INVARIANT MANIFOLDS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Transcription

1 INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS STRCT Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature In this paper, we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations We first introduce a random graph transform and a fixed point theorem for nonautonomous systems Then we show the existence of generalized fixed points which give the desired invariant manifolds 1 INTRODUCTION Invariant manifolds are essential for describing and understanding dynamical behavior of nonlinear and random systems Stable, unstable and center manifolds have been widely used in the investigation of infinite dimensional deterministic dynamical systems In this paper, we are concerned with invariant manifolds for stochastic partial differential equations The theory of invariant manifolds for deterministic dynamical systems has a long and rich history It was first studied by Hadamard [9], then, by Liapunov [12] and Perron [16] using a different approach Hadamard s graph transform method is a geometric approach, while LiapunovPerron method is analytic in nature Since then, there is an extensive literature on the stable, unstable, center, centerstable, and centerunstable manifolds for both finite and infinite dimensional deterministic autonomous dynamical systems (see abin and Vishik [2] or ates et al [3] and the references therein) The theory of invariant manifolds for nonautonomous abstract semilinear parabolic equations may be found in Henry [1] Invariant manifolds with invariant foliations for more general infinite dimensional nonautonomous dynamical systems was studied in Chow et al[6] Center manifolds for infinite dimensional nonautonomous dynamical systems was considered in Chicone and Latushkin [5] Recently, there are some works on invariant manifolds for stochasticrandom ordinary differential equations by Wanner [2], rnold [1], Mohammed and Scheutzow [1], and Schmalfuß [19] Wanner s method is based on the anach fixed point theorem on some anach space containing functions with particular exponential growth conditions, which is essentially the LiapunovPerron approach similar technique has been used by rnold In contrast to this method, Mohammed and Scheutzow have applied a classical technique Date: December 1, Mathematics Subject Classification Primary: 6H15 Secondary: 3H5, 3L55, 3L25, 3D1 Key words and phrases Invariant manifolds, cocycles, nonautonomous dynamical systems, stochastic partial differential equations, generalized fixed points 1

2 2 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS due to Ruelle [1] to stochastic differential equations driven by semimartingals In Caraballo et al [22] an invariant manifold for a stochastic reaction diffusion equation of pitchfork type has been considered This manifold connects different stationary solutions of the stochastic differential equation In Koksch and Siegmund [11] the pullback convergence has been used to construct an inertial manifold for nonautonomous dynamical systems In this paper, we will prove the existence of an invariant manifold for a nonlinear stochastic evolution equation with a multiplicative white noise: (1) where is a generator of a semigroup satisfying a exponential dichotomy condition, is a Lipschitz continuous operator with, and is the noise The precise conditions on them will be given in the next section Some physical systems or fluid systems with noisy perturbations proportional to the state of the system may be modeled by this equation In order to show the existence of an invariant manifold, we will first show this stochastic evolution equation generates a random dynamical system by using a standard technique to transform this equation into a conjugated random evolution equation without a white noise but with random coefficients Then, we will prove the existence of an invariant manifold for the conjugated random evolution equation, and finally we will transform the results back to the original stochastic evolution equation Our method showing the existence of an invariant manifold is different from the methods mentioned above, which is an extension of the result by Schmalfuß [18] We will introduce a random graph transform This graph transform defines a random dynamical system on the space of appropriate graphs One ingredient of a random dynamical system is a cocycle (see the next section) n invariant graph of this graph transform is a generalized fixed point for cocycles generalized fixed point defines an entire trajectory for the cocycle pplying this fixed point theorem to the graph transform dynamical system we can find under a gap condition a fixed point contained in the set of Lipschitz continuous graphs which represent the invariant manifold The main assumption is the gap condition formulated by a linear twodimensional random equation This equation allows us to calculate a priori estimate for the fixed point theorem We note that this linear random differential equation has a nontrivial invariant manifold if and only if the gap condition is satisfied Hence, our results are optimal in this sense We believe that our technique can be applied to other cases that are treated in ates et al [3] We also note that we do not need to use the semigroup given by the skew product flow In Section 2, we recall some basic concepts for random dynamical systems and show that the stochastic partial differential equation (1) generates a random dynamical system We introduce a random graph transform in Section 3 generalized fixed point theorem is presented in Section Finally, we present the main theorem on invariant manifolds in Section 5

3 INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 3 2 RNDOM DYNMICL SYSTEMS We recall some basic concepts in random dynamical systems Let be a probability space flow of mappings is defined on the sample space such that (2)! "#%')( "#,)( for 1 This flow is supposed to be measurable, where 2 is the collection of orel sets on the real line To have this measurability, it is not allowed to replace by its completion 8 see rnold [1] Page 5 In addition, the measure is assumed to be ergodic with respect to " Then 9 : is called a metric dynamical system For our applications, we will consider a special but very important metric dynamical system induced by the rownian motion Let be a twosided Wiener process with trajectories in the space of real continuous functions defined on, taking zero value at This set is equipped with the compact open topology On this set we consider the < = measurable flow " defined by The distribution of this process generates a measure on 2 which is called the Wiener measure Note that this measure is ergodic with respect to the above flow see the ppendix in rnold [1]?> Later on we will consider, instead of the whole, a invariant subset of measure one and the algebra with respect of 2 to set is called invariant if for On we consider the restriction of the Wiener measure also denoted by The dynamics of the system on the state space For our applications it is sufficient to assume that cocycle is a mapping: which is )2 'FGF2 H! I K for L and IF forms a random dynamical system D E measurable such that IJ I # overc the flow is described by a cocycle is a complete metric space I H Then together with the metric dynamical system Random dynamical systems are usually generated by differential equations with random coefficients NM O H or finite dimensional stochastic differential equations QO SR provided that the global existence and the uniqueness can be ensured For details see rnold [1] We call a random dynamical system continuous if the mapping is continuous for U and E I9 H I IJ P IT P

4 = JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS Now we start our investigation on the following stochastic partial differential equation (3) on a separable anach space < Here is a linear partial differential operator is an one dimensional standard Wiener process, and describes formally a white noise Note that is interpreted as a Stratonovich differential However, the existence theory for stochastic evolution equations is usually formulated for Ito equations as in Da Prato and Zabczyk [], Chapter The equivalent Ito equation for (3) is given by () In the following, we assume that the linear (unbounded) operator generates a strongly continuous semigroup on Furthermore, we assume that satisfies the exponential dichotomy with exponents and bound, ie, there exists such that a continuous projection on (i) (ii) the restriction,, is an isomorphism of "! for as the inverse map (iii) define "# % ) '( "!,, "# ' (5) where 1 Denote and Then, 32 onto itself, and we ) ' For simplicity we set 5 For instance, if the operator = is a strongly elliptic and symmetric differential operator on a smooth domain 6 of order under the homogeneous Dirichlet boundary conditions, then the above assumptions are satisfied with 8 In this case has the spectrum << <89: 9 89 Q< << where the space spanned by the associated eigenvectors is equal to 9 For any the associated eigenspace is finite dimensional The space is spanned by the associated eigenvectors for D< < <,< 9 and 9 = 9 We assume that is Lipschitz continuous on?> )I, 5= HI,'= IN with the Lipschitz constant Then, for any initial data I, there exists a unique solution of () For details about the properties of this solution see Da Prato and Zabczyk [], Chapter We also assume that The stochastic evolution equation () can be written in the following mild integral form: I =C DC DC EC F 5=C DC IT almost surely for any IT Note that the theory in [] requires that the associated probability space is complete

5 INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 5 In order to apply the random dynamical systems techniques, we introduce a coordinate transform converting conjugately a stochastic partial differential equation into an infinite dimensional random dynamical system lthough it is wellknown that a large class of partial differential equations with stationary random coefficients and Ito stochastic ordinary differential equations generate random dynamical systems (for details see rnold [1], Chapter 1), this problem is still unsolved for stochastic partial differential equations with a general noise term The reasons are: (i) The stochastic integral is only defined almost surely where the exceptional set may depend on the initial state I and (ii) Kolmogorov s theorem, as cited in Kunita [13] Theorem 11, is only truej for finite dimensional random fields Moreover, the cocycle has to be defined for any However, for the noise term considered here, we can show that () generates a random dynamical system To prove this property, we need the following preparation We consider the onedimensional linear stochastic differential equation: (6) solution of this equation is called an OrnsteinUhlenbeck process Lemma 21 i) There exists a " invariant set 92 sublinear growth: > of measure J one ii) For the = DC C exists and generates a unique stationary solution of (6) given by The mapping = is continuous iii) In particular, we have iv) In addition, for J > C EC for C T = E of full measure with C C Proof i) It follows from the law of iterated logarithm that there exists a set 2, such that for > The set of these s is invariant

6 = 6 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS ii) This can be proven as in Øksendal [15] Page 35 The existence of the integral on the right hand side for follows from the law of iterated logarithm Using the law of iterated logarithm again, the function C DC # is an integrable majorant for C for the continuity at convergence iii) # 3 y the law of iterated logarithm, for such that Hence > > C and CT= Hence follows straightforwardly from Lebesgue s theorem of dominated C EC # and L # C there exists a constant # C which gives the convergence relation in iii) Hence, these convergence relations always define a " invariant set which has a full measure iv) 2 Clearly, from ii) Hence by the ergodic theorem we obtain iv) for This set is also "" invariant Then we set The proof is complete We now replace 2 for by 2 given in Lemma 21 The probability measure is the restriction of the Wiener measure to this algebra, which is also denoted by In the following we will consider the metric dynamical system ' We now back to show that the solution of () defines a random dynamical system To see this, we consider the random partial differential equation IJ () 3 where It is easy to see that for any the function has the same global Lipschitz constant as In contrast to the original stochastic differential equation, no stochastic integral appears here The solution can be interpreted in a mild sense (8) "! P I! # % P(' =C C EC

7 % I I INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS We note that this equation has a unique solution for every appear Hence the solution mapping H I : H I generates a random dynamical system Indeed, the mapping is 32 measurable H I Let be the solution mapping of () which is defined for We now introduce the transform (9) and its inverse transform (1) for I6 and J I I G No exceptional sets 2 JE Lemma 22 Suppose that is the random dynamical system generated by () Then (11) H I < H I H is a random dynamical system For any IT this process is a solution version of () H I Proof pplying the Ito formula to I converse is also true, since and H I J are defined for any is the inverse of, and thus H I : H I? gives a solution of () for each dynamical system Since is measurable with respect to I % gives a solution of () The and It is easy to check that (11) defines a random so is this Similar transformations have been used by Caraballo, Langa and Robinson [22] and Schmalfuß [18] Note that our transform has the advantage that the solution of () generates a random dynamical system for the wise differential equation In Section 5 we will prove the existence of invariant manifolds generated by () These invariant manifolds can be transformed into invariant manifolds for (3) 3 RNDOM GRPH TRNSFORM In this section, we construct a random graph transform The fixed point of this transform gives the desired invariant manifold for the random dynamical system generated by () We first recall that a multifunction of nonempty closed sets, contained in a complete separable metric space )I is a random variable for any IJ Definition 31 random set If we can represent H is called an invariant set if U> " by a graph of a Lipschitz mapping < 1 2 is called a random set if H

8 ! 8 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS such that then 3I I I is called a Lipschitz continuous invariant manifold < Let be a Lipschitz continuous function with Lipschitz constant and also let We consider the system of equations! P = =! P =C DC DC EC (12)! # P!! P 5=C DC DC C on some interval! Note that if (12) has a solution on then a mapping and defines another mapping (13) This latter mapping will serve as the random graph transform Recall that a random variable (1) 6 defines is a generalized fixed point of the mapping if for We assume that is a Lipschitz continuous mapping from to and it takes zero value at zero Conditions for the existence of a generalized fixed point are derived in the next section in the case of a random dynamical system The following theorem describes the relation between generalized fixed points and invariant manifolds Theorem 32 Suppose that is the generalized fixed point of the mapping Then the graph of is the invariant manifold of the random dynamical system generated by () Proof Let be the graph of such that 3I 3I Then for I, we obtain I I I I I, U by the definition of : I if and only if For the measurability statement see Section 5 below I I I y this theorem, we can find invariant manifolds of the random dynamical system generated by () by finding generalized fixed points of the mapping defined in (13) To do so, we will use a generalized fixed point theorem for cocycles and thus we need to show that the above mapping is in fact a random dynamical system For the remainder of this section we will show that defines a random dynamical system We will achieve this in a

9 M M INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 9 few lemmas In the following we denote by # functions from, with value zero at zero, into a anach space the anach space of Lipschitz continuous with the usual (Lipschitz) norm # # ( = = Moreover, denotes the anach space of bounded continuous functions, with value zero at zero and with linearly growth The norm in this space is defined as We first present a result about the existence of a solution of the integral system (12) The proof is quite technical and is given in the Lemma 33 Let be the Lipschitz constant of the nonlinear term # partial differential equation () Then for any H a such that on < H < " " Let be the space of continuous mappings from # some and M! system (12) has a contraction constant less than one Then for M # continuous function M such that # ' < in the random, there exists the integral system (12) has a unique solution into Note that for, the fixed point problem defined by the integral and some Lipschitz # the same contraction constant can be chosen This follows from the structure of the contraction constant see (29) below We would like to calculate a priori estimates for the solution of (12) To do this we need the following lemma and its conclusion on monotonicity will also be used later on Lemma 3 We consider the differential equations (15) with generalized initial conditions (16)! " = = Then this system has a unique solution on! for some interval is independent of Let conditions Then we have! The proof is given in the ppendix This be solutions of (15) but with the generalized initial and " for ' Now we can compare the norms for the solution of (12) and that of (15)(16)

10 = 1 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS be an interval on which the assumptions of the anach fixed point theorem (see the proofs of Lemma 33, 3) are satisfied for (12) and (15)(16) for some Then the norm of the solution of (12) is bounded by the solution of, and 8 being the Lipschitz norm of That is, Lemma 35 Let # (15) (16) with, The proof is given in the ppendix We obtain from Lemma 33 that H H exist for any on some interval We also have # % and! = = # for and! # We have that! because < is a bijection Indeed this mapping is the inverse of I I )I I on One can see this if we plug in <, which is given by, the right hand side of (12) at zero into the projection of the right hand side of (8) for, and vice versa if we plug in this expression into the right hand side of the first equation of (12) On the other hand, we have > = = = = = = = Repeating the arguments of Lemma 35 we obtain = H = for any Hence, we have the following result Lemma 36 The solution of the integral system (12) has the following regularity: # and # In particular, # for sufficiently small Note that by the fixed point argument, Moreover, the comparison result in Lemma 35 remains true and exist only for small To see this, we are going to show that We would like to extent these definitions to if the Lipschitz constant of is bounded by a particular value, then the Lipschitz constant of has the same bound s a preparation we consider the matrix =

11 M I INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 11 which has the eigenvalues (1) These eigenvalues are real and distinct if and only if K=' Then the associated eigenvectors can be written as " We order as 8 The elements are positive Lemma 3 Let given by the fixed point argument for and closed ball # # in will be mapped into itself: #! Proof Let I be the solution of the linear initial value problem and let be the solution operator of M I M be chosen such that (15), (16) have a solution on Then the #! U> I5 I! P Note that and commute Hence differential equation (15) Since we obtain that # )! P is a solution operator of the linear " " )! ) P For the initial conditions (15), (16) )! P and Hence find that # # small depending on such that Since we will equip # choose the state space! we can calculate explicitly for the solution of )! P y the comparison results from Lemmas 35 and36, we # for #! U> # with the norm in Section 5, in the following we will # with the metric 3I HIK= Now we show that the random graph transform defines a random dynamical system

12 % % % 12 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS Theorem 38 Suppose that the gap condition (1) is satisfied Then is welldefined by (13) for any and In addition, together with the metric dynamical system induced by the rownian motion defines a random dynamical system In particular, the following measurability for the operators of the cocycle holds: is 392 measurable for any Proof y Lemma 33, the mapping extend this definition for any To this end we introduce random variables is defined for small " by So we first have to where is defined in (29) below Since is continuous in random variable Hence, "!, and (12) has a unique solution on for We define a sequence by, # and so on Suppose that for some we have that! definition of in (29) implies by Lemma 21 the mapping L is continuous Hence for any and there exists an such that < <<! this is a Then the This is a contradiction, because We can now define (18) # < (H < << < is given by (12) on some inter We show that the right hand side satisfies (12) Suppose that H val for We have < < H < U see Lemma 3 Similarly, H "# H H "# is given by (12) on some interval H H! 5= # # We set U! U

13 = # # (! # ( # # INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 13 y the variation of constants formula on we have for! (19)!! # P 5= ( P =! = # P ( =C! % P(' = # =C! % P(' EC ( = #,)( #! P 5= = 5= C! % P(' EC Now we consider the second equation of (12) with initial condition! # Then at we have for the solution of the second equation which is equal to "# H # "# Hence for H! # 5= #, # U! U we can find! P #, (! = P, C, DC DC C which gives us together with (19) that solves (12) on! and Since so is by Lemma (3) The extension of the definition of!! is correct since we obtain the same value for different whenever For this uniqueness we note that given by the above formula is the inverse of I D I )I which is independent of the choice of and This implied the independence of on y a special choice of (for instance and continuing the above iteration procedure we get (18) y this iteration we also obtain that ' For the measurability, we note that H H H are > measurable because these expressions are given as an wise limit of the iteration of the anach fixed point theorem starting with a measurable expression On the other hand, H H H is continuous Hence by Castaing and Valadier [], Lemma III1, the above terms are measurable with respect to The measurability follows now by the composition formula (18) EC

14 1 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS Remark 39 i) Note that the solution of (15), (16) can be extended to any time interval Then lemma 35, 35 remain true for any ii) Similar to the extension procedure we can show that is defined for any! J and EXISTENCE OF GENERLIZED FIXED POINTS y Theorem 32, the problem of finding invariant manifolds for a cocycle is equivalent to finding generalized fixed points for a related (but different) cocycle In this section, we present a generalized fixed point theorem for cocycles Let and be as in Section 2, except that, in this section, we do not need any measurability assumptions Namely, is an invariant set (of full measure) under the metric dynamical system Let be a cocycle on a complete metric space ' Recall that a mapping H " for ' is called a generalized fixed point of the cocycle if Note that by the invariance of with respect to, the trajectory forms an entire trajectory for The following generalized fixed point theorem for cocycles is similar to the third author s earlier work [19] Theorem 1 Let ' be a complete metric space with bounded metric Suppose that H U> for and that IG H I is! continuous In addition, J we assume the contraction condition: There exists a constant such that for IH )I Then has a unique generalized fixed point in Moreover, the following convergence property holds for any J and IT Proof Let IT For (2) E H I we consider the sequence To see that this sequence is a Cauchy sequence, we compute by using the cocycle property I I H I I I = I = = I H = )I I for We denote the limit of this Cauchy sequence by I U I

15 6 INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 15 If we replace I in (2) by another element 6 we obtain the same limit which follows from This implies that I H " is independent of choice of I Now we prove the convergence property In fact, H I )I H IH " I " = I H " = I I' for where denotes the integer part of Since = I I HI are uniformly bounded for U and IT I9 H I is continuous, for we obtain H H I I I I the values = Next, we show that is, as a matter of fact, a generalized fixed point for Since " Finally, we prove the uniqueness of the generalized fixed point Suppose there is another generalized fixed point 6 Let " and 6 H U J are bounded in and 6 letting H, we have Since and 6 3I I This completes the proof Remark 2 The constant in the above generalized fixed point theorem may be taken as dependent, as long as the following condition is satisfied: >! This latter condition is usually assumed in the situation of ergodicity For applications see for instance Schmalfuß [19] and Duan et al [8] 5 RNDOM INVRINT MNIFOLDS In this final section, we show that the random graph transform, defined in (13), has a generalized fixed point in the state space #! with the metric )I HIK= (21) by using Theorem 1 Thus by Theorem 32, the graph of this generalized fixed point is an invariant manifold of the random dynamical system generated by () We first consider the basic properties of the metric space Lemma 51 The metric space HI = is complete and the metric is bounded # H 3I

16 I I I 16 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS Proof Let 3I Subsequently, HI (22) for Hence, we have for any that I be a Cauchy sequence in Since = = = = for is complete we have I Since the left hand side is uniformly bounded by so is the right hand side of (22) Hence I #! The boundedness assertion is easily seen We now check the assumptions of the generalized fixed point Theorem 1 Let random dynamical system given by the graph transform in (13) be the Theorem 52 Suppose that the gap condition (1) is satisfied Then the < random graph transform defined in (13) has a unique generalized fixed point in where is given in Lemma 3 The graph of this generalized fixed point, namely, )I I H I generated by () is an invariant manifold for the random dynamical system Proof y Lemma 3, Theorem 38 we know that < maps into itself efore we check the contraction condition in Theorem 1 we calculate an estimate for This norm is given where is a solution of (12) for and n estimate for is given by defined in! (15), (16) for y the monotonicity of in we obtain that for is an estimate for any Now we can calculate explicitly which gives us the estimate ) #! E (23) P ' We now check the contraction condition To this end we consider problem (12) for two different elements and we denote the solutions by In particular, we have 5=! = y the Lipschitz continuity of the nonlinear term in the random partial differential equation (), we can estimate > 5= which implies that?> = Similar to Lemma 35 we can estimate (2) by with (25) and '= by, where = = = = = ) = # and is a solution of (15)! P

17 # = = # = # # = INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 1 Indeed, we can estimate the norm of initial condition 5= : 5= = We have a bound for '= = ) E #! P from (23) and # Since as a solu for the above general! We have chosen P tion (15), (16) at is increasing in and the value ized initial conditions (25) is an estimate for (2) for any = ) We now can calculate explicitly For these calculations we have used that the solution operator for the linear problem (15) can be written as ) )! P These calculations of (15) yield with the initial conditions (25) )! P ) )! P = ) ) )! = P In summary, we have for 5= = ) ) % Since!, we thus obtain the contraction condition in Theorem 1 for We obtain similar estimates if we replace by that is continuous at Then these estimates show us So we have found that all assumption of Theorem 1 are satisfied Hence the dynamical system generated by the graph transform has a unique generalized fixed point in The graph of defines a desired invariant manifold for the random dynamical system by Theorem 32 It remains to prove that this manifold is measurable Lemma 53 The manifold is a random manifold I Proof The fixed point is the wise limit of H )I for I and for some in as U I, see Theorem 1 Hence the mapping measurable for any I In order to see that that for any IJ HI = (26) is is a random set we have to verify

18 I I 18 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS is measurable, see Castaing and Valadier [] Theorem III9 Let be a countable dense set of the separable space Then the right hand side of (26) is equal to (2) IK= = < which follows immediately by the continuity of The measurability of (2) follows since is measurable for any Under the additional assumption : we can show that is an unstable manifold denoted by : For any and I there exists an I such that H I I I (28) and I tends to zero We set H )I H 3I H I I Equation (28) is satisfied because I H I )I is the inverse of I H 3I, and because is the fixed point of the graph transform The value H can be estimated by a solution of (15), (16) on! with and and can be calculated explicitly for any Hence H )! HI P (We have to replace by!) We can derive from Lemma 21 iv) EC! for any if is chosen sufficiently large depending on and Hence H tends to zero exponentially On the other hand we have for H )I % H This convergence is exponentially fast We conclude that () 3I Q for is the unstable manifold for However, our intention is to prove that () has an invariant (unstable) manifold On account of conjugacy of () and () by (9) and (1) we will now formulate the following result be the solution version of () generated by (11) Then is the invariant manifold of if and only if is the invariant manifold of Moreover, if is an unstable manifold, then so is Theorem 5 Let by the random dynamical system generated by () and Proof We have the relationship between and given in Lemma 22 H H H U> Note that H does not change the exponential convergence of )I : H H H is unstable It follows that has a sublinear growth rate, see Lemma 21iii) Thus the transform

19 !! INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 19 Remark 55 Note that the main Theorem 52 represents the best possible result in the following sense If we consider the solution of the two dimensional problem (15) then this differential equation generates a non trivial invariant manifold if and only if the gap condition (1) is satisfied Hence we can not formulate stronger general conditions for the existence of global manifolds Here nontrivial means that the dimension of the manifold is less than the dimension of the space PPENDIX PROOFS OF THE LEMMS 33, 3 ND 35 We now give the proof of the technical lemmas 33, 3 and 35 which are based on the usual anach fixed point theorem Proof of Lemma 33: We consider the following operator for some Note that!! Set where % # P(' = = % P(' =C % P(', "% P(', 5=C, depend on It is obvious that if so is U H! DC DC EC DC DC C and fixed point for is a solution of (12) on We check that the contraction condition of the anach fixed point theorem is satisfied We set = 6 = 6 ' y the Lipschitz continuity of : = % 8 #

20 ) ) ) 2 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS Hence we obtain by (5) for! % ) # P(' ( DC DC EC! % ) P('@ ( # # % ) P('! % ) P(' DC DC C! "% ) P(' ( DC DC C! "% ) P(' DC DC # # Choosing (29) sufficiently small, we have! 8 "% ) ( EC % We now can take the supremum with respect to and 8! for the left hand side Hence for sufficiently small the operator is a contraction ) P(' Proof of Lemma 3: The proof of existence and uniqueness is similar to the proof in Lemma 33 The solution can be constructed by successive iterations of (15), (16) If we start with, we get H " U< < < " U << < which gives the conclusion These inequalities also show if exist on! so do H The inequalities for the contraction condition do not contain Proof of Lemma 35: Let, be sequences generated by the successive iterations starting with and # These sequences converge to the solution of (12) and (15) (16) provided sufficiently small We "% ( "% ) ( P(' P(' "% P('@ F % # P(' "% ( "% ) (

21 ) INVRINT MNIFOLDS FOR STOCHSTIC PRTIL DIFFERENTIL EQUTIONS 21 and "% ) ( "% P(' ) ( P(' "% Ł % P(' # ) P(' It is easily seen and that then which gives the REFERENCES [1] L rnold Random Dynamical Systems Springer, New York, 1998 [2] abin and M I Vishik ttractors of Evolution Equations NorthHolland, msterdam, London, New York, Tokyo, 1992 [3] P ates, K Lu, and C Zeng Existence and Persistence of Invariant Manifolds for Semiflows in anach Space, volume 135 of Memoirs of the MS 1998 [] C Castaing and M Valadier Convex nalysis and Measurable Multifunctions LNM 58 Springer Verlag, erlin Heidelberg New York, 19 [5] C Chicone and Y Latushkin Center manifolds for infinite dimensional nonautonomous differential equations J Diff Eqns, 11: , 199 [6] SN Chow, K Lu, and X Lin Smooth foliations for flows in banach space Journal of Differential Equations, 9: , 1991 [] G Da Prato and J Zabczyk Stochastic Equations in Infinite Dimension University Press, Cambridge, 1992 [8] J Duan, K Lu, and Schmalfuß Unstable manifolds for equations with time dependent coefficients 22 In preparation [9] J Hadamard Sur l iteration et les solutions asymptotiques des equations differentielles ull Soc Math France, 29:22 228, 191 [1] D Henry Geometric theory of semilinear parabolic equations, volume 8 of Lecture Notes in Mathematics SpringerVerlag, New York, 1981 [11] N Koksch and S Siegmund Pullback attracting inertial manifolds for nonautonomous dynamical systems Manuscript, 21 [12] M Liapunov Problème géneral de la stabilité du mouvement, volume 1 of nnals Math Studies Princeton, NJ, 19 [13] H Kunita Stochastic Flows and Stochastic Differential Equations Cambridge University Press, Cambridge, 199 [1] SE Mohammed and M K R Scheutzow The stable manifold theorem for stochastic differential equations The nnals of Probability, 2(2): , 1999 [15] Øksendale Stochastic Differential Equations Springer Verlag, erlin Heidelberg New York, third edition, 1992 [16] O Perron Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen Math Z, 29:129 16, 1928 [1] D Ruelle Characteristic exponents and invariant manifolds in hilbert spaces nn of Math, 115:23 29, 1982 [18] Schmalfuß The random attractor of the stochastic Lorenz system ZMP, 8:951 95, 199 [19] Schmalfuß random fixed point theorem and the random graph transformation Journal of Mathematical nalysis and pplications, 225(1):91 113, 1998 [2] Schmalfuß ttractors for the nonautonomous dynamical systems In K Gröger, Fiedler and J Sprekels, editors, Proceedings EQUDIFF99, pages World Scientific, 2 [21] G R Sell Nonautonomous differential equations and dynamical systems mer Math Soc, 12:21 283, 196

22 22 JINQIO DUN, KENING LU, ND JÖRN SCHMLFUSS [22] T Caraballo, J Langa and J C Robinson stochastic pitchfork bifurcation in a reactiondiffusion equation 21 Submitted [23] M I Vishik symptotic ehaviour of Solutions of Evolutionary Equations Cambridge University Press, Cambridge, 1992 [2] T Wanner Linearization random dynamical systems In C Jones, U Kirchgraber and H O Walther, editors, Dynamics Reported, Vol, 23269, SpringerVerlag, New York, 1995 (J Duan) DEPRTMENT OF PPLIED MTHEMTICS, ILLINOIS INSTITUTE OF TECHNOLOGY, CHICGO, IL 6616, US address, J Duan: duan@iitedu (K Lu) DEPRTMENT OF MTHEMTICS, RIGHM YOUNG UNIVERSITY, PROVO, UTH 862, US address, K Lu: klu@mathbyuedu ( Schmalfuß) DEPRTMENT OF SCIENCES, UNIVERSITY OF PPLIED SCIENCES, GEUSER STRSSE, 621 MERSEURG, GERMNY address, Schmalfuß: bjoernschmalfuss@infhmerseburgde