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INVARIANT MANIFOLD REDUCTION AND BIFURCATION

FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

WEI WANG & JINQIAO DUAN

Abstract. Stochastic partial differential equations arise as mathematicalmodels of complex multiscale systems under random influences. Invariantmanifolds often provide geometric structure for understanding stochasticdynamics. In this paper, a random invariant manifold reduction principleis proved for a class of stochastic partial differential equations. The dynam-ical behavior is shown to be described by a stochastic ordinary differentialequation on an invariant manifold, under suitable conditions. The changeof dynamical structures for the stochastic partial differential equations isthus obtained by investigating the stochastic ordinary differential equation.The random cone invariant property is used in the approach. Moreover, theinvariant manifold reduction principle is applied to detect bifurcation phe-nomena and stationary states in stochastic parabolic and hyperbolic partialdifferential equations.

1. Introduction

Stochastic partial differential equations (SPDEs) have drawn more attentionfor their importance as macroscopic mathematical models of complex multi-scale systems under random influences. With the development of randomdynamical systems (RDS) [1] recently, SPDEs have been investigated in thecontext of random dynamical systems, e.g., [7, 9, 10, 12, 27], among others.At this time, many SPDEs are known to generate random dynamical systems.Despite the rapid development of random dynamical systems theory, invariantmanifolds and bifurcations in random systems are still far from being under-stood. There are few rigorous general results and criteria, and in fact, someresults are obtained by numerical simulations or hold only for special models.

2000 Mathematics Subject Classification. Primary 37L55, 35R60; Secondary 60H15,37H20, 34D35.

Key words and phrases. Random invariant manifold reduction, stochastic bifurcation,stochastic partial differential equation (SPDE), multiscale phenomena, pattern formation,stochastic pitchfork bifurcation.

This work was partly supported by the NSF Grants DMS-0209326 & DMS-0542450, andthe Outstanding Overseas Chinese Scholars Fund of the Chinese Academy of Sciences.

1

2 W. WANG & J. DUAN

Invariant manifolds are special invariant sets represented by graphs in func-tion phase spaces where solution processes of SPDEs live. A random invariantmanifold provides a geometric structure to reduce stochastic dynamics. Sto-chastic bifurcation, in a sense, is about the changes in invariant structuresfor random dynamical systems, including the qualitative changes of invariantmanifolds, random attractors, and invariant measures.

The qualitative property of invariant measures for dissipative SPDEs withadditive white noises may be investigated by a multiscale method as in [4,5]. Stochastic amplitude equations, which are stochastic ordinary differentialequations (SODEs) for systems on bounded domains, are used to describe thedynamics of the amplitudes of dominant modes evolving on some slow timescale, near the deterministic bifurcation point. And the unique invariant mea-sure of the corresponding random dynamical system can also be approximatedby that of the amplitude equation. The study of the qualitative property of aninvariant measure appears to be difficult in the case of non-unique invariantmeasures for SPDEs with multiplicative noises.

Caraballo et. al.[8] studied the stochastic bifurcation by detecting the changein random attractors for SPDEs. They considered the following Chafee-Infantereaction-diffusion equation perturbed by a multiplicative white noise in thesense of Stratonovich

du = (∆u+ βu− u3)dt+ u dW (t), in I, u(0) = u0 (1.1)

with the zero Dirichlet boundary condition on an open interval I in R. It isproved that, for β < λ1 (the first eigenvalue of negative Laplace on I withDirichlet boundary), the random attractor consists of the fixed point u =0. If β > λ1, two new random fixed points, ±a(ω), appear. The randomattractor now consists of the interval [−a(ω), a(ω)]. This phenomena is calledthe stochastic pitchfork bifurcation in [8], accompanied by the birth of twonew random fixed points or equilibrium states. But, as [8] pointed out, thereis no criteria of the stability for the new fixed points ±a(ω). The monotonicrandom dynamical system theory, e.g., [3], is used in their approach.

In this paper we consider invariant manifold reduction of SPDEs and thusreduce the bifurcation problem for SPDEs to that of SODEs. As a by product,we obtain the existence of a stochastic pitchfork bifurcation for the parabolicSPDE (1.1) above, together with the criteria of the stability for the accom-panying new equilibrium states ±a(ω); see §5.1 below. Moreover, in §5.2,we apply this random invariant manifold reduction to investigate existence ofthe stationary solution of a hyperbolic SPDE with large diffusivity and highlydamped term.

STOCHASTIC BIFURCATION 3

Our main tool is the random invariant manifold theory and random invari-ant cone property. It is well known that invariant manifold is an importanttool to study the structure of attractor [11, 29]. In this paper we first considerinvariant manifolds for a class infinite dimensional RDS defined by SPDEs,then reduce the random dynamics to the invariant manifolds, and finally ob-tain local bifurcation results of the SPDEs. The basic idea is to reduce aninfinite dimensional random system onto a finite dimensional, asymptoticallycomplete, random invariant manifold (see section 3), and the reduced systemon the invariant manifold is in fact equivalent to a stochastic ordinary dif-ferential system. Then we study the bifurcation of the SPDEs through thestochastic ordinary differential equations (SODEs). In our approach the ran-dom cone invariance concept 4.1, is used to prove asymptotic completenessproperty of the random invariant manifold. Cone invariance is also an im-portant property in the study of the inertial manifold for infinite dimensionaldynamical systems; see [16, 24, 25].

This paper is organized as follows. We state the main result on randominvariant manifold reduction principle for a class of SPDEs, and some back-ground materials in random dynamical systems in section 2. In section 3, weoutline basic setup of the random invariant manifold theory and obtain theexistence of random invariant manifolds. The main result is proved in section4, and applications in stochastic bifurcation and stationary solution are dis-cussed in the final section 5.

2. Main results

We study the stochastic evolutionary equations in the form

du(t) = (νAu(t) + F ǫ(u(t)))dt+ σu(t) dW (t) (2.1)

where A is the generator of a C0 semigroup on Hilbert space (H, | · |) with innerproduct 〈, ·, 〉, F ǫ : R → R is a continuous nonlinear function which Lipschitzconstant Lǫ

F , Fǫ(0) = 0 and W (t) is a standard real valued Wiener process,

ν > 0, σ ∈ R. Moreover, denotes the stochastic differential in the sense ofStratonovich. And ǫ is a small parameter so that F ǫ can be seen as a smallperturbation, that is, we have Lǫ

F → 0 as ǫ→ 0. Suppose that the operator Ahas eigenvalues

0 = λ1 = · · · = λn > λn+1 ≥ · · ·with corresponding eigenvectors e1, · · · , en, en+1, · · · . Denote the kernel of Aby kerA := Hc. Then dimHc = n and let Hs = H⊥

c . Then for u ∈ H we haveu = uc + us with uc ∈ Hc and us ∈ Hs. Define the projections Πc : H → Hc

4 W. WANG & J. DUAN

and Πs : H → Hs as Πcu = uc and Πsu = us. Further we assume Πc and Πs

commute with A. Define the nonlinear map on Hc as

F ǫc : Hc → Hc

u 7→ F ǫc (u) =

n∑

i=1

〈F ǫ(u+ 0), ei〉ei

where 0 is the zero element in Hs. Then we have the following main result ofthe paper.

Theorem A (Invariant manifold reduction)Consider the following stochastic evolutionary equation

du(t) = (νAu(t) + F ǫ(u(t)))dt+ σu(t) dW (t). (2.2)

Assume that this evolutionary equation generates a dissipative random dynam-ical system, that is, the system has a random absorbing set, whose diameter orradius is a tempered random variable. Then for small ǫ, the system has an ndimensional invariant manifold M(ω) such that for any solution u(t) of (2.2)there is a flow U(t) on M(ω)

|u(t)− U(t)| → 0, t→ ∞ almost surely. (2.3)

Moreover the flow on M(ω) is topologically equivalent to the following stochas-tic ordinary differential equation in R

n

du(t) = F ǫc (u(t))dt+ σu(t) dW (t) (2.4)

provided (2.4) is structurally stable for ǫ 6= 0.

Remark 2.1. The random dynamical system φ(t, ω) generated by (2.4) iscalled structurally stable, if for any small perturbation (small in the senseof the usual metric in the space of continuous functions) to F ǫ(x), the per-turbed random dynamical system Φ(t, ω) is topologically equivalent to φ(t, ω).Namely, there exists a random homeomorphism h(ω) so that Φ(t, ω) h(ω) =h(θtω) φ(t, ω). For more detail, we refer to [1].

Remark 2.2. Note that if ν is large enough, the above result still holds evenwhen ǫ is not small. This can be seen from the proof of the main result.

Having the above result, we can study the local bifurcation of (2.1) nearu = 0. In fact the dynamical behavior of the system (2.2) is determined bythe restricted system of (2.2) on M(ω). Furthermore the flow on the invariantmanifold is topologically equivalent to the SODE (2.4) which is structurallystable if ǫ 6= 0. So we can study the dynamics of (2.1) as ǫ passes through 0 bythe system (2.4). And if (2.4) undergoes a stochastic bifurcation as ǫ passesthrough 0, so does (2.1). We will demonstrate this bifurcation analysis method

STOCHASTIC BIFURCATION 5

for a stochastic parabolic partial differential equation (1.1) mentioned above;see §5.1.Another application of the main result is to study the existence of stationary

solution of stochastic dynamical systems. For this issue, we will consider thefollowing stochastic hyperbolic partial differential equation, in §5.2, with largediffusivity and highly damped term on [0, 2π]× (0,+∞)

utt(t, x) + αut(t, x) = ν∆u(t, x) + f(u(t, x), x) + u(t, x) W (t), (2.5)

with

u(0, x) = u0, ut(0, x) = u1, u(t, 0) = u(t, 2π) = 0, t > 0

where ν, α > 0. And f ∈ C2(R,R) is a bounded globally Lipschitzian non-linearity. Note that the stochastic Sine-Gordon equation (f = sin u) and sto-chastic Klein-Gordon equation (f being a polynomial in u) are in our scope.By the same approach for the proof of theorem A, the dynamical behavior ofthe system (2.5) restricted on a finite dimensional invariant manifold, whichattracts all the solution of (2.5), is determined by an SODE. So the existenceof the stationary solution for (2.5) can be obtained by that of the SODE.

Now we present the basic setup and some materials in random dynamicalsystems.Let H be a separable Hilbert space with norm | · | and inner product 〈·, ·〉.

D(A) is dense in H and A : D(A) → H is a linear operator which generates astrong continuous semigroup S(t) on H . Further we suppose S(t) satisfy thefollowing exponential dichotomy. There are two projections on H

Πc : H → Hc, Πs : H → Hs

such that H = Hc +Hs and Πc +Πs = IdH and there exists constants K > 0,0 ≤ α < β, such that

|ΠsS(t)x| ≤ Ke−βt|x|, t ≥ 0, (2.6)

|ΠcS(t)x| ≤ Ke−αt|x|, t ≤ 0 (2.7)

for x ∈ H . We write dimHc = n. Here we consider the general case thatHc may be bigger than the kernel of A. In fact if Hc is the kernel of A,the exponent α can be taken as 0. Further, Πc and Πs are assumed to becommuting with A.For our purpose we use the canonical probability space (Ω0,F0,P) which

consists of the sample paths of W (t), for more see [1]. Consider the following

6 W. WANG & J. DUAN

stochastic evolutionary equations

ut = νAu+ F ǫ(u) + σu W (t), (2.8)

u(0) = u0 ∈ H.

This equation can be written in the following mild integral form

u(t) = eνAtu0 +

∫ t

0

eνA(t−s)F ǫ(u(s))ds+

∫ t

0

σeνA(t−s)u(s) dW (s) (2.9)

By the assumption of A and F ǫ we know that equation (2.9) has a uniquelocal solution u(t, ω; u0) ∈ L2(Ω0, C(0, T ;H)) for some T > 0 in the sense ofprobability. For more about the solution of SPDEs we refer to [15].

For the future reference we give some knowledge of random dynamical sys-tems. First we are given a driven dynamical system (also called a metricdynamical system) θt : (Ω0,F0,P) → (Ω0,F0,P), t ∈ R satisfied

• θ0 = id,• θtθs = θt+s, for all s, t ∈ R,• the map (t, ω) 7→ θtω is measurable and θtP = P for all t ∈ R.

A random dynamical system (RDS) on a Polish space (X ; d) with Borel σ-algebra B over θt on (Ω0,F0,P) is a measurable map

ϕ : R+ × Ω0 × X → X(t, ω, x) 7→ ϕ(t, ω)x

such that

(i) ϕ(0, ω) = id (on X );(ii) ϕ(t+ s, ω) = ϕ(t, θsω)ϕ(s, ω) ∀t, s ∈ R

+ for almost all ω ∈ Ω0 (cocycleproperty).

A RDS ϕ is continuous or differentiable if ϕ(t, ω) : X → X is continuous ordifferentiable (see [1] for more details on RDS). As we consider the canonicalprobability space the metric dynamical system can be defined as

θtω(·) = ω(·+ t)− ω(t), t ∈ R, (2.10)

where ω(·) ∈ Ω0 is a sample path of W(t).For a continuous random dynamical system ϕ(t, ω) : X → X over

(Ω0,F ,P,

(θt)t∈R), we need the following notions to describe its dynamical behavior.

Definition 2.3. A multifunction M = M(ω)ω∈Ω of nonempty closed setsM(ω), ω ∈ Ω, contained in X is called a random set if

ω 7→ infy∈M(ω)

d(x, y)

is a random variable for any x ∈ X .

STOCHASTIC BIFURCATION 7

Definition 2.4. A random set B(ω) is called an tempered absorbing set of ϕif for any bounded set K ⊂ X there exists tK(ω) such that ∀t ≥ tK(ω)

ϕ(t, θ−tω,K

)⊂ B(ω).

and for all ε > 0

limt→∞

e−εtd(B(θ−tω)

)= 0, a.e. ω ∈ Ω,

where d(B) = supx∈B d(x, 0), with 0 ∈ X , is the diameter of B.

For more about random set we refer to [12].

Definition 2.5. A random set M(ω) is called an invariant set for a randomdynamical system ϕ(t, ω, x) if we have

ϕ(t, ω,M(ω)) ⊂M(θtω), for t ≥ 0.

If M(ω) can be written as a graph of a Lipschitz mapping

ψ(·, ω) : X1 → X2

where X = X1 ⊕ X2, such that

M(ω) = x1 + ψ(x1, ω)|x1 ∈ X1.Then M(ω) is called a Lipschitz invariant manifold of ϕ.

For more about the random invariant manifold, see [17].The following definition is important in the study of the random invariant

manifold.

Definition 2.6. Let M(ω) is an invariant manifold of RDS ϕ(t, ω). Theinvariant manifold M is called to have the asymptotic completeness propertyif ∀x ∈ H, there is y ∈ M(ω) such that

|ϕ(t, ω)x− ϕ(t, ω)y| ≤ D|x− y|e−kt, t ≥ 0

where D and k are some positive constants.

Asymptotic completeness property describes the attracting property ofM(ω)for RDS ϕ(t, ω). Then ϕ(t, ω) can be reduced onto M(ω), and the dynamicalbehavior of ϕ(t, ω) is determined by the reduced system on M(ω).

By the assumption that system (2.8) generates a dissipative random dynam-ical system. We suppose the tempered random set B(ω) is an absorbing setfor system (2.8). Then we may only consider the system restricted on B(ω).For convenience we still write the restricted system as (2.8). Then F ǫ is stillLipschitz with constant Lǫ

F and is bounded by M ǫ(ω) which is a temperedrandom variables. And it is easy to see that M ǫ(ω) → 0 a.e. as ǫ → 0.

8 W. WANG & J. DUAN

In order to use the random dynamical theory we introduce the followingstationary process z(ω) which solves

dz + zdt = dW. (2.11)

Solving equation (2.11) with initial value

z(ω) = −∫ 0

−∞eτw(τ)dτ,

we have a unique stationary process for (2.11)

z(θtω) = −∫ 0

−∞eτθτω(τ)dτ = −

∫ 0

−∞eτω(t+ τ)dτ + ω(t).

The mapping t 7→ z(θtω) is continuous. Moreover

limt→±∞

|z(θtω)||t| = 0 and lim

t→±∞

1

t

∫ t

0

z(θτω)dτ = 0 for a.e. ω ∈ Ω0.

We should point out that above property hold in a full probability θt invariantset Ω ⊂ Ω0. For the proof see [17]. In the following part to the end we considerthe probability space (Ω,F ,P) where

F = Ω ∩ U : U ⊂ Ω0.We consider the following random partial differential equation on (Ω,F ,P)

vt = νAv +Gǫ(θtω, v) + z(θtω)v, v(0) = x ∈ H (2.12)

where Gǫ(ω, v) := e−z(ω)F ǫ(ez(ω)v). It is easy to verify that Gǫ has the sameLipschitz constant as F ǫ and is also bounded by M ǫ(ω) since Gǫ(0) = 0. Bythe classical evolutionary equation theory equation (2.12) has a unique solutionv(t, ω; x) which is continuous about x for every ω ∈ Ω. Then

(t, ω, x) 7→ ϕ(t, ω)x := v(t, ω; x)

defines a continuous random dynamical system. We now introduce the trans-form

T (ω, x) = xez(−ω)

and its inverse transformT−1(ω, x) = xez(ω)

for x ∈ H . Then for the random dynamical system v(t, ω; x) generated by(2.12),

(t, ω, x) → T−1(θtω, v(t, ω;T (ω, x)) := u(t, ω; x)

is the random dynamical system generated by (2.8). For more the relationbetween (2.8) and (2.12) we refer to [17].

STOCHASTIC BIFURCATION 9

3. Random invariant manifolds

In this section we prove the existence of a random invariant manifold for(2.12).Let dH be the metric induced by the norm | · |. Then (H, dH) is a complete

separable metric space.Now we briefly give an approach to obtain the random invariant manifold

for system (2.12) by the Lyapunov-Perron method; for detail see [18].Projecting the system (2.12) onto Hc and Hs respectively we have

vc = νAvc + z(θtω)vc + gǫc(θtω, vc + vs) (3.1)

vs = νAvs + z(θtω)vs + gǫs(θtω, vc + vs) (3.2)

wheregǫc(θtω, vc + vs) = ΠcG

ǫ(θtω, vc + vs)

andgǫs(θtω, vc + vs) = ΠsG

ǫ(θtω, vc + vs).

Define the following Banach space for η, −β < η < −α,C−

η = v : (−∞, 0] → H : v is continuous and

supt∈(−∞,0]

e−ηt−∫ t

0z(θsω)ds|v(t)| <∞

with norm|v|C−

η= sup

t∈(−∞,0]

e−ηt−∫ t

0z(θsω)ds|v(t)|.

Define the nonlinear operator T on C−η as

T (v, ξ)(t, ω) = eνAt+∫ t0z(θτω)dτξ +

∫ t

0

eνA(t−τ)+∫ tτz(θςω)dςgǫc(θτω, vc + vs)dτ

+

∫ t

−∞eνA(t−τ)+

∫ t

τz(θςω)dςgǫs(θτω, vc + vs)dτ (3.3)

where ξ = Πcv(0). Then for each v, v ∈ C−η , we have

|T (v, ξ)− T (v, ξ)|C−

η

≤ supt≤0

ν−1KLǫ

F

(∫ t

0

e(−α−η)(t−s)ds+

∫ t

∞e(−β−η)(t−s)ds

)|v − v|C−

η

≤ ν−1KLǫF

( 1

η + β− 1

α + η

)|v − v|C−

η. (3.4)

If

ν−1KLǫF

( 1

η + β− 1

α + η

)< 1, (3.5)

10 W. WANG & J. DUAN

then by the fixed point theorem

v = T (v)

has a unique solution v∗ ∈ C−η . Then we have the following result about

the existence of random invariant manifold for the random dynamical systemϕ(t, ω) generated by (2.12). For the detailed proof see [18].

Proposition 3.1. Suppose the assumptions on A and F ǫ in section 2 and con-dition (3.5) hold. Then there exists a unique n dimensional invariant manifold

M(ω) in C−η for ϕ(t, ω). Moreover, if Mt(ω) = ξ + Πsv

∗(t, ω; ξ) : ξ ∈ Hcwith v∗(t, ω) being a stationary process in C−

η , then M(ω) = M0(ω) and

Mt(ω) = M(θtω), t ≥ 0.

Then by the transform T we have

Corollary 3.2. M(ω) = T−1(ω, M(ω)) is the unique n dimensional invariantmanifold of (2.8).

Remark 3.3. It is easy to see that M(ω) is independent of the choice of η.

4. Reduction to stochastic ordinary differential equations

In this section we prove that the invariant manifold obtained in the last sec-tion has the asymptotic completeness property. Then the dynamical behaviorof (2.12) is determined by the system restricted on the invariant manifold. Fur-thermore the system restricted on the invariant manifold is in fact topologicallyequivalent to a random ordinary differential equation provided the random or-dinary differential equation is structurally stable. This will complete the proofof main result Theorem A stated in §2.First we introduce the cone invariance in the random case. For a positive

random variable δ, define the following random set

Cδ :=(v, ω) ∈ H × Ω : |Πsv| ≤ δ(ω)|Πcv|

.

And the fiber Cδ(ω)(ω) = v : (v, ω) ∈ Cδ is called random cone. For a givenrandom dynamical system ϕ(t, ω) we give the following definition.

Definition 4.1. (Cone invariance) For a random cone Cδ(ω)(ω), there is arandom variable δ ≤ δ almost surely such that for all x, y ∈ H,

x− y ∈ Cδ(ω)(ω)implies

ϕ(t, ω)x− ϕ(t, ω)y ∈ Cδ(θtω)(θtω).

STOCHASTIC BIFURCATION 11

Then the random dynamical system ϕ(t, ω) is called possessing the cone in-variance property for the cone Cδ(ω)(ω).Remark 4.2. Cone invariance is an important property to study the inertialmanifold of infinite dimensional system, see [24, 25, 21]. In [21] the authorsintroduced the nonautonomous cone to study the inertial manifold for nonau-tonomous evolutionary systems.

For the random dynamical system ϕ(t, ω) generated by (2.12), we have thefollowing result.

Proposition 4.3. For small ǫ > 0 random dynamical system ϕ(t, ω) possessesthe cone invariance property for a cone with a deterministic positive constantδ. Moreover if there exists t0 > 0 such that for x, y ∈ H and

ϕ(t0, ω)x− ϕ(t0, ω)y /∈ Cδ(θt0ω),then

|ϕ(t, ω)x− ϕ(t, ω)y| ≤ De−kt|x− y|, 0 ≤ t ≤ t0,

where D = 1 + 1δ2

and k = νβ − LǫF − δ−1Lǫ

F > 0.

Proof. Let v, v be two solutions of (2.12) and p = vc − vc, q = vs − vs, then

p = νAcp + z(θtω)p+ gǫc(θtω, vc + vs)− gǫc(θtω, vc + vs), (4.1)

q = νAsq + z(θtω)q + gǫs(θtω, vc + vs)− gǫs(θtω, vc + vs). (4.2)

From (4.1), (4.2) and by the property of A and F ǫ we have

1

2

d

dt|p|2 ≥ −να|p|2 + z(θtω)|p|2 − Lǫ

F |p|2 − LǫF |p| · |q| (4.3)

and1

2

d

dt|q|2 ≤ −νβ|q|2 + z(θtω)|q|2 + Lǫ

F |q|2 + LǫF |p| · |q|. (4.4)

Then (4.4)−δ2× (4.3), we have

1

2

d

dt(|q|2 − δ2|p|2) ≤ −νβ|q|2 + z(θtω)|q|2 + Lǫ

F |q|2 + LǫF |p| · |q|+

ναδ2|p|2 − z(θtω)δ2|p|2 + δ2Lǫ

F |p|2 + δ2LǫF |p| · |q|.

It is easy to see that if (p, q) ∈ ∂Cδ(ω) (the boundary of the cone Cδ(ω)), then|q| = δ|p| and

1

2

d

dt(|q|2 − δ2|p|2) ≤ (να− νβ + 2Lǫ

F + δLǫF + δ−1Lǫ

F )|q|2.

Note that if ǫ is so small that

να − νβ + 2LǫF + δLǫ

F + δ−1LǫF < 0, (4.5)

12 W. WANG & J. DUAN

then |q|2 − δ2|p|2 is decreasing on ∂Cδ(ω). Thus it is obvious that wheneverx− y ∈ Cδ(ω), ϕ(t, ω)x− ϕ(t, ω)y can not leave Cδ(θtω).We now prove the second claim. If there is t0 > 0 such that ϕ(t0, ω)x −

ϕ(t0, ω)y /∈ Cδ(θt0ω), the cone invariance yields

ϕ(t, ω)x− ϕ(t, ω)y /∈ Cδ(θtω), 0 ≤ t ≤ t0,

that is

|q(t)| > δ|p(t)|, 0 ≤ t ≤ t0.

Then by (4.4) we have

1

2

d

dt|q|2 ≤ −(νβ − Lǫ

F − δ−1LǫF )|q|2, 0 ≤ t ≤ t0.

Hence

|p(t)|2 < 1

δ2|q(t)|2 ≤ 1

δ2e−2kt, 0 ≤ t ≤ t0.

So we have

|ϕ(t, ω)x− ϕ(t, ω)y| ≤ De−kt|x− y|, 0 ≤ t ≤ t0.

This completes the proof of the proposition.

Now we complete the proof of the main result of this paper.

Proof of the main result: Theorem A in §2

It remains to prove the asymptotic completeness of M(ω). We fix a ω ∈ Ω.Consider a solution

v(t, θ−tω) = (vc(t, θ−tω), vs(t, θ−tω))

of (2.12). Since M(ω) is invariant manifold for (2.12), there is a solutionv(t, θ−t+τω; τ, ω) of (2.12) on M(ω), such that

v(t, θ−t+τω; τ, ω) =(vc(t, θ−t+τω; τ, ω), vs(t, θ−t+τω; τ, ω)

)

with

vc(t, θ−t+τω; τ, ω)|t=τ = vc(τ, θ−τω).

Let

vc(0, τ ;ω) := vc(0, θτω; τ, ω)

and

vs(0, τ ;ω) := vs(0, θτω; τ, ω).

STOCHASTIC BIFURCATION 13

By the construction of M(ω)

|vs(0, τ ;ω)| ≤∫ ∞

0

e−νβr−∫ r

0z(θςω)dς |gǫs(θrω, vc + vs)|dr

:= Nν,ǫ(ω).

And Nν,ǫ(ω) is almost surely finite by the tempered property of M ǫ(ω) andz(θtω), see section 2. In fact it is also easy to see that Nν,ǫ(ω) ∼ O(ν−1) asν → ∞ and O(ǫ) as ǫ→ 0.Since vc(t, θ−t+τω; τ, ω)|t=τ = vc(τ, θ−τω), by the cone invariance

v(t, θ−tω)− v(t, θ−t+τω; τ, ω) /∈ Cδ(ω), 0 ≤ t ≤ τ.

Let S(ω) = vc(0, τ ;ω) : τ > 0. Since

|vc(0, τ ;ω)− vc(0, ω)| <1

δ|vs(0, τ ;ω)− vs(0, ω)|

≤ 1

δ

(Nν,ǫ(ω) + |vs(0, ω)|

),

S is a bounded random set in finite dimensional space. So we can pick out asequence τm → ∞ such that

limm→∞

vc(0, τm;ω) = Vc(ω).

Let V (t, θ−tω) = (Vc(t, θ−tω), Vs(t, θ−tω)) be a solution of (2.12) with V (0, ω) =(Vc(ω),Πsv

∗(0, ω;Vc(ω))). Then it is easy to check by contradiction that

v(t, θ−tω)− V (t, θ−tω) /∈ Cδ(ω), 0 ≤ t <∞which means the asymptotic completeness of M(ω). So the dynamical behav-ior of (2.12) is determined by

vc = νAcvc + z(θtω)vc + gǫc(θtω, vc +Πsv∗(t, ω, vc)). (4.6)

Furthermore since Nν,ǫ(ω) ∼ O(ǫ) as ǫ → 0, if random ordinary differentialequation

vc = νAcvc + z(θtω)vc + gǫc(θtω, vc) (4.7)

is structurally stable, the dynamical behavior of the system (4.6) is topologi-cally equivalent to that of the system (4.7) provided ǫ is small enough. Noticethat gǫc(θtω, vc) = ΠcG

ǫ(θtω, vc+0) with 0 ∈ Hs. That is the dynamical behav-ior of the infinite dimensional system (2.8) restricted on M(ω) can be obtainedfrom the following stochastic ordinary differential equation

duc(t) = νAcuc + F ǫc (uc(t)) + σuc(t) dW (t) (4.8)

provided (4.8) is structurally stable. Then the condition Hc = kerA yields themain result with (4.8) becoming (2.4).

14 W. WANG & J. DUAN

Remark 4.4. From the proof of the Theorem A, we see that the theoremis really proved for the random partial differential equation (2.12). In otherwords, we may apply Theorem A to (2.12) as long as appropriate conditionsare satisfied. Moreover, if the stochastic partial differential equation (2.1), orits transformed form, the random partial differential equation (2.12), has thelinear part with finite number of eigenvalues of zero real part and/or positivereal part, we can similarly prove a theorem on reduction to the random in-variant center-unstable manifold. In fact, we will apply this approach to astochastic wave type equation in §5.2.

5. Applications: Stochastic bifurcation and stationary states

In this final section, we consider stochastic bifurcation and stationary so-lution for SPDEs, via a couple of examples. We use the random invariantmanifold reduction principle established in the previous sections.

5.1. Parabolic partial differential equations driven by multiplicative

white noise. We study the stochastic bifurcation in the Chafee-Infante re-action diffusion perturbed by a multiplicative white noise in the sense ofStratonovich

du = (∆u+ δu− u3)dt+ u dW (t), in I, (5.1)

u|∂I = 0, (5.2)

where I ⊂ R is an open interval, δ is a real parameter, and W (t) is a realvalued Wiener process.. Let A = −∆ on I with Dirichlet boundary conditionand 0 < λ1 < λ2 ≤ · · · be the spectrum of A with eigenvectors e1, e2, · · ·respectively. Then we have the following evolutionary system on L2(I)

du = (−Au+ δu− u3)dt+ u dW (t). (5.3)

It is easy to prove that if δ < λ1, u = 0 is the globally exponentially stablestationary solution, see [8]. And if δ > λ1 then u = 0 is unstable. We will provethat system (5.3) undergoes stochastic bifurcation when δ passes through λ1.In fact we will apply our result to system (5.3) by truncating the nonlinearterm u3. Let

F ǫ(u) = −Θ(|u|αǫ

)u3,

where Θ : R → R is a smooth function with

Θ(x) =

0, |x| ≥ 2;1, |x| ≤ 1.

and |u|α = |Aαu|L2(I) with α ∈ (3/4, 1). It is easy to verify that F ǫ(u) isglobally bounded,

|F ǫ(u)| ≤M, ∀u ∈ H2α,

STOCHASTIC BIFURCATION 15

and globally Lipschitz,

|F ǫ(u)− F ǫ(v)| ≤ LǫF |u− v|α, ∀ u, v ∈ H2α

with M ≤ c0ǫ and LǫF ≤ c1ǫ

2 for constants c0, c1.Let us consider the following truncated system

du =[A1u+ (δ − λ1)u+ F ǫ(u)

]dt+ u dW (t), (5.4)

with A1 = −A + λ1. The semigroup generated by the operator A1 satisfies(2.6) and (2.7) with α = 0, β = λ2 − λ1 for some positive constant K anddimHc = 1. Then if ǫ is small, by Theorem A in §2, the system (5.4) has a onedimensional invariant manifold M(ω) which is asymptotically complete andthe dynamical behavior of (5.4) restricted on M(ω) is topologically equivalentto the following one dimensional SODE

du =[(δ − λ1)u+ F ǫ

c (u)]dt + u dW (t) (5.5)

with δ 6= λ1 and

F ǫc (u) =

∫

I

F ǫ(u+ 0)e1dx.

Near u = 0, by the definition of F ǫ, F ǫ(u) = u3. Then F ǫc (u) = au3 with

a =∫Ie41dx > 0 and (5.5) is

du =[(δ − λ1)u+ au3

]dt+ u dW (t). (5.6)

It is well known, see [2] or [14], that near the solution u = 0, system (5.6)undergoes a stochastic pitchfork bifurcation when δ passes through λ1. Ifδ < λ1, u = 0 is a stable stationary solution of system (5.5); if δ > λ1, u = 0is unstable and two new stable stationary solutions appear. Then by TheoremA, near the stationary solution u = 0, system (5.3) also undergoes a stochasticbifurcation when δ passes through λ1. And the bifurcation diagram also canbe described by that of the system (5.6).

Remark 5.1. The amplitude equations, which are SODEs, for stochastic par-tial differential equations also give some information to the dynamical behaviorof the system near a change of stability, see [6] for the amplitude equations ofthe stochastic partial differential equations with small noise on a bounded do-main.

5.2. Hyperbolic partial differential equations with large diffusivity

and highly damped term driven by multiplicative white noise. Weconsider the following hyperbolic equation with Dirichlet boundary conditiondriven by multiplicative white noise,

utt + aut = ν∆u + bu+ f(u) + σu W , in I (5.7)

16 W. WANG & J. DUAN

withu(0) = u0, ut(0) = u1, u(0) = u(2π) = 0

where I is taken as the interval (0, 2π) for simplicity and ν, a > 0. Heref ∈ C1,1(R,R) is bounded with global Lipschitz constant Lf . For examplef(x) = sin x, which yields the Sine-Gordon equation. Here W (t) is a realvalued Wiener process.We study the existence of the stationary solution of (5.7) by reducing the

system to a finite dimensional system on the invariant manifold, which isasymptotically complete. In fact by the same proof of Theorem A in §2,the dynamical behavior of (5.7) restricted on the invariant manifold, is deter-mined by a stochastic ordinary differential equation. Then the existence andstability of the stationary solution of the stochastic hyperbolic equation canbe obtained from that of the stochastic ordinary differential equation.Let H = H1

0 (I)×L2(I). Rewrite the system (5.7) as the following one orderstochastic evolutionary equation in H

du = vdt (5.8)

dv =[− νAu+ bu− av + f(u)

]dt+ σu dW (t) (5.9)

where A = −∆ with Dirichlet boundary condition on I. Note that Theorem A

can not be applied to the system (5.8)-(5.9) directly. But by Remark 4.4, wecan still have the same result of the Theorem A for the stochastic hyperbolicsystem (5.7). It will be clear after we transform (5.8)-(5.9) into the form of(2.12).First we prove that the system (5.8)-(5.9) generates a continuous random

dynamical system in H. Let φ1(t) = u(t), φ2(t) = ut(t) − σu(t)z(θtω) wherez(ω) is the stationary solution of (2.11). Then we have the following randomevolutionary equation

dφ1 =[φ2 + σφ1z(θtω)

]dt (5.10)

dφ2 =[νφ1 + bφ1 − aφ2 + f(φ1)

]dt+

[(σz(θtω)− σaz(θtω)−

σ2z2(θtω))φ1 − σz(θtω)φ2

]dt. (5.11)

By a standard Galerkin approximation procedure as in [29], the system (5.10)-(5.11) is wellposed. In fact we give a prior estimates. Multiplying (5.10) withνφ1 in H1

0 (I) and (5.11) with φ2 in L2(I). Since f is bounded by a simplecalculation we haved

dt

[ν|φ1|2H1

0(I) + |φ2|2L2(D)

]≤ K

(1 + |z(θtω)|2 + |z(θtω)|

)[ν|φ1|2H1

0(I) + |φ2|2L2(I)

]

for appropriate constant K. Then for any T > 0, (φ1, φ2) is bounded inL∞(0, T ;H) which ensures the weak-star convergence by the Lipschitz prop-erty of f , for the detail see [20]. Let Φ(t, ω) = (φ1(t, ω), φ2(t, ω)), then

STOCHASTIC BIFURCATION 17

ϕ(t, ω,Φ(0)) = Φ(t, ω) defines a continuous random dynamical system in H.Notice that the stochastic system (5.8)-(5.9) is conjugated to the random sys-tem (5.10)-(5.11) by the homeomorphism

T (ω, (u, v)) = (u, v + uz(ω)), (u, v) ∈ Hwith inverse

T−1(ω, (u, v)) = (u, v − uz(ω)), (u, v) ∈ H.Then ϕ(t, ω, (u0, v0)) = T (θtω, ϕ(t, ω, T

−1(ω, (u0, v0)))) is the random dynam-ical system generated by (5.8)-(5.9). For more relation about the two systemssee [20].Define

Aν =

(0, −idL2(I)

νA− b, a

), F (Φ) =

(0

f(φ1)

),

Z(θtω) =

(σz(θtω), 0

σz(θtω)− σaz(θtω)− σ2z2(θtω), −σz(θtω)

),

where −idL2(I) is the identity operator on the Hilbert space L2(I).Then (5.10)-(5.11) can be written as

dΦ

dt= −AνΦ+ Z(θtω)Φ + F (Φ) (5.12)

which is in the form of (2.12). Thus by Remark 4.4, we can still apply TheoremA here.The eigenvalues of the operator A are λk = k2 with corresponding eigenvec-

tors sin kx, k = 1, 2, · · · . Then the operator Aν has the eigenvalues

δ±k =a

2±√a2

4+ b− νk2, k = 1, 2, · · ·

and corresponding eigenvectors are (1, δ±k ) sin kx. Define subspace of HHc = span(1, 0)T sin 2x, Hu = span(1, δ−1 )T sin x

and Hcu = Hc ⊕Hu. Write the projections from H to Hc, Hu and Hcu as Pc,Pu and Pcu respectively. We also use the subspaces

H−c = span(1, δ±2 )T sin 2x, H−

u = span(1, δ±1 )T sin 2xwith the projections P−

c and P−u from H to H−

c and H−u respectively. Let

H−cu = H−

c ⊕H−u and H−

s = span(1, δ±k )T sin kx, k ∈ Z+ \ 1, 2.

Here we consider a special case that b = 4ν and ν = a2/4. Then the operatorAν has one zero eigenvalue δ−2 = 0, one negative eigenvalue δ−1 = a

2and the

others are all complex numbers with positive real part. Since (1, δ±k ) sin kx

18 W. WANG & J. DUAN

are not orthogonal, we introduce a new inner product (see [22, 23, 28]), whichdefines an equivalent norm on H. For (u1, v1), (u2, v2) ∈ H−

cu, define

〈(u1, v1), (u2, v2)〉cu = 2(〈−Au1, u2〉0 +

a2

4〈u1, u2〉0 + 〈a

2u1 + v1,

a

2u2 + v2〉0

)

and for (u1, v1), (u2, v2) ∈ H−s define

〈(u1, v1), (u2, v2)〉s = 2(〈Au1, u2〉0 −

a2

4〈u1, u2〉0 + 〈a

2u1 + v1,

a

2u2 + v2〉0

)

where 〈·, ·〉0 is usual inner product in L2(I). Then we introduce the followingnew inner product in H defined by

〈U, U〉H = 〈Ucu, Ucu〉cu + 〈Us, Us〉sfor U = Ucu + Us, U = Ucu + Us ∈ H. And the new norm

‖U‖2H = 〈U, U〉His equivalent to the usual norm ‖ · ‖H1

0×L2(D). Moreover Hcu and Hs = H⊖Hcu

is orthogonal. By the definition of the new norm a simple calculation yieldsthat

Proposition 5.2. (1) ‖U‖H =√2‖v‖L2(D), for U = (0, v) ∈ H.

(2) ‖U‖H ≥√2a2‖u‖L2(D), for U = (u, v) ∈ H.

(3) In terms of the new norm the Lipschitz constant of f is2Lf

a.

For the detail proof we refer to [23].A simple calculation yields that in terms of the new norm the semigroup S(t)

generated by Aν satisfies the exponential dichotomy (2.6)-(2.7) with α = 0,β = a

2and dimHc = dimHcu = 2. Taking η = −β−α

2, condition (3.5) for (5.7)

becomes8KLf

a2< 1 (5.13)

which holds if a is large enough.Define the space C−

η with z(θtω) replaced by Z(θtω). Then by the randominvariant manifold theory in §3 above, we know the system (5.12) has a two di-mensional random invariant manifold M(ω) in C−

η provided a is large enough.Furthermore by the same proof of Theorem A, M(ω) is asymptotically com-plete. Then the dynamical behavior of (5.12), that is, (5.7), restricted onM(ω) is determined by following reduced system which is a finite dimensionalrandom system

dΦcu

dt= −PcuAνΦcu + Z(θtω)Φcu + PcuF (Φcu). (5.14)

STOCHASTIC BIFURCATION 19

provided a is large enough. To see the reduced system more clear, projectingthe system (5.7) to spansin x and spansin 2x by the projection Π1 and Π2

respectively we have the following system

u1 = v1 (5.15)

v1 =3a2

4u1 − av1 +Π1f(u) + σu1 W (t) (5.16)

u2 = v2 (5.17)

v2 = −av2 +Π2f(u) + σu2 W (t). (5.18)

By the new inner product in H, define the following new inner product in R2

as((u, v), (u, v)

)= 2

[a24u1u2 + (

a

2u1 + v1)(

a

2u2 + v2)

].

Then projecting the above system (5.15)-(5.18) to Hcu by the new inner prod-uct in R

2 and noticing Φs is taken as 0 in the reduced system (5.14) we havethe final reduced system

u1 =a

2u1 +

2√10

5Π1f(u1 + u2) + σu1 W (t) (5.19)

u2 = Π2f(u1 + u2) + σu2 W (t) (5.20)

which is a two dimensional stochastic ordinary differential equation. And thedynamical behavior of the system (5.7) restricted on the manifold M(ω) isdetermined by that of the system (5.19)-(5.20) if a is large. So if system (5.19)-(5.20) has a stationary solution, so does the system (5.7). And the stabilityproperty of the stationary solution is also determined by that of (5.19)-(5.20).

Acknowledgements:

The authors thank Dirk Blomker for useful comments, and thank a refereefor the suggestion that improves the result in §5.2.

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(W. Wang) Institute of Applied Mathematics, Chinese Academy of Sciences,

Beijing, 100080, China

E-mail address, W. Wang: wangwei@amss.ac.cn

(J. Duan) Department of Applied Mathematics, Illinois Institute of Tech-

nology, Chicago, IL 60616, USA

E-mail address, J. Duan: duan@iit.edu