, θ−rnω, υn)   2 σdν1 2|s2−s1|1 2 ≤ce Rδ(τ, ω)|s2−s1|1 2. Hence, the sequence {Ψδ(tn, rn−tn, θ−tnω)ψn}n≥Nin Xρ σis equi-continuous from [−ρ, 0] to Xσ. Note that it is obvious that the finite set {Ψδ(tn, rn−tn, θ−tnω)ψn}n<N in Xρ σis equi-continuous, and so is the whole sequence {Ψδ(tn, rn−tn, θ−tnω)ψn}n∈N. Step 3. We prove the existence and equality of two pullback random attractors. By Steps 1-2, it follows from the Ascoli-Arzel`a theorem that the sequence {Ψδ(tn, rn−tn, θ−tnω)ψn}n∈Nis pre-compact in Xρ σ. Thanks to Proposition III.2, Ψδhas a D-pullback random absorbing set Kδ={Kδ(τ, ω)} ∈ D. Using the abstract results established in38 (Theorem 2.23), we derive that Ψδhas a D-pullback random attractor Aδ∈Din Xρ σ, which is the omega-limit set of Kδ. By D⊂e D, we imply that Kδis also a e D-pullback random absorbing set and Kδ∈e D. By the same argument of Proposition III.6 and the above Steps 1-2, we derive that Ψδis e D-pullback asymptotically compact in Xρ σ. It follows from27 that the existence and uniqueness of a e D-pullback random attractor e Aδ∈e Dare obtained, where e Aδis the omega-limit set of Kδ. Therefore, e Aδ=Aδ∈D. IV. UPPER SEMICONTINUITY OF ATTRACTORS AS CORRELATION TIME TENDS TO INFINITY In this section, we mainly discuss the upper semicontinuity of the pullback random attractor Aδfor problem (1) as δ→+∞. For this end, we need to verify convergence of solutions. Lemma IV.1. Suppose the hypotheses E, F1, F2, G1, G2 and (38) hold. Let ϕδ= (uδ, vδ)and ˆϕ= (ˆu, ˆv)be the solutions to (1) and (3) with initial value ψδ= (φδ, υδ)and ˆ ψ= (ˆ φ, ˆυ), respectively. If kψδ−ˆ ψkXρ σ→0 as δ→+∞,more precisely, dXρ σ(ψδ,ˆ ψ) = sup ν∈[−ρ,0] k(φδ, υδ)(ν)−(ˆ φ, ˆυ)(ν)kXσ→0,as δ→+∞,(141) then ϕδconverges to ˆϕin the following sense: lim δ→+∞sup s∈[−ρ,0] kϕδ(t+s, τ, ω, ψδ)−ˆϕ(t+s, τ, ˆ ψ)k2 Xσ= 0,∀t≥τ, ω ∈Ω.(142) Proof. Let Uδ(ν) = uδ(ν, τ, ω, φδ)−ˆu(ν, τ, ˆ φ), V δ(ν) = vδ(ν, τ, ω, υδ)−ˆv(ν, τ, ˆυ)and Wδ(ν) = ϕδ(ν, τ, ω, ψδ)−ˆϕ(ν, τ, ˆ ψ) = (Uδ(ν), V δ(ν)), which is equipped by the norm kWδk2 Xσ=βkUδk2 σ+αkVδk2 σ. We subtract (3) from (1) to obtain Wδ= (Uδ, V δ)satisfies that for ν≥τ,              dUδ i dν + (AUδ)i+λUδ i+αV δ i=Fi(uδ i(ν)) −Fi(ˆui(ν)) + fi(uδ i(ν−(ρ)(ν))) −fi(ˆui(ν−(ρ)(ν))) + Gi(ν, uδ i)Gδ(θνω), dV δ i dν +ςV δ i−βUδ i=fi(vδ i(ν−(ρ)(ν))) −fi(ˆvi(ν−(ρ)(ν))), (143) where Uδ= (Uδ i)i∈Zand Vδ= (Vδ i)i∈Z. Taking the inner product of (143) with (2βξiUδ i,2αξiVδ i)and summing up the product over i∈Z, it follows that d dν (βkUδk2 σ+αkVδk2 σ) + 2κ(βkUδk2 σ+αkVδk2 σ) 32 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems +2 1−ρ∗Zν τ−skUρ(r)k2 σdr +2 1−ρ∗Zτ+T τkˆu0(h−1(r+s)) −ˆu0(r)k2 σdr. Similarly, we deduce Zν τ−skˆvρ(r+s−(ρ)(r+s)) −ˆv0(r)k2 σdr ≤2ρ0 1−ρ∗ sup s∈[−ρ,0] kˆυρ(s)−ˆυ0k2 σ+ 2 Zτ+2ρ τkˆv0(r)−ˆυ0k2 σdr (172) +2 1−ρ∗Zν τ−skVρ(r)k2 σdr +2 1−ρ∗Zτ+T τkˆv0(h−1(r+s)) −ˆv0(r)k2 σdr. It follows from (169)-(172) that kWρ(ν)k2 Xσ≤c2Zν τ−s kWρ(r)k2 Xσdr +c3d∗ Xρ σ(ˆ ψρ,ˆ ψ0)2 + 2βkˆ φ0−ˆu0(τ−s, τ, ˆ φ0)k2 σ+ 2αkˆυ0−ˆv0(τ−s, τ, ˆυ0)k2 σ +c4Zτ+2ρ τ (kˆu0(r)−ˆ φ0k2 σ+kˆv0(r)−ˆυ0k2 σ)dr +c5Zτ+T τkˆu0(h−1(r+s)) −ˆu0(r)k2 σ+kˆv0(h−1(r+s)) −ˆv0(r)k2 σdr +c1Zτ+T τ (kg(r+s)−g(r)k2 σ+kh(r+s)−h(r)k2 σ)dr. (173) Applying the Gronwall lemma to (173), we deduce, for all ν∈[τ−s, τ +T], kWρ(ν)k2 Xσ≤c3ec2Td∗ Xρ σ(ˆ ψρ,ˆ ψ0)2+ 2βec2Tkˆ φ0−ˆu0(τ−s, τ, ˆ φ0)k2 σ+ 2αec2Tkˆυ0−ˆv0(τ−s, τ, ˆυ0)k2 σ +c4ec2TZτ+2ρ τ (kˆu0(r)−ˆ φ0k2 σ+kˆv0(r)−ˆυ0k2 σ)dr +c5ec2TZτ+T τkˆu0(h−1(r+s)) −ˆu0(r)k2 σ+kˆv0(h−1(r+s)) −ˆv0(r)k2 σdr +c1ec2TZτ+T τ (kg(r+s)−g(r)k2 σ+kh(r+s)−h(r)k2 σ)dr. (174) By (162), we imply the first term on the right-hand side of (174) tends to zero as ρ→0.Then we infer from the continuity of ˆu0(·, τ, ˆ φ0),ˆv0(·, τ, ˆυ0)at τand s∈[−ρ, 0] that 2βec2Tkˆ φ0−ˆu0(τ−s, τ, ˆ φ0)k2 σ+ 2αec2Tkˆυ0−ˆv0(τ−s, τ, ˆυ0)k2 σ +c4ec2TZτ+2ρ τ (kˆu0(r)−ˆ φ0k2 σ+kˆv0(r)−ˆυ0k2 σ)dr →0,as ρ→0. Since ˆu0,ˆv0are uniformly continuous over [τ, τ +T+ρ], then the third line of (174) is bounded by c5ec2TZτ+T τkˆu0(h−1(r+s)) −ˆu0(r)k2 σ+kˆv0(h−1(r+s)) −ˆv0(r)k2 σdr →0, as ρ→0. Thanks to g, h ∈L2 loc(R, ℓ2 σ)and s∈[−ρ, 0], the last line of (174) satisfies c1ec2TZτ+T τ (kg(r+s)−g(r)k2 σ+kh(r+s)−h(r)k2 σ)dr →0,as ρ→0. Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 33 Collecting the above estimations, we deduce that for all ν∈[τ−s, τ +T]and s∈[−ρ, 0], kWρ(ν)k2 Xσ→0,as ρ→0.(175) We now consider the other case ν∈[τ, τ −s]. Let µ=ν−τ. Then we obtain ν=µ+τand 0≤µ≤ρ. Therefore, kWρ(ν)k2 Xσ=kˆϕρ(ν+s, τ, ˆ ψρ)−ˆϕ0(ν, τ, ˆ ψ0)k2 Xσ ≤2kˆϕρ(ν+s, τ, ˆ ψρ)−ˆ ψ0k2 Xσ+ 2kˆϕ0(ν, τ, ˆ ψ0)−ˆ ψ0k2 Xσ ≤2 sup s∈[−ρ,0] kˆ ψρ(s)−ˆ ψ0k2 Xσ+ 2kˆϕ0(µ+τ, τ, ˆ ψ0)−ˆ ψ0k2 Xσ. By the continuity of ˆϕ0= (ˆ φ0,ˆυ0)at τ,µ∈[0,−s]and the condition (162), we imply the above inequality goes to zero as ρ→0, which together with (175), yields that for all ν∈[τ, τ +T]and s∈[−ρ, 0], (163) holds true. Lemma V.3. Let the hypotheses E, F1, F2, G1-G3 and (38) be satisfied. If ρn→0,t∈Rand ψn= (φn, υn)∈ Aρn(t)⊂ Xρn σ, then there exist ˆ ψ0= (ˆ φ0,ˆυ0)∈ Xσand an index subsequence {n∗}of {n}such that d∗ Xρn∗ σ(ψn∗,ˆ ψ0) = sup s∈[−ρn∗,0] kψn∗(s)−ˆ ψ0kXσ→0,as n∗→ ∞.(176) Proof. Take a sequence τn→ −∞. By the invariance of Aρn(·), there exists a ˆ ψn:= (ˆ φn,ˆυn)∈ Aρn(τn) such that ψn= Ψρn(t, τn)ˆ ψn.(177) By Aρn∈Dρn, and using the same method as in Step 1 of Lemma V.1, we deduce that {(Ψρn(t, τn)ˆ ψn)(0)}n∈N is pre-compact in Xσ=ℓ2 σ×ℓ2 σ, and thus there exist a ˆ ψ0:= (ˆ φ0,ˆυ0)∈ Xσand an index subsequence {n∗} of {n}such that k(Ψρn∗(t, τn∗)ˆ ψn∗)(0) −ˆ ψ0kXσ→0,as n∗→+∞, which implies that for given any ǫ > 0, there exists N1≥1such that for all n∗≥N1, k(Ψρn∗(t, τn∗)ˆ ψn∗)(0) −ˆ ψ0kXσ≤ǫ. (178) By the arguments as in Step 2 of Lemma V.1, we imply that there exists ι > 0with |s1−s2|< ι such that for all ǫ > 0, k(Ψρn∗(t, τn∗)ˆ ψn∗)(s1)−(Ψρn∗(t, τn∗)ˆ ψn∗)(s2)kXσ≤ǫ. Since ρn∗→0as n∗→+∞, there exists N2≥N1such that ρn∗< ι for all n∗≥N2, then k(Ψρn∗(t, τn∗)ˆ ψn∗)(s)−(Ψρn∗(t, τn∗)ˆ ψn∗)(0)kXσ≤ǫ, (179) for all s∈[−ρn∗,0]. It follows from (177)-(179) that there exists N3≥N2such that kψn∗(s)−ˆ ψ0kXσ=k(Ψρn∗(t, τn∗)ˆ ψn∗)(s)−ˆ ψ0kXσ ≤ k(Ψρn∗(t, τn∗)ˆ ψn∗)(s)−(Ψρn∗(t, τn∗)ˆ ψn∗)(0)kXσ +k(Ψρn∗(t, τn∗)ˆ ψn∗)(0) −ˆ ψ0kXσ≤2ǫ, for all n∗≥N3and s∈[−ρn∗,0], which yields (176) as desired. 34 Dynamical stability of random delayed FitzHugh-Nagumo lattice systems Theorem V.4. Let the hypotheses E, F1, F2, G1-G3,(38) be satisfied. Suppose Aρis the Dρ-pullback attractor of deterministic delayed lattice system (3) and A0is the D0-pullback attractor of deterministic non-delayed lattice system (158). Then Aρconverges to A0, i.e. lim ρ→0d∗ Xρ σ(Aρ(t),A0(t)) = 0,∀t∈R.(180) Proof. If (180) does not hold true, then there exist ǫ > 0, ρn→0and ψn:= (φn, υn)∈ Aρn(t)such that d∗ Xρn σ(ψn,A0(t)) ≥ǫ, ∀n∈N.(181) Thanks to (176) in Lemma V.3, there exist a subsequence ψn(relabeled the same) and an element ˆ ψ0:= (ˆ φ0,ˆυ0)∈ Xσsuch that lim n→∞ sup s∈[−ρn,0] kψn(s)−ˆ ψ0kXσ= 0.(182) We now prove that ˆ ψ∈ A0(t). By the invariance of Aρn, there exists ˆ ψk n:= (ˆ φk n,ˆυk n)∈ Aρn(τk)such that ψn= Ψρn(t, τk)ˆ ψk n,∀n, k ∈N,(183) where τk→ −∞ as k→+∞. By (176) in Lemma V.3, there exist a subsequence of ˆ ψk nand an element ˆ ψk∈ Xσsuch that d∗ Xρn∗ σ(ˆ ψk n∗,ˆ ψk)→0,as n∗→+∞. It follows from a diagonal process that there exists an index subsequence (relabeled the same) of {n∗}such that lim n∗→+∞sup s∈[−ρn∗,0] kˆ ψk n∗(s)−ˆ ψkkXσ→0,∀k∈N.(184) By (163) in Lemma V.2, we have lim n∗→+∞sup s∈[−ρn∗,0] kΨρn∗(t, τk)ˆ ψk n∗(s)−Ψ0(t, τk)ˆ ψkk2 Xσ= 0,∀k∈N, which, together with (182) and (183), implies ˆ ψ0= Ψ0(t, τk)ˆ ψk,∀k∈N.(185) Since Kρnis a pullback Dρn-absorbing set, and by the invariance of Aρn, there exist a ˆτk:= ˆτk(τk,Aρn)≤τk such that Aρn(τk) = Ψρn(τk,ˆτk)Aρn(ˆτk)⊂ Kρn(τk), which shows ˆ ψk n∈ Kρn(τk). Combining (161) and (184), we obtain, for all k∈N, kˆ ψkk2 Xσ= lim n→+∞kˆ ψk n(0)k2 Xσ≤lim sup n→+∞kˆ ψk nk2 Xρ σ≤ kK0(τk)k2 Xσ. As A0(·)is a pullback D0-attracting set, and by (185) and K0∈ D0, we deduce d∗ Xσ(ˆ ψ0,A0(t)) ≤d∗ Xσ(Ψ0(t, τk)ˆ ψk,A0(t)) ≤d∗ Xσ(Ψ0(t, τk)K0(τk),A0(t)) →0, as k→ ∞, which implies ˆ ψ0∈ A0(t). We then infer from (182) that d∗ Xρn∗ σ(ψn,A0(t)) ≤sup s∈[−ρn,0] kψn(s)−ˆ ψ0kXσ+ d∗ Xσ(ˆ ψ0,A0(t)) →0 as n→ ∞. This contradicts with (181). Dynamical stability of random delayed FitzHugh-Nagumo lattice systems 35 ACKNOWLEDGMENT Shuang Yang was supported by the China Scholarship Council (CSC No. 202006990054). This work was done when Shuang Yang visited the Department of Differential Equations and Numerical Analysis at the University of Sevilla. She would like to express her thanks to all people there for their kind hospitality. The research of the first two authors has been partially supported by the National Natural Science Foundation of China (No. 12271444). 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