Slow manifold and parameter estimation for a nonlocal fast-slow stochastic evolutionary system1footnote 11footnote 1The research was partly supported by the NSF grant 1620449 and NSFC grants 11531006 and 11771449.

# Slow manifold and parameter estimation for a nonlocal fast-slow stochastic evolutionary system111The research was partly supported by the NSF grant 1620449 and NSFC grants 11531006 and 11771449.

Hina Zulfiqar School of Mathematics and Statistics, Huazhong University of Science and Technology,
Wuhan 430074, China
& Center for Mathematical Sciences, Huazhong University of Science and Technology
zhinazulfiqar@gmail.com
Ziying He School of Mathematics and Statistics, Huazhong University of Science and Technology,
Wuhan 430074, China
& Center for Mathematical Sciences, Huazhong University of Science and Technology
ziyinghe@hust.edu.cn
Meihua Yang School of Mathematics and Statistics, Huazhong University of Science and Technology,
Wuhan 430074, China
yangmeih@hust.edu.com
and  Jinqiao Duan Department of Applied Mathematics, Illinois Institute of Technology,
Chicago, IL 60616, USA
& Center for Mathematical Sciences, Huazhong University of Science and Technology
duanjq@gmail.com
###### Abstract.

We establish a slow manifold for a fast-slow stochastic evolutionary system with anomalous diffusion, where both fast and slow components are influenced by white noise. Furthermore, we prove the exponential tracking property for the random slow manifold and this leads to a lower dimensional reduced system based on the slow manifold. Also we consider parameter estimation for this nonlocal fast-slow stochastic dynamical system, where only the slow component is observable. In quantifying parameters in stochastic evolutionary systems, this offers an advantage of dimension reduction.
keywords: Nonlocal Laplacian, fast-slow stochastic system, random slow manifold, exponential tracking property, parameter estimation.

## 1. Introduction

{linenomath*}

Nonlocal operators arise in complex phenomena such as anomalous diffusion in geophysical flows [6, 25, 26]. The usual Laplacian operator is the Markov generator of the Gaussian process called Brownian motion (or Wiener process). It may be regarded as the macroscopic manifestation of this normal diffusion phenomenon. Thus we often see the Laplacian operator in models for diffusion-related complex systems appear as stochastic or deterministic evolutionary differential equations. An exceptional, yet imperative nonlocal operator is nonlocal Laplacian operator emerged in non-Gaussian or anomalous diffusion. The nonlocal Laplacian operator is the Markov generator of the non-Gaussian process called symmetric -stable Lvy motion, for , and it is the macroscopic manifestation of this anomalous diffusion phenomenon [1, 15].
For investigating dynamical behaviours of deterministic systems, the theory of invariant manifolds have a satisfactory and popular history. Hadamard [22] introduced it for the first time, then by Lyapunov and Perron [7, 16, 11]. For deterministic system it has been modified by numerous authors [17, 31, 4, 10, 12, 23]. Duan and Wang [18] explained effective dynamics for the stochastic partial differential equations. Schmalfu and Schneider [33] have recently explored random inertial manifolds for stochastic differential equations with two times scales by a fixed point technique, which eliminate the fast variables via random graph transformation. They showed that as the scale parameter approaches to zero the inertial manifold approaches to some other manifold, which can be defined in term of slow manifold. Wang and Roberts [34] explored the averaging of slow manifold for fast-slow stochastic differential equations, in which only fast mode is influenced by white noise. For multi-time-scale stochastic system, the exploration of slow manifolds was conducted by Fu, Liu and Duan [21]. Bai et al. [3] recently established the existence of slow manifold for nonlocal fast-slow stochastic evolutionary equations in which only fast mode influenced by white noise. Ren et al. [30] devised a parameter estimation method based on a random slow manifold for a finite dimensional slow-fast stochastic system.
In the present paper we consider invariant manifold for nonlocal fast-slow stochastic evolutionary system, in which fast and slow mode, both are influenced by white noise. Namely, we explore the following nonlocal fast-slow stochastic evolutionary system.

 (1) ˙uϵ=−1ϵ(−Δ)α2uϵ+1ϵf(uϵ,vϵ)+σ1√ϵ˙W1t, in H1 (2) ˙vϵ=Jvϵ+g(uϵ,vϵ)+σ2˙W2t, in H2 (3) uϵ|(−1,1)c=0,vϵ|(−1,1)c=0,

where, for and

 (−Δ)α2uϵ(x,t)=2αΓ(1+α2)√π|Γ(−α2)|P.V.∫Ruϵ(x,t)−vϵ(y,t)|x−y|1+αdy,

is called fractional Laplacian operator, which is nonlocal operator with the Cauchy principle value considering as the limit of integral over for tends to zero. The Gamma function is defined to be

 Γ(y)=∫∞0ty−1e−tdt,∀y>0.

The fundamental objective of this paper is to establish the existence of random invariant manifold , for small enough with an exponential tracking property in Section 4, and conduct up a parameter estimation by using observations only on the slow system for the above nonlocal fast-slow stochastic evolutionary system in Section 5. Since slow system is lower dimensional as compared to the original stochastic system, so in computational cost this parameter estimator offers a benefit. In addition, it provides advantage of using the observations only on slow components. We verified the results about slow manifold, exponential tracking property and parameter estimation in last section via theoretical example and a example from numerical simulation. Before constructing the invariant manifold, initially we need to prove the existence, and then uniqueness of solution for newly made system (1)-(2). So it is verified in next Section. Many nonlocal problems explained in [5] are just similar to (1)-(3). There are usually two techniques for the establishment of invariant manifolds: Hadamard graph transform method [32, 17] and Lyapunov-Perron method [11, 16, 7] . For the achievement of our objective we used Lyapunov method, which is different from method of random graph transformation. More specifically, it can be shown that this manifold asymptotically approximated to a slow manifold for sufficiently small , under suitable conditions.
The state space for the system is a separable Hilbert space which is a product of Hilbert spaces and . Norm and are considered as the norm of and respectively. The norm of separable Hilbert space is defined by

 ||⋅||=||⋅||1+||⋅||2.

In the system , is a parameter with the bound , which denotes the ratio of two times scales such that
In order to balance the stochastic force and deterministic force, strength of noise in fast equation is chosen to be , while in slow equation is respectively. The interaction functions and are non linearities. Linear operator generates a -semigroup, which satisfy the slow evolution hypothesis given in next section. Consider that is the fast mode, while is the slow mode.
The Weiner process and are time dependent Brownian motions and defined on the probability space and . While natural filtrations and are generated by and respectively. The white noises and are generalized time derivatives of and respectively.
We introduced a few fundamental concepts in random dynamical system and nonlocal or fractional Laplacian operator next section.

## 2. Preliminaries

Consider , which denotes the space for fast variables. While is a separable Hilbert space, which denotes the space for slow variables. Take , which represents the nonlocal fractional Laplacian operator. On the functions that are extended to by zero just as in [24], the above nonlocal fractional Laplacian is applied. Now we review a few basic notions in random systems.

###### Definition 2.1.

([2]) {linenomath*}Let be a probability space and be a flow on , namely a mapping

 θ:R×Ω→Ω,

and satisfies the following conditions

where
the mapping is -measurable, and for all . Then is known as a driving or metric dynamical system.

{linenomath*}

To achieve our objective, we take a special but very important driving dynamical system influenced by Wiener process. Let us consider taking values in state space (Hilbert space) be a two-sided Wiener process, which depends upon time and also varying with . The defined Wiener process has zero value at . Its sample path is the space of real continuous functions on , which is denoted by . Here we take the flow, which is also measurable and defined by

 θrω=ω(⋅+r)−ω(r),ω∈Ω,r∈R.

On , the distribution of above process induces a probability measure, which is known as the Wiener measure. Instead of the whole , we consider an -invariant subset , whose probability-measure is one. With respect to we also take the trace -algebra of . Recall the concept of -invariant, a set is said to be -invariant if for . Here we take the restriction of Weiner measure and we still represent it by
On the state space (which is often taken as a Hilbert space), the dynamics of a stochastic system over a flow is depicted by a cocycle in general.

###### Definition 2.2.

([2]) {linenomath*}A cocycle is defined by a map:

 ϕ:R×Ω×H→H,

which is -measurable and satisfy the conditions

 ϕ(0,ω,y)=y, ϕ(r1+r2,ω,y)=ϕ(r2,θr1ω,ϕ(r1,ω,y)),

for , and . Then , together with metric dynamical system , constructs a random dynamical system.

{linenomath*}

We say that a random dynamical system is differentiable (continuous) if is differentiable (continuous) for and . The metric space contains a collection of closed and nonempty sets , if for every the mapping defined below:

 ω↦infy∈M(ω)||y−y′||H,

is a random variable. Then this family is said to be a random set.

###### Definition 2.3.
{linenomath*}

([15]) A random variable which take values in , is known as stationary orbit (random fixed point) for a random system if

 ϕ(r,ω,y(ω))=y(θrω),a.s.

for all .

{linenomath*}

Now idea of random invariant manifold is given.

###### Definition 2.4.

([21]) {linenomath*}If a random set satisfy the condition

 ϕ(r,ω,M(ω))⊂M(θrω),

for and . Then is known as a random positively invariant set.

{linenomath*}

If can be represent by a Lipschitz mapping graph define as:

 h(ω,⋅):H2→H1,

such that

 M(ω)={(h(ω,v),v):v∈H2},

then is said to be a Lipschitz random invariant manifold. Additionally, if for all , there is an , such that

 ||ϕ(r,ω,y)−ϕ(r,ω,y′)||H⩽c1(y,y′,ω)ec2t||y−y′||H, for all ω∈Ω, and r⩾0.

Where is a positive variable, which depend upon and , while is positive constant, then the random set have exponential tracking property.
Now we review the asymptotic behavior of eigenvalues of , for the spectral problem

 (−Δ)α2φ(y)=λφ(y),y∈(−1,1),

where is extended to by zero just as in [24]. It is known that there exist an infinite sequence of eigenvalues such that

 0<λ1<λ2⩽λ3⩽⋅⋅⋅⩽λk⩽⋅⋅⋅, for k=1,2,3,⋅⋅⋅.

and the corresponding eigenfunctions form a complete orthonormal set in .
Lemma 1. {linenomath*}([24]) For the above spectral problem

 λk=(kπ2−(2−α)π8)α+O(1k),(k→∞).

Lemma 2.{linenomath*} ([36]) The Laplacian operator is a sectorial operator, which satisfies the following upper-bound

 ||eAαt||L2(−1,1)⩽Ce−λ1t, for t⩾0.

Where is independent of and , also is constant.

Note that is a linear bounded operator on , together with it is self-adjoint and compact operator.

## 3. Framework

{linenomath*}

Framework includes a list of assumptions and conversion of stochastic dynamical system to random dynamical system.
Consider the system of stochastic evolutionary equations (1)-(2), which depend upon two time scales as described in previous section 1. We assume the following hypothesis in the fast-slow system (1)-(2).
Hypothesis H1. (Slow evolution): The operator generates -semigroup on , which satisfies

 ||eJtv||2⩽eγ2t||v||2,t⩽0,

with a constant and for all
Hypothesis H2. (Lipschitz condition): Nonlinearities and are continuous and differentiable functions (,

 f:H→H1,g:H→H2,

with

 f(0,0)=0=g(0,0),∂f∂uϵ(0,0)=∂f∂vϵ(0,0)=0=∂g∂uϵ(0,0)=∂g∂vϵ(0,0).

For all , a positive constant , such that

 ||f(uϵ1,vϵ1)−f(uϵ2,vϵ2)||⩽K(||uϵ1−uϵ2||1+||vϵ1−vϵ2||2),

also

 ||g(uϵ1,vϵ1)−g(uϵ2,vϵ2)||⩽K(||uϵ1−uϵ2||1+||vϵ1−vϵ2||2).

Hypothesis H3. (Gap condition): In the system, the term (Lipschitz constant) of the functions and satisfies the bound

 K<λ1γ22λ1+γ2,

where is the decay rate of Laplacian operator .
Now we construct a random evolutionary system corresponding to the stochastic evolutionary system (1)-(2). For the achievement of this objective, we first set up the existence, after that verify the uniqueness of solution for the equations (1)-(2) and for Ornstein-Uhblenbeck equation, which is nonlocal equation.
Lemma 3. Let taking values in Hilbert space be a two sided Weiner process (a time dependent Brownian motion) then under hypothesis , the system (1)-(2) posses a unique mild solution.
Proof. We write the system (1)-(2) in the following form

 (4) (˙uϵ˙vϵ)=(1ϵAα00J)(uϵvϵ)+(1ϵf(uϵ,vϵ)g(uϵ,vϵ))+⎛⎜⎝σ1√ϵ˙W1tσ2˙W2t⎞⎟⎠

By Lemma 1 and , we know that for every in , there exist (dual space of Hilbert space) such that , hence is dissipative and is invertible for any . This implies the range of is whole space .
So by Lumer-Phillips theorem [28], is infinitesimal generator of a semigroup which is strongly continuous. Considering assumption , we can say that the operator

 (1ϵAα00J)

is also an infinitesimal generator of semigroup, which is strongly continuous. Since, taking values in Hilbert space is a two sided Weiner process. Hence by using Theorem 7.2([14],p.188), the above system (1)-(2) posses a unique mild solution.
Lemma 4. The nonlocal stochastic equation

 dρ(t)=Aαρ(t)dt+σ1dW1t, in H1,

where is the Laplacian operator, has the following solution

 ρ(t)=etAαρ0+σ1∫t0eAα(t−s)dW1s, for all t⩾0 and α∈(1,2),

here is -measurable.
By an equation is in means that every term in the equation is in .
Remark 1. [3] The above lemma is not hold for .
Now, let and are two driving dynamical systems as explained in section 2. Define

 Ψ=Ψ1×Ψ2=(Ω1×Ω2,F1⊗F1,P1×P1,θ1t×θ2t)

and

 θtω:=(θ1tω1,θ2tω2)T, for ω:=(ω1,ω2)T∈Ω1×Ω2:=Ω.

Consider the stochastic evolutionary equations

 (5) dηϵ(t)=1ϵAαηϵdt+σ1√ϵdW1t, (6) dξ(t)=Jξdt+σ2dW2t.

Lemma 5. ([33]) Assume that the hypothesis (H1) holds. Then equations (5) and (6) have continuous stationary solutions and respectively.
Introduce a random transformation

 (7) (UϵVϵ):=ν(ω,uϵ,vϵ)=(uϵ−ηϵ(θ1tω1)vϵ−ξ(θ2tω2)),

so the original stochastic evolutionary system (1)-(2) is converted to a random evolutionary system as

 (8) dUϵ=1ϵAαUϵdt+1ϵf(Uϵ+ηϵ(θ1tω1),Vϵ+ξ(θ2tω2))dt, (9) dVϵ=JVϵdt+g(Uϵ+ηϵ(θ1tω1),Vϵ+ξ(θ2tω2))dt.

The nonlinear functions and in (8)-(9) satisfies the same Lipschitz condition. Let be the solution of (8)-(9) with initial value . By the classical theory for evolutionary equations [19], under the hypothesis (H1-H3) the system (8)-(9) with initial data has unique global solution for every just like to [9].
Hence the solution mapping of random evolutionary system (8)-(9)

 Φϵ(t,ω,(U0,V0)T):=(Uϵ(t,ω,(U0,V0)T),Vϵ(t,ω,(U0,V0)T))T,

indicates a random dynamical system. Furthermore,

 ϕϵ(t,ω):=Φϵ(t,ω)+(ηϵ(θ1tω1),ξ(θ2tω2)),t⩾0,ω∈Ω.

is the random system, which is generated by the original stochastic system (1)-(2).

## 4. Slow manifold

{linenomath*}

In this part, we construct the slow manifold existence and verify the exponential tracking property for the random evolutionary system (8)-(9). Define

 Mϵ(ω)≜{(Hϵ(ω,V),V)T:V∈H2}.

We prove that the random set is invariant manifold defined as a graph of Lipschitz mapping with the help of Lyapunov-Perron method. To study system (8)-(9), we construct the following Banach spaces in term of functions with a weighted geometrically sup norm [35]. For any :
is continuous and
is continuous and
with the norms
and
Similarly we define,
is continuous and
is continuous and
with the norms
and
Let be the product Banach spaces , with the norm

 ||z||C∓β=||uϵ||C1,∓β+||vϵ||C2,∓β,z=(uϵ,vϵ)T∈C∓β.

For convenience, let be a number, which satisfy

 (10) 0<μ=γ22λ1+γ2<1, and K<μλ1<λ1, also −μ+λ1>K.

Now, for the achievement of our goal, the following Lemma from [9] is needed.
Lemma 6. Consider the so-called Lyapunov-Perron equation

for , where and are the solutions of system (8)-(9) with initial value satisfying the form

 (Uϵ(t,ω;V0)Vϵ(t,ω;V0)) =(1ϵ∫t−∞eAα(t−s)/ϵf(Uϵ(s,ω;V0)+ηϵ(θ1sω1),Vϵ(s,ω;V0)+ξ(θ2sω2))dseJtV0+∫t0eJ(t−s)g(Uϵ(s,ω;V0)+ηϵ(θ1sω1),Vϵ(s,ω;V0)+ξ(θ2sω2))ds).

Then
Theorem 1. (Slow manifold) Suppose that (H1-H3) hold and is small enough. Then random evolutionary system (8)-(9) possessing a Lipschitz random slow manifold represented by a graph

 Mϵ(ω)={(Hϵ(ω,V),V)T:V∈H2),

where

 Hϵ(⋅,⋅):Ω×H2→H1,

is a graph mapping with Lipschitz constant satisfying

 LipHϵ(ω,⋅)⩽K(−μ+λ1)[1−K(1−μ+λ1+ϵϵγ2+μ)].

Remark 2. Since there is a relation among the solutions of stochastic evolutionary system (1)-(2) and random system (8)-(9), so if the system (1)-(2) satisfies the conditions described in Theorem 1, then it also possess Lipschitz random invariant manifold

 ˘Mϵ(ω)=Mϵ(ω)+(ηϵ(ω1),ξ(ω2))={(˘Hϵ(ω,V),V)T:V∈H2},

where

 ˘Hϵ(ω,V)=Hϵ(ω,V)+(ηϵ(ω1),0)).

Remark 3. If and are and a large spectrum gap condition holds then it can be proved similar to [16], that manifolds are -smooth.
Proof. To establish an invariant manifold corresponding to the random dynamical system (8)-(9), we first consider the integral equation

 (11) (Uϵ(t)Vϵ(t))=(1ϵ∫t−∞eAα(t−s)/ϵf(Uϵ(s)+ηϵ(θ1sω1),Vϵ(s)+ξ(θ2sω2))dseJtV0+∫t0eJ(t−s)g(Uϵ(s)+ηϵ(θ1sω1),Vϵ(s)+ξ(θ2sω2))ds),t⩽0.

Step 1. Let be the solution of (11) with initial value . We prove that is the unique solution of (11), by using of Banach fixed point theorem. For this,
define two operators and by means of

 Iϵ1(z(⋅))[t]=1ϵ∫t−∞eAα(t−s)/ϵf(u(s)+ηϵ(θ1sω1),v(s)+ξ(θ2sω2))ds,
 Iϵ2(z(⋅))[t]=eJtV0+∫t0eJ(t−s)g(u(s)+ηϵ(θ1sω1),v(s)+ξ(θ2sω2))ds,

for , and the Lyapunov-Perron transform is

 Iϵ(z(⋅))=(Iϵ1(z(⋅))Iϵ2(z(⋅)))T=(Iϵ1(z(⋅)),Iϵ2(z(⋅))).

For verify that maps into itself. Take , such that:

 ||Iϵ1(z(⋅))[t]||C1,−β =||1ϵ∫t−∞eAα(t−s)/ϵf(u(s)+ηϵ(θ1sω1),v(s)+ξ(θ2sω2))ds||1 ⩽1ϵsupt⩽0{e−β(t−s)∫t−∞e−λ1(t−s)/ϵ||f(u(s)+ηϵ(θ1sω1),v(s) +ξ(θ2sω2))||1ds} ⩽Kϵsupt⩽0{e−β(t−s)∫t−∞e−λ1(t−s)/ϵ(||u(s)||1+||v(s)||2)ds}+C1 ⩽Kϵsupt⩽0{∫t−∞e(−λ1/ϵ−β)(t−s)ds}||z||C−β+C1 =Kϵβ+λ1||z||C−β+C1,

and

 ||Iϵ2(z(⋅))[t]||C2,−β =||eJtV0+∫t0eJ(t−s)g(u(s)+ηϵ(θ1sω1),v(s)+ξ(θ2sω2))ds||2 ⩽supt⩽0{e−β(t−s)∫0teγ2(t−s)||g(u(s)+ηϵ(θ1sω1),v(s) +ξ(θ2sω2))||2ds}+supt⩽0{e−βteγ2t||V0||2} ⩽Ksupt⩽0(∫0te(γ2−β)(t−s)ds)(||u(s)||1+||v(s)||2)+C2+||V0||2 =Kγ2−β||z||C−β+C2+||V0||2 =Kγ2−β||z||C−β+C3.

Hence, by combining with the definition of , we obtain that

 ||Iϵ(z)||C−β⩽[Kϵβ+λ1+Kγ2−β]||z||C−β+C4,

where are constants and

 υ(K,β,λ1,γ2,ϵ)=Kϵβ+λ1+Kγ2−β.

This implies that maps into itself.
Furthermore, to show that the map is contractive.
Take ,

 ||Iϵ1(z)−Iϵ1(¯z)||C1,−β ⩽1ϵsupt⩽0{e−β(t−s)∫t−∞e−λ1(t−s)/ϵ||f(u(s)+ηϵ(θ1sω1),v(s) +ξ(θ2sω2))−f(¯u(s)+ηϵ(θ1sω1),¯v(s)+ξ(θ2sω2))||1ds} ⩽Kϵsupt⩽0{e−β(t−s)∫t−∞e−λ1(t−s)/ϵ(||u(s)−¯u(s)||1 +||v(s)−¯v(s)||2)ds} ⩽Kϵsupt⩽0{∫t−∞e(−λ1ϵ−β)(t−s)ds}||z−