# Representation of Manifolds for the Stochastic Swift-Hohenberg Equation with Multiplicative Noise

Authors: Yanfeng Guo; Donglong Li
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

Representation of approximation for manifolds of the stochastic Swift-Hohenberg equation with multiplicative noise has been investigated via non-Markovian reduced system. The approximate parameterizations of the small scales for the large scales are given in the process of seeking for stochastic parameterizing manifolds, which are obtained as pullback limits of some backward-forward systems depending on the time-history of the dynamics of the low modes in a mean square sense through the nonlinear terms. When the corresponding pullback limits of some backward-forward systems are efficiently determined, the corresponding non-Markovian reduced systems can be obtained for researching good modeling performances in practice.


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## Body

## 1. Introduction

Recently, more and more authors have paid attention to considering the approximation problems of manifolds for the stochastic partial differential equations (SPDEs). For decades, various approximating methods have been given to solve these problems, such as amplitude equations approach [1–4] and the manifolds-based approaches [5–11].In this paper, approximation of manifolds for the stochastic Swift-Hohenberg equation with multiplicative noise will be investigated in Stratonovich sense [12]. It is well known that there have been some authors to consider the approximation of manifolds in large probability sense [1, 2, 13] and they have obtained some results until now. In addition, approximation in parameterizing manifold for a stochastic Swift-Hohenberg equation with additive noise have been investigated by us in [14]. Furthermore, it is needed to consider the problems in Stratonovich sense. Until now, there have been few consideration from the point of view of approximation in parameterizing manifold under the pathwise sense for the stochastic Swift-Hohenberg equation with multiplicative noise. The ideas in [14] can be used to consider the approximation of manifold for some stochastic equations with multiplicative noise. Because the different difficulties come from different noise terms, there are some different methods and techniques in studying stochastic equations with multiplicative noise. Here, we investigate the corresponding problems for the stochastic Swift-Hohenberg equation with multiplicative noise with pathwise and obtain some new results for it. The results obtained in this paper are different from those in [14], although there are some similar sentences in some manuscripts. It is well known that various noises cause various stochastic processes for stochastic equations with different noises. The main differences from results in [14] are given by some formulas with various mathematics meanings, in which some different stochastic functions are used. Because the different difficulties mainly come from various noise terms, there are some new difficulties coming from the multiplicative noise solved in our manuscript. So, some different techniques are used in studying stochastic equations with multiplicative noise.We will extend the strategy introduced in [5, 6] to the stochastic Swift-Hohenberg equation [12] with multiplicative noise and obtain the approximation of parameterizing manifolds and corresponding non-Markovian reduced system. The key idea is mainly based on the approximate parameterizations of the small scales for the large scales via the stochastic parameterizing manifolds. Random manifolds will improve the partial knowledge of the solutions of SPDEs in mean square error, when it is compared with its projection onto the resolved modes. Approximation of parameterizing manifolds can be obtained by representing the modes with high wave numbers as the pullback limit depend on the time-history of the modes with low wave numbers for the corresponding backward-forward systems. Some conditions with nonresonance conditions below are given and weaker than those in the classical stochastic invariant manifold theory (see [7, 15, 16] and references therein). On the base of these approximations of parameterizing manifolds, when the corresponding pullback limits of some backward-forward systems are efficiently determined, the corresponding non-Markovian stochastic reduced systems are given to reach good modeling performances in practice and take the form of stochastic differential equations with random coefficients, which convey memory effects via the history of the Wiener process and arise from the nonlinear interactions between the low modes embedded in the noise bath. These random coefficients show an exponential decay of correlations, whose rate depends explicitly on the gaps of the nonresonance conditions. In fact, it is possible to achieve very good parameterizing quality for the stochastic Swift-Hohenberg equation with multiplicative noise from our results. And the performances from the reduced system can be numerically assessed for a corresponding optimal or suboptimal control problem.The paper is organized as follows. In Section2, we give our functional framework, some definitions about parameterizing manifolds and some properties of some stochastic processes being used. We have devoted Section 3 to studying the representation of approximation of parameterizing manifolds as pullback limits of the corresponding backward-forward systems for a stochastic Swift-Hohenberg equation with multiplicative noise. In Section 4, on the basis of the approximation of parameterizing manifolds, the non-Markovian stochastic reduced systems involving random coefficients are obtained for the stochastic Swift-Hohenberg with multiplicative noise.


## 2. Preliminaries

The functional framework spaces are a pair of Hilbert spaces (H1, H) such that H1 is compactly and densely embedded in H. Let A:H1⟶H be a sectorial operator [16] such that −A is stable in the sense that its spectrum satisfies Reσ−A<0. And we consider interpolated spaces Hα between H1 and H with α∈0,1 along with the perturbations of the linear operator −A given by a one parameter family Bλ of bounded linear operators from Hα to H, depending continuously on λ. Define Lλ=−A+Bλ, which maps H1 into H.A local stochastic Swift-Hohenberg equation with multiplicative noise in Stratonovich sense [1] is written as follows:
(1)du=λu−1+Δ2u−u3dt+σu∘dWt≔Lλu+Fudt+σu∘dWt,with Dirichlet boundary conditions u0,t;ω=ul,t;ω=0,t>0 and initial condition ux,0;ω=u0x,x∈0,l, where λ is a parameterizing variable, σ is positive, and u0 is some appropriate initial datum with H=L20,l and H1=H40,l∩H020,l; Wt is a standard real valued one-dimensional Brownian motion [17] with paths in C0ℝ,ℝ; and Ω being endowed with its corresponding Borel σ-algebra ℱ, its filtration ℱt, the Wiener measure ℙ.LetFu=−u3, which is a continuous triple nonlinear mapping from Hα into H, where α>1/3. Obviously, function Fu is a mapping from H1 into H. Assume Lλ=−A+Bλ, where Bλ=λ and A=1+Δ2 is closed self-adjoint linear operator with dense domain DA in H=L2D. The operator Lλ is self-adjoint with an orthonormal basis of eigenfunctions ek=2/lsinkπx/lk∈ℕ in H with corresponding eigenvalues βkλ=λ−1−k2π2/l22k∈ℕ.Then, problem (1) can be rewritten as
(2)du=Lλu+Fudt+σu∘dWt,with initial condition ux,0;ω=u0x,x∈0,l and Dirichlet boundary conditions u0,t;ω=ul,t;ω=0,t>0. Now, we investigate the random dynamical systems of system (2) in the sense of parameterizing manifolds in [6, 17]. The stochastic parameterizing manifolds are mainly considered for local stochastic Swift-Hohenberg equation with multiplicative noise (2). Firstly, a stochastic parameterizing manifold M is seen as the graph of a random function hpm, which is a mapping from Hc to Hα,s and provides approximation parameterizations of the high part ust,ω=Psut,ω by using of the low part uct,ω=Pcut,ω. The scalar Langevin equation,
(3)dz+zdt=σdW,is given. A unique stationary solution zθtω of this equation is called the stationary Ornstein-Uhlenbeck (OU) process. By simply integrating on the both sides of (3), the identity
(4)∫0tzσθsωds+zσθtω=zσω+σWtω,∀t∈ℝ,holds, which is important for representation of approximation.


## 3. Representation of Manifolds with Multiplicative Noise

Making use of the method in [6], we investigate the local stochastic Swift-Hohenberg (equation (2)) with multiplicative noise in Stratonovich sense. One considers the following backward-forward system associated with SPDE (2).
(5)du^c1=Lλcu^c1ds+σu^c1∘dWs,s∈−T,0,(6)u^c1s,ω∣s=0=ξ∈Hc,(7)du^s1=Lλsu^s1+PsFu^c1s−T,ωds+σu^s1∘dWs−T,s∈0,T,(8)u^s1s,θ−Tω∣s=0=0,where Lλc≔PcLλ and Lλs≔PsLλ. From system (5), (6), (7), and (8), we know that the initial value of u^c1 is represented in fiber ω and the initial value of u^s1 is prescribed in fiber θ−Tω.It is possible to obtain the solution of system (5) and (6) by using a backward-forward integration procedure due to the partial coupling between the equations constituting this system, where u^c1 forces the evolution equation of u^s1 but not reciprocally. In addition, since uc1 is emanated backward from ξ in Hc and forces the equation ruling the evolution of u^s1, thus u^s1 depends naturally on ξ. One emphasizes this dependence as u^s1ξ in the whole paper.The nonresonance conditions should be given in following theorem, under which the pullback limit ofu^s1ξ exists. Now, representation of an analytical description of such parameterizing manifolds will be provided. In particularly, it emphasizes the dependence on the part of the noise path of the manifolds.Theorem 1.
Consider the SPDE (2) in the functional setting of Section 2, with F assumed to be a trilinear function. Let ℐ≔1,⋯,m with m=dimHc.
Suppose alsoβnλ<0 for all n>m. Furthermore, assume that the following nonresonance conditions for all i1,i2,i3∈ℐ3, n>m,
(9)if<Fei1,ei2,ei3,en>≠0,thenβi1+βi2+βi3−βn>0hold. Then, the pullback limit of the solution u^s1ξT,θ−Tω;0 of (7) and (8) exists and is given by
(10)h^λ1ξ,ω=limT→+∞u^s1ξT,θ−Tω;0=∫−∞0e−Lλsτ+2σWτωPsFeLλcτξdτ,∀ξ∈Hc,ω∈Ω,where u^c1s,ω;ξ is the solution of (5) and (6)
(11)u^c1s,ω;ξ=eLλcs+σWsωξ.
Moreover,h^λ1 has the following analytic expression:
(12)h^λ1ξ,ω=∑n=m+1∞∑i1=1m∑i2=1m∑i3=1mξi1ξi2ξi3Mn,λi1i2i3ω<Fei1,ei2,ei3,en>en,where ξi=<ξ,ei>,i=1,⋯,m, and
(13)Mn,λi1i2i3ω=∫−∞0eτβi1λ+βi2λ+βi3λ−βnλ+2σWτωdτ.Proof.
Firstly, from (5), (6), (7), and (8), one introduces two processes uc1 and us1 for u^c1 and u^s1 as follows:
(14)uc1s,ω;ξ=e−zσθsωu^c1s,ω;ezσθsωξ,s∈−T,0,ξ∈Hc,us1ξs,θ−Tω;0=e−zσθs−Tωu^s1ezσωξs,θ−Tω;0,s∈0,T.
Here, via the above transformation processes, the backward-forward system (5), (6), (7), and (8) is transformed into the following system of random differential equations:
(15)duc1ds=Lλcuc1+zσθsωuc1,s∈−T,0,(16)uc1s,ω∣s=0=ξ∈Hc,(17)dus1ds=Lλsus1+zσθs−Tωus1+e2zσθs−TωPsFuc1s−T,ω,s∈0,T,(18)us1s,θ−Tω∣s=0=0.
Using the variation of constants method, we can formally obtain the solution of (15) and (16), which is followed by making use of an integration by parts performed to the resulting stochastic convolution terms
(19)uc1s,ω;ξ=eLλcs+∫0szσθrωdrξ.
Similarly, the solution of (17) and (18) can be also obtained at T, which is formed
(20)us1ξT,θ−Tω;0=∫−T0e−τLλs+2zσωdr+2σWτωPsFeτLλcξdτ,where uc1⋅,ω;ξ is taken as a form of (19). When T⟶+∞, since condition (9), the limit of (20) exists, which is formed
(21)hλ1ξ,ω=limT→+∞us1ξT,θ−Tω;0=∫−∞0e−τLλs+2zσωdr+2σWτωPsFeτLλcξdτ.
Secondly, one investigates the analysis presentation of this limit. Propose that(22)hλ1ξ,ω=∑n=m+1∞hλ1,nξ,ωen,where
(23)hλ1,nξ,ω=∑i1,i2,i3=1me2zσωξi1ξi2ξi3Mn,λi1i2i3ω<Fei1,ei2,ei3,en>.
Here,ij=1,⋯,m and ξij=<ξ,eij>,j=1,2,3, and
(24)Mn,λi1i2i3ω=∫−∞0e∑j=13β1jλ−βnλs+2σWsωds.
According to the same assumptions and the inverse transformation, (11) can be immediately obtained from (19), and the analytic expression of h^λ1 has the form of (12).Furthermore, the approximationh^λ1 can be provided by the above theorem, which constitutes a parameterizing manifold function of SPDE (2). Moreover, the random coefficients Mn,λ have decaying property of correlations when it is checked by similar calculations performing for the proof of Lemma 5.1 in [6], which are solutions of auxiliary SDEs
(25)dM=1−∑j=13βijλ−βnλMdt−σM∘dWt.Remark 2.
Here, the random coefficientsMn,λi1i2i3ω satisfied the stochastic equation (25) and are different from Mi in [14], since the stochastic processes’ transformations are various for stochastic equations with multiplicative noises. So, Mn,λi1i2i3ω in this paper and Mi in [14] have different representations by formulas. In this paper, the transformations of stochastic processes are more difficult than those in [14]. So, the random coefficients Mn,λi1i2i3ω are much more complex than those in [14]. Then, these differences hold in the whole paper.


## 4. PM-Based Non-Markovian Reduced System with Multiplicative Noise

In this section, the PM-based non-Markovian reduced system of problem (2) is investigated in two cases that are in two subspaces, Hc=spane1 or Hc=spane1,e2, respectively.When one projects (2) into the subspace Hc, it yields that
(26)duc=Lλcuc+PcFuc+usdt+σu∘dWt,where uc=Pcu with Pc being the canonical projector on subspace Hc. By replacing ust,ω=Psut,ω with (12), the pullback limit h^λ1ξ,θtω, one yields the following reduced system
(27)dξ=Lλcξ+PcFξ+h^λ1ξ,θtωdt+σξ∘dWt,which provides an approximation of the SPDE dynamics projected onto the low modes.From (12), the random coefficients of eixi=1,2,⋯ contained in the expansion of h^λ1 exhibit the decaying property of correlations. Therefore, extrinsic memory effects in the Stratonovich sense are conveyed by the drift part of (27), making such reduced systems be non-Markovian (see [18, 19]).The analytic form ofh^λ1 from (12) can be used. The nonlinear interactions Fni1i2i3=<Fei1,ei2,ei3,en>, have the following form. When m=1,
(28)F3i1i2i3=12l,n=3,Fni1i2i3=0,n=2orn≥4.Whenm=2,
(29)Fni1i2i3=12l,wheni1+i2+i3=nori1−i2−i3=n,Fni1i2i3=−12l,wheni1+i2−i3=nori1−i2+i3=n,Fni1i2i3=0,n≥7.Firstly, we investigate the system in casem=1. Since approximation of parameterizing manifolds have been obtained in Section 3, then one yields that
(30)h^λ1ξ,ω=12lξ13M3,λ111ωe3,where ξ1=<ξ,e1>, and
(31)M3,λ111ω=∫−∞0e3β1λ−β3λs+2σWsωds.In this case, the approximation is simple. However, it is not enough to present the performances of the corresponding dynamics. Furthermore, parameterizing manifolds in two-dimensional case for low mode are considered, which perform more dynamics than in the above case.Secondly, whenm=2, then one can obtain the more complex results than in the case of m=1.Here,(32)h^λ1ξ,ω=∑n−m+1∞∑i1,i2,i3=1mξi1ξi2ξi3Mn,λi1i2i3ω<Fei1,ei2,ei3,en>en=12lξ13M3,λ111−3ξ1ξ22M3,λ122e3+32lξ12ξ2M4,λ112e4+32lξ1ξ22M5,λ122e5+12lξ23M6,λ222e6,where ξ1=ξ,e1, ξ2=ξ,e2, and
(33)M3,λ111ω=∫−∞0e3β1λ−β3λs+2σWsωds,M3,λ122ω=∫−∞0eβ1λ+2β2λ−β3λs+2σWsωds,M4,λ112ω=∫−∞0e2β1λ+β2λ−β4λs+2σWsωds,M5,λ122ω=∫−∞0eβ1λ+2β2λ−β5λs+2σWsωds,M6,λ222ω=∫−∞0e3β2λ−β6λs+2σWsωds.However, it is complex to use directly the analytic formula ofh^λ1 to obtain the vector PcFξ+h^λ1ξ,θtω as ξ is various in Hc in spite of any case in fact. So, we can use u^s1ξt,ωt+T,θTω;0 to take the place of h^λ1ξ,θtω on the fly along a trajectory ξt,ω of interest, where u^s1 is given by integrating on both sides of the backward-forward system (5), (6), (7), and (8), when T is chosen sufficiently large [6]. Then, it is natural to study the reduced system as follows:
(34)dξt=Lλcξt+PcFξt+u^s1ξt,ωt+T,θTω;0dt+σξt∘dWt,ξ0,ω=ϕ,t>0,where ϕ is appropriately chosen according to the SPDE initial datum and u^s1ξt,ω is given from the following system:
(35)du^c1=Lλcu^c1ds+σu^c1∘dWs,u^c1s,ω∣s=t=ξt,ω,s∈t−T,t,du^s1=Lλsu^s1+PsFu^c1s−T,ωds+σu^s1∘dWs−T,u^s1s,θ−Tω∣s=t=0,s∈t,t+T.Now, we give the corresponding non-Markovian systems from the above system. Investigating them in two casesm=1 and m=2.Firstly, whenm=1, one denotes ξ1t,ω=ξ1t,ωe1, with ξ1t,ω=<ξt,ω,e1>. Then, the system can be written as in coordinate form
(36)dξ1=β1λξ1−32lξ13−ξ12y31+ξ1y312dt+σξ1∘dWt,t>0,with ξ10,ω=ϕ,e1, where ξt=ξ1t,ωe1 and yj1,j=2,⋯, are given from the following system:
(37)dy11=β1λy11ds+σy11∘dWs,s∈t−T,t,dy21=β2λy21ds+σy21∘dWs−T,s∈t,t+T,dy31=β3λy31+12ly11s−T,ω3ds+σy31∘dWs−T,s∈t,t+T,dyj1=βjλyj1ds+σyj1∘dWs−T,s∈t,t+T,j=4,⋯,with y11s,ω∣s=t=ξ1t,ω,yj1s,θ−Tω∣s=t=0,j=2⋯.Secondly, whenm=2, one denotes ξt,ω=ξ1t,ωe1+ξ2t,ωe2, with ξit,ω=<ξt,ω,ei>,i=1,2. Then, for t>0, the corresponding system can be written as in coordinate form
(38)dξ1=β1λξ1−32lξ13−3lξ1ξ22+32lξ12−ξ22y31+2ξ1ξ2y41+ξ22y51+3ξ1ly31y51+y41y61−∑i=36yi12+3ξ2ly31y61−∑i=46yi−11yi1−1ly312y51+2y31y41y61−12ly31y31y51+y41y61dt+σξ1∘dWt,dξ2=β2λξ1−3lξ12ξ2−32lξ23+32lξ12y41+2ξ1ξ2−y31+y51+ξ22y61+3ξ1ly31y61−∑i=46yi−11yi1−3ξ2l∑i=36yi12−32ly312y41+2y31y41y51+y31y51y61+y412y61dt+σξ2∘dWt,with ξ10,ω=ϕ,e1, ξ20,ω=ϕ,e2, where ξt=ξ1t,ωe1+ξ2t,ωe2 and yj1,j=3,⋯, are given from following system
(39)dy11=β1λy11ds+σy11∘dWs,s∈t−T,t,dy21=β2λy21ds+σy21∘dWs,s∈t−T,t,dy31=β3λy31+12ly11s−T,ω3−3y11s−T,ωy21s−T,ω2ds+σ3y31∘dWs−T,s∈t,t+T,dy41=β4λy41+32ly11s−T,ω2y21s−T,ωds+σy41∘dWs−T,s∈t,t+T,dy51=β5λy41+32ly11s−T,ωy21s−T,ω2ds+σy51∘dWs−T,s∈t,t+T,dy61=β6λy61+12ly21s−T,ω3ds+σy61∘dWs−T,s∈t,t+T,dyj1=βjλyj1ds+σyj1∘dWs−T,s∈t,t+T,j=7,⋯,with y11s,ωs=t=ξ1t,ω,y21s,ωs=t=ξ2t,ω,yj1s,θ−Tωs=t=0,j=3,⋯..From the above equations, the representations of approximation for manifold and the corresponding reduced non-Markovian systems for stochastic Swift-Hohenberg equation with multiplicative noise are obtained. And the performances given by the above non-Markovian reduced system should have approximate dynamics on theHc modes in modeling of the pathwise SPDE (2). It is more important that one should give partial dynamics in approximation sense on the Hc modes in modeling of the pathwise SPDEs in practice. The performances from the reduced system may be numerically assessed for a corresponding optimal or suboptimal control problems in the deduced processes. The numerical results will be further shown in the future. The processes deduced in this manuscript offered an idea in order to further investigate the approximation of stochastic manifold for some quantum stochastic equations with multiplicative noise.



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