Semimartingale attractors for Allen-Cahn SPDEs driven by space-time white noise I: existence and finite dimensional asymptotic behavior

arXiv:1005.3825v1 [math.PR] 20 May 2010 SEMIMARTINGALE ATTRACTORS FOR ALLEN-CAHN SPDES DRIVEN BY SPACE-TIME WHITE NOISE I: EXISTENCE AND FINITE DIMENSIONAL ASYMPTOTIC BEHAVIOUR H. ALLOUBA AND J.A. LANGA Abstract. We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa [4]. In this article we focus on second order SPDEs of the Allen-Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and a corresponding random PDE of Allen-Cahn type, we prove the existence of semimartingale global attractors for these equations. We also give some results on the finite dimensional asymptotic behavior of the solutions. In particular, we show the finite fractal dimension of this random attractor and give a result on determining modes, both in the forward and the pullback sense. 1. Introduction and organization of the article The analysis of qualitative properties of ordinary and partial differential equations is the key point in dynamical system theory. When a phenomenon from Physics, Chemistry, Biology, Economics can be described by a system of differential equations (in which the existence of global solutions can be assured), one of the most interesting problems is to describe the asymptotic behavior of the system when time grows to infinity. The study of the asymptotic dynamics of the system gives us relevant information about “the future” of the phenomenon described in the model. In this context, the concept of global attractor has become a very useful tool to describe the long-time behavior of many important differential equations (see, among others, Ladyzhenskaya [25], Babin and Vishik [9], Hale [24], Temam [32], Robinson [30]). A new difficulty appears when a random term is added to the deterministic equation, a white noise for instance, and the resulting stochastic partial differential equation must be treated in a different way. Firstly, the equation becomes non-autonomous, which makes necessary the introduction of a two-sided time dependent process instead of a semigroup. Moreover, the strong dependence on the random term adds another difficulty. The rapidly growing theory of random dynamical systems (Arnold [8]) has become the appropriate tool for the study of many important random and stochastic equations. In this framework, Crauel and Flandoli [13] (see also Schmalfuss [31]) introduced the concept of a random attractor as a proper generalization of the corresponding deterministic global attractor. The theory of random attractors is turning out to be very helpful in the understanding of the long-time dynamics of some stochastic ordinary and partial differential equations. On the other hand, one of the most important results in the Date: April 26, 2004. 1991 Mathematics Subject Classification. 37H10; 37H15; 35B42. Key words and phrases. semimartingale attractors; stochastic Allen-Cahn equations; spacetime white noise. 1 2 H. ALLOUBA AND J.A. LANGA theory of global attractors for deterministic PDEs claims that the fractal, and so the Hausdorff, dimension of this set is finite (Constantin and Foias [11], Contantin et al. [12], Ladyzhenskaya [26]; see also the books of Temam [32] and Robinson [30]). That is, although the trajectories depend on an infinite number of degrees of freedom, the finite dimensionality of the attractors leads to the idea that the asymptotic behavior can be described by a finite number of time-dependent coordinates. This makes, for example, really interesting the study of the dynamics on the global attractor. There are also some results which generalize the finite-dimensionality of attractors to the stochastic case (Debussche [18], [19]). In this paper we show how all the theory of finite dimensional random attractors can be generalized to the situation in which the partial differential equation is affected by a space-time white noise, and we characterize this randomness in the attractor as one coming from semimartingale-type solutions (see Definition 2.1). Some of these results were recently sketched in Allouba and Langa [4]. Here, we prove in details the existence of a finite dimensional random attractor associated to the random dynamical system corresponding to a space-time white noise driven stochastic PDE of Allen-Cahn type; and we give a determining modes result for such a SPDE, both in the forward and pullback sense. In the course of our proof, we also give detailed proofs and discussions of existence, uniqueness, and regularity (both weak and strong) results for our SPDE as well as for an associated Allen-Cahn type random PDE. The lack of regularity caused by our driving space-time white noise causes several difficulties in the SPDEs we study. These difficulties are not present in the traditional case of noises that are only white in time (see Remark 3.2 and Remark 3.3 below). Before spelling out the organization of this paper, we wanted to highlight two key features of this work: i) Our solutions are weak semimartingales (see Definition 2.1 and Section 3.4 below), and this characterizes the randomness in our attractors as one coming from some type of semimartingale solutions (not simply random processes); thus we call our random attractors semimartingale attractors. This characterization is crucial and will lead to several new stochastic analytic aspects of these random attractors, like the notion of semimartingale decomposition of semimartingale attractors (e.g., [5]). ii) As in Walsh [33], we regard space-time white noise as a continuous orthogonal martingale measure, which we think will lead to a richer structure of the noise, and so to new aspects of the SPDE under consideration, even compared to cylindrical noise. One such aspect is the notion of semimartingale measure attractors (to which we devote a separate paper), which is built upon the notion of semimartingale measure introduced in Allouba [3]. The paper is organized as follows: in the next Section we write the general theory of random attractors and give the definition of weak semimartingales; Section 3 develops the existence, uniqueness, and regularity (both weak and strong) of solutions for a stochastic PDE of Allen-Cahn type with space-time white noise and for a corresponding random PDE; we follow by proving the existence of a semimartingale attractor associated to these equations. Finally, we show the dependence of the asymptotic behavior of the model on a finite number of degrees of freedom, by proving, with probability one, the finite fractal dimensionality of the semimartingale attractor and some results on determining modes, both in the forward and the SEMIMARTINGALE ATTRACTORS FOR SPACE-TIME WHITE NOISE ALLEN-CAHN SPDES3 pullback sense. Some conclusions are then given, placing the results here in the context of our ongoing research program. We also include some technical results in a final Appendix. Throughout this article we will denote by Ka constant that may change its value from line to line. 2. Semimartigale global attractors 2.1. Definitions. Proceeding toward a precise statement of our results, let us recall some definitions associated with random attractors. Let (Ω,F,P) be a probability space and {θt: Ω →Ω, t ∈R}a family of measure preserving transformations such that (t, ω)7→ θtωis measurable, θ0= id, θt+s=θtθs, for all s, t ∈R. The flow θttogether with the probability space (Ω,F,P,(θt)t∈R) is called a measurable dynamical system. Furthermore, we suppose that the shift θtis ergodic. Arandom dynamical system (RDS) (Arnold [8]) on a complete metric (or Banach) space (B, d) with Borel σ-algebra B, over θon (Ω,F,P) is a measurable map R+×Ω×B∋(t, ω, ξ)7→ Φ(t, ω)ξ∈Bsuch that P–a.s. i) Φ(0, ω) = id (on B) ii) Φ(t+s, ω) = Φ(t, θsω)◦Φ(s, ω),∀t, s ∈R+(cocycle property). A RDS is continuous (differentiable) if Φ(t, ω) : B→Bis continuous (differentiable). A random set K(ω)⊂Bis said to absorb the set D⊂Bif there exists a random time tD(ω) such that t≥tD(ω)→Φ(t, θ−tω)D⊂K(ω),P–a.s. K(ω) is forward invariant if Φ(t, ω)K(ω)⊆K(θtω),for all t∈R+,P–a.s. Now, let dist(·,·) denote the Hausdorff semidistance dist(B1, B2) = sup ξ1∈B1 inf ξ2∈B2 d(ξ1, ξ2), B1, B2⊂B. A random set A(ω)⊂Bis said to be a random attractor associated with the RDS Φ if P–a.s. i) A(ω) is compact and, for all ξ∈B,the map ξ7→ dist(ξ, A(ω)) is measurable, ii) Φ(t, ω)A(ω) = A(θtω),∀t≥0 (invariance), and iii) for all D⊂Bbounded (and nonrandom) limt→∞ dist(Φ(t, θ−tω)D, A(ω)) = 0. Remark 2.1. Note that Φ(t, θ−tω)ξcan be interpreted as the position at t= 0 of the trajectory which was in ξat time −t. Thus, the attraction property holds from t=−∞. We have the following theorem about existence of random attractors due to Crauel ([15], Theorem 3.3): Theorem 2.1. There exists a global random attractor A(ω)iff there exists a random compact set K(ω)attracting every bounded nonrandom set D⊂B. Moreover, Crauel [15] proved that random attractors are unique and, under the ergodicity assumption on θt, there exists a deterministic compact set K⊂Bsuch that P−a.s. the random attractor is the omega-limit set of K, that is, A(ω) = \ n≥0[ t≥n Φ(t, θ−tω)K. Our SPDEs solutions are weak semimartingale, which we now define. 4 H. ALLOUBA AND J.A. LANGA Definition 2.1. We call a random field U(t, x, ω), x∈G⊂Rd, a weak semimartingale sheet (or simply a weak semimartingale) if there exists a p≥0 such that the L2scalar product (U(t), ϕ) is a semimartingale in time for each fixed ϕ∈Cp c(G). If Ais a random attractor corresponding to a SPDE whose solutions are weak semimartingales, then Ais called a semimartingale attractor. 2.2. Finite dimensional asymptotic behavior. Here we obtain some results on the finite dimensional asymptotic behavior of trajectories associated to a random dynamical system, which we will apply below to the solutions for Allen-Cahn type SPDEs in (3). 2.2.1. The random squeezing property. Suppose the existence of a random compact absorbing set K(ω) such that, for some random variable r(ω), we have that P-a.s. K(ω)⊂B(0, r(ω)).Moreover, suppose that the r(ω) is a tempered random variable, that is, P-a.s. lim t→+∞ r(θtω) eǫt = 0, for all ǫ > 0. Let P:B→PBbe a finite-dimensional orthogonal projector and let Q=I−P be its counterpart. In what follows, the main hypothesis (H) is the following: Suppose there exist 0 < δ < 1 and a random variable c(ω) with finite expectation, (1) EP(c(ω)) <ln(1/δ), such that, for τ∈R (2) |Q(Φ(1, θτω)u−Φ(1, θτω)v)| ≤ δexp Zτ+1 τ c(θsω)ds|u−v|, for all u, v ∈K(θτω),where |·| denotes the norm in B. This property is called the random squeezing property (RSP) in Flandoli and Langa [21], and it was first used in Debussche [18] to prove that the random attractor associated to a RDS has finite Hausdorff dimension P-a.s. Proposition 2.1. ([18],[21]) Suppose that (1),(2) hold. Then, P-a.s. df(A(ω)) <+∞, where df(K). = lim sup ǫ→0 log Nǫ(K) log(1/ǫ) denotes the fractal dimension of a compact set K⊂B,where Nǫ(K)is the minimum number of balls of radius ǫnecessary to cover K. 2.2.2. Forward and Pullback determining modes. The following theorem shows the dependence of the asymptotic behavior, starting with two initial data, on a finite number of degrees of freedom (Langa [28], Theorem 2, and Flandoli and Langa, Theorem 2): Theorem 2.2. Suppose (1) and (2) hold. Then, SEMIMARTINGALE ATTRACTORS FOR SPACE-TIME WHITE NOISE ALLEN-CAHN SPDES5 a) (Forward determining modes) given u0, v0∈B, suppose that for some α≥0, we have, P-a.s., that lim t→+∞|P(Φ(t, ω)u0−Φ(t, ω)v0)| ≤ α. Then, lim t→+∞|Φ(t, ω)u0−Φ(t, ω)v0| ≤ α. b) (Pullback determining modes) On the other hand, if t∈Rand for all r≤t,P-a.s., and u0, v0∈B lim s→+∞|P(Φ(r+s, θ−sω)u0−Φ(r+s, θ−sω)v0)| ≤ α, then, for all r≤t, lim s→+∞|Φ(r+s, θ−sω)u0−Φ(r+s, θ−sω)v0| ≤ α. Note that in b) we need a convergence in all final times r≤tto get the result. In the next result we will write a weaker hypothesis for this result. Remark 2.2. a) If α= 0 we would have a classical determining modes result (cf. Foias and Prodi [22]). b) Due to the fact that the pullback convergence to the random attractor implies the forward convergence to this set in probability (Crauel and Flandoli [13]), i.e., for all ǫ > 0 lim t→+∞ P(ω∈Ω : dist(Φ(t, ω)D, A(θtω)) > ǫ) = 0, we get that our hypotheses in the previous theorem implies those in Chueshov et al. [10], Theorem 2.3, so that the assertion there also holds in our case. Using Proposition 2 in Langa [28], we also get the pullback convergence in the previous theorem under a weaker condition. Theorem 2.3. ([28]) Suppose that u(ω), v(ω)are two random variables on the attractor A(ω)such that P–a.s. Φ(t, ω)u(ω)6= Φ(t, ω)v(ω),for all t∈R+and lim s→+∞|P0(Φ(t+s, θ−sω)u(θ−sω)−Φ(t+s, θ−sω)v(θ−sω))|= 0 whenever u(ω)6=v(ω),P–a.s. (where P0is a projection which is injective between St∈RA(θtω)and its image, see Langa and Robinson [27] for the existence of such (dense) set of projections). Then, for all r≤twe have that lim s→+∞|P0(Φ(r+s, θ−sω)u(θ−sω)−Φ(r+s, θ−sω)v(θ−sω))|= 0,P–a.s. 3. Generalized Allen-Cahn SPDEs and Random PDEs 3.1. Definitions. In this part we consider the SPDE (3)      ∂U ∂t = ∆xU+f(U) + ∂2W ∂t∂x,(t, x)∈ OL. = (0,+∞)×(0, L); U(t, 0) = U(t, L) = 0,0< t < ∞; U(0, x) = u0(x),0< x < L; 6 H. ALLOUBA AND J.A. LANGA where W(t, x) is the Brownian sheet corresponding to the driving space-time white noise, written formally as ∂2W/∂t∂x. As noted earlier, we treat white noise as a continuous orthogonal martingale measure, which we denote by W. The drift f:R→Ris of the form: (4) f(u) = 2p−1 X k=0 akuk,with p∈N,and a2p−1<0, It is not difficult to prove the following elementary inequalities for f(see Temam [32]), which we use in the proof of Theorem 3.2: i) There exists K > 0 such that f′(v)≤K, ∀v∈R. ii) There exist c1, c0>0 such that f(v)v≤ −c1v2p+c0,∀v∈R. iii) There exist k1, k0>0 such that |f(v)| ≤ k1|v|2p−1+k0,∀v∈R. We denote the SPDE (3) by eAC(f, u0). We collect here definitions and conventions that are used throughout this article (see Walsh [33] for a whole setting of this type of SPDEs; see also Allouba [2, 3]). Filtrations are assumed to satisfy the usual conditions (completeness and right continuity), and any probability space (Ω,F,{Ft},P) with such a filtration is termed a usual probability space. Definition 3.1 (Strong and Weak Solutions to eAC(f, u0)).We say that the pair (U, W) defined on the usual probability space (Ω,F,{Ft},P) is a continuous or L2valued solution to the stochastic PDE eAC(f, u0) if Wis a space-time white noise on CL. =R+×[0, L]; the random field U(t, x) is Ft-adapted (U(t, ·)∈ Ft∀t), with either U∈C(CL;R) (a continuous solution) or U∈C(R+;L2(0, L)) (an L2-valued solution); and the pair (U, W) satisfies either one of the following two formulations: (TFF) the test function formulation (U(t)−u0, ϕ)−Zt 0 (U(s), ϕ′′)ds =Zt 0 (f(U(s), ϕ)ds +ZL 0Zt 0 ϕ(x)W(ds, dx); 0 ≤t < ∞,a.s. P, for every ϕ∈ΘL 0 . ={ϕ∈C∞(R;R) : ϕ(0) = ϕ(L) = 0},where (·,·) is the L2inner product on [0, L],or (GFF) the Green function formulation U(t, x) = ZL 0Zt 0 f(U(s, y))Gt−s(x, y)dsdy +ZL 0Zt 0 Gt−s(x, y)W(ds, dy) +ZL 0 Gt(x, y)u0(y)dy; 0 ≤t < ∞a.s. P, where Gt(x, y) is the fundamental solution to the deterministic heat equation (ut=uxx) with vanishing boundary conditions. A solution is said to be strong if the white noise Wand the usual probability space (Ω,F,{Ft},P) are fixed a priori and Ftis the augmentation of the natural filtration for Wunder P. It is termed a weak solution if we are allowed to choose the usual probability space and the white noise Won it, without requiring that the filtration be the augmented natural filtration of W. We say pathwise uniqueness holds for eAC(f, u0) if whenever (U(1),W) and (U(2),W) are two solutions to eAC(f, u0) on the same probability space (Ω,F,{Ft},P), and with respect to the same white noise W, then PU(1)(t, x) = U(2)(t, x); 0 ≤t < ∞, x ∈[0, L]= 1. SEMIMARTINGALE ATTRACTORS FOR SPACE-TIME WHITE NOISE ALLEN-CAHN SPDES7 We often simply say that Usolves eAC(f, u0) (weakly or strongly) to mean the same thing as above. Remark 3.1. As it is well known (Walsh [33]), if the drift and diffusion coefficients are locally bounded random fields (in our case they trivially are for continuous solutions since the diffusion coefficient a≡1 and the drift fis clearly locally Lipschitz under our conditions in (4), then the two formulations (GFF) and (TFF) are equivalent. 3.2. Existence and uniqueness of solutions. Let β≥0; let Zβ(t, x) be the pathwise-unique strong solution to (3) with f(Zβ) = −βZβand Zβ(0, x)≡0, which is H¨older continuous with αtime = 1/4−ǫin time and αspace = 1/2−ǫin space, ∀ǫ > 0 (a standard result as in [33] pp. 321-322). Let Vβ=U−Zβ, for any solution Uto (3). We see then that Vβsatisfies Vβ(t, x) = U(t, x)−ZL 0Zt 0 Gt−s(x, y) [W(ds, dy)−βZβ(s, y)dsdy] =ZL 0 Gt(x, y)u0(y)dy +ZL 0Zt 0 [f(Vβ+Zβ(s, y)) + βZβ(s, y)] Gt−s(x, y)dsdy . =ZL 0 Gt(x, y)u0(y)dy +Iβ(t, x) = M(t, x) + Iβ(t, x). That is, Vβsolves the random PDE: (5)      ∂Vβ ∂t = ∆xVβ+f(Vβ+Zβ) + βZβ,(t, x)∈ OL; Vβ(t, 0) = Vβ(t, L) = 0,0< t < ∞; Vβ(0, x) = u0(x), x ∈[0, L]. Our first result gives detailed existence, uniqueness, and comparative regularity results of our SPDE eAC (f, u0) in (3) and the associated random PDE (5). Theorem 3.1. Suppose fsatisfies (4). (i) (Strong Regularity)If u0: [0, L]→Ris Lipschitz continuous and deterministic. Then, the SPDE eAC(f, u0)has a strong, pathwise-unique, a.s. αH¨older continuous solution with αt= 1/4−ǫin time and αx= 1/2−ǫin space, for all ǫ > 0. On the other hand, under the same conditions on u0, the random PDE (5)has an a.s. C1,2((0,∞)×(0, L); R)unique solution. (ii) (Weak Regularity)For all 0≤s < T , we have: a) if u0∈L2(0, L), there exist a.s.˜unique solutions Uand Vto eAC(f, u0) and (5), respectively, such that V∈C([s, ∞); L2(0, L)) ∩L2(s, T ;H1 0(0, L)) ∩L2p(s, T ;L2p(0, L)), and U∈C([s, ∞); L2(0, L)); b) if u0∈H1 0(0, L), then the a.s. unique solutions Uand Vare such that V∈C([s, ∞); H1 0(0, L)) ∩L2(s, T ;H2(0, L)) ∩L2p(s, T ;L2p(0, L)), and U∈C([s, ∞); L2(0, L)). and hence, V∈C([s+ε, ∞); H1 0(0, L)) ∩L2(s+ε, ∞;H2(0, L)),for every u0∈L2(0, L)and ε > 0. 8 H. ALLOUBA AND J.A. LANGA Remark 3.2. In addition to the existence, uniqueness, and regularity for the SPDE eAC(f, u0), our proof of Theorem 3.1 gives detailed strong, as well as weak, regularity results for the random PDE (5) associated with our SPDE (3). The strong regularity results are for completeness, and they are not needed for the rest of the paper. Two points are worth emphasizing: 1. solutions to the random PDE (5) are typically much smoother than solutions to the Allen-Cahn SPDE eAC(f, u0) and 2. while increasing the regularity of the initial function u0has a considerable effect on smoothing out the random PDE solution (if u0is Lipschitz then the solution Vis in C1,2((0,∞)×(0, L)); the most regularity we can guarantee for the Allen-Cahn SPDE solution is H¨older continuity (with H¨older exponents 1/4 in time and 1/2 in space) regardless of how smooth the initial data is. This of course is a direct result of the fact that the driving noise is white in both space and time. In the case of a time only white noise, the solution Uto the Allen-Cahn SPDE driven by such noises is spacially much smoother than our solutions (typically at least in H1(0, L), e.g., see [13]). For more on the effects of our rougher noise on the proof of the existence of the attractor see Remark 3.3 below. Proof. (of Theorem 3.1) We note that when p= 1 in (4) fis Lipschitz and the strong existence, pathwise uniqueness, and H¨older regularity for eAC(f, u0) follow from standard results (see [33]). We now turn to the case p > 1. For simplicity and without loss of generality, we assume β= 0. Let Z. =Z0and V. =V0. Clearly, the existence and uniqueness for eAC(f, u0) is equivalent to the existence and uniqueness for the corresponding random PDE (5). This is because Zis the pathwise-unique strong solution (see [33]) to the standard heat SPDE and V+Zis a solution to eAC(f, u0) if and only if V solves (5). Furthermore Z(t, x) is a.s. α-H¨older-continuous with αt= 1/4−ǫin time and αx= 1/2−ǫin space, for all ǫ > 0 (see [33]), and it vanishes at 0 and L. For the rest of the proof, we fix ω∈Ω, and treat the path-by-path deterministic version of our random PDE (5). Following the proof of Theorem 1.1 in Temam [32], Chapter III—and for the usual Sobolev spaces H1 0(0, L) := {v∈H1(0, L), v(0) = v(L) = 0} and H2(0, L)—we have P-a.s. that there is a unique continuous (in (t, x)) solution Vto (5) satisfying (5) if u0: [0, L]→Ris deterministic and continuous. This implies that |f(V+Z)| ≤ C1<∞on [0, t]×[0, L]; thus I0(t, ·)∈C1(0, L) with |DI0(t, x)| ≤ CC1t1 2(the smoothness for I0is obtained throughout as in Theorems 2 to 5 in Chapter 1 of [23]) and hence V(t, ·)∈C1(0, L) for every t(the first term in (5), M, is in C2(0, L) whenever u0is continuous on [0, L]). If additionally u0is Lipschitz on [0, L]; then f(V+Z) is H¨older continuous on [0, L], uniformly locally in t. To see this, remember that when u0is Lipschitz on [0, L] then, with Mas defined as in (5), we have M∈C2(0, L) and (6) DM(t, x) = ZL 0 u0(y)∂ ∂xGt(x, y)dy ≤K. The bound in (6) again follows from standard analysis methods as in Chapter 1 in [23] (see also Lemma A.3 below for a probabilistic proof of this fact on Rd,d≥1). The bound in (6) and the bound that we have for DI0(t, x) imply that V, and hence f(V+Z), is H¨older continuous on [0, L], uniformly locally in t. This, in turns implies that I0(t, ·)∈C2(0, L) and hence V(t, ·)∈C2(0, L) for every t. The temporal regularity for Vis proved similarly and we omit it, and we obtain that V∈C1,2((0,∞)×(0, L); R). It is then clear that U(t, x) = V(t, x) + Z(t, x) is the SEMIMARTINGALE ATTRACTORS FOR SPACE-TIME WHITE NOISE ALLEN-CAHN SPDES9 pathwise-unique (because uniqueness holds a.s. for both Vand Z) strong solution (because the white noise Wis fixed throughout) of (3), and that Uis Pa.s. H¨older continuous under our conditions on u0with αt= 1/4−ǫin time and αx= 1/2−ǫ in space, for all ǫ > 0 (since both Vand Zare) In addition, we also get P-a.s. that for all 0 ≤s < T : a) if u0∈L2(0, L), there exists a unique solution V∈C([s, ∞); L2(0, L)) ∩L2(s, T ;H1 0(0, L)) ∩L2p(s, T ;L2p(0, L)), b) if u0∈H1 0(0, L), then there exists a unique solution V∈C([s, ∞); H1 0(0, L)) ∩L2(s, T ;H2(0, L)) ∩L2p(s, T ;L2p(0, L)), and hence, V∈C([s+ε, ∞); H1 0(0, L)) ∩L2(s+ε, ∞;H2(0, L)),for every u0∈ L2(0, L) and ε > 0.Again, the assertions about the existence, uniqueness and weak regularity of U(part ii) a) and b) in Theorem 3.1) easily follow from the corresponding results for V(parts a) and b) above), the regularity of Z, and the fact that U(t, x) = V(t, x) + Z(t, x).  3.3. Growth rates for Zβ.In this subsection, we obtain asymptotic growth rates of interest related to Zβ. Lemma 3.1. Let Zβbe as in the proof of Theorem 3.1. Then Zβmay be rewritten as Zβ(t, x) = ZL 0Zt 0 Gβ,t−s(x, y)W(ds, dy), where Gβis the fundamental solution to the noiseless version of (3) with f(Zβ) = −βZβ, with Dirichlet boundary conditions ([33]). Let ˆ Zβ(t). = sup0≤x≤LZβ(t, x). Then, i) For each 0< p < 3and 0≤γ < 1∧(3 −p), there exists a constant K > 0 such that (7) ZL 0Zt 0 Gp β,t−s(x, y)dsdy ≤K(x∧(L−x))γt(3−p−γ)/2;x∈(0, L), t > 0, β ≥0. ii) P[ˆ Zβ(t)> t1 4+ǫ]≤Kt−ǫ→0as t→ ∞, for every ǫ > 0and every β≥0 for some universal constant K > 0. iii) If Zϕ β(t). = (Zβ(t), ϕ)−Zt 0 (Zβ(s), ϕ′′)ds +Zt 0 (βZβ(s), ϕ)ds; 0 ≤t < ∞, then, for every β≥0and ϕ∈ΘL 0,Zϕ β(t)/t →0as t→ ∞ P-a.s. Proof. i) The Green function, Gβis easily found to be Gβ,t(x, y) = e−βt √4πt ∞ X n=−∞exp −(2nL +y−x)2 4t−exp −(2nL +y+x)2 4t. It is clearly enough to prove the estimate on Gβfor β= 0, and we will denote G0by simply G. Now, let Bxbe the scaled Brownian motion 16 H. ALLOUBA AND J.A. LANGA Consequently, |v|Lp≤  RL 0Lp p′|Dv|p Lpdy1 p≤L|Dv|Lp;p > 1, L|Dv|L1. The proof is complete.  The second inequality gives us a bound on the Laplacian of a function integrated against an odd power of the same function: Lemma A.2 (Laplacian and Odd Power Integral Inequality).Suppose v∈C2((0, L); R), for some L > 0, with v(0) = v(L) = 0; then ZL 0 ∂2v ∂x2·v2p−1dx ≤ −(2p−1) p2L|v|2p L2pfor all p≥1. If v∈C1((0, L); R), for some L > 0, with v(0) = 0; then −ZL 0 ∂v ∂x ·∂v2p−1 ∂x dx ≤ −(2p−1) p2L|v|2p L2pfor all p≥1. Proof. Let ube the function given by u(x). =Zx 0∂vp ∂y 2 dy; 0 ≤x≤L. Then u′(x) = ∂vp ∂x 2and we have, using Lemma A.1, that ZL 0∂vp ∂y 2 dy =|u′|L1≥1 L|u|L1=1 LRL 0Rx 0∂vp ∂y 2dydx(15) ≥1 LRL 0Rx 0 ∂vp ∂y dy2dx(16) =1 L|v|2p L2p.(17) Therefore, ZL 0 ∂2v ∂x2·v2p−1dx =−RL 0 ∂v ∂x ·∂v2p−1 ∂x dx =−(2p−1) RL 0vp−1·∂v ∂x 2dx =−2p−1 p2RL 0∂vp ∂y 2dy ≤ −2p−1 p2L|v|2p L2p. where the last inequality follows from (17).  We now give a probabilistic proof of (6) in the case of the heat equation on Rd; i.e., when [0, L] is replaced with Rdand Gt(x, y) is replaced with the fundamental solution to the heat equation on Rd,pt(x, y). Lemma A.3. With the notations above, we have (18) ZRd u0(y)∂ ∂xpt(x, y)dy ≤K. for some universal constant K > 0whenever u0is Lipschitz. SEMIMARTINGALE ATTRACTORS FOR SPACE-TIME WHITE NOISE ALLEN-CAHN SPDES17 Proof. Let Bx=nBx t . =√2˜ Bx/√2 t; 0 ≤t < ∞o, where ˜ Bx=n˜ Bx t; 0 ≤t < ∞ois a standard d-dimensional Brownian motion starting at x∈Rd. Then, pt(x, y) is the density of the scaled Brownian motion Bxon Rd,pt(x, y), we have |DjM(t, x)|=ZRd u0(y)∂ ∂xj pt(x, y)dy =ZRd u0(y)−(xj−yj) 2t(4πt)−d/2e−|x−y|2/4tdy =−1 2tEh(xj−Bj,x t)u0(Bx t)i≤1 tE(xj−Bj,x t) (u0(Bx t)−u0(x)) ≤1 thE(xj−Bj,x t)2E(u0(Bx t)−u0(x))2i1/2 ≤K1 thE(xj−Bj,x t)2E|Bx t−x|2i1/2≤K2, where Dj=∂/∂xjand Bj,x is the j-th component of the d-dimensional Bx, 1 ≤ j≤d; and where we have used elementary facts about the Brownian motion Bx, H¨older inequality, and the Lipschitz condition on u0to get (19).  References [1] Adams, R., Sobolev spaces (Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975). [2] H. Allouba, Uniqueness in law for the Allen-Cahn SPDE via change of measure, C.R. Acad. Sci. 330, no. 5(2000) 371-376. [3] H. Allouba, Different types of SPDEs in the eyes of Girsanov’s theorem, Stochastic Anal. Appl. 16 (1998), no. 5, 787–810. [4] H. Allouba and J.A. Langa, Semimartingale attractors for generalized Allen-Cahn SPDEs driven by space-time white noise, C. R. Acad. Sci. Paris, Ser. I 337 (2003), 201-206. [5] H. Allouba and J.A. Langa, Semimartingale attractors for Allen-Cahn SPDEs driven by space-time white noise II: semimartingale decomposition and regularity. In preparation. [6] H. Allouba, J.A. Langa, Semimartingale attractors for Allen-Cahn equations: from SDDEs to SPDEs . In preparation. [7] H. Allouba, J.A. Langa, Semimartingale attractors for Kuramoto-Sivashinsky and CahnHilliard SPDEs driven by d-dimensional space-time white noise. In preparation. [8] Arnold, L., Random dynamical systems (Springer Monographs in Mathematics, Springer, Berlin, 1998). [9] A.V. Babin, M.I. Vishik, Attractors of partial differential equations and estimate of their dimension, Russian Math. Surveys 38 (1983), 151-213. [10] I.D. Chueshov, J. Duan, B. Schmalfuss, Determining functionals for random partial differential equations, NoDEA 10 (2003), 431-454. [11] P. Constantin, C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, ,Comm. Pure Appl. Math. 38 (1985) 127. [12] P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1985). [13] H. Crauel, F. Flandoli, Attractors for random dynamical systems, Prob. Th. and Related Fields 100 (1994), 365-393. [14] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dyn. Diff. Eq. 9(1997), 307-341. [15] H. Crauel, Global random atractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura App. 176, no. 4 (1999), 57-72. [16] G. Da Prato, A. Debussche, R. Temam, Stochastic Burgers’ equation, NoDEA 1(1994), 389-402. [17] G. Da Prato, J. Zabczyk, J., Stochastic equation in infinite dimension (Encyclopedia of Mathematics and its Applications, CUP, Cambridge, 1992). [18] A. Debussche, On the finite dimensionality of random attractors, Stoch. Anal. and Appl. 15, no. 4 (1997) 473-491. 18 H. ALLOUBA AND J.A. LANGA [19] A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pure App., 77 (9), no. 10 (1998) 967-988. [20] A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Exponential attractors for dissipative evolution equations (RAM, Wiley, Chichester, 1994). [21] F. Flandoli, J.A. Langa, Determining modes for dissipative random dynamical systems, Stoch. and Stoch. Reports, 66 (1999), 1-25. [22] C. Foias, G. Prodi, Sur le comportement global des solutions non-stationnaires des equations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967) 1-34. [23] A. Friedman, Partial differential equations of parabolic type (Prentice-Hall, Englewood Cliffs, NJ 1964). [24] J. Hale, Asymptotic Behavior of Dissipative Systems , Math. Surveys and Monographs (AMS, Providence 1988). [25] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations (Accademia Nazionale dei Lincei, Cambridge University Press, Cambridge 1991). [26] O.A. Ladyzhenskaya (1982) On finite dimensionality of bounded invariant sets for the NavierStokes equations and some other dissipative systems, Zapiskii nauchnich seminarovs LOMI 115, 137155. [27] J.A. Langa, J.C. Robinson, A finite number of point observations which determine a nonautonomous fluid flow, Nonlinearity 14, no. 4 (2001), 673-682. [28] J.A. Langa, Finite dimensional limiting dynamics of random dynamical systems, Dynamical Systems, an International J. 18 (1) (2003), 57-68. [29] D. Revuz, and M. Yor, Continuous martingales and Brownian motion (Springer, New York, 1999). [30] J.C. Robinson, Infinite-dimensional dynamical systems (Cambridge University Press, Cambridge, 2001). [31] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour, eds W. Reitmann, T. Riedrich and N Koksch, (1992) 185-192. [32] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Second Edition, Springer-Verlag, New York, 1997). [33] J.B. Walsh An introduction to stochastic partial differential equations. Ecole d’Et´e de Probabilit´es de Saint-Flour XIV. Lecture Notes in Math. 1180. (Springer, New York, 1986). Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA, [email protected] Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico,, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain, lang[email protected]