Discretization of Asymptotically Stable Stationary Solutions of Delay Differential Equations with a Random Stationary Delay

DISCRETIZATION OF ASYMPTOTICALLY STABLE STATIONARY SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS WITH A RANDOM STATIONARY DELAY TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL Abstract. We prove the existence of a stationary random solution to a delay random ordinary differential system which attracts all other solutions in both pullback and forwards senses. The equation contains a one-sided dissipative Lipschitz term without delay, while the random delay appears in a globally Lipschitz one. The delay function only needs to be continuous in time. Moreover, we also prove that the split implicit Euler scheme associated to the random delay differential system generates a discrete time random dynamical system which also possesses a stochastic stationary solution with the same attracting property, and which converges to the stationary solution of the delay random differential equation pathwise as the stepsize goes to zero. 1. Introduction We consider ordinary delay differential equations of the form (1) d dtx(t) = F(x(t))+G(x(t−τ(t, ω))) with a random delay τ(t, ω) which has continuous sample paths with bounded values τ(t, ω)∈[τ∗, τ∗] where 0 ≤τ∗≤τ∗are finite deterministic numbers and ω∈ Ω for a given probability space (Ω,F,P). In particular, we will be more interested in the case where the delay is a stationary stochastic process, for example τ(t, ω) = 1 + cos2Ut(ω) for an Ornstein-Uhlenbeck process Ut. The function F:Rd→Rdis assumed to be continuously differentiable and satisfy a one-sided dissipative Lipschitz condition (2) hx−y, F (x)−F(y)i ≤ −Lkx−yk2 dfor all x, y ∈Rd, where L > 0 and k·kdis the Euclidean norm on Rd, and function G:Rd→Rdis assumed to satisfy the global Lipschitz condition (3) kG(x)−G(y)kd≤LGkx−ykdfor all x, y ∈Rd. Caraballo et al. [4] have considered the existence of nonautonomous pullback attractors for deterministic time-variable delays. Their results apply pathwise here Key words and phrases. random delay, pullback attractor, stationary solution, split implicit Euler scheme (2000) Mathematics Subject Classification: 34D45, 34K20, 34K25, 47H20, 58F39, 73K70. Partly supported by Ministerio de Educaci´on y Ciencia (Spain) and FEDER (European Community), project MTM2005-01412. Peter Kloeden has been also supported by the same Ministerio under the grant SAB2004-0146, within the Programa de Movilidad del Profesorado universitario espa˜nol y extranjero. 1 2 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL too. Our work here differs in that we use the cocycle rather than process formulation of nonautonomous dynamical systems, which allows us to use the theory of random dynamical systems. In addition, we do not assume that the delay term Gis bounded and we use Razumikhin-Lyapunov theory to establish dissipativity of the system which allows to weaken the differentiability assumption on the delay function in [4] to be only continuous. We will also investigate the effect of time discretization with a constant time step ∆>0 for which N∆:= τ∗/∆ is an integer. Let tn=n∆ for n≥ −N∆and define Nn(∆, ω) := bτ(θtnω)/∆c(integer part) for n≥0. We apply to the random DDE (1) the split implicit Euler scheme (SIES) (4) xn+1 =xn+F(xn+1) ∆ + G(xn−Nn(∆,ω)) ∆, n = 0,1,2,· · · , which is essentially a difference equation in Rdof random order or delay requiring initial data (5) x−N∆, x−N∆+1,· · · , x−1, x0. We will take x−j:= φ0(−j∆) for j= 0,1, · · · ,N∆, where φ0∈C([−τ∗,0],Rd) is the initial value under consideration of the random DDE (1). Our main result is the following. Theorem 1.1. In addition to the previous assumptions, suppose LG< L and that the delay function is given by τ(t, ω) = τ(θtω), where θis a metric dynamical system on Ωrepresenting the noise (see Section 2 below) and t7→ τ(θtω)is a continuous function from Rinto [τ∗, τ∗]for each ω∈Ω. Then, the DDE (1) and the SIES (4) generate, respectively, continuous and discrete time random dynamical systems, each of which has a stochastic stationary solution which pathwise attracts in both forward and pullback senses all other trajectories of their respective equations. Moreover, the numerical stochastic stationary solution converges to that of the DDE pathwise as the stepsize converges to zero. In Section 2 we recall some background on random dynamical systems. The existence of a stationary solution for the delay system (1) which attracts all other solutions in both the pullback and forwards senses is proved in Section 3. A Razumikhin-Lyapunov argument is exploited in the proofs so that no differentiability of the delay function is needed. Finally, we prove in Section 4 that the split implicit Euler scheme (4) replicates the behaviour of the random DDE (1). The convergence of the numerical attractor to the stationary solution to the DDE (1) is finally established in Section 5. 2. Random dynamical systems We need some background notation and results. Let (Ω,F,P) be a probability space and let (X, dX) be a metric space and let T+=R+for a continuous time system and T+=Z+for a discrete time system. Arnold [1] defined a random dynamical system (RDS) (θ, Φ) on Ω × X in terms of a metric dynamical system θon Ω, which represents the noise driving the system, and a cocycle mapping Φ : T+×Ω× X → X , which represents the dynamics in the RANDOM DELAYS 3 state space Xand satisfies the properties 1). Φ(0, ω, φ0) = φ0for all φ0∈ X and ω∈Ω; 2). Φ(s+t, ω, φ0) = Φ(s, θtω, Φ(t, ω, φ0)) for all s,t≥0; φ0∈ X and ω∈Ω; 3). (t, φ0)7→ Φ(t, ω, φ0) is continuous for each ω∈Ω; 4). ω7→ Φ(t, ω, φ0) is F-measurable for all (t, φ0)∈T+× X . For example, for an Ito stochastic differential equation on Rddriven by a twosided scalar Wiener process Wt, i.e. defined for all t∈R,θis defined by θtω(·) = ω(t+·)−ω(·) on a canonical sample space Ω = C0(R,R) (the space of continuous functions from Rinto itself and vanishing at zero) and Φ on the the space X=Rd is defined by Φ(t, ω, x0) = X(t; 0, ω, x0), the solution of the SDE starting at x0at time 0. See [1] for more details. In what follows, we consider as our probability space the canonical one described above. Although a more general expression for the random delay could be possible, in this paper we only consider a time dependent random delay of the form τ(t, ω) := τ(θtω) where τ: Ω →[τ∗, τ∗],for some finite τ∗≥τ∗≥0,is a given F-measurable function. Then, the solutions of the random delay DE (1) generate a continuous time RDS Φ(·,·,·) on the Banach state space X=C([−τ∗,0],Rd) with the norm kφkX= max−τ∗≤s≤0kφ(s)kd. To fix our notation, we denote by x(·;t0, ω, φ) the solution of the following problem (6) (d dtx(t) = F(x(t)) + G(x(t−τ(θtω))), t ≥0, x(t0+s) = φ(s), s ∈[−τ∗,0], where φ∈ X. For t≥0, we denote by xt(·;t0, ω, φ) the element in C([−τ∗,0],Rd) defined by xt(s;t0, ω, φ) = x(t+s;t0, ω, φ), s ∈[−τ∗,0]. Then, Φ is defined as follows: Φ : R+×Ω× X −→ X (t, ω, φ)7→ Φ(t, ω, φ) := xt(·; 0, ω, φ). The cocycle mapping properties 1–3 above follow pathwise as in the nonrandom variable case considered in [4], while the measurability property 4 follows from the fact that the projection x(t) = Φ(t, ω, φ0)(0) in Rdof the solution Φ(t, ω, φ0) in X satisfies the integral equation x(t) = x(0) + Zt 0 F(x(s)) ds +Zt 0 G(x(s−τ(θsω))) ds, the right hand side of which is measurable in ωin view of the measurability of τand θt, the continuity of the solution and the mapping G. Similarly, the split implicit Euler scheme (4) with constant stepsize ∆ applied to the random delay DE (1) generates a discrete time RDS on the state space R(1+N∆)dwith dimension depending on the stepsize ∆. 4 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL A family b A={A(ω), ω ∈Ω}of nonempty measurable compact subsets of Xis called Φ-invariant if Φ(t, ω, A(ω)) = A(θtω) for all t≥0 and is called a random attractor if in addition it is pathwise pullback attracting in the sense that H∗ X(Φ(t, θ−tω, D(θ−tω)), A(ω)) →0 as t→+∞ for all suitable (i.e. in a given attracting universe) families of b D={D(ω), ω ∈Ω} of nonempty measurable bounded subsets of X. Here H∗ Xis the Hausdorff semidistance between the closed and bounded subsets of X. The following result [1] ensures the existence of a random attractor for RDS on a metric space. Theorem 2.1. Let (θ, Φ) be a continuous or discrete time RDS on Ω×X such that Φ(t, ω, ·) : X → X is a compact operator for each fixed t > 0and ω∈Ω. If there exists a family b B={B(ω), ω ∈Ω}of nonempty measurable closed and bounded subsets of Xand a Tb D,ω ≥0such that Φ(t, θ−tω, D(θ−tω)) ⊂B(ω),∀t≥Tb D,ω for all families b D={D(ω), ω ∈Ω}in the given attracting universe, then the RDS (θ, Φ) has a random attractor b A={A(ω), ω ∈Ω}with the component subsets defined for each ω∈Ωby A(ω) = \ s>0[ t≥s Φ(t, θ−tω, B(θ−tω)) dX . The family B(ω) is called a pullback absorbing family for the RDS. Note that if the random attractor consists of singleton sets, i.e A(ω) = {X∗(ω)} for some random variable X∗with X∗(ω)∈ X , then X∗(t, ω) := X∗(θtω) is a stationary stochastic process on X. 3. Stationary stochastic solutions of the DDE We first show that the delay system (1) is pathwise “extremely stable” in the terminology of Yoshizawa. Thus all solutions converge together – then we will use the theory of random dynamical systems to show that they converge to a stochastic stationary solution. To prove this fact we will use a Razumikhin-Lyapunov type argument which is an improvement of some results in [11] which is also mentioned in [12]. We also need the following lemma. Lemma 3.1. Let x(·)and y(·)be two solutions to the random DDE (1). Then, denoting by R(t) := kx(t)−y(t)kd, it follows that (7) d dtR2(t)≤ −2LR2(t)+2LGR(t)R(t−τ(t)),∀t≥0. RANDOM DELAYS 5 Proof. Let us fix ω∈Ω and write τ(t) instead of τ(θtω).Observe that, thanks to our assumptions, if x(·) and y(·) are two solutions to the random DDE (1), we have d dtkx(t)−y(t)k2 d= 2 x(t)−y(t),d dtx(t)−d dty(t) = 2 hx(t)−y(t), F(x(t)) −F(y(t))i +hx(t)−y(t), G(x(t−τ(t))) −G(y(t−τ(t)))i ≤ −2Lkx(t)−y(t)k2 d + 2LGkx(t)−y(t)kdkx(t−τ(t)) −y(t−τ(t)kd.  We now have the following result. Theorem 3.2. If LG< L, then kx(t)−y(t)kd→0pathwise as t→ ∞ for any two solutions x(·)and y(·)of the random DDE (1). Proof. Let us fix ω∈Ω and write τ(t) instead of τ(θtω).Now, we consider a fixed solution y(·) to Eq. (1) defined globally in time, and take another arbitrary solution x(·) satisfying the same condition. First, we need to write a differential system which has zero and x(·)−y(·) as solutions. Then, we will prove that zero is a global attractor. Indeed, let us notice that d dt (x(t)−y(t)) = F(x(t)) −F(y(t)) + G(x(t−τ(t))) −G(y(t−τ(t))), t ≥0 =F(y(t) + x(t)−y(t)) −F(y(t)) +G(y(t−τ(t)) + x(t−τ(t)) −y(t−τ(t))) −G(y(t−τ(t))) =ˆ F(t, x(t)−y(t)) + ˆ G(t, x(t−τ(t)) −y(t−τ(t))), where the functions ˆ Fand ˆ Gare defined as ˆ F(t, v) = F(y(t) + v)−F(y(t)), t ≥0, v ∈Rd, ˆ G(t, v) = G(y(t−τ(t))+v)−G(y(t−τ(t))), t ≥0, v ∈Rd. Denoting u(t) = x(t)−y(t),it follows that d dtu(t) = ˆ F(t, u(t)) + ˆ G(t, u(t−τ(t))) =f(t, ut), t ≥0,(8) where f(t, φ) = ˆ F(t, φ(0)) + ˆ G(t, φ(−τ(t))),for φ∈ X,t≥0. Now we are going to apply Theorem 4.2 in Hale & Lunel [12] taking into account that the assumption in this theorem which imposes on fin (8) to take R+×(bounded sets of X) into bounded sets in Rdhas been weakened by Burton in [2] to satisfy only that ftakes bounded and closed sets from R+×X into bounded sets of Rd,which in our case is obviously fulfilled thanks to the continuity of Fand 6 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL the Lipschitz continuity of G. Then, denoting R(t) = kx(t)−y(t)kd=ku(t)kd, we have from Lemma 3.1 d dtR2(t)≤ −2LR2(t)+2LGR(t)R(t−τ(t)), t ≥0. Taking now V(v) = ||v||2 dfor v∈Rd, it follows that V(u(t)) = R2(t) and d dtV(u(t)) = d dtR2(t) ≤ −2LR2(t)+2LGR(t)R(t−τ(t)) ≤ −2(L−LG)R2(t), provided V(u(t−τ(t))) = R2(t−τ(t)) < R2(t) = V(u(t)). On the other hand, if we consider again the Lyapunov-Razumikhin function V(v) = ||v||2 dfor v∈Rd,and the real function p(s) = q2s, s ∈R+for some q > 1 such that L−qLG>0 (which is possible since L−LG>0), we then have d dtV(u(t)) = d dtR2(t) ≤ −2LR2(t)+2LGR(t)R(t−τ(t)) ≤ −2(L−qLG)R2(t), provided p(V(u(t))) = q2R2(t)> V (u(t−τ(t))) = R2(t−τ(t)).Consequently, by Theorem 4.2 in [12] it follows that u= 0 is a global attractor for the problem (8), and the proof is complete.  We adapt the definition in Hale ([10], page 104) of uniformly ultimately bounded. Definition 3.3. The random DDE (1) is said to be pathwise uniformly ultimately bounded if there is a β > 0such that for each ωand α > 0there is a T0(ω, α)>0 such that the solution of (1) satisfies kx(t;t0, ω, φ)kd≤βfor t≥t0+T0(ω, α) for all t0∈Rand φ∈ X with kφkX≤α. The following theorem will also be proved using Razumikhin-Lyapunov arguments. Theorem 3.4. If LG< L, then the random DDE (1) is pathwise uniformly ultimately bounded. Proof. ¿From assumptions (2) and (3) with y= 0 we have (9) hx, F(x)i≤−Lkxk2 d+kxkdkF(0)kdfor all x∈Rd, and (10) kG(x))kd≤LGkxkd+kG(0)kdfor all x∈Rd. RANDOM DELAYS 7 Writing τ(t) again instead of τ(θtω) and using the inequalities (9)–(10) we have d dtkx(t)k2 d= 2 x(t),d dtx(t) = 2 hx(t), F(x(t))i+ 2 hx(t), G(x(t−τ(t)))i ≤ −2Lkx(t)k2 d+ 2kx(t)kdkF(0)kd+ 2kx(t)kd(LGkx(t−τ(t))kd+kG(0)kd) =−2Lkx(t)k2 d+ 2LGkx(t)kdkx(t−τ(t))kd+ 2kx(t)kd(kF(0)kd+kG(0)kd) and, for a small enough γ > 0 such that L−LG−γ > 0, (11) d dtR2(t)≤ −2(L−γ)R2(t)+2LGR(t)R(t−τ(t)) + C0 where R(t) := kx(t)kdand C0:= 1 2γ(kF(0)kd+kG(0)kd)2. Now we apply Theorem 4.3 in Hale [12] for each ω. To this end, we take q > 1 such that L−LGq−γ > 0,define p(s) = q2sfor s > 0,and w(s) = 2(L−LGq−γ)s for s > 0.Now, we observe that d dtR2(t)≤ −2(L−γ)R2(t)+2LGR(t)R(t−τ(t)) + C0 ≤ −2(L−γ−LGq)R2(t) + C0 provided R2(t−τ(t)) < p(R2(t)).In addition, there exists H≥0 such that d dtR2(t)≤ −2(L−γ−LGq)R2(t) + C0 ≤ −2(L−γ−LGq)R(t), provided that R(t)≥H. This completes the proof.  Since we have uniform ultimate boundedness and, as one can prove by straightforward computations that (12) x(t;t0, θt1ω, φ) = x(t+t1;t0+t1, ω, φ), for t, t0, t1∈R,ω∈Ω, φ ∈ X, then we have that T0(θtω, α)≡T0(ω, α) for all t∈ R. Indeed, observe that the uniform ultimate boundedness implies that ||x(t;t0, ω, φ)||d≤βfor t≥t0+T0(ω, α). Applying this definition for θt1ωinstead of ω, we have ||x(t;t0, θt1ω, φ)||d≤βfor t≥t0+T0(θt1ω, α), but, thanks to (12), we should have ||x(t;t0, θt1ω, φ)||d=||x(t+t1;t0+t1, ω, φ)||d≤βfor t≥t0+T0(θt1ω, α) and ||x(t+t1;t0+t1, ω, φ)||d≤βfor t+t1≥t0+t1+T0(ω, α), so T0(θt1ω, α) = T0(ω, α) for any t1∈R. Hence we have both pullback and forward ultimate boundedness, i.e. absorbing sets, for each fixed ωand we can take B(ω) to be the ball of radius βabout the origin in the space X. 8 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL The solution mapping Φ(t, ω, ·) of our pathwise random DDE (1) is compact from the space Xinto itself for each t > 0 and ω∈Ω. This follows from the fact that the solutions are Lipschitz in time even if the initial value function φ0∈C([−τ∗,0],R) is only continuous, i.e. (13) kx(t+ρ)−x(t)kd≤Zt+ρ t kF(x(s))kdds +Zt+ρ t kG(x(s−τ(s)))kdds ≤Kρ, for t,ρ≥0, where K:= maxx∈CkF(x)kd+ maxx∈CkG(x)kdand Cis a compact subset of Rdwhich contains the DDE solution over the time interval [0, T]. Hence the pathwise random DDE (1) has a random attractor which pathwise attracts all bounded subsets in both the pullback and forward senses. By Theorem 3.2 all solutions converge together as time increases, which means that the random attractor consists of singleton sets, i.e. is a stochastic stationary solution X∗(·)∈ X. It is pathwise asymptotically attracting in both the pullback and forward senses. 4. Stationary stochastic solutions of the SIES Our aim here is to show that the split implicit Euler scheme (SIES) (4) replicates the behaviour of the random DDE (1) projected into Rd, in particular that it has a unique entire solution {x∗ n(ω), n ∈Z}, which attracts all other solutions pathwise in both forwards and pullback senses, and is stationary in the sense that the N∆d-dimensional random vectors X∗ n,∆with realisations X∗ n,∆(ω) =x∗ n−N∆(ω), x∗ n−N∆+1(ω),· · · , x∗ n(ω)>are stationarily distributed in R(1+N∆)d. Moreover , we will show that the X∗ 0,∆converge in an appropriate sense to X∗(ω) for each ω∈Ω as ∆ →0. For this we will show that the SIES generates a discretetime random dynamical system on R(1+N∆)d. We need discrete time analogues of Theorems 3.2 and 3.4. As in their proofs we have the following inequalities. For notational brevity we write Nn(ω) for Nn(∆, ω) when ∆ is fixed. Lemma 4.1. Any two solutions xnand ynof the SIES (4) satisfy kxn+1 −yn+1kd≤1 1 + L∆kxn−ynkd+LG∆ 1 + L∆kxn−Nn(ω)−yn−Nn(ω)kd, kxn+1kd≤1 1 + L∆kxnkd+LG∆ 1 + L∆kxn−Nn(ω)kd +∆ 1 + L∆(kF(0)kd+kG(0)kd) for n= 0,1,2,· · · . RANDOM DELAYS 9 Proof. The proof of the first inequality follows from the fact that kxn+1 −yn+1k2 d=hxn+1 −yn+1, xn−yni+hxn+1 −yn+1, F(xn+1)−F(yn+1)i∆ +xn+1 −yn+1, G(xn−Nn(ω))−G(yn−Nn(ω))∆ ≤ kxn+1 −yn+1kdkxn−ynkd−L∆kxn+1 −yn+1k2 d +LG∆kxn+1 −yn+1kdkxn−Nn(ω)−yn−Nn(ω)kd, while the proof of the second inequality follows from the fact that kxn+1k2 d=hxn+1, xni+hxn+1 −0, F(xn+1)−F(0)i∆ + hxn+1, F(0)i∆ +xn+1 −0, G(xn−Nn(ω))−G(0)∆ + hxn+1, G(0)i∆ ≤ kxn+1kdkxnkd−L∆kxn+1k2 d+kxn+1kdkF(0)kd∆ +LG∆kxn+1kdkxn−Nn(ω)kd+kxn+1kdkG(0)kd∆.  Theorem 4.2. Suppose that LG< L. Then kxn−ynkd→0as n→ ∞ for two solutions xnand ynof the SIES (4). Proof. Define Rn:= kxn−ynkd≥0 for n≥ −N∆. The first inequality of Lemma 4.1 then reads Rn+1 ≤γ(∆)Rn+γ(∆)LG∆Rn−Nn(ω), n ≥1, where γ(∆) := 1 1 + L∆. Suppose that the initial values R−j∈[0, B] for j= 0, 1, . . .,N∆and define η(∆) := γ(∆)(1 + LG∆), so clearly γ(∆) < η(∆) <1. Then R1≤γ(∆)R0+γ(∆)LG∆R−N0(ω)≤η(∆)B < B. Repeating this argument inductively we obtain Rn≤η(∆)B < B for all n≥1. Consider the norm kxk∞:= max j=0,1,...,N∆ kxjkd on R(1+N∆)and define Rn:= (Rn−N∆,· · · , Rn)>∈R(1+N∆)for n≥0. Then kR0k∞≤Bby assumption, hence kRnk∞≤Bfor n= 1, · · · ,N∆. Thereafter we have kRN∆+1k∞≤γ(∆)kRN∆k∞+γ(∆)LG∆kRN∆k∞=η(∆)kRN∆k∞ and it follows inductively that kRN∆+jk∞≤η(∆)jkRN∆k∞≤η(∆)jB for j≥1. Thus kRnk∞→0 as n→ ∞, which completes the proof of the theorem.  Theorem 4.3. Suppose that LG< L. Then the SIES (4) is pathwise uniformly ultimately bounded.