A KHASMINSKII TYPE AVERAGING PRINCIPLE FOR STOCHASTIC REACTION DIFFUSION EQUATIONS. BY SANDRA CERRAI Università di Firenze

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1 The Annals of Applied Probability 29, Vol. 19, No. 3, DOI: /8-AAP56 Institute of Mathematical Statistics, 29 A KASMINSKII TYPE AVERAGING PRINCIPLE FOR STOCASTIC REACTION DIFFUSION EQUATIONS BY SANDRA CERRAI Università di Firenze We prove that an averaging principle holds for a general class of stochastic reaction diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite-dimensional systems. 1. Introduction. Consider the deterministic system with a finite number of degrees of freedom d ˆX ε (t) = εb( ˆX ε (t), Ŷ ε (t)), ˆX ε () = x R n, (1.1) dt dŷ ε dt (t) = g( ˆX ε (t), Ŷ ε (t)), Ŷ ε () = y R k for some parameter <ε 1 and some mappings b : R n R k R n and g : R n R k R k. Under reasonable conditions on b and g, it is clear that as the parameter ε goes to zero, the first component ˆX ε (t) of the perturbed system (1.1) converges to the constant first component x of the unperturbed system, uniformly with respect to t in any bounded interval [,T], with T>. But in applications that is more interesting is the behavior of ˆX ε (t) for t in intervals of order ε 1 or even larger. Actually, it is indeed on those time scales that the most significant changes happen, such as exit from the neighborhood of an equilibrium point or of a periodic trajectory. With the natural time scaling t t/ε, if we set X ε (t) := ˆX ε (t/ε) and Y ε (t) := Ŷ ε (t/ε), (1.1) can be rewritten as dx ε (t) = b(x ε (t), Y ε (t)), X ε () = x R n, (1.2) dt dy ε dt (t) = 1 ε g(x ε(t), Y ε (t)), Y ε () = y R k and with this time scale the variable X ε is always referred as the slow component and Y ε as the fast component. In particular, the study of system (1.1) in time intervals of order ε 1 is equivalent to the study of system (1.2) on finite time intervals. Received August 27; revised July 28. AMS 2 subject classifications. 615, 34C29, 37L4. Key words and phrases. Stochastic reaction diffusion equations, invariant measures, ergodic and strongly mixing processes, averaging principle. 899

2 9 S. CERRAI Now, assume that for any x R n there exists the limit (1.3) 1 T b(x) = lim b(x,y x (t)) dt, T T where Y x (t) is the fast motion with frozen slow component x R n dy x (t) = g(x,y x (t)), Y x () = y. dt Such a limit exists, for example, in the case the function Y x (t) is periodic. Moreover, assume that the mapping b : R n R n satisfies some reasonable assumption, for example, it is Lipschitz continuous. In this setting, the averaging principle says that the trajectory of X ε can be approximated by the solution X of the so-called averaged equation d X dt (t) = b( X(t)), X() = x, uniformly in t [,T], foranyfixedt>. This means that by averaging principle a good approximation of the slow motion can be obtained by averaging its parameters in the fast variables. The theory of averaging, originated by Laplace and Lagrange, has been applied in its long history in many fields as, for example, celestial mechanics, oscillation theory and radiophysics, and for a long period it has been used without a rigorous mathematical justification. The first rigorous results are due to Bogoliubov (cfr. [3]) and concern both the case of uncoupled systems and the case of g(x,y) = g(x). Further developments of the theory, for more general systems, were obtained by Volosov, Anosov and Neishtadt (to this purpose, we refer to [23] and [28]) and a good understanding of the involved phenomena was obtained by Arnold et al. (cfr. [1]). A further development in the theory of averaging, which is of great interest in applications, concerns the case of random perturbations of dynamical systems. For example, in system (1.1), the coefficient g maybeassumedtodependalsoonapa- rameter ω, for some probability space (, F, P), so that the fast variable is a random process, or even the perturbing coefficient b may be taken random. Of course, in these cases, one has to reinterpret condition (1.3) and the type of convergence of the stochastic process X ε to X. One possible way is to require (1.3) with probability 1, but in most cases this assumption turns out to be too restrictive. More reasonable is to have (1.3) either in probability or in the mean, and in this case one expects to have convergence in probability of X ε to X. As far as averaging for randomly perturbed systems is concerned, it is worthwhile to quote the important work of Brin, Freidlin and Wentcell (see [4, 12 14]) andalsothework of Kifer and Veretennikov (see, e.g., [2, 17 19] and[27]).

3 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 91 An important contribution in this direction has been given by Khasminskii with his paper [16] which appeared in In this paper, he has considered the following system of stochastic differential equations: l dx ε (t) = A(X ε (t), Y ε (t)) dt + σ r (Xi ε (t), Y ε (t)) dw r (t), r=1 (1.4) X ε () = x, dy ε (t) = 1 ε B(Xε (t), Y ε (t)) dt + 1 l ϕ r (X ε (t), Y ε (t)) dw r (t), ε r=1 Y ε () = y for some l-dimensional Brownian motion w(t) = (w 1 (t),..., w l (t)). In this case, the perturbation in the slow motion is given by the sum of a deterministic part and a stochastic part εb(x,y)dt = εa(x,y)dt + εσ(x,y)dw(t) and the fast motion is described by a stochastic differential equation. In [16], the coefficients A : R l 1 R l 2 R l 1 and σ : R l 1 R l 2 M(l l 1 ) in the slow motion equation are assumed to be Lipschitz continuous and uniformly bounded in y R l 2. The coefficients B : R l 1 R l 2 R l 2 and ϕ : R l 1 R l 2 M(l l 2 ) in the fast motion equation are assumed to be Lipschitz continuous, so that in particular the fast equation with frozen slow component x, l dy x,y (t) = B(x,Y x,y (t)) dt + ϕ r (x, Y x,y (t)) dw r (t), Y x,y () = y, r=1 admits a unique solution Y x,y,foranyx R l 1 and y R l 2. Moreover, it is assumed that there exist two mappings Ā : R l 1 R l 1 and {a ij } : R l 1 M(l l 1 ) such that 1 T EA(x, Y x,y (1.5) (t)) dt Ā(x) T α(t )(1 + x 2 ) and for any i = 1,...,l 1 and j = 1,...,l 2, 1 T l E σ r i T σ j r (x, Y x,y (t)) dt a ij (x) α(t )(1 + x 2 ) r=1 for some function α(t ) vanishing as T goes to infinity. In his paper, Khasminskii shows that an averaging principle holds for system (1.4). Namely, the slow motion X ε (t) converges in weak sense, as ε goes to zero, to the solution X of the averaged equation dx(t) = Ā(X(t)) dt + σ(x(t))dw(t), X() = x, where σ is the square root of the matrix {a ij }.

4 92 S. CERRAI The behavior of solutions of infinite-dimensional systems on time intervals of order ε 1 is at present not very well understood, even if applied mathematicians do believe that the averaging principle holds and usually approximate the slow motion by the averaged motion, also with n =. As far as we know, the literature on averaging for systems with an infinite number of degrees of freedom is extremely poor (to this purpose it is worth mentioning the papers [25] by Seidler Vrkoč and [21] by Maslowskii Seidler Vrkoč, concerning with averaging for ilbert-space valued solutions of stochastic evolution equations depending on a small parameter, and the paper [2] by Kuksin and Piatnitski concerning with averaging for a randomly perturbed KdV equation) and almost all has still to be done. In the present paper, we are trying to extend the Khasminskii argument to a system with an infinite number of degrees of freedom. We are dealing with the following system of stochastic reaction diffusion equations on a bounded domain D R d, with d 1, u ε t (t, ξ) = A 1u ε (t, ξ) + b 1 (ξ, u ε (t, ξ), v ε (t, ξ)) (1.6) + g 1 (ξ, u ε (t, ξ), v ε (t, ξ)) wq 1 (t, ξ), t v ε t (t, ξ) = 1 ε [A 2v ε (t, ξ) + b 2 (ξ, u ε (t, ξ), v ε (t, ξ))] + 1 g 2 (ξ, u ε (t, ξ), v ε (t, ξ)) wq 2 (t, ξ), ε t u ε (,ξ)= x(ξ), v ε (,ξ)= y(ξ), ξ D, N 1 u ε (t, ξ) = N 2 v ε (t, ξ) =, t, ξ D for a positive parameter ε 1. The stochastic perturbations are given by Gaussian noises which are white in time and colored in space, in the case of space dimension d>1, with covariances operators Q 1 and Q 2. The operators A 1 and A 2 are second order uniformly elliptic operators, having continuous coefficients on D, and the boundary operators N 1 and N 2 can be either the identity operator (Dirichlet boundary condition) or a first order operator satisfying a uniform nontangentiality condition. In our previous paper [8], written in collaboration with Mark Freidlin, we have considered the simpler case of g 1 andg 2 1,andwehaveprovedthatan averaging principle is satisfied by using a completely different approach based on Kolmogorov equations and martingale solutions of stochastic equations, which is more in the spirit of the general method introduced by Papanicolaou, Strook and Varadhan in their paper [24] of ere, we are considering the case of general reaction coefficients b 1 and b 2 and diffusion coefficients g 1 and g 2, and the method based on the martingale approach seems to be very complicated to be applied. We would like to stress that both here and in our previous paper [8] weare considering averaging for randomly perturbed reaction diffusion systems, which are of interest in the description of diffusive phenomena in reactive media, such

5 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 93 as combustion, epidemic propagation and diffusive transport of chemical species through cells and dynamics of populations. owever, the arguments we are using adapt easily to more general models of semilinear stochastic partial differential equations. Together with system (1.6), for any x,y := L 2 (D), we introduce the fast motion equation v t (t, ξ) =[A 2v(t,ξ) + b 2 (ξ, x(ξ), v(t, ξ))]+g 2 (ξ, x(ξ), v(t, ξ)) wq2 (t, ξ), t v(,ξ)= y(ξ), ξ D, N 2 v(t,ξ) =, t, ξ D with initial datum y and frozen slow component x, whose solution is denoted by v x,y (t). The previous equation has been widely studied, as far as existence and uniqueness of solutions are concerned. In Section 3, we introduce the transition semigroup Pt x associated with it and, by using methods and results from our previous paper [7], we study its asymptotic properties and its dependence on the parameters x and y (cfr. also [5] and[6]). Under this respect, in addition to suitable conditions on the operators A i and Q i and on the coefficients b i and g i,fori = 1, 2 (see Section 2 for all hypotheses), in the spirit of Khasminskii s work, we assume that there exist a mapping α(t ), which vanishes as T goes to infinity, and two Lipschitz-continuous mappings B 1 : and Ḡ : L(L (D), ) such that for any choice of T>, t andx,y 1 t+t E B 1 (x, v x,y (s)), h ds B 1 (x), h T t (1.7) α(t )(1 + x + y ) h for any h,and 1 t+t E G 1 (x, v x,y (s))h, G 1 (x, v x,y (s))k ds Ḡ(x)h, Ḡ(x)k T t (1.8) α(t )(1 + x 2 + y 2 ) h L (D) k L (D) for any h, k L (D). ere, B 1 and G 1 are the Nemytskii operators associated with b 1 and g 1, respectively. Notice that unlike B 1 and G 1 which are local operators, the coefficients B and Ḡ are not local. Actually, they are defined as general mappings on, and also in applications, there is no reason why they should be composition operators. In Section 3, we describe some remarkable situations in which conditions (1.7) and (1.8) are fulfilled: for example, when the fast motion admits a strongly mixing invariant measure μ x, for any fixed frozen slow component x, and the diffusion coefficient g 1 of the slow motion equation is bounded and nondegenerate.

6 94 S. CERRAI Our purpose is showing that under the above conditions the slow motion u ε converges weakly to the solution ū of the averaged equation u t (t, ξ) = A 1u(t, ξ) + B(u)(t, ξ) + Ḡ(u)(t, ξ) wq1 (1.9) (t, ξ), t u(,ξ)= x(ξ), ξ D, N 1 u(t, ξ) =, t, ξ D. More precisely, we prove that for any T> (1.1) L(u ε ) L(ū) in C([,T]; ) as ε (see Theorem 6.2). Moreover, in the case the diffusion coefficient g 1 in the slow equation does not depend on the fast oscillating variable v ε, we show that the convergence of u ε to ū is in probability, that is, for any η> (1.11) lim P( u ε ū C([,T ];) >η ) = ε (see Theorem 6.4). In order to prove (1.1), we have to proceed in several steps. First of all, we show that the family {L(u ε )} ε (,1] is tight in P (C([,T]; ))and this is obtained by a priori bounds for processes u ε in a suitable ölder norm with respect to time and in a suitable Sobolev norm with respect to space. We would like to stress that as we are only assuming (1.7)and(1.8) and not a law of large numbers, we also need to prove a priori bounds for the conditioned momenta of u ε. Once we have the tightness of the family {L(u ε )} ε (,1], we have the weak convergence of the sequence {L(u εn )} n N,forsomeε n, to some probability measure Q on C x ([,T]: ). The next steps consist in identifying Q with L(ū) and proving that limit (1.1) holds. To this purpose, we introduce the martingale problem with parameters (x, A 1, B,Ḡ, Q 1 ) and we show that Q is a solution to such martingale problem. As the coefficients B and Q are Lipschitz-continuous, we have uniqueness, and hence we can conclude that Q = L(ū). This in particular implies that for any ε n the sequence {L(u εn )} n N converges weakly to L(ū), and hence (1.1) holds. Moreover, in the case g 1 does not depend on v ε,bya uniqueness argument, this implies convergence in probability. In the general case, the key point in the identification of Q with the solution of the martingale problem associated with the averaged (1.9) is the following limit 2 lim E ( ) L sl ϕ(u ε (r), v ε (r)) L av ϕ(u ε (r)) F t1 dr =, ε E t 1 where L sl and L av are the Kolmogorov operators associated, respectively, with the slow motion equation, with frozen fast component, and with the averaged equation, and {F t } t is the filtration associated with the noise. Notice that it is sufficient to check the validity of such a limit for any cylindrical function ϕ and any t 1 t 2 T. The proof of the limit above is based on the Khasminskii

7 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 95 argument introduced in [16], but it is clearly more delicate than in [16], as it concerns a system with an infinite number of degrees of freedom (with all well-known problems arising from that). In the particular case of g 1 not depending on v ε, in order to prove (1.11), we do not need to pass through the martingale formulation. For any h D(A 1 ), we write u ε (t), h = x,h + u ε (s), A 1 h ds + B 1 (u ε (s)), h ds + G 1 (u ε (s))h, dw Q 1 1 (s) + R ε (t), where R ε (t) := B 1 (u ε (s), v ε (s)) B 1 (u ε (s)), h ds and we show that for any T> (1.12) lim E sup R ε (t) =. ε t [,T ] Thanks to the Skorokhod theorem and to a general argument due to Gyöngy and Krylov (see [15]), this allows us to obtain (1.11). 2. Assumptions and preliminaries. Let D be a smooth bounded domain of R d, with d 1. Throughout the paper, we shall denote by the ilbert space L 2 (D), endowed with the usual scalar product, and with the corresponding norm. The norm in L (D) will be denoted by. We shall denote by B b ( ) the Banach space of bounded Borel functions ϕ : R, endowed with the sup-norm ϕ := sup ϕ(x). x C b ( ) is the subspace of uniformly continuous mappings and Cb k ( ) is the subspace of all k-times differentiable mappings, having bounded and uniformly continuous derivatives, up to the kth order, for k N. Cb k ( ) is a Banach space endowed with the norm k k ϕ k := ϕ + sup D i ϕ(x) L i ( ) =: ϕ + [ϕ] i, i=1 x where L 1 ( ) := and, by recurrence, L i ( ) := L(, L i 1 ( )),foranyi>1. Finally, we denote by Lip( ) the set of functions ϕ : such that ϕ(x) ϕ(y) [ϕ] Lip( ) := sup <. x,y x y x y i=1

8 96 S. CERRAI We shall denote by L( ) the space of bounded linear operators in and we shall denote by L 2 ( ) the subspace of ilbert Schmidt operators, endowed with the norm Q 2 = Tr[Q Q]. The stochastic perturbations in the slow and in the fast motion equations (1.6) are given, respectively, by the Gaussian noises w Q 1/ t(t, ξ) and w Q 2/ t(t, ξ), for t andξ D, which are assumed to be white in time and colored in space, in the case of space dimension d>1. Formally, the cylindrical Wiener processes w Q i (t, ξ) are defined as the infinite sums w Q i (t, ξ) = Q i e k (ξ)β k (t), i = 1, 2, where {e k } k N is a complete orthonormal basis in, {β k (t)} k N is a sequence of mutually independent standard Brownian motions defined on the same complete stochastic basis (, F, F t, P) and Q i is a compact linear operator on. The operators A 1 and A 2 appearing, respectively, in the slow and in the fast motion equation, are second-order uniformly elliptic operators, having continuous coefficients on D, and the boundary operators N 1 and N 2 can be either the identity operator (Dirichlet boundary condition) or a first-order operator of the following type d β j (ξ)d j + γ(ξ)i, ξ D j=1 for some β j,γ C 1 ( D) such that inf β(ξ),ν(ξ) >, ξ D where ν(ξ) is the unit normal at ξ D (uniform nontangentiality condition). The realizations A 1 and A 2 in of the differential operators A 1 and A 2,endowed, respectively, with the boundary conditions N 1 and N 2, generate two analytic semigroups e ta 1 and e ta 2, t. In what follows, we shall assume that A 1, A 2 and Q 1, Q 2 satisfy the following conditions. YPOTESIS 1. For i = 1, 2, there exist a complete orthonormal system {e i,k } k N in and two sequences of nonnegative real numbers {α i,k } k N and {λ i,k } k N, such that If d = 1, we have (2.1) A i e i,k = α i,k e i,k, Q i e i,k = λ i,k e i,k, k 1. κ i := sup λ i,k <, ζ i := k N α β i i,k e i,k 2 <

9 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 97 for some constant β i (, 1),andifd 2, we have (2.2) κ i := λ ρ i i,k e i,k 2 <, ζ i := α β i i,k e i,k 2 < for some constants β i (, + ) and ρ i (2, + ) such that β i (ρ i 2) (2.3) < 1. ρ i Moreover, (2.4) inf α 2,k =: λ>. k N REMARK In several cases as, for example, in the case of space dimension d = 1, and in the case of the Laplace operator on a hypercube, endowed with Dirichlet boundary conditions, the eigenfunctions e k are equibounded in the sup-norm and then conditions (2.1) and(2.2) become κ i = λ ρ i i,k <, ζ i = α β i i,k < for positive constants β i,ρ i fulfilling (2.3). In general, e i,k k a i, k N for some a i. Thus, the two conditions in (2.2) become κ i := λ ρ i i,k k2a i <, ζ i := α β i i,k k2a i <. 2. For any reasonable domain D R d, one has α i,k k 2/d, k N. Thus, if the eigenfunctions e k are equibounded in the sup-norm, we have ζ i c k 2βi/d. α β i i,k This means that in order to have ζ i <, we need β i > d 2. In particular, in order to have also κ i < and condition (2.3) satisfied, in space dimension d = 1 we can take ρ i =+, so that we can deal with white noise, both in time and in space. In space dimension d = 2, we can take any ρ i < and in space dimension d 3, we need ρ i < 2d d 2.

10 98 S. CERRAI In any case, notice that it is never required to take ρ i = 2,whichmeanstohave a noise with trace-class covariance. To this purpose, it can be useful to compare these conditions with ypotheses 2 and 3 in [6]. As far as the coefficients b 1,b 2 and g 1,g 2 are concerned, we assume the following conditions. YPOTESIS The mappings b i : D R 2 R and g i : D R 2 R are measurable, both for i = 1andi = 2, and for almost all ξ D the mappings b i (ξ, ) : R 2 R and g i (ξ, ) : R 2 R are Lipschitz-continuous, uniformly with respect to ξ D. Moreover, (2.5) 2. It holds sup sup ξ D σ 2,ρ 2 R σ 1 R σ 2 ρ 2 sup b 2 (ξ,, ) <. ξ D b 2 (ξ, σ 1,σ 2 ) b 2 (ξ, σ 1,ρ 2 ) σ 2 ρ 2 where λ is the constant introduced in (2.4). 3. There exists γ<1 such that (2.6) =: L b2 <λ, sup g 2 (ξ, σ ) c(1 + σ 1 + σ 2 γ ), σ = (σ 1,σ 2 ) R 2. ξ D REMARK 2.2. Notice that condition (2.6) on the growth of g 2 (ξ, σ 1, ) could be replaced with the condition for some η sufficiently small. and In what follows we shall set L g2 := sup ξ D σ 1 R Moreover, we shall set sup [g 2 (ξ, σ 1, )] Lip η ξ D σ 1 R sup σ 2,ρ 2 R σ 2 ρ 2 g 2 (ξ, σ 1,σ 2 ) g 2 (ξ, σ 1,ρ 2 ). σ 2 ρ 2 B i (x, y)(ξ) := b i (ξ, x(ξ), y(ξ)) [G i (x, y)z](ξ) := g i (ξ, x(ξ), y(ξ))z(ξ)

11 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 99 for any ξ D, x,y,z and i = 1, 2. Due to ypothesis 2, the mappings (x, y) B i (x, y), are Lipschitz-continuous, as well as the mappings and (x, y) G i (x, y) L( ; L 1 (D)) (x, y) G i (x, y) L(L (D); ). Now, for any fixed T>andp 1, we denote by T,p the space of processes in C([,T]; L p ( ; )), which are adapted to the filtration {F t } t associated with the noise. T,p is a Banach space, endowed with the norm ( u T,p = sup t [,T ] E u(t) p ) 1/p. Moreover, we denote by C T,p the subspace of processes u L p ( ; C([,T]; )), endowed with the norm ( ) 1/p u CT,p = E sup u(t) p. t [,T ] With all notation we have introduced, system (1.6) can be rewritten as the following abstract evolution system du ε (t) =[A 1 u ε (t) + B 1 (u ε (t), v ε (t))] ds + G 1 (u ε (t), v ε (t)) dw Q 1 (t), u ε () = x (2.7) dv ε (t) = 1 ε [A 2v ε (t) + B 2 (u ε (t), v ε (t))] ds + 1 ε G 2 (u ε (t), v ε (t)) dw Q 2 (t), v ε () = y. As known from the existing literature (see, e.g., [9]), according to ypotheses 1 and 2 for any ε>andx,y and for any p 1andT>there exists a unique mild solution (u ε,v ε ) C T,p C T,p to system (1.6). This means that there exist two processes u ε and v ε in C T,p, which are unique, such that and u ε (t) = e ta 1 x + v ε (t) = e ta 2/ε y + 1 ε e (t s)a 1 B 1 (u ε (s), v ε (s)) ds + e (t s)a 1 G 1 (u ε (s), v ε (s)) dw Q 1 (s) e (t s)a 2/ε B 2 (u ε (s), v ε (s)) ds + 1 e (t s)a2/ε G 2 (u ε (s), v ε (s)) dw Q 2 (s). ε

12 91 S. CERRAI 2.1. The fast motion equation. For any fixed x, we consider the problem v t (t, ξ) = A 2v(t,ξ) + b 2 (ξ, x(ξ), v(t, ξ)) + g 2 (ξ, x(ξ), v(t, ξ)) wq 2 (2.8) (t, ξ), t v(,ξ)= y(ξ), ξ D, N 2 v(t,ξ) =, t, ξ D. Under ypotheses 1 and 2, such a problem admits a unique mild solution v x,y C T,p,foranyT>andp 1, and for any fixed frozen slow variable x and any initial condition y (for a proof, see, e.g., [1], Theorem 5.3.1). By arguing as in the proof of [7], Theorem 7.3, is it possible to show that there exists some δ 1 > such that for any p 1 (2.9) E v x,y (t) p c p(1 + x p + e δ1pt y p ), t. In particular, as shown in [7], this implies that there exists some θ>such that for any a> (2.1) sup E v x,y (t) D(( A2 ) θ ) c a (1 + x + y ). t a Now, for any x, we denote by Pt x the transition semigroup associated with problem (2.8),whichisdefinedby Pt x ϕ(y) = Eϕ(vx,y (t)), t, y for any ϕ B b ( ). Due to (2.1), the family {L(v x,y (t))} t a is tight in P (, B( )) and then by the Krylov Bogoliubov theorem, there exists an invariant measure μ x for the semigroup Pt x. Moreover, due to (2.9) foranyp 1, we have (2.11) z p μx (dz) c p (1 + x p ) (for a proof see [8], Lemma 3.4). As in [7], Theorem 7.4, it is possible to show that if λ is sufficiently large and/or L b2, L g2, ζ 2 and κ 2 are sufficiently small, then there exist some c,δ 2 > such that (2.12) sup E v x,y 1 (t) v x,y 2 (t) ce δ2t y 1 y 2, t x for any y 1,y 2. In particular, this implies that μ x is the unique invariant measure for Pt x and is strongly mixing. Moreover, by arguing as in [8], Theorem 3.5 and Remark 3.6, from (2.11) and(2.12), we have (2.13) P t x ϕ(z)μ x (dz) c(1 + x + y )e δ2t [ϕ] Lip( ) for any x,y and ϕ Lip( ). In particular, this implies the following fact.

13 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 911 LEMMA 2.3. Under the above conditions, for any ϕ Lip( ), T>, x, y and t 1 t+t E ϕ(v x,y (s)) ds ϕ(z)μ x (dz) T c ( (2.14) ϕ (x, y) + ϕ() ), t T where (2.15) ϕ (x, y) := [ϕ] Lip( ) (1 + x + y ). PROOF. ( 1 E T We have +T t = 1 T 2 = 2 T 2 ϕ(v x,y (s)) ds ϕ x ) 2 +T +T t t +T +T t r E ( ϕ(v x,y (s)) ϕ x)( ϕ(v x,y (r)) ϕ x) dsdr E ( ϕ(v x,y (s)) ϕ x)( ϕ(v x,y (r)) ϕ x) dsdr, where ϕ x := ϕ(z)μ x (dz). From the Markovianity of v x,y (t),forr s, we have E ( ϕ(v x,y (s)) ϕ x)( ϕ(v x,y (r)) ϕ x) = E [( ϕ(v x,y (r)) ϕ x) ( P s r ϕ(v x,y (r)) ϕ x)], so that in view of (2.9) and(2.13), ( 1 t+t ) 2 E ϕ(v x,y (s)) ds ϕ x T t c +T +T ( [ϕ]lip( T 2 ) (E v x,y (r) 2 )1/2 + ϕ() + ϕ x ) t r ( E[P s r ϕ(v x,y (r)) ϕ x ] 2) 1/2 dsdr +T +T c ( ϕ T 2 (x, y) + ϕ() + ϕ x ) ϕ (x, y) t r c ( ϕ (x, y) + ϕ() + ϕ x ) ϕ (x, y) T with ϕ (x, y) defined as in (2.15). As from (2.11), we have ϕ x [ϕ] Lip( ) (1 + x ) + ϕ(), e δ 2(s r) dsdr we can conclude that (2.14) holds.

14 912 S. CERRAI 2.2. The averaged coefficients. In the next hypotheses, we introduce the coefficients of the averaged equation, and we give conditions which assure the convergence of the slow motion component u ε to its solution. For the reaction coefficient, we assume the following condition. YPOTESIS 3. There exists a Lipschitz-continuous mapping B : such that for any T>, t andx,y,h 1 t+t E B 1 (x, v x,y (s)), h ds B(x),h T t (2.16) α(t )(1 + x + y ) h for some function α(t ) such that lim α(t ) =. T Concerning the diffusion coefficient, we assume the following condition. YPOTESIS 4. There exists a Lipschitz-continuous mapping Ḡ : L(L (D); ) such that for any T>, t, x,y and h, k L (D) 1 t+t E G 1 (x, v x,y (s))h, G 1 (x, v x,y (s))k ds Ḡ(x)h, Ḡ(x)k T t (2.17) α(t )(1 + x 2 + y 2 ) h k for some α(t ) such that lim α(t ) =. T 3. The averaged equation. In this section, we describe some relevant situations in which ypotheses 3 and 4 are verified and we give some notation and some results about the martingale problem and the mild solution for the averaged equation The reaction coefficient B. Forany fixed x,h, the mapping y B 1 (x, y), h R is Lipschitz-continuous. Then if we define B(x) := B 1 (x, z)μ x (dz), x, thanks to (2.13) we have that limit (2.16) holds, with α(t ) = c/ T.

15 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 913 Due to (2.16), for any x 1,x 2,y,h, we have B 1 (x 1 ) B 1 (x 2 ), h 1 T = lim T T E B 1 (x 1,v x1,y (s)) B 1 (x 2,v x2,y (s)), h ds. Then as the mapping B 1 : is Lipschitz continuous, we have B 1 (x 1 ) B 1 (x 2 ), h c h lim sup T 1 T T = c h ( x 1 x 2 + lim sup T ( x1 x 2 + E v x 1,y (s) v x 2,y (s) ) ds 1 T T ) E ρ(s) ds, where ρ(t) := v x1,y (t) v x2,y (t), foranyt. In the next lemma, we show that under suitable conditions on the coefficients there exists some constant c>such that for any T> 1 T T E ρ(s) ds c x 1 x 2. Clearly, this implies the Lipschitz continuity of B 1. (3.1) LEMMA 3.1. Assume that L b2 λ + L g 2 ( β2 e ( ρ 2 λ(ρ 2 + 2) ) β2 (ρ 2 2)/(2ρ 2 ) ζ (ρ 2 2)/(2ρ 2 ) 2 κ 2/(2ρ 2) 2 ) 1/2 β2 (ρ 2 2)/(2ρ 2 ) =: M < 1. Then under ypotheses 1 and 2, there exists c> such that for any x 1,x 2, y and t> 1 t E v x1,y (s) v x2,y (s) ds c x 1 x 2. t PROOF. Ɣ(t) := We set ρ(t) := v x 1,y (t) v x 2,y (t) and define e (t s)a 2 [G 2 (x 1,v x 1,y (s)) G 2 (x 2,v x 2,y (s))] dw Q 2 (s) and set (t) := ρ(t) Ɣ(t). Foranyη (,λ/2), we can fix c 1,η > such that 1 d 2 dt (t) 2 λ (t) 2 + ( ) c x 1 x 2 + L b2 ρ(t) (t) ( ) λ 2 η (t) 2 + L2 b 2 2λ ρ(t) 2 + c 1,η x 1 x 2 2.

16 914 S. CERRAI This implies ( (t) c ) 1,η x 1 x L2 b 2 e (λ 2η)(t s) ρ(s) 2 λ 2η λ ds, so that for any ε>andη<λ/2 ( E ρ(t) 2 (1 + ε) 1 + 2c ) 1,η x 1 x 2 2 λ 2η (3.2) + (1 + ε)l2 b 2 e (λ 2η)(t s) E ρ(s) 2 λ ds (1 + ε) + E Ɣ(t) 2 ε. Thanks to ypothesis 1, foranyj L(L (D), ) L(, L 1 (D)), with J = J and for any s, we have e sa 2 JQ = λ 2 2,k esa 2 Je 2,k 2 ( λ ρ 2 2,k e 2,k 2 k N ) 2/ρ2 ( κ 2/ρ 2 2 sup e sa 2 Je 2,k 4/ρ 2 e 2,k 4/ρ 2 ence, thanks to (2.4), we obtain (3.3) e sa 2 Je 2,k 2ρ 2/(ρ 2 2) e 2,k 4/(ρ 2 2) ( ) (ρ2 2)/ρ 2 e sa 2 Je 2,k 2. e sa 2 JQ ( ) (ρ2 2)/ρ κ 2/ρ 2 2 J 4/ρ 2 2 L(L (D), ) e 4λ/ρ 2s e sa 2 Je 2,k 2. ) (ρ2 2)/ρ 2 We have e sa 2 Je 2,k 2 = e sa 2 Je 2,k,e 2,h 2 = e 2,k,Je sa 2 e 2,h 2 h=1 h=1 = Je 2,h 2 e 2α2,hs e λs J 2 L(L (D), ) e 2,h 2 e α2,hs. h=1 h=1 Then as for any β>, ( ) β β (3.4) e αt t β α β, α,t >, e

17 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 915 if we take β 2 as in condition (2.2), we get ( ) e sa 2 Je 2,k 2 β2 β2s β 2 e λs J 2 e ( β2 e and from (3.3) we can conclude (3.5) e sa 2 JQ ( β2 e This means that if we set and if we take L(L (D), ) h=1 ) β2ζ2 s β 2 e λs J 2 L(L (D), ) e 2,h 2 α β 2 2,h ) β2 (ρ 2 2)/ρ 2ζ (ρ 2 2)/ρ 2 2 κ 2/ρ 2 2 s β 2(ρ 2 2)/ρ 2 e λ(ρ 2+2)/ρ 2 s J 2 L(L (D), ). K 2 := ( β2 e ) β2 (ρ 2 2)/ρ 2ζ (ρ 2 2)/ρ 2 2 κ 2/ρ 2 2 J := G 2 (x 1,v x 1,y (s)) G 2 (x 2,v x 2,y (s)) for any <η<λ/2, E Ɣ(t) 2 = E e (t s)a 2 [G 2 (x 1,v x1,y (s)) G 2 (x 2,v x2,y (s))q 2 ] 2 2 ds K 2 (t s) β 2(ρ 2 2)/ρ 2 e λ(ρ 2+2)/ρ 2 (t s) E ( ) 2 c x 1 x 2 + L g2 ρ(s) ds c(1 + η) x 1 x 2 2 η + (1 + η)l 2 g 2 K 2 (t s) β 2(ρ 2 2)/ρ 2 e λ(ρ 2+2)/ρ 2 (t s) E ρ(s) 2 ds, last inequality following from the fact that, according to (2.3), β 2 (ρ 2 2)/ρ 2 < 1. Now, if we plug the inequality above into (3.2), for any ε>and<η<λ/2, we obtain ( E ρ(t) 2 (1 + ε) 1 + 2c 1,η λ 2η ) x 1 x c + (1 + ε)l2 b 2 e (λ 2η)(t s) E ρ(s) 2 λ ds (1 + ε) + (1 + η)l 2 g ε 2 K 2 ( )( 1 + ε 1 + η (t s) β 2(ρ 2 2)/ρ 2 e λ(ρ 2+2)/ρ 2 (t s) E ρ(s) 2 ds ε η ) x 1 x 2 2

18 916 S. CERRAI and hence, if we integrate with respect to t both sides, from the Young inequality we get where (3.6) E ρ(s) 2 ds [( 1 + 2c ) 1,η + c λ 2η ε + (1 + ε)[ L 2 b2 λ ( 1 + η η e (λ 2η)s ds )] (1 + ε)t x 1 x η L 2 g ε 2 K 2 s β 2(ρ 2 2)/ρ 2 e λ(ρ 2+2)/ρ 2 s ds E ρ(s) 2 ds c η,ε t x 1 x M η,ε [ M η,ε := (1 + ε) L 2 b 2 λ(λ 2η) η ε ( ζ (ρ 2 2)/ρ 2 2 κ 2/ρ 2 2 E ρ(s) 2 ds, ( ) β2 L 2 (ρ β2 2 2)/ρ 2 g 2 e ρ 2 λ(ρ 2 + 2) Now, by taking the minimum over ε>. we get E ρ(s) 2 ds c η, εt x 1 x M2 η where L b2 ) 1 β2 (ρ 2 2)/ρ 2 ]. E ρ(s) 2 ds, ) β2 (ρ 2 2)/(2ρ 2 ) M η := λ(λ + ( β2 1 + ηl g2 2η) e ( ) ζ (ρ 2 2)/(2ρ 2 ) 2 κ 2/(2ρ 2) ρ 1/2 β2 (ρ 2 2)/2ρ λ(ρ 2 + 2) Then as in (3.1) we have assumed that M < 1, we can fix η (,λ/2) such that M η < 1, and hence This implies c η, ε E ρ(s) 2 ds 1 M 2 η t x 1 x t ( ) 1/2 c3, η E ρ(s) ds t 1 M 2 η x 1 x 1 and the proof of the lemma is complete. ]

19 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS The diffusion coefficient Ḡ. If we assume that the function g 1 : D R 2 R is uniformly bounded, the mapping G 1 is well defined from into L( ). Moreover, for any fixed x,h,k the mapping z G 1 (x, z)h, G 1 (x, z)k R, is Lipschitz continuous. Thus, under the assumptions described above, if we take (3.7) S(x)h, k = G 1 (x, z)h, G 1 (x, z)k μ x (dz), we have that S : L( ) and, due to (2.13), for any T >, t and x,y,h,k 1 t+t E G 1 (x, v x,y (s))h, G 1 (x, v x,y (s))k ds S(x)h, k T t (3.8) α(t )(1 + x 2 + y 2 ) h k for some function α(t ) going to zero as T. It is immediate to check that S(x) = S(x) and S(x), for any x.then as is well known, there exists an operator Ḡ(x) L( ) such that Ḡ(x) 2 = S(x). If we assume that there exists δ> such that inf ξ D σ R 2 g 1 (ξ, σ ) δ, we have that S(x) δ 2, and hence Ḡ(x) δ. In particular, Ḡ(x) is invertible and Next, we notice that for any x 1,x 2 (3.9) Ḡ(x) 1 L( ) 1 δ. S(x 1 )S(x 2 ) = S(x 2 )S(x 1 ). Actually, according to (3.7) foranyh, k, S(x 1 )S(x 2 )h, k = G 1 (x 1, z)s(x 2 )h, G 1 (x 1,z)k μ x 1 (dz) = G 2 1 (x 2, w)h, G 2 1 (x 1,z)k μ x 2 (dw)μ x 1 (dz) = G 2 1 (x 1, z)h, G 2 1 (x 2,w)k μ x 1 (dz)μ x 2 (dw) = S(x 2 )S(x 1 )h, k. In particular, from (3.9) foranyx 1,x 2, we have (3.1) Ḡ(x 1 )Ḡ(x 2 ) = Ḡ(x 2 )Ḡ(x 1 ).

20 918 S. CERRAI Now, as g 1 (ξ, ) : R 2 R is bounded and Lipschitz-continuous, uniformly with respect to ξ D, wehavethatg 2 1 (ξ, ) : R2 R is Lipschitz-continuous as well, uniformly with respect to ξ D. This implies that for any x 1,x 2,y, h L (D) and k [G 2 1 (x 1,v x 1,y (s)) G 2 1 (x 2,v x 2,y (s))]h, k so that according to (3.8), ( S(x 1 ) S(x 2 ) ) h, k c ( x 1 x 2 + v x 1,y (s) v x 2,y (s) ) h k, c h k lim sup T 1 T T = c h k ( x 1 x 2 + lim sup T ( x1 x 2 + E v x 1,y (s) v x 2,y (s) ) ds 1 T T ) E v x1,y (s) v x2,y (s) ds. Then thanks to Lemma 3.1, we can conclude that S : L(L (D), ) is Lipschitz-continuous. This implies that Ḡ : L(L (D), ) is Lipschitz-continuous as well. Actually, thanks to (3.1) and to the fact that Ḡ(x 1 ) + Ḡ(x 2 ) is invertible, for any h L (D) and k, (3.11) Then as [Ḡ(x 1 ) Ḡ(x 2 )]h, k = [S(x 1 ) S(x 2 )]h, [Ḡ(x 1 ) + Ḡ(x 2 )] 1 k. (3.12) [Ḡ(x 1 ) + Ḡ(x 2 )] 1 L( ) 1 2δ, we obtain 1 [Ḡ(x 1 ) Ḡ(x 2 )]h, k c x 1 x 2 h 2δ k and this implies the Lipschitz-continuity of Ḡ : L(L (D), ). We conclude by showing that the operator Ḡ introduced in ypothesis 4 satisfies a suitable ilbert Schmidt property which assures the well-posedness of the stochastic convolution e (t s)a 1 Ḡ(u(s)) dw Q 1 (s), t in L p ( ; C([,T]; )),foranyp 1andT > and for any process u C([,T]; L p ( ; )). LEMMA 3.2. Assume ypotheses 1, 2 and 4. Then for any t> and x 1, x 2, we have e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]Q 1 2 c(t) x 1 x 2 t β 1(ρ 1 2)/(2ρ 1 ) for some continuous increasing function c(t).

21 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 919 PROOF. According to ypothesis 1, wehave e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]Q = e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]Q 1 e 1,k 2 ( λ ρ 1 1,k e 1,k 2 ( ) 2/ρ1 e 1,k 4/(ρ 1 2) e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]e 1,k 2ρ 1/(ρ 1 2) c sup e 1,k 4/ρ 1 eta 1 [Ḡ(x 1 ) Ḡ(x 2 )]e 1,k 4/ρ 1 k N ( ) (ρ1 2)/ρ 1 e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]e 1,k 2 and then as Ḡ : L(L (D), ) is Lipschitz-continuous, we conclude (3.13) e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]Q ( c(t) x 1 x 2 4/ρ 1 e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]e 1,k 2 ) (ρ1 2)/ρ 1 ) (ρ1 2)/ρ 1 for some continuous increasing function c(t). Now, by using again the Lipschitz-continuity of Ḡ : L(L (D), ), we have e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]e 1,k 2 = e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]e 1,k,e 1,h 2 h=1 = [Ḡ(x 1 ) Ḡ(x 2 )]e 1,h 2 e 2tα 1,h h=1 c x 1 x 2 2 e 2tα 1,h e 1,h 2 h=1 and then, if we take β 1 as in ypothesis 1, we obtain e ta 1 [Ḡ(x 1 ) Ḡ(x 2 )]e 1,k 2 c x 1 x 2 2 t β 1 h=1 α β 1 1,h e 1,h 2 c x 1 x 2 2 t β 1.

22 92 S. CERRAI Thanks to (3.13), this implies our thesis Martingale problem and mild solution of the averaged equation. Since the mappings B : and Ḡ : L(L (D); ) are both Lipschitzcontinuous and Lemma 3.2 holds, for any initial datum x the averaged equation (3.14) du(t) =[A 1 u(t) + B(u(t))] dt + Ḡ(u(t)) dw Q 1 (t), u() = x, admits a unique mild solution ū in L p (, C([,T]; )),foranyp 1andT> (for a proof, see, e.g., [6], Section 3). This means that there exists a unique adapted process ū L p (, C([,T]; )) such that for any t T ū(t) = e ta 1 x + e (t s)a 1 B(ū(s)) ds + e (t s)a 1 Ḡ(ū(s)) dw Q 1 (s) or equivalently, ū(t), h = x,h + [ ū(s), A 1 h + B(ū(s)), h ] ds + Ḡ(ū(s)) dw Q 1 (s), h for any h D(A 1 ). Now, we recall the notion of martingale problem with parameters (x, A 1, B,Ḡ, Q 1 ).Foranyfixedx, we denote by C x ([,T]; ) the space of continuous functions ω : [,T] such that ω() = x and we denote by η(t) the canonical process on C x ([,T]; ), which is defined by η(t)(ω) = ω(t), t [,T]. Moreover, we denote by E t the canonical filtration σ(η(s),s t),fort [,T], and by E the canonical σ -algebra σ(η(s),s T). DEFINITION 3.3. A function ϕ : R is a regular cylindrical function associated with the operator A 1 if there exist k N, f Cc (Rk ), a 1,...,a k and N N such that ϕ(x) = f( x,p N a 1,..., x,p N a k ), x, where P N is the projection of onto span e 1,1,...,e 1,N and {e 1,n } n N is the orthonormal basis diagonalizing A 1 and introduced in ypothesis 1.

23 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 921 In what follows, we shall denote the set of all regular cylindrical functions by R( ).Foranyϕ R( ) and x, we define L av ϕ(x) := 1 2 Tr[(Ḡ(x)Q 1 ) D 2 ϕ(x)ḡ(x)q 1 ] + A 1 Dϕ(x), x + Dϕ(x), B(x) (3.15) = 1 k Dij 2 2 f( x,p Na 1,..., x,p N a k ) i,j=1 Ḡ(x)Q 1 P N a i, Ḡ(x)Q 1 P N a j k + D i f( x,p N a 1,..., x,p N a k ) i=1 ( x,a 1 P N a i + B(x),P N a i ). L av is the Kolmogorov operator associated with the averaged equation (3.14). Notice that the expression above is meaningful, as for any i = 1,...,k and N Q 1 P N a i = λ 1,k a i,e 1,k e 1,k L (D) N A 1 P N a i = α 1,k a i,e 1,k e 1,k. DEFINITION 3.4. A probability measure Q on (C x ([,T]; ),E) is a solution of the martingale problem with parameters (x, A 1, B,Ḡ, Q 1 ) if the process ϕ(η(t)) L av ϕ(η(s))ds, t [,T] is an E t -martingale on (C x ([,T]; ),E, Q),foranyϕ R( ). As the coefficients B and Ḡ are Lipschitz-continuous, the solution Q to the martingale problem with parameters (x, A 1, B,Ḡ, Q 1 ) exists, is unique and coincides with L(ū) (to this purpose see [9], Chapter 8, and also [26], Theorems 5.9 and 5.1). 4. A priori bounds for the solution of system (1.6). In the present section, we prove uniform estimates, with respect to ε (, 1], for the solution u ε of the slow motion equation and for the solution v ε of the fast motion equation in system (1.6). As a consequence, we will obtain the tightness of the family {L(u ε )} ε (,1] in C([,T]; ),foranyt>.

24 922 S. CERRAI In what follows, for the sake of simplicity, we denote by θ D(( A1 ) θ ). Moreover, for any ε>, we denote the norm (4.1) Ɣ 1,ε (t) := e (t s)a 1 G 1 (u ε (s), v ε (s)) dw Q 1 (s), t. LEMMA 4.1. Under ypotheses 1 and 2, there exists θ > and p 1 such that for any ε>, T>, p> p and θ [, θ] (4.2) E sup Ɣ 1,ε (t) p θ c T,p,θ t T T ( 1 + E uε (r) p + E v ε(r) p ) ds for some positive constant c T,p,θ which is independent of ε>. PROOF. where and By using a factorization argument, for any α (, 1/2), we have Ɣ 1,ε (t) = c α (t s) α 1 e (t s)a 1 Y ε,α (s) ds, s Y ε,α (s) := (s r) α e (s r)a 1 G 1,ε (s) dw Q 1 (r) G 1,ε (s) := G 1 (u ε (s), v ε (s)). For any p>1/α and θ>, we have ( ) p 1 sup Ɣ 1,ε (s) p θ c α,p s (α 1)p/(p 1) ds Y ε,α (s) p θ ds s t (4.3) = c α,p t αp 1 Y ε,α (s) p θ ds. According to the Burkholder Davis Gundy inequality, we have E Y ε,α (s) p θ ds ( s c p E (s r) 2α ) ( A 1 ) θ e (s r)a p/2 1 G 1,ε (s)q dr ds. By the same arguments as those used in the proof of Lemma 3.1, wehave ( A 1 ) θ e (s r)a 1 G 1,ε (s)q sup ( A 1 ) θ e (s r)a 1 G 1,ε (s)e 1,k 4/ρ 1 e 1,k 4/ρ 1 k N

25 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 923 κ 2/ρ 1 1 ( ( A 1 ) θ e (s r)a 1 G 1,ε (s)e 1,k 2 ) (ρ1 2)/ρ 1 c θ κ 2/ρ 1 1 (s r) 4θ/ρ 1 G 1,ε (s) 4/ρ 1 L(L (D), ) ( ) (ρ1 2)/ρ1 ( A 1 ) θ e (s r)a 1 G 1,ε (s)e 1,k 2. By proceeding again as in the proof of Lemma 3.1 we have ( A 1 ) θ e (s r)a 1 G 1,ε (s)e 1,k 2 and then thanks to (3.4), we get Therefore, if we set G 1,ε (s) 2 L(L (D), ) e 1,k 2 αθ 1,k e α 1,k(s r) ( A 1 ) θ e (s r)a 1 G 1,ε (s)e 1,k 2 ( ) β1 + θ β1 +θ ζ 1 (s r) (β1+θ) G 1,ε (s) 2 L(L e (D), ). and fix θ > such that K 1,θ := c θ ( β1 + θ e for any θ [, θ], we have E Y ε,α (s) p ds ) (β1 +θ)(ρ 1 2)/ρ 1ζ (ρ 1 2)/ρ 1 1 κ 2/ρ 1 1 β 1 (ρ 1 2) + θ(ρ 1 + 2) ρ 1 < 1 ( s c p K p/2 1,θ E (s r) (2α+(β 1(ρ 1 2)+θ(ρ 1 +2))/ρ 1 ) ence, if we choose ᾱ>, such that G 1,ε (r) 2 L(L (D), ) dr ) p/2 ds. 2ᾱ + β 1(ρ 1 2) + θ(ρ 1 + 2) ρ 1 < 1

26 924 S. CERRAI and p> p := 1/ᾱ, by the Young inequality this yields for t [,T] E Y ε,ᾱ (s) p θ ds ( c p K p/2 1,θ ) p/2 s (2ᾱ+(β 1(ρ 1 2)+θ(ρ 1 +2))/ρ 1 ) ds E G 1,ε (s) p L(L (D), ) ds ( c T,p 1 + E uε (s) p + E v ε(s) p ) ds. Thanks to (4.3), this implies (4.2). Now, we can prove the first a priori bounds for the solution u ε of the slow motion equation and for the solution v ε of the fast motion equation in system (1.6). PROPOSITION 4.2. Under ypotheses 1 and 2, for any T>and p 1, there exists a positive constant c(p,t ) such that for any x,y and ε (, 1] (4.4) E sup t [,T ] u ε (t) p c(p,t )(1 + x p + y p ) and (4.5) T E v ε (t) p dt c(p,t )(1 + x p + y p ). Moreover, there exists some c T > such that (4.6) sup t [,T ] E v ε (t) 2 c T (1 + x 2 + y 2 ). PROOF. Let ε>andx,y be fixed once for all and let Ɣ 1,ε (t) be the process defined in (4.1). If we set 1,ε (t) := u ε (t) Ɣ 1,ε (t), wehave d dt ( 1,ε(t) = A 1 1,ε (t) + B 1 1,ε (t) + Ɣ 1,ε (t), v ε (t) ), and then for any p 2wehave 1 d p dt 1,ε(t) p = A 1 1,ε (t), 1,ε (t) 1,ε (t) p 2 + ( B 1 1,ε (t) + Ɣ 1,ε (t), v ε (t) ) 1,ε () = x B 1 (Ɣ 1,ε (t), v ε (t)), 1,ε (t) 1,ε(t) p 2 + B 1 (Ɣ 1,ε (t), v ε (t)), 1,ε (t) 1,ε (t) p 2 c p 1,ε (t) p + c p B 1 (Ɣ 1,ε (t), v ε (t)) p c p 1,ε (t) p + c p( 1 + Ɣ1,ε (t) p + v ε(t) p ).

27 This implies that AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 925 1,ε (t) p ec pt x p + c p e c p(t s) ( 1 + Ɣ 1,ε (s) p + v ε(s) p ) ds, so that, for any t [,T], u ε (t) p c p Ɣ 1,ε (t) p + c pe cpt x p + c p e c p(t s) ( 1 + Ɣ 1,ε (s) p + v ε(s) p ) ds c T,p (1 + x p + sup Ɣ 1,ε (s) p + v ε (s) p ). ds s t According to (4.2) (with θ = ), we obtain E sup u ε (s) p c T,p(1 + x p ) + c T,p s t ( + c T,p 1 + E sup r s and hence by comparison, (4.7) E sup u ε (s) p c T,p s t Now, we have to estimate If we define E v ε (s) p ds u ε (r) p ) ds (1 + x p + E v ε (s) p ). ds E v ε (s) p ds. Ɣ 2,ε (t) := 1 e (t s)/εa 2 G 2 (u ε (s), v ε (s)) dw Q 2 (s) ε and set 2,ε (t) := v ε (t) Ɣ 2,ε (t),wehave d dt 2,ε(t) = 1 [ ( A2 2,ε (t) + B 2 uε (t), 2,ε (t) + Ɣ 2,ε (t) )], 2,ε () = y. ε ence, as before, for any p 1, we have 1 d p dt 2,ε(t) p = 1 ε A 2 2,ε (t), 2,ε (t) 2,ε (t) p ( B2 uε (t), 2,ε (t) + Ɣ 2,ε (t) ) ε B 1 (u ε (t), Ɣ 2,ε (t)), 2,ε (t) 2,ε(t) p ε B 2(u ε (t), Ɣ 2,ε (t)), 2,ε (t) 2,ε (t) p 2 λ L b 2 2ε 2,ε (t) p + c p ( 1 + uε (t) p ε + Ɣ 2,ε(t) p ).

28 926 S. CERRAI By comparison this yields (4.8) v ε (t) p c p 2,ε (t) p + c p Ɣ 2,ε (t) p c p e p(λ L b 2 )/(2ε)t y p + c p e p(λ L b 2 )/(2ε)(t s) ( 1 + u ε (s) p ε + Ɣ 2,ε(s) p ) ds + c p Ɣ 2,ε (t) p. Therefore, by integrating with respect to t, we easily obtain (4.9) v ε (s) p ds c p (ε y p + Ɣ 2,ε (s) p ds + u ε (s) p ). ds + 1 According to the Burkholder Davis Gundy inequality and to (3.5), we have (4.1) E Ɣ 2,ε (s) p ( s ) p/2 c p ε p/2 E e (s r)/εa 2 G 2 (u ε (r), v ε (r))q dr ( s ( ) c p K p/2 s r β2 2 ε p/2 (ρ 2 2)/ρ 2e λ(ρ E 2 +2)/(ερ 2 )(s r) ε G 2 (u ε (r), v ε (r)) 2 L(L (D), ) dr ) p/2 so that (4.11) ( s ( ) c p K p/2 s r β2 2 ε p/2 (ρ 2 2)/ρ 2e λ(ρ E 2 +2)/(ερ 2 )(s r) ε E Ɣ 2,ε (s) p ds c p ( ) 1 + u ε (r) 2 + v ε(r) 2γ ) p/2 dr, ( 1 + E uε (s) p + E v ε(s) pγ ) ds. Due to (4.9), this allows to conclude E v ε (s) p ds c p (ε y p + ( 1 + E uε (s) p ) t ds + E v ε (s) pγ ds + 1 ) andthenasγ is assumed to be strictly less than 1, if ε (, 1] and t [,T], we obtain E v ε (s) p ds 1 E v ε (s) p 2 ds + c p y p + c p E u ε (s) p ds + c p,t. This yields (4.12) E v ε (s) p ds c p y p + c p E sup u ε (r) p ds + c p,t. r s

29 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 927 ence, if we plug (4.12) into(4.7), we get E sup u ε (s) p c T,p(1 + x p + y p ) + c T,p s t E sup u ε (r) p ds r s and from the Gronwall lemma (4.4) follows. Now, in view of estimates (4.4) and (4.11), from (4.9), we obtain (4.5). Finally, let us prove (4.6). From (4.1), with p = 2, we get sup E Ɣ 2,ε (t) 2 c 2 y 2 + c 2 t T ( 1 + sup t T and then if we substitute in (4.8), we obtain ( ) E v ε (t) 2 c y 2 + sup E u ε (t) 2 t T As γ<1, for any η>, we can fix c η > such that E u ε (t) 2 + sup t T c 2 sup E v ε (t) 2γ η sup E v ε (t) 2 + c η. t T t T Therefore, if we take η 1/2, we obtain ( 1 E v ε (t) 2 c 2 2 sup t T and (4.6) follows from (4.4). 1 + y 2 + sup t T ) E v ε (t) 2γ + c 2 sup E v ε (t) 2γ. t T E u ε (t) 2 Next, we prove uniform bounds for u ε in L (,T; D(( A 1 ) α )), forsome α>. PROPOSITION 4.3. Under ypotheses 1 and 2, there exists ᾱ> such that for any T>, p 1, x D(( A 1 ) α ), with α [, ᾱ), and y ) (4.13) sup ε (,1] E sup u ε (t) p α c T,α,p(1 + x p α + y p ) t T for some positive constant c T,α,p. PROOF. Assume that x D(( A 1 ) α ),forsomeα. We have u ε (t) = e ta 1 x + e (t s)a 1 B 1 (u ε (s), v ε (s)) ds + e (t s)a 1 G 1 (u ε (s), v ε (s)) dw Q 1 (s).

30 928 S. CERRAI If α<1/2, t T and p 2 e (t s)a 1 B 1 (u ε (s), v ε (s)) ds ( c p,α (t s) α B 1 (u ε (s), v ε (s)) ds p α ( c p,α (t s) α( ) ) p 1 + u ε (s) + v ε (s) ds ) c p,α (1 + sup u ε (s) p T (1 α)p s T ( T ) p/2 ( T + c p,α s 2α ds v ε (s) p )T ds (p 2)/2, so that, thanks to (4.4) and(4.5), (4.14) E sup e (t s)a 1 B 1 (u ε (s), v ε (s)) ds t T p α ) p c T,α,p (1 + x p + y p ). Concerning the stochastic term Ɣ 1,ε (t), due to Lemma 4.1 andto(4.4), there exists θ > such that for any α θ and p 1 (4.15) E sup Ɣ 1,ε (t) p α c T,α,p(1 + x p + y p ). t T ence, if we choose ᾱ := θ 1/2, thanks to (4.14) and(4.15), for any p 2and α<ᾱ we have E sup u ε (t) p α sup e ta 1 x p α + E sup t e (t s)a p 1 B 1 (u ε (s), v ε (s)) ds t T t T t T α + E sup Ɣ 1,ε (t) p α t T c T,α,p (1 + x p α + y p ). Next, we prove uniform bounds for the increments of the mapping t [,T] u ε (t). PROPOSITION 4.4. Under ypotheses 1 and 2, for any α>, there exists β(α) > such that for any T>, p 2, x D(( A 1 ) α ) and y it holds (4.16) sup E u ε (t) u ε (s) p c T,α,p t s β(α)p ( x p α + y p + 1), ε (,1] s,t (,T].

31 PROOF. AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 929 Forany t,h, with t,t + h [,T],wehave u ε (t + h) u ε (t) = (e ha 1 I)u ε (t) + + +h t +h t e (t+h s)a 1 B 1 (u ε (s), v ε (s)) ds e (t+h s)a 1 G 1 (u ε (s), v ε (s)) dw Q 1 (s). In view of (4.13), if we fix α [, ᾱ) and p 1, we have (4.17) E (e ha 1 I)u ε (t) p c ph αp E u ε (t) p α c T,α,ph αp (1 + x p α + y p ). In view of (4.4) and(4.5), +h E e (t+h s)a p 1 B 1 (u ε (s), v ε (s)) ds (4.18) t +h ch p 1 ( 1 + E uε (s) p + E v ε(s) p ) ds t ) T c T h (1 p + sup E u ε (s) p + ch p 1 E v ε (s) p ds s T c T,p (1 + x p + y p )hp 1. Finally, for the stochastic term, by using (3.5), for any t T and p 1, we have +h E e (t+h s)a 1 G 1 (u ε (s), v ε (s)) dw Q p 1 (s) t c p E (+h t (+h c p K p/2 1 E Then, if we take p 1 such that ) e (t+h s)a p/2 1 G 1 (u ε (s), v ε (s))q ds t (t + h s) β 2(ρ 2 2)/ρ 2 G 1 (u ε (s), v ε (s)) 2 L(L (D), ) ds ) p/2. β 2 (ρ 2 2) ρ 2 p p 2 < 1 for any p p, we have +h E e (t+h s)a 1 G 1 (u ε (s), v ε (s)) dw Q 1 (s) t c T,p h (p 2)/2 β 2(ρ 2 2)/ρ 2 p/2 T p ( 1 + E uε (s) p + E v ε(s) p ) ds

32 93 S. CERRAI and, thanks to (4.4) and(4.5), we conclude +h E e (t+h s)a 1 G 1 (u ε (s), v ε (s)) dw Q p 1 (s) t (4.19) c T,p (1 + x p + y p )h(1 2/ p β 2(ρ 2 2)/ρ 2 )p/2. Therefore, collecting together (4.17), (4.18) and(4.19), we obtain E u ε (t + h) u ε (t) p c T,α,p h αp (1 + x p α + y p ) + c T,p ( h (1 2/ p β 2 (ρ 2 2)/ρ 2 )p/2 + h p 1) (1 + x p + y p ) and, as we are assuming h 1, (4.16) follows for any p p by taking { β(α) := min α, 1 ( 1 2 p 2 β )} 2(ρ 2 2). ρ 2 Estimate (4.16) forp< p follows from the ölder inequality. As a consequence of Propositions 4.3 and 4.4, we have the following fact. COROLLARY 4.5. Under ypotheses 1 and 2, for any T>, x D(( A 1 ) α ), with α>, and y the family {L(u ε )} ε (,1] is tight in C([,T]; ). PROOF. Let α>befixedandletx D(( A 1 ) α ) and y. According to (4.16), in view of the Garcia Rademich Rumsey theorem, there exists β > such that for any p 1 sup E u ε p ε (,1] C β c T,p(1 + x p ([,T ];) α + y p ). Due to Proposition 4.3, this implies that for any η> we can find R η > such that P(u ε K Rη ) 1 η, ε (, 1], where, by the Ascoli Arzelà theorem, K Rη is the compact subset of C([,T]; ) defined by { K Rη := u C([,T]; ): u C β ([,T ];) + sup u(t) α R η }. t [,T ] This implies that the family of probability measures {L(u ε )} ε (,1] is tight in C([,T]; ). We conclude this section by noticing that with arguments analogous to those used in the proof of Propositions 4.2, 4.3 and 4.4, we can obtain a priori bounds also for the conditional second momenta of the -norms of u ε and v ε.

33 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 931 PROPOSITION 4.6. Assume ypotheses 1 and 2. Then for any s<t T and any ε (, 1] the following facts holds. 1. There exists ᾱ> such that for any x D(( A 1 ) α ), with α [, ᾱ], and y E( u ε (t) 2 α F s) c T,α ( 1 + uε (s) 2 α + v ε(s) 2 ), P-a.s. for some constant c T,α independent of ε. 2. For any x,y (4.2) E( v ε (t) 2 F s) c T ( 1 + uε (s) 2 + v ε(s) 2 ), P-a.s. for some constant c T independent of ε. 3. For any α>, there exists β(α) > such that for any x D(( A 1 ) α ) and y E ( u ε (t) u ε (s) 2 F s) ct,α (t s) 2β(α)( u ε (s) 2 α + v ε(s) 2 +1), P-a.s. for some constant c T,α independent of ε. 5. The key lemma. We introduce the Kolmogorov operator associated with the slow motion equation, with frozen fast component, by setting for any ϕ R( ) and x,y L sl ϕ(x,y) = 1 2 Tr[Q 1G 1 (x, y)d 2 ϕ(x)g 1 (x, y)q 1 ] + A 1 Dϕ(x), x + Dϕ(x), B 1 (x, y) (5.1) = 1 k Dij 2 2 f( x,p Na 1,..., x,p N a k ) i,j=1 G 1 (x, y)q 1,N a i,g 1 (x, y)q 1,N a j k + D i f( x,p N a 1,..., x,p N a k ) i=1 ( ) x,a 1,N a i + B 1 (x, y), P N a i. LEMMA 5.1. Assume ypotheses 1 4 and fix x D(( A 1 ) α ), with α>, and y. Then for any ϕ R( ) and t 1 <t 2 T, 2 lim E ( (5.2) ) L sl ϕ(u ε (r), v ε (r)) L av ϕ(u ε (r)) F t1 dr =. ε E t 1 PROOF. By using the Khasminskii idea introduced in [16], we realize a partition of [,T] into intervals of size δ ε >, to be chosen later on, and for each ε>

34 932 S. CERRAI we denote by ˆv ε (t) the solution of the problem (5.3) ˆv ε (t) = e (t kδ ε)a 2 /ε v ε (kδ ε ) + 1 ε + 1 ε kδ ε e (t s)a 2/ε B 2 (u ε (kδ ε ), ˆv ε (s)) ds kδ ε e (t s)a 2/ε G 2 (u ε (kδ ε ), ˆv ε (s)) dw Q 2 (s), for k =,...,[T/δ ε ]. In what follows, we shall set ζ ε := δ ε /ε. Step 1. Now, we prove that there exist κ 1,κ 2 > such that if we set ( ζ ε = log 1 ) κ1, ε κ 2 then (5.4) lim sup ε t [,T ] E ˆv ε (t) v ε (t) 2 =. If we fix k =,...,[T/δ ε ] and take t [kδ ε,(k+ 1)δ ε ),wehave so that v ε (t) = e (t kδ ε)a 2 /ε v ε (kδ ε ) + 1 ε E ˆv ε (t) v ε (t) ε t [ kδ ε,(k+ 1)δ ε ) kδ ε e (t s)a 2/ε B 2 (u ε (s), v ε (s)) ds kδ ε e (t s)a 2/ε G 2 (u ε (s), v ε (s)) dw Q 2 (s), 2δ ε ε 2 E B 2 (u ε (kδ ε ), ˆv ε (s)) B 2 (u ε (s), v ε (s)) 2 ds kδ ε + 2 t ε E e (t s)a2/ε [G 2 (u ε (kδ ε ), ˆv ε (s)) G 2 (u ε (s), v ε (s))] dw Q 2 2 (s). kδ ε For the first term, we have δ t ε ε 2 E B 2 (u ε (kδ ε ), ˆv ε (s)) B 2 (u ε (s), v ε (s)) 2 ds kδ ε (5.5) c ( ζ ε E uε (kδ ε ) u ε (s) 2 ε + E ˆv ε(s) v ε (s) 2 ) ds. kδ ε For the second term, by proceeding as in the proof of Proposition 4.2, we obtain E e (t s)a2/ε [G 2 (u ε (kδ ε ), ˆv ε (s)) G 2 (u ε (s), v ε (s))] dw Q 2 2 (s) kδ ε ( ) t s β2 (ρ 2 2)/ρ 2e λ(ρ c 2 +2)/(ερ 2 )(t s) (5.6) kδ ε ε ( E u ε (kδ ε ) u ε (s) 2 + E ˆv ε(s) v ε (s) 2 ) ds.

35 AVERAGING PRINCIPLE FOR REACTION DIFFUSION EQUATIONS 933 In view of (4.16), we have 1 t [( ) t s β2 (ρ 2 2)/ρ 2e ] λ(ρ 2 +2)/(ερ 2 )(t s) + ζ ε E u ε (kδ ε ) u ε (s) 2 ε ε ds kδ ε c T ε kδ ε [( ) t s β2 (ρ 2 2)/ρ 2e ] λ(ρ 2 +2)/(ερ 2 )(t s) + ζ ε ε (s kδ ε ) 2β(α) ds(1 + x 2 α + y 2 ) c T δ 2β(α) ε (1 + ζ 2 ε )(1 + x 2 α + y 2 ). Moreover, 1 t [( ) t s β2 (ρ 2 2)/ρ 2e ] λ(ρ 2 +2)/(ερ 2 )(t s) + ζ ε E ˆv ε (s) v ε (s) 2 ε kδ ε ε ds c ( ε β 2 (ρ 2 2)/ρ 2 + ζ ε δ β 2(ρ 2 2)/ρ ) 2 ε ε (t s) β 2(ρ 2 2)/ρ 2 E ˆv ε (s) v ε (s) 2 ds. kδ ε Then thanks to (5.5) and(5.6), we obtain (5.7) E ˆv ε (t) v ε (t) 2 c T δε 2β(α) (1 + ζε 2 )(1 + x 2 α + y 2 ) + cε β 2(ρ 2 2)/ρ 2 1 ( 1 + ζ 1+β 2(ρ 2 2)/ρ ) 2 ε kδ ε (t s) β 2(ρ 2 2)/ρ 2 E ˆv ε (s) v ε (s) 2 ds. Now, we recall the following simple fact (for a proof, see, e.g., [11]). LEMMA 5.2. such that then If M,L,θ are positive constants and g is a nonnegative function g(t) M + L (t s) θ 1 g(s)ds, t t t, g(t) M + ML θ (t t ) θ + L 2 1 r θ 1 (1 r) θ 1 dr (t s) 2θ 1 g(s)ds, t Notice that if we iterate the lemma above n-times, we find g(t) c 1,n,θ M ( 1 + L 2n 1 (t t ) 2n 1 ) + c 2,n,θ L 2n for some positive constants c 1,n,θ and c 2,n,θ. t (t s) 2n θ 1 g(s)ds t t.