Slow manifold for a nonlocal stochastic evolutionary system with fast and slow components

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Slow manifold for a nonlocal stochastic evolutionary system with fast and slow components ✩ Lu Bai a , Xiujun Cheng a , Jinqiao Duan b , Meihua Yang a,∗ a School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China b Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

Received 3 July 2015; revised 2 April 2017

Abstract This work is devoted to investigating invariant manifolds for a fast–slow stochastic evolutionary system with nonlocal diffusion. We establish the slow reduction via a random slow manifold, which captures slow dynamics of the original stochastic fast–slow system. A simple example is shown to illustrate this slow reduction method. © 2017 Elsevier Inc. All rights reserved. Keywords: Nonlocal Laplacian operator; Random dynamical systems; Random invariant manifolds; Slow manifolds; Reduced system

1. Introduction Nonlocal partial differential equations with the nonlocal Laplacian operator have attracted a lot of attention recently. The usual Laplacian operator  may be thought as macroscopic manifestation of Brownian motion, as known from the Fokker–Plank equation for a stochastic differential equation with a Brownian motion (a Gaussian process), whereas the nonlocal Laplacian operator (−)α/2 arises in non-Gaussian stochastic systems. For a stochastic differ✩ This work was partly supported by the NSF grant 1620449, NSFC grants (11571125, 11531006, 11371367, 11271290), and Central University Fundamental Research Fund (Grant 2015QT005). * Corresponding author. E-mail address: [email protected] (M. Yang).

http://dx.doi.org/10.1016/j.jde.2017.06.003 0022-0396/© 2017 Elsevier Inc. All rights reserved.

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ential system with a symmetric α-stable Lévy motion (a non-Gaussian stochastic process) Lαt for α ∈ (0, 2), the corresponding Fokker–Planck equation contains the nonlocal Laplacian operator (−)α/2 . See [1,18] for a discussion about this microscopic–macroscopic relation. Nonlocal Laplacian operator appears in complex systems, such as certain heat transfer processes in fractal and disordered media, and fluid flows and acoustic propagation in porous media [28]. A nonlocal diffusion equation also arises in pricing derivative securities in financial markets [4]. The fast–slow stochastic evolutionary systems are appropriate mathematical models for various multi-scale systems under random influences. In this paper, we consider the following fast–slow system of stochastic equations, 1 1 σ u˙  = − (−)α/2 u + f (u , v  ) + √ W˙ t (x),    v˙  = Bv  + g(u , v  ), u |(−1,1)c = 0, 

in H1 ,

(1.1)

in H2 ,

(1.2)

v |(−1,1)c = 0,

(1.3)



where, for x ∈ R and α ∈ (0, 2), 2α ( 1+α 2 ) P.V. u (x, t) = √ π|(− α2 )|



α/2 

(−)

R

u (x, t) − u (y, t) dy, |x − y|1+α

is the so-called nonlocal Laplacian operator, with the Cauchy principal value (P.V.) taken as the limit ofthe integral over R\(x − ε, x + ε), as ε → 0. The Gamma function  is defined by ∞ (r) = 0 t r−1 e−t dt for every r > 0; for more information see [8,36]. The system (1.1)–(1.3) is defined in a separable Hilbert space H = H1 × H2 , a product space of separable Hilbert spaces H1 , H2 , with norm of  · 1 and  · 2 , respectively. The norm for H is  ·  =  · 1 +  · 2 . The Wiener process Wt (x) is defined on the probability space , with natural filtration Ft generated by Wt (x). Other quantities in the system (1.1)–(1.3) are:  is a small positive parameter (0 <   1) representing the ratio of the two time scales, we usually say that u ∈ H1 is the “fast” component, while v  ∈ H2 is the “slow” component; f, g are nonlinearities; σ is noise intensity (a positive parameter); and B is a linear operator satisfying the Condition (I) (Slow evolution), which will be specified in Section 2. The precise conditions on these quantities will be given in Conditions (I)–(III), see Section 2. The main goal for this paper is to establish the slow reduction via a random slow manifold with an exponential tracking property, for sufficiently small  and α ∈ (1, 2). For α ∈ (0, 1], our results do not apply (see Remark 1). The original system can be reduced to a stochastic evolutionary equation with a modified nonlinear term, which is useful for describing slow dynamics of the original fast–slow stochastic system. We first use a stationary solution ([15]) of the associated nonlocal Ornstein–Uhlenbeck equation to convert the original stochastic system to a system with random coefficients, facilitating the investigation of a stochastic slow manifold. Then, we construct the slow invariant manifold by Lyapunov–Perron method (see [11,17]). The key assumption of this approach is that the Lipschitz constant of the nonlinear term in fast component is small enough comparing with the decay rate of the nonlocal Laplacian operator. Invariant manifolds play a significant role in analyzing dynamical behaviors of deterministic systems. It was first introduced by Hadamard [21], then by Lyapunov [27] and Perron [31] for

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Table 1 The numerical approximations of λ1 for various α obtained in [25]. α

0.01

0.1

0.2

0.5

1

1.5

1.8

1.9

1.99

λ1

0.997

0.973

0.957

0.970

1.158

1.597

2.048

2.243

2.442

deterministic systems. It has been further developed by many authors for infinite dimensional deterministic systems; see, e.g., [3,10,12,23]. Recently, there has been intense interest in investigating invariant manifolds for infinite dimensional stochastic systems; see [5,7,14,13,16,17,26, 29] among others. For SDEs with two time scales, Schmalfuß and Schneider [35] have recently investigated random inertial manifolds that eliminate the fast variables, by a fixed point technique based on a random graph transformation [34]. They show that the inertial manifold tends to another so-called slow manifold as the scaling parameter goes to zero. Wang and Roberts [38] further studied the qualitative analysis for the behavior of the slow manifold for fast–slow SDEs on the long time scales. The paper is organized as follows. In Section 2, we list assumptions for the fast–slow system and recall basic results in nonlocal Laplacian operator, random dynamical systems and random invariant manifolds. In Section 3, we convert the original stochastic system to a random system. In Section 4, we establish the existence of a random slow invariant manifold M  possessing an exponential tracking property and a reduced system on the slow manifold. In Section 5, we consider an approximation for random slow manifolds, and a illustrative example is given in Section 6. 2. Preliminaries In this section, we recall several basic information about nonlocal Laplacian operator and random dynamical systems. For more details, see [2,16,17,19,22,39]. We take H1 = L2 (−1, 1), H2 is a separable Hilbert space and denote by Aα the nonlocal Laplacian operator −(−)α/2 , which applied to functions extended over all R by being zero outside of (−1, 1). Now we recall the eigenvalues of Aα in L2 (−1, 1) (their leading term analytical approximations). Lemma 1. ([25]) The eigenvalues of the following spectral problem (−)α/2 e(x) = λe(x), x ∈ (−1, 1), e|(−1,1)c = 0,

(2.1)

where e(·) ∈ L2 (−1, 1), are λj =

jπ 2

−

1 (2 − α)π α + O( ) (j → ∞). 8 j

(2.2)

Moreover, 0 < λ1 < λ2 ≤ · · · ≤ λj ≤ · · ·, for j = 1, 2, · · · . See Table 1 for numerical approximations of the first eigenvalue, and Fig. 1 for the comparison of numerical vs. analytical approximations of this eigenvalue.

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Fig. 1. Comparison of the analytical approximations λ˜ 1 = ( π2 − (2−α)π )α (blue curve) and the numerical approximations 8 of λ1 (red plus sign). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Note that (−Aα )−1 is a bounded linear operator on L2 (−1, 1), and also it is a compact, selfadjoint operator. Owing to Hilbert–Schmidt Theorem [25,33], the eigenfunctions ej (x) of Aα form an orthonormal basis in L2 (−1, 1). These eigenfunctions ej (x) have approximate expressions and are plotted in [25]. Lemma 2. ([39]) The nonlocal Laplacian operator Aα is a sectorial operator, satisfying eAα t L2 (−1,1) ≤ Ce−λ1 t , where C > 0 is a positive constant, independent of t and λ1 . In fast–slow system (1.1)–(1.2), we assume the following conditions. Conditions: (I) (Slow evolution) The linear operator B is a generator of a C0 -group eBt on H2 satisfying eBt v2 ≤ Ce−γ t v2 ,

t ≤ 0,

for all v ∈ H2 , with a constant γ ≥ 0. (II) (Lipschitz condition) Nonlinear functions f and g are continuously differentiable functions, f : H → H1 ,

g : H → H2 ,

with f (0, 0) = g(0, 0) = 0,

fu (0, 0) = fv (0, 0) = gu (0, 0) = gv (0, 0) = 0.

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There exists a positive constant K such that for all (ui , vi ) ∈ H1 × H2 f (u1 , v1 ) − f (u2 , v2 )1 ≤ K(u1 − u2 1 + v1 − v2 2 ), and g(u1 , v1 ) − g(u2 , v2 )2 ≤ K(u1 − u2 1 + v1 − v2 2 ). (III) (Gap condition) The Lipschitz constant K of the nonlinear terms in system is smaller than the decay rate λ1 of Aα , that is, K < λ1 . Before introducing the random dynamical system concept, let us describe the driving dynamical system. Definition 1. Let ( , F , P) be a probability space and θ = {θt }t∈R be a flow on which is defined as a mapping θ : R × → , and satisfies • θ0 = id , • θs θt = θs+t for all s, t ∈ R, • the mapping (t, ω) → θt ω is (B(R) × F , F )-measurable and θt P = P for all t ∈ R. Then the quadruple ( , F, P, θ ) is called a driving dynamical system. We will work on the driving dynamical system represented by the Wiener process Wt . Here, the Wiener process Wt is a L2 -valued Q-Wiener process, Wt (x) =

∞  √ qj wj (t)ej (x), j =1

where 1 wj (t) = √ Wt (x), ej (x) qj are independent, scalar Wiener processes taking  values in R, and nonnegative constants qj ’s are such that the trace of Q satisfying Tr(Q) = ∞ j =1 qj < +∞. To be more precise, let = C0 (R, H1 ) be the continuous paths ω(t) on R with values in H1 such that ω(0) = 0. This set is equipped with the compact-open topology. Let F be the associated Borel σ -field and P be the Wiener measure. Then we identify ω(t) with Wt (ω) = ω(t). The operators θt forming the flow are given by the Wiener shift θt ω(·) = ω(· + t) − ω(t), ω ∈ , t ∈ R.

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Note that the measure P is invariant with respect to the above flow and then the quadruple ( , F, P, θ ) is a driving dynamical system. Definition 2. ([2]) Let (H, dH ) be a Hilbert space with Borel σ -field B(H). A cocycle is a mapping φ : R+ × × H → H,  which is B(R+ ) × F × B(H), B(H) -measurable such that φ(0, ω, z) = z, φ(t + s, ω, z) = φ(t, θs ω, φ(s, ω, z)), for t, s ∈ R+ , ω ∈ , and z ∈ H. Then φ, together with the driving system ( , F, P, θ ), forms a random dynamical system (RDS). A RDS is called continuous (differentiable) if z → φ(t, ω, z) is continuous (differentiable) for t ≥ 0 and ω ∈ . A family of nonempty closed sets M = {M(ω)} contained in a metric space (H,  · H ) is called a random set if for every z ∈ H the mapping ω → inf z − z H , z∈M(ω)

is a random variable. Definition 3. ([18]) A random variable z(ω), taking values in H, is called a stationary orbit for a random dynamical system φ if φ(t, ω, z(ω)) = z(θt ω),

a.s.

for every t . Now we introduce the conception of random invariant manifold. Definition 4. ([20]) A random set M(ω) is called a random positively invariant set if φ(t, ω, M(ω)) ⊂ M(θt ω),

t ≥0

and ω ∈ .

If M can be represented as a graph of a Lipschitz mapping h(·, ω) : H2 → H1 , such that M(ω) = {(h(v), v) : v ∈ H2 }, then M(ω) is called a Lipschitz random invariant manifold. If, in addition, for every z ∈ H, there exists an z ∈ M(ω) such that for all ω ∈ ,

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φ(t, ω, z) − φ(t, ω, z )H ≤ c1 (z, z , ω)e−c2 t z − z H , t ≥ 0, where c1 is a positive random variable depending on z and z , while c2 is a positive constant, then M(ω) is said to have an exponential tracking property. 3. Converting to a random dynamical system In this section, we transform the stochastic evolutionary system (1.1)–(1.2) into a random evolutionary system which generates a RDS, as in [17]. For this purpose, we first establish the existence and uniqueness of solutions for the system (1.1)–(1.2) and the nonlocal Ornstein– Uhlenbeck equation. Lemma 3. Under conditions (I)–(III), the system (1.1)–(1.2) has a unique mild solution. Proof. We rewrite the system (1.1)–(1.2) in the form

u˙  v˙ 



=

1  Aα

0 B

0



u v



+

1    f (u , v )  g(u , v  )



 +

σ ˙ √ W  t

0

.

By Lemma 1 and  > 0, we know that  1 Aα u, u ≤ 0. Thus, 1 Aα is a infinitesimal generator of a strongly continuoussemigroup [30], and taking into account assumption (I) we obtain that the 1 A 0 is a infinitesimal generator of a strongly continuous semigroup. Now, by operator  α 0 B Theorem 7.4 ([15], p. 186) the result follows. 2 Lemma 4. When α ∈ (1, 2), the nonlocal stochastic equation dξ(t) = Aα ξ(t)dt + σ dWt ,

in H1 ,

(3.1)

with Aα the nonlocal Laplacian operator, has the solution t ξ(t) = e

tAα

ξ0 + σ

eAα (t−s) dWs ,

for all

t ≥ 0,

0

where ξ(0) = ξ0 is F0 -measurable. Proof. Note that Aα is a sectorial operator [39], then it generates a C0 -semigroup S(·) in H1 . t Hence we only need to prove that 0 S(r)2 0 dr < +∞. Here L02 denotes the space of Hilbert– L2

Schmidt operators. Because t S(r)2L0 dr 2 0

=

t  ∞ 0 i,j =1

√ |erAα qj ej , ei |2 dr

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=

t  ∞

qj e−2λj r dr

0 j =1

=

∞  qj , 2λj j =1

where

∞

j =1 qj

< +∞, and

∞

1 j =1 λj

(but divergent when α ∈ (0, 1]), we

=

∞

1 j =1 ( j π − (2−α)π )α +O( 1 ) is convergent 2 j t 8 conclude that 0 S(r)2 0 dr < +∞. L

when α ∈ (1, 2)

2

Then by Theorem 5.4 ([15], p. 121), we infer that when α ∈ (1, 2), the equation (3.1) has a solution t ξ(t) = e

tAα

ξ0 + σ

eAα (t−s) dWs ,

for all

t ≥ 0.

0

The proof is complete.

2

Remark 1. This lemma will not hold for α ∈ (0, 1], as seen in the proof that when α ∈ (0, 1].

∞

1 j =1 λj

is divergent

Furthermore, for a fixed , the equation 1 σ dη(t) = Aα η(t)dt + √ dWt  

(3.2)

also has a solution. It is known from [6,16] that the following process η is a stationary solution of equation (3.2),

ησ (ω) =

σ √ 

0

e−Aα s/ dWs  σ η (ω).

−∞

Moreover,

ησ (θt ω) =

σ √ 

t eAα (t−s)/ dWs  σ η (θt ω). −∞

Similarly, η(ω) is the stationary solution of the linear system dη(t) = Aα η(t)dt + σ dWt , with

(3.3)

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9

e−Aα s dWs  σ η(ω),

ησ (ω) = σ −∞

and t ησ (θt ω) = σ

eAα (t−s) dWs  σ η(θt ω). −∞

Denote Wt (ψ ω)  √1 Wt (ω). This is also a Wiener Process, and has the same distribution as Wt (ω), with ψ : → . Then we obtain the following relations by the help of a transformation s = s/, 1 η (θt ω) = √ 

t



t e

Aα (t−s/)

dWs =

−∞

−∞

 1 

eAα (t−s ) d √ Ws  (ω) = η(θt ψ ω), 

(3.4)

and 1 η (ω) = √ 

0



e

−Aα s/

0 dWs =

−∞

−∞

 1 

e−Aα s d √ Ws  (ω) = η(ψ ω). 

(3.5)

Due to s = r − t, we have the relation of η (θt ω) and η (ω) in distribution: 1 η (θt ω) = √ 

t



e −∞

Aα (t−r)/

1 dWr = √ 

0

e−Aα s/ dWs = η (ω).

(3.6)

−∞

Equations (3.4) and (3.5) indicate that η (θt ω) and η (ω) are identically distributed with η(θt ψ ω) and η(ψ ω), respectively. It follows from (3.5) and (3.6) that η (θt ω) and η(ψ ω) have the same distribution. By a random transformation

U V



:= ν(ω, u, v) =

u − σ η (ω) v

 ,

the original evolutionary system (1.1)–(1.2) is converted to the following random evolutionary system 1 1 dU  = Aα U  dt + f (U  + σ η (θt ω), V  )dt,   dV  = BV  dt + g(U  + σ η (θt ω), V  )dt.

(3.7) (3.8)

The state space for system (3.7)–(3.8) is H. Rescaling the time by τ = t/ and using (3.4), system (3.7)–(3.8) becomes

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dU  (τ ) = Aα U  (τ )dτ + f (U  (τ ) + σ η(θτ ψ ω), V  (τ ))dτ,

(3.9)

dV  (τ ) = BV  (τ )dτ + g(U  (τ ) + σ η(θτ ψ ω), V  (τ ))dτ.

(3.10)

These can be rewritten as the integral form τ U (τ ) = 

eAα (τ −s) f (U  (τ ) + σ η(θτ ψ ω), V  (τ ))ds,

(3.11)

−∞

τ V (τ ) = V (0) + 

[BV  (τ ) + g(U  (τ ) + σ η(θτ ψ ω), V  (τ ))]ds.



(3.12)

0

 Let Z  (t, ω, Z0) = U  (t, ω, U0 , V0 ), V  (t, ω, U0 , V0 ) be the solution of (3.7)–(3.8) with initial data Z0 := U  (0), V  (0) = (U0 , V0 ). Then the solution operator of the random evolutionary system (3.7)–(3.8)    t, ω, (U0 , V0 ) = U  (t, ω, U0 , V0 ), V  (t, ω, U0 , V0 ) , defines a random dynamical system. Furthermore φ  (t, ω) :=  (t, ω) + (σ η (θt ω), 0), t ≥ 0, ω ∈ , is the random dynamical system generated by the original fast–slow system (1.1)–(1.2). For convenience, we introduce some notations. Let μ be a positive number satisfying λ1 − μ > K.

(3.13)

In order to specify a space where slow dynamical orbits live, we introduce the following Banach spaces which are the working spaces for random slow manifolds. For a real positive number β:  Cβ1,−



= ϕ : (−∞, 0] → H1 is continuous and sup e

ϕ(t)1 < ∞ ,

t≤0

 Cβ2,−

−βt

= ϕ : (−∞, 0] → H2 is continuous and sup e

 −βt

ϕ(t)2 < ∞ ,

t≤0

with the norms ϕC 1,− = sup e−βt ϕ(t)1 , ϕC 2,− = sup e−βt ϕ(t)2 , β

β

t≤0

t≤0

respectively. Let Cβ− be the product Banach spaces Cβ− := Cβ1,− × Cβ2,− , with the norm zC − = uC 1,− + vC 2,− , z = (u, v) ∈ Cβ− . β

β

β

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4. Slow manifold and slow reduction In this section, we establish the existence of a slow manifold for the random evolutionary system (3.7)–(3.8) introduced in Section 3. Define   − M  (ω)  Z0 ∈ H : Z  (·, ω, Z0 ) ∈ C−μ/ . We use Lyapunov–Perron method to prove that M  (ω) is an invariant manifold described by the graph of a Lipschitz mapping. For this purpose, the following lemma which follows from [16] is needed.  − Lemma 5. Suppose that Z  (·, ω) = U  (·, ω), V  (·, ω) is in C−μ/ . Then Z  (t, ω) is the solu tions of (3.7)–(3.8) with initial data Z0 = (U0 , V0 ) if and only if Z (·, ω) satisfies

U  (t)

V  (t)

=

 1 t Aα (t−s)/ f (U  (s) + σ η (θ ω), V  (s))ds s  −∞ e  t B(t−s) Bt   g(U (s) + σ η (θs ω), V  (s))ds e V0 + 0 e

.

Theorem 1 (Slow manifold). Assume that (I)–(III) hold and  is sufficiently small. Then there exists a Lipschitz random invariant manifold of the random dynamical system defined by (3.7)–(3.8). It is M  (ω) =

   h (ω, V0 ), V0 : V0 ∈ H2 ,

as the graph of the following Lipschitz mapping h (·, ·) : × H2 → H1 , with Lipschitz constant satisfying Lip(h ) ≤

K  (λ1 − μ) 1 − K λ1 1−μ + 

 μ−γ

 .

Proof. Step 1. To construct an invariant manifold for the random evolutionary system (3.7)–(3.8), we first consider integral equation

U  (t) V  (t)

=

 1 t Aα (t−s)/ f (U  (s) + σ η (θ ω), V  (s))ds s  −∞ e  t B(t−s) Bt   g(U (s) + σ η (θs ω), V  (s))ds e V0 + 0 e

, t ≤ 0.

(4.1)

 A solution of (4.1) is denoted by Z  (t, ω, Z0 ) = U  (t, ω, V0 ), V  (t, ω, V0 ) . We use Banach fixed point theorem to prove that Z  (t, ω, Z0 ) is the unique solution of (4.1). Introduce the oper1,− 2,− − − ators J1 : C−μ/ → C−μ/ and J2 : C−μ/ → C−μ/ by means of J1 (z(·))[t] =

1 

t eAα (t−s)/ f (u(s) + σ η (θs ω), v(s))ds, −∞

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t J2 (z(·))[t] = eBt V0

+

eB(t−s) g(u(s) + σ η (θs ω), v(s))ds, 0

for t ≤ 0 and define the mapping J  by J (z(·)) := 

J1 (z(·))

J2 (z(·))

.

− − Firstly, it can be verified that J  maps C−μ/ into itself. Taking z = (u, v) ∈ C−μ/ , the following estimates hold:

1  

t eAα (t−s)/ f (u(s) + σ η (θs ω), v(s))ds1 −∞

t

1 ≤ eμt/ 

e−λ1 (t−s)/ f (u(s) + σ η (θs ω), v(s))1 ds

−∞

K ≤ eμt/ 

t

 e−λ1 (t−s)/ u(s)1 + v(s)2 ds

−∞

K 

≤



t

 e(−λ1 /+μ/)(t−s) ds zC −

−μ/

−∞

K zC − , −μ/ λ1 − μ

= and

t e V0 +

eB(t−s) g(u(s) + σ η (θs ω), v(s))ds2

Bt

0

0

e−γ (t−s) g(u(s) + σ η (θs ω), v(s))2 ds + e−γ t · eμt/ V0 2

≤ eμt/ t

0 ≤ Ke

μt/

e−γ (t−s) (u(s)1 + v(s)2 )ds + V0 2

t

0 ≤K

e

(−γ +μ/)(t−s)

 ds zC −

t

=

K zC − + V0 2 . −μ/ μ − γ

−μ/

+ V0 2

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Hence, J1 (z)C 1,− −μ/

1 = sup  t≤0  ≤

t eAα (t−s)/ f (u(s) + σ η (θs ω), v(s))ds1 −∞

K zC − , −μ/ λ1 − μ

(4.2)

and t J2 (z)C 2,− −μ/

= sup e V0 +

eB(t−s) g(u(s) + σ η (θs ω), v(s))ds2

Bt

t≤0

≤

0

K zC − + V0 2 . −μ/ μ − γ

(4.3)

Combining with the definition of J  , we deduce that 

J (z)C − 

−μ/

 K K ≤ + zC − + V0 2 , −μ/ λ1 − μ μ − γ

and denote by κ(K, λ1 , γ , μ, ) =

K K + . λ1 − μ μ − γ

− Which implies that J  maps C−μ/ into itself. Then, we show that the mapping J  is contractive. − Taking z = (u, v), z¯ = (u, ¯ v) ¯ ∈ C−μ/ ,

J1 (z) − J1 (¯z)C 1,−

−μ/



t

1 ≤ sup eμt/  t≤0

e

−λ1 (t−s)/

 f (u(s) + η (θs ω), v(s)) − f (u(s) ¯ + σ η (θs ω), v(s)) ¯ 1 ds 



−∞



K ≤ sup eμt/  t≤0

t e

−λ1 (t−s)/







u(s) − u(s) ¯ ¯ 1 + v(s) − v(s) 2 ds

−∞

K ≤ sup  t≤0 = and



 t e

(−λ1 /+μ/)(t−s)

−∞

K z − z¯ C − , −μ/ λ1 − μ

ds z − z¯ C −

−μ/

(4.4)

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 J2 (z) − J2 (¯z)C 2,− −μ/

0

≤ K sup e

μt/

t≤0

e

 ds z − z¯ C −

−μ/

t

 0 ≤ K sup

e

t≤0

=

e

−γ (t−s) −μs/

(−γ +μ/)(t−s)

 ds z − z¯ C −

−μ/

t

K z − z¯ C − . μ/ μ − γ

(4.5)

Which implies that J  (z) − J  (¯z)C −

−μ/

≤ κ(K, λ1 , γ , μ, )z − z¯ C −

−μ/

,

where κ(K, λ1 , γ , μ, ) =

K K K + → , λ1 − μ μ − γ λ1 − μ

as  → 0. Then there is a sufficiently small parameter 0 > 0 such that κ(K, λ1 , γ , μ, ) < 1,

for

 ∈ (0, 0 ].

− Therefore, the mapping J  is contractive in C−μ/ , and then (4.1) has a unique solution  −    Z (t, ω, V0 ) = U (t, ω, V0 ), V (t, ω, V0 ) in C−μ/ . Furthermore,

Z  (·, ω, V1 ) − Z  (·, ω, V2 )C −

−μ/

= U  (·, ω, V1 ) − U  (·, ω, V2 )C 1,− + V  (·, ω, V1 ) − V  (·, ω, V2 )C 2,− −μ/

≤

−μ/

K Z  (·, ω, V1 ) − Z  (·, ω, V2 )C − −μ/ λ1 − μ K + Z  (·, ω, V1 ) − Z  (·, ω, V2 )C − + V1 − V2 2 −μ/ μ − γ

= κ(K, λ1 , γ , μ, )Z  (·, ω, V1 ) − Z  (·, ω, V2 )C −

−μ/

+ V1 − V2 2 .

Thus, the following estimate holds: Z  (·, ω, V1 ) − Z  (·, ω, V2 )C −

−μ/

≤

1 V1 − V2 2 , 1 − κ(K, λ1 , γ , μ, )

(4.6)

for all ω ∈ , V1 , V2 ∈ H2 . Step 2. Using the unique solution in Step 1 to construct the mapping h , 1 h (ω, V0 ) = 

0



−∞

 e−Aα s/ f U  (s, ω, V0 ) + σ η (θs ω), V  (s, ω, V0 ))ds,

(4.7)

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then owing to (4.6) there holds 1 K   V1 − V2 2 , γ1 − μ 1 − κ(K, λ1 , γ , μ, )

h (ω, V1 ) − h (ω, V2 )1 ≤ 

for all V1 , V2 ∈ H2 , ω ∈ . It then follows from Lemma 5 that   M  (ω) = h (ω, V0 ), V0 : V0 ∈ H2 . Step 3. Prove that M  (ω) is a random set. We need to show that for every z = (u, v) ∈ H, ω → inf (u, v) − (h (ω, J z ), J z )

z ∈H

(4.8)

is measurable; see Theorem III.9 in Castaing and Valadier ([9], p. 67). Let Hc be a countable dense set of the separable space H. Then the right hand side of (4.8) is equal to inf (u, v) − (h (ω, J z ), J z ),

z ∈Hc

(4.9)

which follows immediately from the continuity of h (ω, ·). The measurability of every expression under the infimum of (4.8) follows since ω → h (ω, J z ) is measurable for every z ∈ H. Step 4. Show that M  (ω) is positively invariant. That is for each Z0 = (U0 , V0 ) ∈ M  (ω), we show that Z  (s, ω, Z0 ) ∈ M  (θs ω) for all s ≥ 0. Note that for each fixed s ≥ 0, Z  (t + s, ω, Z0 ) is a solution of 1 1 dU  = AU  dt + f (U  + σ η (θt (θs ω)), V  )dt,   dV  = BV  dt + g(U  + σ η (θt (θs ω)), V  )dt, with initial datum Z(0) = (U (0), V (0)) = Z  (s, ω, Z0 ). Thus, Z  (t + s, ω, Z0 ) = Z  (t, θs ω, − − , then Z  (t, θs ω, Z  (s, ω, Z0 )) ∈ C−μ/ . Therefore, Z  (s, ω, Z0 )). Since Z  (·, ω, Z0 ) ∈ C−μ/   Z (s, ω, Z0 ) ∈ M (θs ω). This completes the proof. 2 Similar to [20], we have the following result. Lemma 6 (Exponential tracking property). Assume that the assumptions (I)–(III) hold and  > 0 is small enough. Then the Lipschitz invariant manifold for (3.7)–(3.8) obtained in Theorem 1 has the exponential tracking property in the following sense: there exist constants C1 , C2 > 0, for each Z0 = (U0 , V0 ) ∈ H there is a Z¯ 0 = (U¯0 , Y¯0 ) ∈ M  (ω) such that  (t, ω, Z0 ) −  (t, ω, Z¯ 0 ) ≤ C1 e−C2 t Z0 − Z¯ 0 , t ≥ 0. Remark 2. More specific, for any solution Z  = (U  , V  ) of (3.7)–(3.8), there is an orbit Z˜  = (U˜  , V˜  ) on M  satisfies the evolutionary equation V˙˜  = B V˜  + g(h (θt ω, V˜  ) + σ η (θt ω), V˜  ), such that

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Z  − Z˜   ≤

1−



e−μt/ K λ1 −μ

+

K μ−γ

 Z0 − Z˜ 0 , t > 0,

where Z0 = (U  (0), V  (0)), Z˜ 0 = (U˜  (0), V˜  (0)). Remark 3. By the relation between solutions of system (1.1)–(1.2) and system (3.7)–(3.8), if the system (1.1)–(1.2) satisfies the conditions of Theorem 1, then it also has a Lipschitz random invariant manifold M˜  (ω) = M  (ω) + (σ η (ω), 0) = {(h˜  (ω, V0 ), V0 ) : V0 ∈ H2 )}, where h˜  (ω, V0 ) = h (ω, V0 ) + σ η (ω). Applying Remark 1 and Remark 2, we can obtain a reduced system on the slow manifold which captures the dynamical behavior of the original fast–slow system (1.1)–(1.2). Theorem 2 (Slow reduction). Assume that (I)–(III) hold and  is sufficiently small. For every solution z (t) = (u (t), v  (t)) to (1.1)–(1.2), there exists an orbit z˜  (t) = (h (ω, v˜  (t)) + σ η (ω), v˜  (t)), lying on M˜  (ω), which is governed by the following system v˙˜  = B v˜  + g(h (θt ω, v˜  ) + σ η (θt ω), v˜  ), such that, for almost all ω and t ≥ 0, z (t, ω) − z˜  (t, ω) ≤

1−



e−μt/ K λ1 −μ

+

K μ−γ

 z0 − z˜ 0 , t ≥ 0,

where z0 = (u (0), v  (0)), z˜ 0 = (u˜  (0), v˜  (0)). 5. Approximation of a random slow manifold In this section, we approximate the slow manifold for sufficiently small . Expand the solution of (3.9) in the form as in [37,32]: U  (τ ) = U ,0 (τ ) + U ,1 (τ ) + · · · ,

(5.1)

U  (0) = h (ω, V0 ) = h(0) (ω, V0 ) + h(1) (ω, V0 ) + · · · .

(5.2)

with the initial condition

By Taylor expansion, it follows from (3.12) and (5.1) that,

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17

g(U  (τ ) + σ η(θτ ψ ω), V  (τ )) = g(U ,0 (τ ) + σ η(θτ ψ ω), V0 ) + gu (U ,0 (τ ) + σ η(θτ ψ ω), V0 )(U ,1 (τ ) + · · · ) τ +gv (U

,0

(τ ) + σ η(θτ ψ ω), V0 )

[BV  (τ ) + g(U  (τ ) + σ η(θτ ψ ω), V  (τ ))]ds 0

= g(U

,0

(τ ) + σ η(θτ ψ ω), V0 ) + [gu (U ,0 (τ ) + σ η(θτ ψ ω), V0 )U ,1 (τ ) τ

+gv (U

,0

(τ ) + σ η(θτ ψ ω), V0 )



BV0 + g(U ,0 (s) + σ η(θs ψ ω), V0 ) ds]

0

+··· ,

(5.3)

and f (U  (τ ) + σ η(θτ ψ ω), V  (τ )) = f (U ,0 (τ ) + σ η(θτ ψ ω), V0 ) + [fu (U ,0 (τ ) + σ η(θτ ψ ω), V0 )U ,1 (τ ) τ +fv (U

,0

(τ ) + σ η(θτ ψ ω), V0 )



BV0 + g(U ,0 (s) + σ η(θs ψ ω), V0 ) ds]

0

+··· .

(5.4)

Substituting (5.1) and (5.4) into (3.11) and equating the terms with the same power of , we conclude ⎧ ,0 ⎪ ⎨ dU (τ ) = A U ,0 (τ ) + f (U ,0 (τ ) + σ η(θ ψ ω), V ), α τ  0 dτ ⎪ ⎩ ,0 U (0) = h(0) (ω, V0 ), and ⎧ ⎪ dU ,1 (τ ) ⎪ ⎪ = [Aα + fu (U ,0 (τ ) + σ η(θτ ψ ω), V0 )]U ,1 (τ ) + fv (U ,0 (τ ) ⎪ ⎪ dτ ⎪ ⎪ ⎪ ⎨ τ  + σ η(θτ ψ ω), V0 ) BV0 + g(U ,0 (s) + σ η(θs ψ ω), V0 ) ds, ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ ,1 (1) U (0) = h (ω, V0 ). Solving the two equations for U ,0 (τ ) and U ,1 (τ ), we get τ U

,0

(τ ) = e

h (ω, V0 ) +

Aα τ (0)

0

 eAα (τ −s) f (U ,0 (s) + σ η(θs ψ ω), V0 ) ds,

(5.5)

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18

and U

,1

(τ ) = e

τ

Aα + τ

0

τ

fu (U ,0 (s)+σ η(θs ψ ω),V0 )ds (1)

h (ω, V0 ) +

eAα (τ −s)

0 fu (U ,0 (r)+σ η(θr ψ ω),V0 )dr

·e s fv (U (τ ) + σ η(θτ ψ ω), V0 ) s   BV0 + g(U ,0 (r) + σ η(θr ψ ω), V0 ) drds. · ,0

(5.6)

0

Substituting (3.4) and (5.4) into (4.7), then 1 h (ω, V0 ) = 

0



 e−Aα s/ f U  (s) + σ η(θs ω), V  (s))ds

−∞

0 =

 e−Aα s f U  (s) + σ η(θs ψ ω), V  (s))ds

−∞

0 =

e

−Aα s

0 f (U

,0

(s) + σ η(θs ψ ω), V0 )ds + 

−∞

−∞



fu (U s

·

e−Aα s



,0

(s) + σ η(θs ψ ω), V0 )U

,1

(s) + fv (U ,0 (s) + σ η(θs ψ ω), V0 )

 BV0 + g(U ,0 (r) + σ η(θr ψ ω), V0 ) dr ds + O( 2 ).

0

The second equality holds by using a transformation s = s/ and omitting the prime. And matching the powers of h (ω, V0 ) in , we find that 0 h (ω, V0 ) = (0)

e−Aα s f (U ,0 (s) + σ η(θs ψ ω), V0 )ds,

(5.7)

−∞

and h(1) (ω, V0 ) 0 =

 e−Aα s fu (U ,0 (s) + σ η(θs ψ ω), V0 )U ,1 (s) + fv (U ,0 (s) + σ η(θs ψ ω), V0 )

−∞

s · 0



 BV0 + g(U ,0 (r) + σ η(θr ψ ω), V0 ) dr ds.

(5.8)

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That is, the random slow manifold M  (ω) = O( 2 ) is represented by

19

   h (ω, V0 ), V0 : V0 ∈ H2 of (3.7)–(3.8) up to

h (ω, V0 ) = h(0) (ω, V0 ) + h(1) (ω, V0 ) + O( 2 ).

(5.9)

Therefore, by Remark 3 and (5.9), the system (1.1)–(1.2) has a slow manifold M˜  (ω) = {(h˜  (ω, V0 ), V0 ) : V0 ∈ H2 )}, where h˜  (ω, V0 ) = h(0) (ω, V0 ) + h(1) (ω, V0 ) + σ η (ω) + O( 2 ). Moreover, we obtain the slow reduced approximate random system, ˜ + h(1) (θt ω, v) ˜ + σ η (θt ω), v˜  ), v˙˜  = B v˜  + g(h(0) (θt ω, v)

(5.10)

for  sufficiently small. 6. An example Finally, we present a simple example, demonstrating the results proved in the preceding section. Example 1. Consider the following fast–slow system 1 K1 σ u˙  = Aα u + (cos v  − 1) + √ W˙ t ,   

1 v˙ = −v + K2 sin 



 u (x)dx , 

in H1 = L2 (−1, 1),

in H2 = R,

(6.1)

(6.2)

−1

where Wt is specified in the previous section. Let us take K = max {K1 , K2 } to be less than λ1 (gap condition). The interaction functions f (u , v  ) = K1 (cos v  − 1) and g(u , v  ) = 1 K2 sin( −1 u dx) are Lipschitz continuous with a Lipschitz constant K > 0 and satisfy the condition (III). In order to numerically plot the random slow manifold u = h (ω, v), we choose a one dimensional system for v (therefore, we set the dependence on u via its integration with respect to x in equation (6.2)). It is difficult to visualize higher dimensional slow manifolds, but see [24] for some insights. The (6.1)–(6.2) can be transformed into the following random system 1 K1 (cos V  (t) − 1), U˙  (t) = Aα U  (t) +  

1      ˙ [U (t) + σ η (θt ω)]dx . V (t) = −V (t) + K2 sin −1

(6.3)

(6.4)

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20

Denote V  (0) = V0 ∈ R. Then for small enough , the random dynamical system generated by (6.3)–(6.4) has a slow invariant manifold M  (ω) = {(h (ω, V0 ), V0 ) : V0 ∈ R}, where K1 h (ω, V0 ) = 

0



 e−Aα s/ cos V  (s, ω, V0 ) − 1 ds.

−∞

By (5.7) and (5.8), we get 0 h (ω, V0 ) = K1 (0)

e

−Aα s

0 (cos V0 − 1)ds = K1 (cos V0 − 1)

−∞

e−Aα s ds,

−∞

and 0

se−Aα s ds

h (ω, V0 ) = K1 V0 sin V0 (1)

−∞

0  s − K1 K2 sin V0

e

−Aα s

r sin (K1 cos V0 − K1 ) eAα (r−l) dl

−∞ 0

σ +√ 

1

r

−∞

 eAα (r−l/) dWl drds.

−1 −∞

Then the approximation of h˜  (ω, V0 ) (with error O( 2 )) is hˆ  (ω, V0 ) = h(0) (ω, V0 ) + h(1) (ω, V0 ) + σ η (ω).

(6.5)

Finally, we obtain the approximate slow reduced random system, for  small enough, v˙˜  = −v˜  + K2 sin

1 [h(0) (θt ω, v˜  ) + h(1) (θt ω, v˜  )]dx −1

σ +√ 

1

t e

Aα (t−s)/

 dWs (x) dx .

(6.6)

−1 −∞

We have conducted numerical simulations to illustrate the slow manifold and the slow reduction. This is achieved by finite difference schemes for solving (6.1)–(6.2), and by Euler’s scheme

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21

Fig. 2. Deterministic slow manifold at the cross-section x = 0.5 for Example 1: σ = 0 (noise absent), α = 1.5,  = 0.001, λ K1 = K2 = 21 .

Fig. 3. Approximate random slow manifold at the cross-section x = 0.5 for Example 1 with noise intensity σ = 0.1: hˆ  (ω, v) = h(0) (ω, v) + h(1) (ω, v) + σ η (ω) (green curve) together with a few solution orbits, for α = 1.5,  = 0.001, λ K1 = K2 = 21 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

for solving (6.6), as in [24]. Here we consider the Wiener process Wt (x) with qj = 1/λj , for j = 1, 2, 3, · · · . Fig. 2 shows the deterministic slow manifold at the cross-section x = 0.5, for α = 1.5,  = 0.001, K1 = K2 = λ21 . Figs. 3 and 4 show one sample of the approximate random slow manifold at the cross-section x = 0.5 for noise intensity σ = 0.1 and σ = 0.3, respectively. In order to illustrate the validity of the slow reduction, we plotted one sample of v˜  (t) for slow reduction system (6.6) (red curve) and the slow component v  (t) of the original system (6.1)–(6.2) (blue curve), in the case with noise intensity σ = 0.1 (see Fig. 5).

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Fig. 4. Approximate random slow manifold at the cross-section x = 0.5 for Example 1 with noise intensity σ = 0.3: hˆ  (ω, v) = h(0) (ω, v) + h(1) (ω, v) + σ η (ω) (green curve) together with a few solution orbits, for α = 1.5,  = 0.001, λ K1 = K2 = 21 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Validity of slow reduction for Example 1 with noise intensity σ = 0.1: Comparison of v˜  (t) for slow reduction system (6.6) (red curve) and the slow component v  (t) of the original system (6.1)–(6.2) (blue curve), for α = 1.5, λ  = 0.001, K1 = K2 = 21 . (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

References [1] D. Applebaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge University Press, Cambridge, 2009. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. [3] B. Aulbach, T. Wanner, The Hartman–Grodom theorem for Carathéodory-type differential equations in Banach spaces, Nonlinear Anal. 40 (2000) 91–104.

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