Non Random Invariant Sets For Some Systems of Parabolic Stochastic

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Abstract. In this article we investigate a class of non-autonomous, semilinear, parabolic sys- tems of stochastic partial differential equations defined on a smooth, ...
Non Random Invariant Sets For Some Systems of Parabolic Stochastic Partial Differential Equations I. D. Chueshov∗

and

P.-A. Vuillermot†

Abstract In this article we investigate a class of non-autonomous, semilinear, parabolic systems of stochastic partial differential equations defined on a smooth, bounded domain O ⊂ Rn and driven by an infinite-dimensional noise defined from an L2 (O)-valued Wiener process; in the general case the noise can be colored relative to the space variable and white relative to the time variable. We first prove the existence and the uniqueness of a solution under very general hypotheses, and then establish the existence of invariant sets along with the validity of comparison principles under more restrictive conditions; the main ingredients in the proofs of these results consist of a new proposition concerning Wong-Zaka¨ı approximations and of the adaptation of the theory of invariant sets developed for deterministic systems. We also illustrate our results by means of several examples such as certain stochastic systems of Lotka-Volterra and Landau-Ginzburg equations that fall naturally within the scope of our theory.

1

Introduction

There are many works concerning the existence of invariant sets for systems of deterministic partial differential equations (see, for instance, [8], [19], [34], [50] and their references), and it is a well established fact that such invariant sets along with the validity of comparison principles play an important role in the analysis of the qualitative properties of solutions (see, for instance, [2], [29], [35], [41], [45], [49] and their references). There are comparatively fewer related results regarding stochastic partial differential equations; in fact, whereas there exist several papers devoted to the proof of the so-called stochastic invariance for certain stochastic evolution equations (see, for instance, [7], [38] [40], [51], [59] and their references), the analysis of comparison principles has been essentially limited to single parabolic stochastic partial differential equations (see, for instance, [10], [11], [22], [23], [28], [30], [39] and their numerous references), although there exists a recent theory of monotone random dynamical systems which applies to stochastic ordinary differential equations (see [20] and the references therein). An important exception is the paper [6], in which the author proves a comparison theorem that pertains to some systems of parabolic stochastic partial differential equations. We also mention the paper [18] devoted to applications of the methods of monotone random systems to a class of parabolic systems of stochastic partial differential equations (or SPDE’s for short). In this article we prove the existence of non-random invariant sets and establish comparison results for a class of parabolic systems of m stochastic partial differential equations in ∗ Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody sq., Kharkov, 61077, Ukraine, [email protected] † D´ epartement de Math´ ematiques, Universit´ e Henri-Poincar´ e, Nancy I, B.P. 239, F-54506 Vandœuvre-l` esNancy cedex, France, [email protected]

1

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CHUESHOV AND VUILLERMOT

m unknowns, where m ∈ N+ is arbitrary. As we shall see, the complication of our analysis will stem from the fact that the systems we consider are defined on bounded domains of Rn where n ∈ N+ is also arbitrary, and involve very general second-order, non-random, anisotropic, linear elliptic operators in their principal parts as well as drift-nonlinearities which may depend on the gradient of the unknown functions. More precisely, we consider the following system of semilinear Itˆ o parabolic SPDE’s $ %   dul (x, t) = −Al (x, t, D)ul (x, t) + f l (x, t, u(x, t), Du(x, t) dt (1.1) &∞  l + q · g (x, t, u(x, t)) dW (t), l = 1, . . . m, x ∈ O, t > 0, j j j j=1

in a bounded smooth domain O ⊂ Rn with the boundary and initial conditions B l (x, D)ul (x, t) = 0, x ∈ ∂O, t > 0,

ul (0, x) = ul0 (x), x ∈ O,

l = 1, . . . m.

(1.2)

$ % Here u = (u1 , . . . , um ) : O × R+ %→ Rm is a vector function, Al (x, t, D), B l (x, D) is an elliptic second order regular pair of differential operators for each l = 1, . . . m and {Wj (t)} is a family of independent standard scalar Wiener processes on the canonical Wiener space (Ω, F , P), dWj (t) is the corresponding Itˆo differential. The nonnegative parameters qj are normalization factors. Hypotheses concerning the nonlinear functions will be listed below. Our primary motivation for investigating the existence of invariant sets and the validity of comparison principles for problems of the above form is that such systems are quite relevant to the mathematical analysis of a variety of physical processes; in fact, particular cases of them as well as their deterministic counterparts have been used over the years to model, for instance, certain migration phenomena in population dynamics and population genetics via Lotka-Volterra type nonlinearities, as well as certain nucleation phenomena in the theory of superconductivity through the Ginzburg-Landau equations (see, for instance, [1], [6], [11], [13], [17], [21], [22], [23], [29], [35], [55], [56] and their references). The key ingredient of the proof of our main results is a Wong–Zaka¨ı type approximation theorem for the stochastic problem (1.1) and (1.2). This theorem claims the possibility to construct smooth (with respect to both spatial and time variables) convergent approximations for the stochastic problem (1.1) and (1.2) and therefore allows us to invoke results from the theory of deterministic parabolic equations for the investigations of properties of the solutions to (1.1) and (1.2). In the finite-dimensional case this kind of approximation theorems is well-known (see, e.g., [37], [43], [57], [58] and also the survey [54] and the references therein). There is a substantial number of publications devoted to Wong–Zaka¨ı type approximations of infinite-dimensional stochastic equations (see, e.g., [9], [14], [32], [33], [51], [53], [54] and the literature cited there). However the assumptions in these publications are rather restrictive and they do not cover the case of stochastic systems like (1.1) and (1.2). Our hypotheses concerning the coefficients and the nonlinear terms are essentially more general. However we assume the existence of uniformly bounded (in some sense) approximate solutions. The main situation in which we can prove the existence of these solutions is the case when the corresponding deterministic problem possesses a bounded invariant set in Rm . We use this approximation theorem to prove our main theorem on the existence of deterministic sets in Rm which are invariant with respect to the stochastic system (1.1) and (1.2). A similar problem was discussed earlier by P.Kotelenez [38] for a single parabolic equation in the case when f does not depend on Du. Another consequence of the approximation theorem is a comparison principle. Under an additional requirement of cooperativity of the nonlinear terms we prove that the solutions to the stochastic problem (1.1) and (1.2) depend on the initial data in a monotone way. For single stochastic parabolic equations this principle was discussed by many authors (see, e.g., [28], [30], [38], [39] and the references

NON-RANDOM INVARIANT SETS

3

therein). In contrast with these known results our comparison theorem deals with systems of equations and allows for derivatives in the drift term f . The paper is organized as follows. In Section 2 we introduce notations, definitions and basic hypotheses. There we also establish several preliminary propositions concerning elliptic operators and nonlinear terms. In Section 3 we prove the existence and uniqueness theorem for stochastic problem (1.1) and (1.2). Section 4 is central to the paper. It contains our main technical tool. This is a Wong–Zaka¨ı type approximation theorem for the stochastic problem (1.1) and (1.2). We first study properties of smooth approximations of the problem (1.1) and (1.2). Then we introduce the concept of ε-residual approximate solutions to the problem and prove their existence. We invoke there some ideas developed in [9] in the abstract setting of bounded operators. Using these considerations we conclude the proof of the approximation theorem. Section 5 contains our main results. We give conditions on f l (x, t, u, η) and gjl (x, t, u) under which problem (1.1) and (1.2) possesses solutions u(x, t, ω) that belong to some deterministic set D ⊂ Rm for all (x, t) ∈ O × R+ and for almost all ω ∈ Ω. (see Theorem 5.6). We rely crucially on the Wong–Zaka¨ı type approximation theorem which is proved in Section 4. The result on the existence of invariant sets for (1.1) and (1.2) extends to stochastic systems the theorem proved by H.Amann [2] in the deterministic case. Moreover the results from [2] lay in the background of our considerations. We invoke these results to obtain the corresponding result for the smooth regularizations of system (1.1) and (1.2). We also prove a comparison principle (see Theorem 5.8) for the problem considered in the case of cooperative f and gj . In Section 6 we give several examples concerning certain stochastic Lotka-Volterra or Ginzburg-Landau equations that fall within the scope of our theory.

2 2.1

Main hypotheses and preliminaries Function spaces and linear evolution families

Let O be a bounded domain in Rn whose boundary ∂O is an (n − 1)-dimensional C 2+µ manifold for some µ ∈ (0, 1), such that O lies locally on one side of the boundary ∂O. For any integer k ≥ 0 and for 1 ≤ p ≤ ∞ we denote by Wpk (O) the Sobolev space Wpk (O) = {u ∈ Lp (O) : Dα u ∈ Lp (O) for all |α| ≤ k} , where D = Dx = (D1 , . . . , Dn ) is the gradient operator and α = (α1 , . . . , αn ) is a multiindex of nonnegative integers, |α| = α1 + . . . + αn . We will denote by * · *k,p the norm in Wpk (O). We also define the Sobolev space Wps (O) for positive real supersripts s +∈ N by     ' Wps (O) = u ∈ Wpk (O) : *u*ps,p ≡ *u*pk,p + Iσ,p (Dα u) < ∞ ,   |α|=k

where s = k + σ with k ∈ N and 0 < σ < 1 and + |u(x) − u(y)|p Iσ,p (u) = dxdy. n+σp O×O |x − y|

Let C k (O) be the space of k times continuously differentiable functions on O. For 0 < µ < 1 we denote by C k+µ (O) the subset of C k (O) which consists of functions whose derivatives of order k are locally µ-H¨ older continuous. Below we will use the continuity of the following embeddings: n Wps (O) ⊂ C σ (O) if s − > σ, 1 < p < ∞, s, σ ≥ 0 (2.1) p

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CHUESHOV AND VUILLERMOT

(if σ is not an integer, then the embedding holds also for σ = s − n/p) and C k+σ (O) ⊂ Wpk+β (O) if k ∈ N, 0 ≤ β < σ ≤ 1, 1 < p < ∞.

(2.2)

The embeddings (2.1) and (2.2) remain true for multi-component spaces Wps (O; Rm ) and C σ (O; Rm ) which we define by the formulas Wps (O; Rm ) = Wps (O) × . . . × Wps (O)

and C σ (O; Rm ) = C σ (O) × . . . × C σ (O).

We refer to [12] and [52] for the proofs and for other facts from the theory of Sobolev spaces (see also a short survey in [27]). For every l = 1, . . . m we define a linear differential operator Al (x, t, D) by the formula Al (x, t, D) = −

n '

alki (x, t)Dk Di +

k,i=1

n '

alk (x, t)Dk + al0 (x, t),

l = 1, . . . , m.

(2.3)

k=1 µ

(A) We suppose that alki (x, t), alk (x, t), al0 (x, t) ∈ C µ, 2 (O × [0, T ]) with 0 < µ < 1 for any T > 0, alki = alik and al0 (x, t) ≥ 0, where l = 1, . . . m and k, i = 1, . . . n. Here and below µ we denote by C µ, 2 (O × [0, T ]) the space of functions which are uniformly µ-H¨older continuous with respect to x and µ2 -H¨older continuous with respect to t. Moreover we suppose that each Al (x, t, D) is a uniformly elliptic operator, i.e. there exists ν0 > 0 such that n '

alki (x, t)ξk ξi ≥ ν0 |ξ|2 ,

(x, t) ∈ O × R+ , ξ ∈ Rn , l = 1, . . . , m.

k,i=1

We also consider the boundary operators B l (x, D) of the form B l (x, D) = bl0 (x) + δ l

n '

blk (x)Dk ,

l = 1, . . . , m,

(2.4)

k=1

and we assume that (B) δ l ∈ {0, 1}, blk (x), bl0 (x) ∈ C 1+µ (∂O) with bl0 (x) ≥ 0 such that bl0 (x) ≡ 1 when δ l = 0 and bl = (bl1 , . . . , blm ) is an outward pointing, nowhere tangent vector field on ∂O. Now for any 1 < p < ∞ we consider the Lp (O, Rm )-realization of the elliptic collection {A (x, t, D), B l (x, D); l = 1, . . . , m}. This collection generates the family of closed operators {A(t), t ∈ R+ } in Lp (O, Rm ) defined on the domain , 2 D(A(t)) = Wp,B ≡ u = (u1 , . . . , um ) ∈ Wp2 (O, Rm ) : Bu = 0 on ∂O , l

where Bu = (B 1 (x, D)u1 (x), . . . , B m (x, D)um (x)), by the formula

A(t)u = (Al (x, t, D)u1 (x), . . . , Am (x, t, D)um (x)). For each 1 < p < ∞ and t ≥ 0 the operators A(t) possess (see, e.g., [4] or [42]) the properties (i) there exists λ0 ≥ 0 such that {Reλ ≥ λ0 } belongs to the resolvent set of −A(t); (ii) for any T > 0 there exists a constant K > 0 such that −1

* [(A(t) − A(s)] (A(τ ) + λ0 )

µ

* ≤ K|t − s| 2 for all t, s, τ ∈ [0, T ]

where µ is the H¨ older exponent of the coefficients of the operators A(x, t, D).

(2.5)

NON-RANDOM INVARIANT SETS

5

The operators A(t) are also generators of analytic semigroups of compact operators in Lp (O, Rm ) for each t ∈ [0, T ] and for any 1 < p < ∞. Therefore for every real α and 1 < p < ∞ we can define the fractional powers (A(t) + λ0 )α of the operators A(t) + λ0 [42]. For 0 ≤ α ≤ 1 the domain Xα,p ≡ D((A(t) + λ0 )α ) is independent of t and possesses (see, e.g., [3] or [27] and the references therein) the property β γ Wp,B (O, Rm ) ⊂ Xα,p ⊂ Wp,B (O, Rm ) with 0 < γ < 2α < β < 2 (2.6) . / s provided γ, β +∈ 1p , 1, 1 + p1 . Here for s += 1 and 0 < s < 2 we suppose Wp,B (O, Rm ) =

s s s s −1 s Wp,B + δ l and with Wp,B 1 (O) × . . . × Wp,B m (O) with Wp,B l (O) = Wp (O) for s < p l (O) = s l −1 l l {u ∈ Wp (O) : B u = 0} for s > p + δ , where δ = 0 if the boundary operator B l is of Dirichlet type and δ l = 1 otherwise. We note that for 0 ≤ α ≤ 1 the domains

Xα,p = D((A(t) + λ0 )α )

(2.7)

are Banach spaces with the norm * · *Xα,p = *((A(0) + λ0 )α · *Lp . These norms possess the property (see [4] or [42]): *u*Xβ,p ≤ c(α, β, γ, p) · *u*λXα,p · *u*1−λ Xγ,p with λ =

γ−β , γ−α

0 ≤ α ≤ β ≤ γ ≤ 1,

(2.8)

where u ∈ Xγ,p and c(α, β, γ, p) denotes some positive constant, 1 < p < ∞. Below we also assume that the L2 -realization of the operators A(t) possesses the following property. (A∗ ) there exist constants γ > 0 and c ∈ R independent of t ∈ [0, T ] such that (A(t)u, u)X0,2 + (u, A(t)u)X0,2 ≥ γ · *u*2W 1 (O) − c · *u*2X0,2 2

(2.9)

2 for all u ∈ W2,B (O, Rm ).

We note that inequality (2.9) amounts to assuming G˚ arding-type inequalities for the uniformly elliptic operators defined by (2.3); such inequality is central to the proofs of most of the theorems of this article and are not automatically satisfied in our general setting as we assume neither more structure regarding the coefficients in (2.4), nor a priori relations between these coefficients and those in (2.3) (see, e.g., the discussion in [3]). Inequality (2.9) is satisfied, however, in many important situations that involve Dirichlet or conormal boundary conditions. For instance, the following proposition provides us with sufficient conditions on the coefficients of the boundary operators B l which guarantee (A∗ ). Proposition 2.1 Let (A) and (B) be valid. Assume that the coefficients blk (x) of the boundary operators B l in the case δ l = 1 have the structure blk (x) =

n '

l βik (x)νi (x),

x ∈ ∂O,

(2.10)

i=1

where ν(x) = (ν1 (x), . . . , νn (x)) is the outer normal to ∂O at the point x and the functions l βik (x) ∈ C 1 (O) have the properties l l βik (x) = βki (x),

n '

i,k=1

l βik (x)ξi ξk ≥ δ ·

n '

ξi2 ,

x ∈ O, ξ ∈ Rn ,

i=1

with some positive δ. Then the operators A(t) possesses property (2.9).

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CHUESHOV AND VUILLERMOT

& l Proof. Let A˜l (x, D) = − nk,i=1 Di βki (x)Dk , where l = 1, . . . , m. It is clear that l l l ˜ {A (x, t, D)−.· A (x, D), B (x, D)} is an elliptic pair for some . > 0 small enough. Therefore the corresponding operators A& (t) = A(t)−.· A˜ generate an analytic semigroup in L2 (O, Rm ) for every t ∈ [0, T ]. Therefore (see, e.g., [42]) these operators possess the property (A& (t)u, u)X0,2 + (u, A& (t)u)X0,2 ≥ −c · *u*2X0,2 ,

2 u ∈ W2,B (O, Rm ).

The structure of the symbol A˜l (x, D) and either the Dirichlet boundary conditions in the case δ l = 0 or property (2.10) when δ l = 1 allow us to prove (2.9) after integration by parts. This completes the proof of Proposition 2.1. Denote by {U (t, s) : 0 ≤ s ≤ t < ∞} the linear evolution family generated in X0,p = Lp (O, Rm ) by the collection of operators {A(t) : t ≥ 0}. We refer to [4] and to the references therein for a general theory of evolution families in Banach spaces. Proposition 2.2 Under conditions (A) and (B) the family {U (t, s)} possesses the properties (U1) U (t, t) = I and U (t, s) = U (t, τ )U (τ, s) for 0 ≤ s ≤ τ ≤ t < ∞; (U2) for any 0 ≤ s ≤ t < ∞ we have ∂t U (t, s)h = −A(t)U (t, s)h,

∂s U (t, s)h = U (t, s)A(s)h,

h ∈ X1,p ;

(U3) for any interval [0, T ] and for any 0 ≤ α ≤ β < 1+µ/2, where µ is the H¨ older exponent of the coefficients of the operators A(x, t, D) (see also (2.5) above), there exists C0 > 0 such that *[A(t) + λ0 ]β U (t, s)[A(s) + λ0 ]−α * ≤ C0 (t − s)α−β for any 0 ≤ s ≤ t ≤ T, where and also in (U4) and (U5) * · * stands for the norm in the space of bounded operators in X0,p = Lp (O, Rm ); (U4) for any interval [0, T ] and for any 0 ≤ α ≤ 1, 0 ≤ β ≤ γ < 1 + µ/2 with µ as above and with 0 ≤ γ − α ≤ 1 and arbitrary θ < β, there is C1 > 0 such that *[A(ξ) + λ0 ]α [U (t + τ, s) − U (t, s)] [A(s) + λ0 ]−β * ≤ C1 τ γ−α (t − s)θ−γ for all 0 ≤ s < t ≤ T , ξ ∈ [0, T ] and 0 < τ ≤ 1; (U5) for any interval [0, T ] and for any 0 ≤ α < β < 1 and θ > 0 small enough, there is C2 > 0 such that for all t, τ ∈ [s, T ] and ξ ∈ [0, T ] we have *[A(ξ) + λ0 ]α [U (t, s) − U (τ, s)] [A(s) + λ0 ]−β * ≤ C1 (t − τ )β−α−θ . Proof. The properties (U3) and (U4) easily follow from Theorem 1 and Theorem 2 proved in [44]. The proofs of the other properties can be found in [4] or [42]. Below we also invoke the following property of the evolution family U (t, τ ). Proposition 2.3 Let (A), (B) and (A∗ ) be valid. Then there exists a constant CT independent of t, s ∈ [0, T ] such that + t *U (τ, s)h*2W 1 (O,Rm ) dτ ≤ CT *h*2X0,2 for all 0 ≤ s ≤ t ≤ T, h ∈ X0,2 . (2.11) s

2

NON-RANDOM INVARIANT SETS

7

Proof. Using (U2) for every h ∈ X1,2 we have d *U (τ, s)h*2X0,2 = (A(τ )U (τ, s)h, U (τ, s)h)X0,2 + (U (τ, s)h, A(τ )U (τ, s)h)X0,2 , dt where τ ≥ s. Therefore (2.9) implies d *U (τ, s)h*2X0,2 ≥ γ*U (τ, s)h*2W 1 (O,Rm ) − c*U (τ, s)h*2X0,2 . 2 dt Hence γ

+

s

t

*U (τ, s)h*2W 1 (O,Rm ) 2

dτ ≤

*U (t, s)h*2X0,2

+c

+

s

t

*U (τ, s)h*2X0,2 dτ.

2 Since X1,2 = W2,B (O, Rm ) is dense in X0,2 = L2 (O, Rm ), using (U3) with α = β = 0 we obtain (2.11).

2.2

Nonlinear terms

Now we turn to the nonlinear situation. Our primary hypotheses are as follows. (F0) We assume that f ≡ (f 1 , . . . , f m ) : O × [0, T ] × D × Rnm %→ Rm is a measurable function possessing the properties: |f (x, t, u, η)| ≤ c · (1 + |u| + |η|) and |f (x, t, u, η) − f (x, t, v, ξ)| ≤ c · (|u − v| + |η − ξ|)

(2.12)

with some constant c > 0 for all (x, t) ∈ O × [0, T ] and (u, η), (v, ξ) ∈ Rm × Rnm . (G0) We assume that gj ≡ (gj1 , . . . , gjm ) : O × [0, T ] × Rm %→ Rm is a measurable function possessing the properties: |gj (x, t, u)| ≤ c(1 + |u|) for all (x, t, u) ∈ O × [0, T ] × Rm and |gj (x, t, u) − gj (x, t, v)| ≤ c|u − v| for all x ∈ O, u, v ∈ Rm and t ∈ [0, T ], with some constant c > 0 independent of j. (G1) The function gj ≡ (gj1 , . . . , gjm ) : O × [0, T ] × Rm %→ Rm is bounded, possesses the first order derivatives with respect to x, t and u = (u1 , . . . , um ) and the functions gj , ∂gj /∂xk , ∂gj /∂t and ∂gj /∂ul are uniformly bounded and Lipschitz with respect to x and u, i.e. there exists a constant M independent of j such that |h(x, t, u)| ≤ M for all (x, t, u) ∈ O × [0, T ] × Rm

(2.13)

|h(x, t, u) − h(y, τ, v)| ≤ M (|t − τ | + |x − y| + |u − v|)

(2.14)

and for all (x, u), (y, v) ∈ O × Rm and t, τ ∈ [0, T ], where h is either gj , or ∂gj /∂t, or ∂gj /∂xk , or ∂gj /∂ul , k = 1, . . . , n, l = 1, . . . , m. In the Dirichlet case we also assume that the functions g(x, t, u) are compatible with the boundary conditions, i.e. if B l (x) is the Dirichlet type boundary operator (bl0 (x) ≡ 1, δ l = 0) then g l (x, t, u) = 0 for x ∈ ∂O, ul = 0.

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CHUESHOV AND VUILLERMOT

Below we will denote by f (t, ·) and gj (t, ·) the nonlinear (Nemytskii) operators u(x) %→ f (x, t, u(x), Du(x)) and u(x) %→ gj (x, t, u(x)). The following assertion contains some useful properties of these operators. Proposition 2.4 Let (A) and (B) be valid and Xα,p be defined by (2.7). Assume that f and gj satisfy (F0), (G0) and (G1). Then (i) for any p ≥ 1 and α > 1/2 there exists a constant C = C(p, α) such that $ % *f (t, u)*X0,p ≤ C 1 + *u*Xα,p f or any u ∈ Xα,p , t ∈ [0, T ],

(2.15)

and

*f (t, u) − f (t, v)*X0,p ≤ C*u − v*Xα,p f or any u, v ∈ Xα,p , t ∈ [0, T ];

(2.16)

(ii) for any α < 1/2 we have that gj (t, u) ∈ Xα,p provided u ∈ Xβ,p with β > α and there exists a constant C = C(α, β) independent of j such that $ % *gj (t, u)*Xα,p ≤ C 1 + *u*Xβ,p f or any u ∈ Xβ,p , t ∈ [0, T ], (2.17) and

*gj (t1 , u1 ) − gj (t2 , u2 )*Xα,p ≤ C*u1 − u2 *Xβ,p $ % $ % +C |t1 − t2 | + *u1 − u2 *C(O) · 1 + *u1 *Xβ,p + *u2 *Xβ,p

(2.18)

for any u1 , u2 ∈ Xβ,p and t1 , t2 ∈ [0, T ]. If 1/2 ≤ α < 1/2 + 1/(2p) we also have that gj (t, u) ∈ Xα,p provided u ∈ Xβ,p with β > α. Moreover, there exists a constant C = C(α, β) independent of j such that $ % *gj (t, u)*Xα,p ≤ C 1 + *u*Xβ,p + *u*C 1 (O) · *u*Xβ−1/2,p , t ∈ [0, T ], (2.19) for any u ∈ Xβ,p with arbitrary β > α and 1/2 ≤ α < 1/2 + 1/(2p), p > 1. Proof. The first part follows directly from (F0) and (2.6). Using the Lipschitz condition for gj we have 0 1 *gj (t, u)*Wps ≤ C 1 + *u*Wps , 0 ≤ s < 1.

Therefore (2.6) and the compatibility condition for gj give that gj (t, u) ∈ Xα,p for 0 ≤ 2α < 1 provided u ∈ Xβ,p with β > α. This implies (2.17) for 2α < s < 2β < 1. We present gj (t1 , u1 ) − gj (t2 , u2 ) in the form + 1 ∂ gj (t1 , u1 ) − gj (t2 , u2 ) = gj (λt1 + (1 − λ)t2 , u1 )dλ · (t1 − t2 ) ∂t 0 + 1 2 3 + ∇u gj (t2 , λu1 + (1 − λ)u2 ), u1 − u2 dλ ≡ G1j · (t1 − t2 ) + G2j . 0

From (2.17) with ∂gj /∂t instead of gj we obtain $ % *G1j *Xα,p ≤ C 1 + *u1 *Xβ,p for any u1 ∈ Xβ,p , t ∈ [0, T ].

Further, from the definition of the norm in Wps (O, Rm ) for 0 < s < 1 we have that 0 1 *v · w*Wps ≤ C *v*C(O) · *w*Wps + *v*Wps · *w*C(O) , v, w ∈ Wps ∩ C(O).

(2.20)

NON-RANDOM INVARIANT SETS

9

Therefore using this inequality and (2.17) we obtain that 0 0 1 1 *G2j *Wps ≤ C *u1 − u2 *Wps + 1 + *u1 *Wps + *u2 *Wps · *u1 − u2 *C(O) . The estimates for G1j and G2j implies (2.18). In a similar way we  n ' m ' *gj (t, u)*Xα,p ≤ C*gj (t, u)*Wps ≤ C 1 + *u*Wp1 +

i=1 l,k=1

obtain

 6 ∂g k ∂ul 6 6 j 6  6 l · 6 ∂u ∂xi Wps−1

for 1 < 2α < 1 + 1/p and s ∈ (2α, 1 + 1/p). Therefore (2.20) implies (2.19).

3

Existence and uniqueness of mild solutions

In this section we prove the existence and uniqueness of mild solutions to problem (1.1) and (1.2). In spite of the fact that stochastic reaction–diffusion systems have been studied by many authors (see, e.g., [15], [16], [25], [26], [31], [39] and the references therein), our Theorem 3.2 is new because (i) it deals with a general time-dependent elliptic system (Al (t), B l (t)) and (ii) drift nonlinearities may depend on the gradient of the unknown functions. We consider problem (1.1) and (1.2) in a scale of Hilbert spaces connected with X0,2 = L2 (O; Rm ). Here, in addition to (A), we assume that the family of operators A(t) possesses property (A∗ ) which implies relation (2.11). We note that relations similar to (2.11) play an essential role in the theory of infinite-dimensional stochastic equations (see, e.g., [25] for the case of homogeneous evolution operators U (t, s) = U (t − s)). We denote by {Wj (t) : j = 1, 2, . . .} an infinite family of mutually independent standard scalar Wiener processes on the probability space (Ω, F , P) with the corresponding rightcontinuous increasing family {Ft : t ∈ R+ } of sub-σ-fields of F each containing P-null sets. These processes are Gaussian and {Ft }-adapted. They possess the properties Wj (0) = 0,

EWj (t) = 0,

EWj (t)Wi (τ ) = δji · min(t, τ ), t, τ ≥ 0.

Here and below E is the mean value operator on (Ω, F , P). If {ξj (t, ω) = (ξj1 (x, t, ω), . . . , ξjm (x, t, ω)) : j ∈ Z+ } are predictable processes with values in L2 (O; Rm ) with the property   + T ∞ + + T '  E *ξ(τ )*2HS dτ ≡ E |ξj (x, τ, ·)|2 dx dτ < ∞, (3.1)  0  0 O j=1

we can define the stochastic Itˆ o integral +

t

/ξ(τ ), dW (τ )0 ≡

s

∞ + ' j=1

t

ξj (x, τ, ω)dWj (τ, ω),

0 ≤ s < t ≤ T,

s

which is a predictable process with values in L2 (O; Rm ). The point is that we can consider the family {Wj (t)} as a Wiener process on the Hilbert space 02 (Z+ ) (with values in some extension of 02 (Z+ )). Then under condition (3.1) the sequence {ξj } defines a Hilbert Schmidt operator ξ(t, ω) from 02 (Z+ ) into L2 (O; Rm ) by the formula ξ(t, ω)k =

∞ ' j=1

ξj (x, t, ω)kj ,

k = (k1 , k2 , . . .) ∈ 02 (Z+ )

10

CHUESHOV AND VUILLERMOT

for almost every (t, ω) ∈ [0, T ] × O. Therefore we can apply the standard approach (see, e.g., [9] or [25]) to the definition of stochastic Itˆo integrals. This integral possesses the property 6+ t 62 6 6 6 E 6 /ξ(τ ), dW (τ )06 6 s

=

L2 (O)

∞ + '

t

s

j=1

E*ξj (τ )*2L2 (O) dτ.

(3.2)

Below we will also use the following generalization of this formula 6+ t 6r 6 6 6 E 6 /ξ(τ ), dW (τ )06 6 s

≤ cr ·

L2 (O)

   +   

t

s

r    r2  2r  2  ∞   '  2  *ξj (τ )*L2 (O) , E  dτ   j=1 

(3.3)

where r ≥ 2. For the proof see [25], for example. We also note that under condition (2.11) we can define the stochastic integral (Iξ)(t) =

+

t

/U (t, τ )ξ(τ ), dW (τ )0 ≡

0

∞ + ' j=1

t

U (t, τ )ξj (x, τ, ω)dWj (τ, ω),

0 ≤ t ≤ T,

0

as a predictable process belonging to L2 (Ω × [0, T ]; V ), where V = W21 (O, Rm ), such that E

+

T 0

*(Iξ)(t)*2V

dt ≤ C · E

+

0

T

∞ '

*ξj (τ )*2L2 (O) dτ.

(3.4)

j=1

The proof of this property follows from Proposition 2.3. We refer to [25] and also to [48] for the discussion in the case of homogeneous evolution operators U (t, s) = U (t − s) (see also [24], where a similar property was used in the case of abstract evolution families). Now we are in position to introduce a concept of mild solution to problem (1.1) and (1.2). Definition 3.1 A random function u(x, t, ω) = (u1 (x, t, ω), . . . , um (x, t, ω)) is said to be a 1 mild solution to problem (1.1) and (1.2) in the space V ≡ W2,B (O, Rm ) on the interval [0, T ], if u(t) ≡ u(x, t, ω) ∈ C(0, T ; L2(Ω × O)) is a predictable process such that +

T 0

E*u(t)*2V dt < ∞

and satisfies the integral equation u(t) = U (t, 0)u0 +

+

0

t

U (t, τ )f (τ, u(τ )) dτ +

∞ ' j=1

qj ·

+

t

U (t, τ )gj (τ, u(τ )) dWj (τ, ω), (3.5)

0

where we assume that all integrals in (3.5) exist. Our main result in this section is the following existence and uniqueness theorem for problem (1.1) and (1.2). Theorem 3.2 Assume that the properties (A), (A∗ ), (B) and (F0), (G0) are valid. Let & 2 1 m j qj < ∞ and V = W2,B (O, R ). Assume that u0 = u0 (x, ω) is F0 -measurable, belongs to Xα,2 ∩ V for every 0 < α < 1/2 and almost all ω ∈ Ω and such that E*u0 *2V < ∞. Then

NON-RANDOM INVARIANT SETS

11

problem (1.1) and (1.2) has a unique (up to equivalence) mild solution u(t) ≡ u(x, t, ω) in the space V on any interval [0, T ]. This solution possesses the property + T sup E*u(t)*2Xβ,2 + E *u(t)*2V dt ≤ C f or all β ∈ [0, 1/2). (3.6) t∈[0,T ]

0

Moreover for any β ∈ [0, 1/2) we have that . /1/2 E*u(t) − u(s)*2Xβ,2 ≤ C|t − s|1/2−β−θ ,

t, s ∈ [0, T ],

(3.7)

with arbitrary small θ > 0.

Proof. Since we deal with a nonautonomous elliptic part and with drift nonlinearities containing the gradients of the unknown functions, we cannot apply the standard results from [25] or [31]. Nevertheless the properties of the evolution operator U (t, s) allow us to apply a fixed point method. Let 0 < T ∗ ≤ T and VT ∗ be a space of progressively measurable processes v(t) ≡ v(t, ω) on [0, T ∗ ] with values in V such that @+ ∗ A1/2 T 0 1 2 2 |v|VT ∗ ≡ E *v(t)*V + L · *v(t)*X0,2 dt < ∞. 0

The positive parameters T ∗ and L will be chosen below. The idea to include an additional term with a free parameter L in the norm of the space VT ∗ is borrowed from [48]. We also refer to [24], where a similar idea was applied in the case of a parabolic SPDE with dynamical boundary conditions. In the space VT ∗ we define a mapping B by the formula Bt [Bv] (t) = U (t, 0)u0 + 0 U (t, τ )f (τ, v(τ )) dτ +



&∞

j=1 qj

·

Bt 0

U (t, τ )gj (τ, v(τ )) dWj (τ, ω)

U (t, 0)u0 + [B1 v] (t) + [B2 v] (t),

t ∈ [0, T ∗].

It is easy to see that B maps VT ∗ into itself. Let *v*2V,1 = *v*2V + L · *v*2X0,2 . Since *v*2V,1 ≤ C (1 + L) *v*2Xα,2 for any 1/2 + δ/2 < α < 1, from (U3) in Proposition 2.2 and from (2.12) we have C+ t D2 dτ 2 E* [B1 v] (t) − [B1 u] (t)*V,1 ≤ C (1 + L) E *v(τ ) − u(τ )*V . α 0 (t − τ ) A simple calculation shows that D2 + T ∗ C+ t + ∗ ϕ(τ )dτ (T ∗ )2−2α T dt ≤ |ϕ(τ )|2 dτ α (1 − α)2 0 0 0 (t − τ ) for any α ∈ [0, 1) and ϕ ∈ L2 (0, T ∗ ). Therefore we obtain 1/2

|B1 v − B1 u|VT ∗ ≤ C (1 + L)

T ∗ 1−α |v − u|VT ∗ ,

v, u ∈ VT ∗ .

(3.8)

Further, using (3.2) and (3.4) we obtain + T∗ 6 ' + t 62 6 6 |B2 v − B2 u|2VT ∗ = E qj U (t, τ ) [gj (τ, v(τ )) − gj (τ, u(τ ))] dWj (τ )6 dt 6 0

j

0

V

12

CHUESHOV AND VUILLERMOT

+L · E

+

T∗

0

6' + t 62 6 6 qj U (t, τ ) [gj (τ, v(τ )) − gj (τ, u(τ ))] dWj (τ )6 6

≤C

X0,2

0

j

'

qj2

·E

j

+C · L ·

'

qj2 · E

j

+

+

T∗

0

dt

*gj (τ, v(τ )) − gj (τ, u(τ ))*2X0,2 dτ

T∗

dt 0

+

0

t

*gj (τ, v(τ )) − gj (τ, u(τ ))*2X0,2 dτ.

Therefore from (G0) we have $ % |B2 v − B2 u|2VT ∗ ≤ C L−1 + T ∗ |v − u|VT ∗ ,

v, u ∈ VT ∗ .

It follows from this formula and from (3.8) that we can choose the constants T ∗ and L in a such way that the mapping B becomes a contraction in VT ∗ . This allow us to obtain a unique solution to (3.5) on the interval [0, T ∗ ]. These parameters T ∗ and L do not depend on the initial data and therefore we can extend the solution to any interval [0, T ] by the standard method. Now we prove (3.6). From (3.5), property (3.2) and from (U3) with α = β = 0 we have E*u(t)*2X0,2 ≤ CE*u0 *2X0,2 +C ·

+

0

t

∞ ' $ % 1 + E*u(τ ))*2V dτ + C qj2 j=1

+ t0 0

1 + E*u(τ )*2X0,2

1



Therefore since u ∈ VT , we have that supt∈[0,T ] E*u(t)*2X0,2 ≤ C. Using this relation we can now find in a similar way that   + t + t ∞ ' $ % dτ  E*u(t)*2Xβ,2 ≤ C E*u0 *2Xβ,2 + 1 + E*u(τ ))*2V dτ + qj2 2β (t − τ ) 0 0 j=1

for β ∈ [0, 1/2). This implies (3.6). Further, from (3.5) for t ≥ s we have + t u(t) − u(s) = (U (t, s) − I) u(s) + U (t, τ )f (τ, u(τ )) dτ s

+

∞ ' j=1

+

t

qj U (t, τ )gj (τ, u(τ )) · dWj (τ, ω).

(3.9)

s

It follows from (U5) that *(U (t, s) − I)u(s)*Xβ,2 ≤ C|t − s|α−β−θ *u(s)*Xα,2 ,

0≤β 0. Therefore, as above using (U3) and also (3.2) we obtain

+C2

@+

s

t

.

E*u(t) − u(s)*2Xβ,2

/1/2

. /1/2 ≤ C1 |t − s|α−β−θ E*u(s)*2Xα,2

dτ (1 + E*u(τ )*2V )1/2 dτ + (t − τ )β

C+

s

t

dτ (1 + E*u(τ )*2X0,2 )dτ (t − τ )2β

D1/2 A

where 0 ≤ β ≤ α < 1/2. Therefore using (3.6) we obtain (3.7). The proof of Theorem 3.2 is complete.

NON-RANDOM INVARIANT SETS

13

Remark 3.3 Assume that the functions gj (x, t, u) are of the form gj (x, t, u) = ej (x) · g(x, t, u), where g(x, t, u) satisfies (G0) and {ej (x)} is a family of continuous functions on O such that |ej (x)| ≤ C with some constant independent of j. In this case we can rewrite equations (1.1) in the form  $ %  dul (x, t) = −Al (x, t, D)u(x, t) + f l (x, t, u(x, t), Du(x, t) dt (3.11)  + g l (x, t, u(x, t)) dW (x, t), l = 1, . . . m, x ∈ O, t > 0, & where W (x, t) = ∞ is a Wiener process in L2 (O) with the covariance operj=1 qj Wj (t)ej (x) &∞ ator Q given by the formula Qh = j=1 qj2 (h, ej )ej for h ∈ L2 (O). Theorem 3.2 implies the existence and the uniqueness of a mild solution u(t) for problem (3.11) provided hypotheses (A), (A∗ ), (B) and (F0) hold. We also note that using the method presented in [46] and [47] one can prove that the solution u(t) also satisfies a variational form of equation (3.11).

4

Approximation theorem

We first introduce the concept of a smooth predictable approximation of a Wiener process. Definition 4.1 Let {W (t), t ≥ 0} be a standard scalar Wiener process on the probability space (Ω, F , P) with filtration {Ft : t ∈ R+ }. A family {W ε (t), t ≥ 0}ε>0 of random processes is said to be a smooth predictable approximation of the process {W (t), t ≥ 0} if (i) for each ε > 0 the function W ε (t, ω) is measurable with respect to (t, ω) and continuously differentiable with respect to t for almost all ω; (ii) W ε (t) is Ft -measurable for every ε > 0 and t > 0, W ε (0) = 0 almost surely and EW ε (t) = 0 for t ≥ 0 and E|W ε (t)|2 − E|W ε (s)|2 = t − s, if t ≥ s ≥ ε; ˙ ε (t) ≡ (iii) the derivative W

d ε dt W (t)

(4.1)

is F[t−ε,t] -measurable for each t ≥ ε, ε > 0 and

˙ ε (t)|q ≤ Cq ε−q/2 for all q ≥ 1; E|W

(4.2)

here and below F[a,b] denotes the σ-algebra generated by the family of random variables {W (t) − W (s) : t, s ∈ [a, b]}; (iv) for any Ft−2ε -measurable random process a(t) possessing the property ess sup E|a(t)|2 < ∞

for any

T >0

[0,T ]

we have that C+ t D2 + t ˙ ε (τ ) dτ − E a(τ )W a(τ )dW (τ ) s

c1 ≤ · ε

+

s

t−ε



+

s ξ+ε

E|a(τ ) − a(ξ)|2 dτ + c2 · ε max E|a(τ )|2 ξ

τ ∈[s,t]

(4.3)

for all t ≥ s ≥ 2ε, where c1 and c2 are positive constants. We note that a slightly different concept of smooth approximations for Wiener processes was introduced in [37]. However approximations considered in [37] are not predictable. Our main example of a smooth predictable approximation is presented in the following assertion.

14

CHUESHOV AND VUILLERMOT

Proposition 4.2 Let W ε (t) =

+



ϕε (t − τ )W (τ ) dτ ≡ 0

+

t

ϕε (t − τ )W (τ ) dτ,

(4.4)

max(0,t−ε)

where ϕε (t) = ε−1 ϕ(t/ε) and ϕ(t) is a function with the properties ϕ(t) ∈ C 1 (R),

supp ϕ(t) ⊂ [0, 1],

+

1

ϕ(t) dt = 1.

(4.5)

0

Then the family {W ε (t)}ε>0 defined by (4.4) is a smooth predictable approximation of the Wiener process {W (t), t ≥ 0}. Proof. It is clear that W ε (t) given by (4.4) is Gaussian with continuously differentiable trajectories such that W ε (0) = 0, EW ε (t) = 0 and + ∞ + ∞ E|W ε (t)|2 = dτ1 dτ2 ϕε (t − τ1 ) · ϕε (t − τ2 ) · min(τ1 , τ2 ) 0

=2

+

0



ϕε (t − τ1 )dτ1

0

+

τ1

τ2 · ϕε (t − τ2 )dτ2

0

for t ≥ ε. Simple calculations gives E|W ε (t)|2 = 2

+

ε

ϕε (η)dη

0

+

ε

(t − ξ) · ϕε (ξ) dξ.

η

From this formula it is easy to find that d E|W ε (t)|2 = − dt

+

ε

0

d dη dη

C+

η

ε

D2 ϕε (ξ) dξ = 1,

t ≥ ε.

˙ ε (t) can be written as a stochastic Therefore we have (4.1). It is clear that the derivative W integral of the form + t ˙ ε (t) = W ϕε (t − η)dW (η), t ≥ ε. t−ε

˙ ε (t) is F[t−ε,t] -measurable. A simple calculation leads to (4.2) for q = 2 and Hence W consequently for all q ≥ 1. Let us prove (4.3). We suppose that a(τ ) = 0 outside [s, t] and denote by ∆(s, t) the left-hand side of (4.3). It is clear from the stochastic Fubini theorem (see, e.g., [25, p.109]) that F + t + t E+ ξ+ε ˙ ε (τ ) dτ = a(τ ) · W a(τ )ϕε (τ − ξ)dτ dW (ξ). s

s−ε

ξ

Therefore from (3.3) we obtain ∆(s, t) =

+

t

dξE

s−ε



φ20 · ε

+

@+

t

s−ε



A2

ξ+ε

a(τ )ϕε (τ − ξ)dτ − a(ξ) ξ

+

ξ

ξ+ε

E|a(τ ) − a(ξ)|2 dτ,

NON-RANDOM INVARIANT SETS where φ20 = sition.

B1 0

15

[ϕ(ξ)]2 dξ. This implies property (4.3) and concludes the proof of the propo-

. We note that it is also easy to see that the family W ε (t) = provides another example of a predictable approximation.

1 ε

Bt

W (τ )dτ max(0,t−ε)

/

Using smooth predictable approximations {Wjε (t), t ≥ 0}ε>0 of the Wiener processes {Wj (t), t ≥ 0} we can introduce a predictable smoothing of Itˆo’s problem (3.5) as a family of random equations of the form u(t) = U (t, 0)u0 +

+

t

U (t, τ )f (τ, u(τ )) dτ +

0

∞ '

qj

+

t

0

j=1

˙ jε (τ, ω) dτ. (4.6) U (t, τ )gj (τ, u(τ )) · W

Our main result in this section is the following Wong-Zaka¨ı type approximation theorem. & Theorem 4.3 Let hypotheses (A), (A∗ ), (B), (F0), (G0) and (G1) be valid. Let j qj < ∞ and {Wjε (t), t ≥ 0}0 1 and for this solution there exists a constant C independent of ε such that sup E*uε (t)*rLp (O) ≤ C

for all

p > 1.

(4.7)

t∈[0,T ]

Then lim

ε→0

+

0

T

E*ˆ u(t) − uε (t)*2W 1 (O;Rm ) dt = 0. 2

(4.8)

Here u ˆ(t) is the solution to the problem (3.5) in the sense of Definition 3.1 with 1 ' 2 fˆ(x, t, u, η) = f (x, t, u, η) + · q hj (x, t, u) 2 j j

(4.9)

instead of f , where hj (x, t, u) = (h1j (x, t, u), . . . , hm j (x, t, u)) with hlj (x, t, u) = /gj , ∇u 0gjl (x, t, u) ≡

m ' i=1

gji (x, t, u)

∂gjl (x, t, u) . ∂ui

(4.10)

Remark 4.4 We note that infinite dimensional versions of the Wong–Zaka¨ı theorem were proved earlier in an abstract setting in [9] and [53] under assumptions which look rather restrictive from point of view of SPDE’s. For example, the theorem proved by Twardowska [53] can be applied for one spatial dimension only. Our assumptions concerning the coefficients and the nonlinear terms are essentially more general. However we were forced to assume some additional properties of solutions to the smoothing of the corresponding Itˆo’s PDE (cf. (4.7)). Nevertheless we show in Section 5 that assumption (4.7) is valid for a rather wide class of stochastic systems. We also mention the recent paper [51] devoted to Wong–Zaka¨ı approximations of abstract parabolic equations with autonomous linear parts. The hypotheses of this paper do not allow one to consider drift terms which depend on the gradients of unknown functions. We split the proof of Theorem 4.3 into several steps.

16

CHUESHOV AND VUILLERMOT

4.1

Associated random problem

In this subsection we consider a random version of problem (1.1) and (1.2). The main goal is to obtain several estimates for mild solutions of the predictable smoothing (4.6) which possess the property of uniform boundness (4.7). We invoke these estimates in the next subsection to prove that these solutions satisfy the corresponding Itˆo equation with a small error. We start with a bit more general situation. Let {sj (t, ω) : j ∈ N, t ∈ [0, T ]} be a family of random processes with 21 µ-H¨older continuous trajectories on the probability space (Ω, F , P). We assume that the functions sj (t, ω) are measurable with respect to (t, ω) and for some r > 2 we have ' 1/r Sr ≡ s¯j,r < ∞, where s¯j,r ≡ sup (E|sj (t)|r ) . (4.11) t∈[0,T ]

j

As above E is the mean value operator on (Ω, F , P). We consider the following system of semilinear parabolic random PDE’s   ∂t ul (x, t) = −Al (x, t, D)ul (x, t) + f l (x, t, u(x, t), Du(x, t)) (4.12) &∞ l  + l = 1, . . . m, x ∈ O, t > 0, j=1 gj (x, t, u(x, t)) · sj (t, ω),

in the domain O ⊂ Rn with the boundary and initial conditions: B l (x, D)ul (x, t) = 0, x ∈ ∂O, t > 0,

ul (0, x) = ul0 (x), x ∈ O,

l = 1, . . . m.

(4.13)

Our main assumption in this subsection is the following statement on the existence of solutions to problem (4.12) and (4.13). (S) The problem (4.12) and (4.13) has a bounded mild solution in the following sense: there exists a function u(t) = u(x, t, ω) from the class C(0, T ; Lr (Ω; Xα,p )) for all 0 ≤ α < 1 and p > 1 which satisfies the integral equation u(t) = U (t, 0)u0 +

+

t

U (t, τ )f (τ, u(τ )) dτ +

0

∞ + ' j=1

t

U (t, τ )gj (τ, u(τ ))·sj (τ, ω)dτ (4.14)

0

for all t ∈ [0, T ]. Moreover, for this solution there exists a constant C independent of the processes sj (τ, ω) such that sup E*u(t)*rLp (O) ≤ C

for all p > 1.

(4.15)

t∈[0,T ]

Our first result in this subsection is the following assertion. Proposition 4.5 Assume that (A), (B), (F0) and (G0) are valid. Let u0 (x) ≡ u0 (x, ω) ∈ 2 CB (O, Rm ) almost surely and E*u0 *rC 2 (O) < R with some R > 0. Assume that statement (S) is fulfilled for such initial data. Let u(t) ≡ u(x, t, ω) be the solution to problem (4.12) and (4.13) given by (S). If gj (x, t, u) satisfy (2.13) for all j, then for each 1 < p < ∞ and 0 ≤ α < 1 the following properties hold: (i) there is a constant C1 such that E*u(t)*rXα,p ≤ C1 (1 + Srαλr ),

t ∈ [0, T ],

1 < λ ≤ α−1 ;

(4.16)

NON-RANDOM INVARIANT SETS

17

(ii) for any 0 < δ < 1 − α there is a constant C2 such that E*u(t) − u(t − h)*rXα,p ≤ C2 h(1−α−δ)r (1 + Srr )

f or all

t ∈ [h, T ];

(4.17)

(iii) if, in addition, gj (x, t, u) satisfy (2.14), then for any 0 < δj ≤ δ0 with δ0 > 0 small enough there is a constant C3 such that . /2/r . / r/2 E*u(t) − u(t − h)*Xα,p ≤ C3 h1−α−δ1 1 + Sr1/2+δ2 + hγ Sr (1 + Srγ )

(4.18)

for all t ∈ [h, T ], where γ = min(α, µ/2).

Proof. Below constants in estimates may vary from formula to formula. It follows from representation (4.14), relation (U3) and from the uniform boundness of gj that   + t   ' *u(t)*Xα,p ≤ C1 *u0 *Xα,p + C2 (t − τ )−α *f (τ, u(τ ))*X0,p + |sj (τ )| dτ. (4.19)   0 j

Therefore using (2.15) we have

*u(t)*Xα,p ≤ C0 + C1 *u0 *C 2 (O) + C2

+

t

0

*u(τ )*Xα,p dτ + C3 (t − τ )α

+

t

(t − τ )−α

0

'

|sj (τ )| dτ

j

with 1/2 < α < 1. This implies 0

E*u(t)*rXα,p

11/r

≤ C1 (1 + Sr ) + C2

+

t

0

0

E*u(τ ))*rXα,p (t − τ )α

11/r



for all t ∈ [0, T ]. Now the Gronwall type argument implies 0 11/r E*u(t)*rXα,p ≤ C(1 + Sr )

for all t ∈ [0, T ]

(4.20)

with α ∈ (1/2, 1) and hence with any α ∈ [0, 1). From inequalities (2.8) we have λ *u(τ )*Xα,p ≤ C*u(τ )*1−λ X0,p · *u(τ )*Xα∗ ,p ,

0 ≤ α ≤ α∗ < 1, λ = α/α∗ .

Hence from (4.15) and (4.20) we obtain 0 11−λ 0 1λ E*u(τ )*rXα,p ≤ C E*u(τ )*rX0,p · E*u(τ )*rXα∗ ,p ≤ C(1 + Srλr ).

Thus for any 0 ≤ α < 1 and p ≥ 1 we obtain (4.16). Further, from (4.14) for t ≥ s we have u(t) − u(s) =

(U (t, s) − I) u(s) + +

∞ + ' j=1

+

t

U (t, τ )f (τ, u(τ )) dτ

s

t

U (t, τ )gj (τ, u(τ )) · sj (τ, ω)dτ.

(4.21)

s

It follows from (U5) that *(U (t, s) − I)u(s)*Xα,p ≤ C|t − s|β−α *u(s)*Xβ+θ,p ,

O≤α 0 is arbitrary small. Hence relation (4.20) implies (4.17) provided we choose β and θ in an appropriate way. Now we prove (4.18). If α = 0, then relation (4.18) follows from (4.17). Let α > 0. Using (4.21) we obtain that *u(t) − u(t − h)*Xα,p ≤ w1 + w2 + w3 , (4.23) where w1 = *(U (t, t − h) − I)u(t − h)*Xα,p ,

w2 = Cβ

+

t t−h

$ % (t − τ )−α 1 + *u(τ ))*Xβ,p dτ

with arbitrary β > 1/2, and w3 = C

'+

t

(t − τ )−(α−γ) *gj (τ, u(τ ))*Xγ,p |sj (τ )| dτ,

0 ≤ γ ≤ α.

t−h

j

From (2.17) we obtain that .

r/2

Ew3

/2/r

≤ CSr ·

+

t t−h

0 11/r (t − τ )−(α−γ) 1 + E*u(τ )*rXβ,p dτ

provided γ < 1/2 and γ < β < 1/2. Using (4.16) and choosing β close to γ we get . /2/r r/2 Ew3 ≤ Cδ h1−α+γ Sr (1 + Srγ+δ ),

(4.24)

. /2/r r/2 Ew2 ≤ Cδ h1−α (1 + Sr(1/2+δ) )

(4.25)

where 0 < γ ≤ min(α, µ/2) and δ > 0 is arbitrary small. Using (4.16) we also have that

for any 0 ≤ α < 1 and arbitrary δ > 0. Now we consider w1 . It is clear from (4.14) that w1 ≤ w12 + w12 + w13 , where w11 = * [U (t, 0) − U (t − h, 0)] u0 *Xα,p , w12 =

+

t−h

* [U (t, τ ) − U (t − h, τ )] f (τ, u(τ ))*Xα,p dτ,

0

w13 =

'+ j

Using (U5) we have

t−h

* [U (t, τ ) − U (t − h, τ )] gj (τ, u(τ ))*Xα,p · |sj (τ, ω)| dτ.

0

. /2/r r/2 Ew11 ≤ Cδ h1−α−δ Rr ,

(4.26)

NON-RANDOM INVARIANT SETS

19

where δ > 0 is arbitrary. From (U4) with β = 0, γ = 1−δ ∗ , θ = −δ ∗ /2, where δ ∗ ∈ (0, 1−α), we obtain + t−h $ % ∗ ∗ w12 ≤ Ch1−δ −α (t − h − τ )−1+δ /2 1 + *u(τ ))*X1/2+δ∗ ,p dτ, 0

with arbitrary 0 < δ ∗ < 1 − α. Therefore using (4.16) as for w2 we obtain . /2/r r/2 Ew12 ≤ Cδ h1−α−δ1 (1 + Sr1/2+δ2 )

(4.27)

with arbitrary δ1 > 0 and δ2 > 0. In a similar way, from (U4) we have w13 ≤ Chγ−α

'+ j

t−h

(t − h − τ )θ−γ *gj (τ, u(τ ))*Xβ,p · |sj (τ, ω)| dτ,

0

where 0 ≤ β ≤ γ < 1 + µ/2, 0 ≤ γ − α ≤ 1, θ < β. Using (2.17) we obtain + . /2/r r/2 γ−α Ew13 ≤ Ch Sr ·

0

t−h

0 11/r (t − h − τ )θ−γ 1 + E*u(τ )*rXβ∗ ,p dτ,

with arbitrary β ∗ ∈ (β, 1) provided β < 1/2. Hence . /2/r 0 1 ∗ ∗ r/2 Ew13 ≤ Chγ−α Sr · 1 + Srβ +δ

with arbitrary δ ∗ > 0. If we choose β ∗ = min(α, µ/2) − δ ∗ , β = β ∗ − δ ∗ , θ = β ∗ − 2δ ∗ and γ = 1 + min(α, µ/2) − 4δ ∗ , we obtain . /2/r 0 1 r/2 Ew13 ≤ Ch1+min(α,µ/2)−α−δ Sr · 1 + Srmin(α,µ/2)

(4.28)

with arbitrary δ > 0. Thus from (4.26)–(4.28) we have .

r/2

Ew1

/2/r

0 1 ≤ C1 h1−α−δ1 (1 + Sr1/2+δ2 ) + C2 h1+min(α,µ/2)−α−δ3 Sr · 1 + Srmin(α,µ/2)

with arbitrary δi > 0. This inequality and also (4.23), (4.24), (4.25) imply (4.18).

The main application of Proposition 4.5 deals with the case when sj (t) are derivatives with respect to t of smooth predictable approximations of Wiener processes Wj (t). Proposition 4.5 implies the following assertion for this case. & Proposition 4.6 Assume that (A), (B), (F0), (G0) and (G1) are valid. Let j qj < ∞ and {Wjε (t)}, 0 < ε < 1, be smooth predictable approximations of Wiener processes 2 Wjε (t) (see Definition 4.1). Let u0 (x) ≡ u0 (x, ω) ∈ CB (O, Rm ) almost surely and r E*u0 *C 2 (O) < R with some R > 0 and r > 2. Assume that the corresponding predictable smoothing (4.6) of Itˆ o’s problem (3.5) has a mild solution uε (t) = uε (x, t, ω) such that uε (t) ∈ C(0, T ; Lr (Ω; Xα,p )) for all 0 ≤ α < 1 and p > 1 and for this solution there exists a constant C independent of ε such that (4.7) is valid. Then the following estimates hold. (i) for any 1 < p < ∞, 0 ≤ α < 1 there is a constant C = C(α, p) such that .

E*uε (t)*rXα,p

/1/r

≤ C · ε−αλ/2 ,

t ∈ [0, T ],

1 < λ ≤ α−1 ;

(4.29)

20

CHUESHOV AND VUILLERMOT

(ii) for any 1 < p < ∞, 0 ≤ α < 1, γ ∗
0 is a fixed number; (iii) for any 1 < p < ∞, 0 ≤ α < 1/2, γ I< of j such that

1 4

min{α, µ} there is a constant C independent

. /3/r r/3 E*gj (t, uε (t)) − gj (t − τ1 , uε (t − τ2 ))*Xα,p ≤ Cε1/2+eγ −α ,

(4.31)

for any τ1 , τ2 ∈ [0, k1 ε], where k1 > 0 is a fixed number and t ∈ [max(τ1 , τ2 ), T ]; (iv) for any δ > 0 there exists ν > 0 such that for every 1 < p < ∞ and 0 ≤ α < 1/2 + ν there is a constant C independent of j such that . /2/r r/2 E*gj (t1 , uε (t2 ))*Xα,p ≤ Cε−α/2−δ ,

t1 , t2 ∈ [0, T ].

(4.32)

& ˙ ε (t), then it follows from (4.2) that Sr ≤ Cr ε−1/2 Proof. If we denote sj (t) = qj · W j j qj for any r ≥ 1 in this case. Therefore relations (4.29) and (4.30) are direct consequences of relations (4.16) and (4.18) from Proposition 4.5. From (2.1) and (2.6) we have that *v*C(O) ≤ C*v*Xδ,p for δ > n/(2p). Therefore using (2.18) we obtain . / r3 . / 3r r/3 r/3 ∆ε ≡ E*gj (t, uε (t)) − gj (t − τ1 , uε (t − τ2 ))*Xα,p ≤ C1 E*uε (t) − uε (t − τ2 )*Xβ,p C . / r2 D 0 1 r1 r/2 +C2 ε + E*uε (t) − uε (t − τ2 )*Xδ,p · 1 + E*uε (t)*rXβ,p + E*uε (t − τ2 )*rXβ,p

with δ ∈ (n/(2p), 1), β > α and p > n. Hence, if we choose δ = α/2, then (4.29) and (4.30) imply (4.31) with p > n/α. Since Xβ,p ⊂ Xβ,p# for 1 ≤ p' ≤ p, (4.31) holds for all p > 1. In a similar way from (2.1) and (2.6) we have that *v*C 1 (O) ≤ C*v*Xγ,p for γ > 1/2 + n/(2p). Therefore from (2.19) and (4.29) we obtain .

r/2

E*gj (t1 , uε (t2 ))*Xα,p

/2/r

0 1 ≤ C ε−βλ/2 + ε−γλ/2 · ε−(2β−1)λ/4 ,

where 1/2 + n/(2p) < γ < 1, p > n, β > α, λ > 1. It is easy to see that after choosing these parameters in an appropriate way we obtain (4.32) for α ∈ [1/2, 1/2 + 1/(2p)) and p ≥ p0 (δ). As above, this implies property (iv) for 1/2 ≤ α < 1/2 + ν, where ν = [2p0 (δ)]−1 . The same argument involving (2.17) instead of (2.19) gives (4.32) for α ∈ (0, 1/2).

4.2

Approximate ε-residual solutions

In this subsection we carry out the main preliminary step in the proof of Theorem 3.2. We assume here the existence and some properties of solutions to the predictable smoothing (4.6) of problem (1.1) and (1.2). We do not use the detailed information concerning the elliptic pairs {Al , B l }. Our assumptions in this section deal with the coefficients f and gj and with the family of operators U (t, s) only. Concerning these operators we assume that (U∗ ) {U (t, s) : 0 ≤ s ≤ t ≤ T } is an evolution family possessing the properties (U1)-(U5) with p = 2.

NON-RANDOM INVARIANT SETS

21

We start with the following concept of approximate ε-residual solution. Definition 4.7 A family {uε (t) : t ∈ [0, T ], 0 < ε < 1} of predictable random processes with values in ∩{Xα,2 : 0 < α < 1} is said to be an approximate ε-residual solution to Itˆo’s problem (3.5) on the interval [0, T ] if (i) for almost all ω ∈ Ω, all ε ∈ (0, 1) and 0 ≤ α < 1 the function uε (t, ω) belongs to the space C([0, T ]; Xα,2 ); (ii) uε (0, ω) = u0 (ω) and there exist positive constants k0 , 3 and β > 1/2 such that for all k0 ε ≤ s ≤ t ≤ T we have   uε (t) 

=

U (t, s)uε (s) + +

&∞

j=1 qj ·

Bt s

Bt s

U (t, τ )f (τ, uε (τ )) dτ (4.33)

U (t, τ )gj (τ, uε (τ )) dWj (τ ) + θ(ε, t),

where all integrals exist and θ(ε, t) ≡ θ(ε, t, s, x, ω) possesses the property . / sup E*θ(ε, t)*2Xβ,2 : k0 ε ≤ s ≤ t ≤ T ≤ Cε- .

(4.34)

The following assertion can be considered as a Wong–Zaka¨ı type theorem for approximate solutions. & Theorem 4.8 Let (U∗ ), (F0), (G0) and (G1) be valid and j qj < ∞. Assume that for some smooth predictable approximations {Wjε (t)} of Wiener processes {Wj (t)} the predictable smoothing (4.6) of the problem (1.1) and (1.2) have mild solutions {uε (t)}. Assume that there exists r > 2 such that (i) for all ε ∈ (0, 1) and 0 ≤ α < 1 the function uε (t, ω) belongs to the space C([0, T ]; Lr (Ω, Xα,2 )); (ii) for any 0 < α < 1 and 1 < λ ≤ α−1 there exist positive constants C, κ and Cκ such that 0 11/r E*uε (t)*rXα,2 ≤ Cε−αλ/2 , t ∈ [0, T ], (4.35) and for any fixed k0 > 0 we have

and

0 0

E*uε (t) − uε (t − τ )*rXα,2

11/r

E*uε (t) − uε (t − τ )*2r L4 (O)

≤ Cκ ε1/2+κ−α ,

11/(2r)

≤ C · ε1/2−δ ,

t ∈ [τ, T ],

(4.36)

t ∈ [τ, T ],

(4.37)

for any τ ∈ [0, k0 ε] with arbitrary small δ > 0;

(iii) for any δ > 0 there exists ν > 0 such that for every 0 ≤ α < 1/2 + ν there exists a positive constant C independent of j such that 0 11/r E*gj (t1 , uε (t2 ))*rXα,2 ≤ Cε−α/2−δ ,

t1 , t2 ∈ [0, T ];

(4.38)

22

CHUESHOV AND VUILLERMOT

(iv) for any 0 < α < 1/2 there exist positive constants κ and Cκ independent of j such that for any fixed k1 > 0 we have 0 11/r E*gj (t, uε (t)) − gj (t − τ1 ), uε (t − τ2 )*rXα,2 ≤ Cκ ε1/2+κ−α (4.39) for any τ1 , τ2 ∈ [0, k1 ε] and t ∈ [max(τ1 , τ2 ), T ].

Then {uε (t) : ε > 0} is an approximate ε-residual solution on the interval [0, T ] to Itˆ o’s problem + t + t ∞ ' u(t) = U (t, s)u0 + U (t, τ )fˆ(τ, u(τ )) dτ + qj · U (t, τ )gj (τ, u(τ )) dWj (τ ) (4.40) 0

0

j=1

where fˆ is given by (4.9) Remark 4.9 It is easy to see that the hypotheses (i)–(iv) of Theorem 4.8 hold for some r > 2 provided the conditions of Proposition 4.6 are valid for some r > 8. Our proof of Theorem 4.8 is rather technical. We divide it into several steps. We rely on ideas presented in [9] for a class of abstract stochastic equations with bounded operators in a Banach space. Below we omit the subscript ε near u. 4.2.1

Splitting of θ(ε, t)

If u(t) is a mild solution to problem (4.6), then it follows from (4.6), and (4.33) with fˆ instead of f that the value θ(ε, t) can be presented in the form θ(ε, t) = θ1 (ε, t) + θ2 (ε, t) + θ3 (ε, t), where

Bt · s U (t, τ ) [gj (τ, u(τ − 2ε)) − gj (τ, u(τ ))] dWj (τ ), 0B &∞ t ˙ε θ2 (ε, t) = q · j j=1 s U (t, τ )gj (τ, u(τ − 2ε))W j (τ ) dτ

θ1 (ε, t) =

(4.41)

&∞

j=1 qj



Bt s

1 U (t, τ )gj (τ, u(τ − 2ε))dWj (τ )

(4.42)

Bt &∞ . ˙ε θ3 (ε, t) = j=1 qj · s U (t, τ ) [gj (τ, u(τ )) − gj (τ, u(τ − 2ε))] W j (τ ) dτ − where hj (t, uε (t)) given by (4.10). 4.2.2

qj2 2

Bt s

/ U (t, τ )hj (τ, u(τ )) dτ ,

An estimate for θ1 (ε, t)

Using (3.2) it easy to find that E*θ1 (ε, t)*2Xα,2

=

∞ '

qj2

j=1

·

+

t

s

E*U (t, τ ) (gj (τ, u(τ − 2ε)) − gj (τ, u(τ ))) *2Xα,2 dτ.

Therefore from (U3) we have E*θ1 (ε, t)*2Xα,2 ≤ C

∞ ' j=1

qj2 ·

+

s

t

(t − τ )−2(α−β) E*gj (τ, u(τ − 2ε)) − gj (τ, u(τ ))*2Xβ,2 dτ,

NON-RANDOM INVARIANT SETS

23

with 0 < β < 1/2 and α − β < 1/2. Hence from (4.39) we have E*θ1 (ε, t)*2Xα,2 ≤ Cε2κ ·

∞ '

qj2 for each 0 < α < 1, 2ε ≤ s ≤ t ≤ T,

(4.43)

j=1

where κ > 0 is the same as in (4.39). 4.2.3

An estimate for θ2 (ε, t)

From (4.42) we have that 0 11/2 E*θ2 (ε, t)*2Xα,2 @ + C+ t D2 A1/2 + t ∞ ' ˙ jε (τ ) dτ − = qj E dx aj (x, τ )W aj (x, τ )dWj (τ ) , O

j=1

s

s

where aj (x, τ ) = [(A(0) + λ0 )α U (t, τ )gj (τ, u(τ − 2ε))] (x). It is clear that aj (x, τ ) is Fτ −ε measurable. Therefore we can apply property (4.3) of predictable approximations to conclude that @ A1/2 + t−ε + ξ+ε ∞ ' $ %1/2 1 ∗ E*θ2 (ε, t)*Xα,2 ≤C· qj ε · bj + · dξ bj (τ, ξ)dτ , (4.44) ε s ξ j=1 where

b∗j = max E*U (t, τ )gj (τ, u(τ − 2ε))*2Xα,2 τ ∈[s,t]

and bj (τ, ξ) = E*U (t, τ )gj (τ, u(τ − 2ε)) − U (t, ξ)gj (ξ, u(ξ − 2ε))*2Xα,2 . From (U3) we have 0 1 bj (τ, ξ) ≤ C · (t − τ )−α E*gj (τ ) − gj (ξ)*2Xα/2,2 + E*(U (τ, ξ) − I)gj (ξ)*2Xα/2,2 ,

where we denote gj (τ ) = gj (τ, u(τ − 2ε)) for shortness. Relation (4.39) implies that E*gj (τ ) − gj (ξ)*2Xα/2,2 ≤ C · ε2κ for ξ ≤ τ ≤ ξ + ε.

(4.45)

From (U5) and (4.38) we have ∗

E*(U (τ, ξ) − I)gj (ξ)*2Xα/2,2 ≤ Cε2β−α−δ E*gj (ξ)*2Xβ,2 ≤ Cεβ−α−δ

∗∗

with δ ∗ > 0 and δ ∗∗ > 0 arbitrary small and β ∈ (α/2, 1/2). Consequently we can choose admissible β and δ ∗∗ such that ∗

E*(U (τ, ξ) − I)gj (ξ)*2Xα/2,2 ≤ Cεκ , for all 0 ≤ α < 1 with κ∗ > 0.

(4.46)

In a similar way from (U5) and (4.38) we obtain that b∗j ≤ C max E*gj (τ, u(τ − 2ε))*2Xβ∗ ,2 ≤ Cε−β τ ∈[0,T ]



−δ

α < β ∗ < 1/2 + ν(δ)

with δ > 0 arbitrary small. Consequently it follows from (4.44), (4.45), (4.46) that there exist ν0 > 0 and γ > 0 such that  2 ∞ ' 2 γ  E*θ2 (ε, t)*Xα,2 ≤ C · ε · qj  , 0 < α < 1/2 + ν0 , 2ε ≤ s ≤ t ≤ T. (4.47) j=1

24

CHUESHOV AND VUILLERMOT An estimate for θ3 (ε, t)

4.2.4

Technically this is the most difficult step in the proof of Theorem 4.8. We divide it into several substeps. Splitting of θ3 (ε, t): Using the equality gj (τ, u(τ )) − gj (τ, u(τ − 2ε)) + 1J K = ∇u gj (τ, ξ · u( τ ) − (1 − ξ) · u(τ − 2ε)), u(τ )) − u(τ − 2ε) dξ 0

with

m 2 3 ' ∂gj ∇u gj (τ, u), v = (u) · vi , ∂ui i=1

u = (u1 , . . . , um ) ∈ Rm , v = (v1 , . . . , vm ) ∈ Rm ,

and the relation u(τ )) − u(τ − 2ε) = (U (τ, τ − 2ε) − I)u(τ − 2ε) + τ + τ ∞ ' ˙ kε (ζ), dζ, + U (t, ζ)f (ζ, u(ζ)) dζ + qk · U (τ, ζ)gk (ζ, u(ζ)) · W τ −2ε

τ −2ε

k=1

we can present θ3 (ε, t) in the form θ3 (ε, t) = θ31 (ε, t) + θ32 (ε, t) + θ33 (ε, t) + θ34 (ε, t), where θ31 (ε, t)

=

∞ '

qj ·

j=1

with

+

t

U (t, τ ) s

+

1

0

(4.48)

˙ jε (τ ) dτ, ρj (τ, ξ) dξ W

2 3 ρj (τ, ξ) = ∇u gj (τ, ξu(τ ) − (1 − ξ)u(τ − 2ε)) − ∇u gj (τ, u(τ − 2ε)), u(τ )) − u(τ − 2ε) ,

and

θ32 (ε, t)

=

θ33 (ε, t) = θ34 (ε, t) =

∞ ' j=1 ∞ '

j=1 ∞ ' j=1

qj

+

t

s

qj ·

+

2 3 ε ˙ (τ ) dτ, U (t, τ ) ∇u gj (τ, u(τ − 2ε)), (U (τ, τ − 2ε) − I)u(τ − 2ε) W j t

s

qj ·

+

s

t

2 U (t, τ ) ∇u gj (τ, u(τ − 2ε)),

+

0

τ

τ −2ε

3 ε ˙ (τ ) dτ, U (τ, ζ)f (ζ, u(ζ)) dζ W j

2 31 ˙ ε (τ ) − qj ∇u gj (τ, uε (τ )), gj (uε (τ )) dτ U (t, τ ) gj∗ (τ, ε)W j 2

with ∞ ' 2 gj∗ (τ, ε) = ∇u gj (τ, uε (τ − 2ε)), qk · k=1

+

τ

τ −2ε

3 ˙ ε (ζ) dζ . U (τ, ζ)gk (ζ, u(ζ)) W k

An estimate for θ31 (ε, t): It is clear from (U3) that + t + 10 G ∞ 0 11/2 ' H11/2 ˙ ε (τ )|2 E*θ31 (ε, t)*2Xα,2 ≤ qj · (t − τ )−α E *ρj (τ, ξ)*2X0,2 · |W dξ dτ. j j=1

s

0

NON-RANDOM INVARIANT SETS

25

Since ∇u gj are uniformly Lipschitz with respect to u we have from (4.37) that 0 11/r 0 11/r E*ρj (τ, ξ)*rX0,2 ≤ C E*u(τ ) − u(τ − 2ε))*2r ≤ Cε1−2δ . L4 (O)

0 11/(2r∗ ) ˙ j (τ )|2r∗ Since by (4.2) E|W ≤ Cε−1/2 , where r∗ = r · (r − 2)−1 , we obtain 0

E*θ31 (ε, t)*2Xα,2

11/2



≤ Cεδ ·

∞ '

qj ,

2ε ≤ s ≤ t ≤ T,

α ∈ (0, 1),

(4.49)

j=1

where δ ∗ = 1/2 − 2δ > 0. An estimate for θ32 (ε, t): We have E*θ32 (ε, t)*2Xα,2 =

∞ '

j,k=1

where

and

+

JIjk (t, s) =

+

t



s

t

qj qk · EJIjk (t, s),

˙ jε (τ )W ˙ kε (τ ∗ ) dτ ∗ (bj (τ ), bk (τ ∗ ))Xα,2 W

s

2 3 bj (τ ) = U (t, τ ) ∇u gj (τ, u(τ − 2ε)), (U (τ, τ − 2ε) − I)u(τ − 2ε) .

The symmetry of the integrands implies that E*θ32 (ε, t)*2Xα,2 =

∞ '

qj qk · EJjk (t, s),

j,k=1

where Jjk (t, s) = 2

+

+

t



s

τ

˙ jε (τ )W ˙ kε (τ ∗ ). dτ ∗ (bj (τ ), bk (τ ∗ ))Xα,2 W

s

If t ≥ s + 2ε, we can present Jjk (t, s) in the form 1 2 3 Jjk (t, s) = Jjk (t, s) + Jjk (t, s) + Jjk (t, s),

where 1 Jjk (t, s) = 2 2 Jjk (t, s) = 2

+

3 Jjk (t, s) = 2

Since

+



s

t



s+2ε

+

+

s+2ε

+

˙ ε (τ )W ˙ ε (τ ∗ ), dτ ∗ (bj (τ ), bk (τ ∗ ))Xα,2 W j k

s

τ −2ε

˙ ε (τ )W ˙ ε (τ ∗ ), dτ ∗ (bj (τ ), bk (τ ∗ ))Xα,2 W j k

s

t



s+2ε

ηjk (τ ) ≡ 2

τ

+

τ

τ −2ε

+

τ −2ε

s

˙ jε (τ )W ˙ kε (τ ∗ ). dτ ∗ (bj (τ ), bk (τ ∗ ))Xα,2 W

˙ kε (τ ∗ ) dτ ∗ (bj (τ ), bk (τ ∗ ))Xα,2 W

˙ ε (τ ) which is is Fτ −2ε -measurable, we have that ηjk (τ ) is statistically independent of W k F[τ −ε,τ ]-measurable. Hence we have 2 EJjk (t, s) =

+

t

s+2ε

˙ ε (τ ) = 0. dτ Eηjk (τ ) · EW j

26

CHUESHOV AND VUILLERMOT

Therefore 1 3 EJjk (t, s) = EJjk (t, s) + EJjk (t, s).

Using property (4.2) of the predictable approximation and the H¨older inequality we have that + s+2ε + τ 0 11/r 0 11/r C 1 |EJjk (t, s)| ≤ · dτ dτ ∗ E*bj (τ )*rXα,2 E*bk (τ ∗ )*rXα,2 . ε s s Since ∇u gj (x, t, u) is uniformly bounded, properties (U3) and (U5) imply that *bj (τ )*Xα,2 ≤

C C · εβ−θ *(U (τ, τ − 2ε) − I)u(τ − 2ε)* ≤ *u(τ − 2ε)*Xβ,2 X 0,2 (t − τ )α (t − τ )α

with β ∈ (0, 1) and arbitrary small θ > 0. Consequently from (4.35) we have

Therefore

0 11/r E*bj (τ )*rXα,2 ≤ C · εβ−θ−βλ/2 · (t − τ )−α ,

1 < λ ≤ β −1 .

1 |EJjk (t, s)| ≤ C · ε2(β−θ)−βλ−1 · I1 (t, s, ε)

with β ∈ (0, 1) and arbitrary small θ > 0, where I1 (t, s, ε) =

+

s+2ε

s

dτ (t − τ )α

+

τ

dτ ∗ ≤ Cε1−α . (t − τ ∗ )α

s

(4.50)

Thus for any α ∈ (0, 1) we can choose β ∈ (0, 1), 1 < λ ≤ β −1 and θ > 0 such that 1 |EJjk (t, s)| ≤ C · εκ



0 < α < 1,

2ε ≤ s + 2ε ≤ t ≤ T,

(4.51)

with some positive κ∗ . In a similar way we have that 3 |EJjk (t, s)| ≤ C · ε2(β−θ)−βλ−1 · I3 (t, s, ε)

with β ∈ (0, 1), 1 < λ ≤ β −1 and arbitrary small θ > 0, where I3 (t, s, ε) =

+

t

s+2ε

dτ (t − τ )α

+

τ −2ε

τ

dτ ∗ ≤ Cε1−α . (t − τ ∗ )α

(4.52)

2ε ≤ s + 2ε ≤ t ≤ T,

(4.53)

Therefore as above we have 3 |EJjk (t, s)| ≤ C · εκ



0 < α < 1,

with positive κ∗ . In the same manner we can obtain an estimate like (4.51) and (4.53) for |EJjk (t, s)| in the case when t ∈ [s, s + 2ε]. This remark together with relations (4.51) and (4.53) imply that ∗



E*θ32 (ε, t)*2Xα,2 ≤ Cεκ · 

with some positive κ∗ .

∞ ' j=1

2

qj  ,

0 < α < 1,

2ε ≤ s ≤ t ≤ T,

(4.54)

NON-RANDOM INVARIANT SETS

27

An estimate for θ33 (ε, t): We have 0 11/2 E*θ33 (ε, t)*2Xα,2 C L+ + t ∞ ' −α ≤C qj · (t − τ ) E s

j=1

≤ Cε

−1/2

∞ '

τ

τ −2ε

+

qj ·

+

t

−α

(t − τ )

s

j=1

*U (τ, ζ)f (ζ, u(ζ)) dζ*2X0,2

τ

τ −2ε

˙ ε (τ )|2 · |W j

N 0 11/r O r 1 + E*u(ζ)*X1/2+δ,2 dζ dτ.

MD1/2



Therefore (4.35) implies that 0

E*θ33 (ε, t)*2Xα,2

11/2

  ∞ ' ∗ ≤C qj  εδ ,

0 < α < 1,

2ε ≤ s ≤ t ≤ T,

(4.55)

j=1

where δ ∗ = 1/2 − λ/4 − δλ/2 with 1 < λ < (1/2 + δ)−1 . Splitting of θ34 (ε, t): We present this value in the form θ34 (ε, t) = θ341 (ε, t) + θ342 (ε, t) + θ343 (ε, t). Here θ341 (ε, t)

=

∞ '

+

qj ·

s

j=1

with g ∗∗ (τ, ε) = θ342 (ε, t)

∞ '

qk ·

k=1 ∞ '

=

+

2 3 ε ˙ (τ ) dτ U (t, τ ) ∇u gj (τ, u(τ − 2ε)), g ∗∗ (τ, ε) W j

τ

τ −2ε

qj ·

j=1

with

t

g ∗∗∗ (τ, ε) =

+

t

s

∞ '

˙ ε (ζ) dζ, (gk (ζ, u(ζ)) − gk (τ − 2ε, u(τ − 2ε))) W k

2 3 ε ˙ (τ ) dτ U (t, τ ) ∇u gj (τ, u(τ − 2ε)), g ∗∗∗ (τ, ε) W j

qk ·

k=1

and θ343 (ε, t)

=

∞ '

qj ·

+

+

τ −2ε

t

U (t, τ )

s

j=1

τ

E

where hj (τ, x, u) is given by (4.10),

˙ ε (ζ) dζ (U (τ, ζ) − I)gk (ζ, u(ζ)) W k

∞ '

k=1

h∗jk (τ, ε)ψjk (τ )

F qj − hj (τ, u(τ )) dτ 2

h∗jk (τ, ε) = /∇u gj (τ, uε (τ − 2ε)), gk (τ − 2ε, u(τ − 2ε))0 and ψjk (τ ) =

+

τ

τ −2ε

˙ kε (ζ)W ˙ jε (τ ) dζ. W

An estimate for θ341 (ε, t): From (4.39) we have . /2/(2+r) 1+r/2 E*g ∗∗ (τ, ε)*X0,2 + τ 0 ∞ 1 1r ' ≤ Cε−1/2 qk · E*gk (ζ, u(ζ)) − gk (τ − 2ε, u(τ − 2ε))*rX0,2 dζ ≤ Cε1−δ

k=1 ∞ '

qk

k=1

τ −2ε

(4.56) (4.57)

28

CHUESHOV AND VUILLERMOT

with arbitrary small δ > 0. Since + ∞ . /1/2 ' 41 2 E*θ3 (t, ε)*Xα,2 ≤C qj

0 1 2 0 11/r∗ dτ 1+r/2 2+r ∗∗ ˙ jε (τ )|r∗ E*g (τ, ε)* E| W X 0,2 (t − τ )α

t

s

j=1

with r∗ = 2(r + 2)(r − 2)−1 , we obtain the estimate . /1/2 E*θ341 (t, ε)*2Xα,2 ≤ Cε1/2−δ

E

∞ '

qk

k=1

F2

,

0 < α < 1,

2ε ≤ s ≤ t ≤ T.

(4.58)

An estimate for θ342 (ε, t): Since *g ∗∗∗ (τ, ε)*X0,2 ≤ C

∞ '

qk ·

+

τ

˙ ε (ζ)| dζ, * (U (τ, ζ) − I) gk (ζ, u(ζ))*X0,2 |W k

τ −2ε

k=1

we obtain from (U5) that *g

∗∗∗

(τ, ε)*X0,2 ≤ C

∞ '

qk ·

+

τ

τ −2ε

k=1

β ˙ kε (ζ)| dζ. (τ − ζ) *gk (ζ, u(ζ))*Xβ∗ ,2 |W

where 0 < β < β ∗ < 1. Consequently using (4.38) we have .

1+r/2

E*g ∗∗∗ (τ, ε)*X0,2

2 / 2+r

≤ Cε1/2+β/2−δ



∞ '

qk

k=1

with arbitrary δ ∗ > 0 and β ∈ (0, 1/2 + ν(δ ∗ )), ν(δ ∗ ) > 0. Hence using the estimate . / 12 '+ E*θ342 *2Xα,2 ≤C j

/ 2 . / 1∗ qj dτ . 1+r/2 2+r ∗∗∗ ˙ ε (τ )|r∗ r E*g (τ, ε)* E| W j X0,2 (t − τ )α

t

s

with r∗ = 2(r + 2)(r − 2)−1 , we can choose β and δ ∗ > 0 such that .

E*θ342 (t, ε)*2Xα,2

/1/2

≤ Cε1/4

E

∞ '

k=1

qk

F2

,

0 < α < 1,

2ε ≤ s ≤ t ≤ T.

(4.59)

Splitting of θ343 (ε, t): Let h∗jk (τ, ε) and ψjk (τ ) be given by (4.56) and (4.57). We present the value θ343 (ε, t) in the form θ343 (ε, t) = θ3431 (ε, t) + θ3432 (ε, t) + θ3433 (ε, t) + θ3434 (ε, t)) + θ3435 (ε, t), where t ≥ s ≥ 4ε and ∞

θ3431 (ε, t)

1' 2 = q · 2 j=1 j

θ3432 (ε, t) =

∞ '

j,k=1

+

qj qk ·

t

U (t, τ ) [hj (τ − 2ε, u(τ − 4ε)) − hj (τ, u(τ ))] dτ,

s

+

s

t

$ % U (t, τ ) h∗jk (τ, ε) − h∗jk (τ − 2ε, ε)) ψjk (τ ) dτ,

NON-RANDOM INVARIANT SETS

θ3433 (ε, t)

=

'

qj2

·

+

t

s

j

θ3434 (ε, t) =

$ % U (t, τ ) h∗jj (τ − 2ε, ε) − hj (τ − 2ε, u(τ − 4ε)) ψjj (τ ) dτ, '

qj qk ·

θ3435 (ε, t) =

' j

qj2 ·

+

+

t

s

j(=k

and

29

U (t, τ )h∗jk (τ − 2ε, ε) · ψjk (τ ) dτ,

C D 1 U (t, τ )hj (τ − 2ε, u(τ − 4ε)) · ψjj (τ ) − dτ. 2

t

s

Below we need the following properties of the processes ψjk (τ ). Lemma 4.10 The functions ψjk (τ ) given by (4.57) are F[τ −3ε,τ ]-measurable and possess the properties Eψjk (τ ) = 0 if j += k, and Eψjj (τ ) = 1/2, τ ≥ 2ε, (4.60) {E|ψjk (τ )|q }

1/q

≤ C for all j, k ∈ Z+ , q ≥ 1,

τ ≥ 2ε.

(4.61)

Proof. Since Wk (t) and Wj (t) are independent for k += j, we have Eψjk (τ ) = 0 in this case. Now we consider $ % ε ˙ j (τ ). Eψjj (τ ) = E Wjε (τ ) − Wjε (τ − 2ε) W ˙ ε (τ ) are independent (see Definition 4.1) we have Since Wjε (τ − 2ε) and W j ˙ ε (τ ) = 1 · d E|W ε (τ )|2 . Eψjj (τ ) = EWjε (τ ) · W j j 2 dτ Therefore (4.1) implies (4.60). To prove (4.61) we note that + τ 0 11/(2q) 0 11/(2q) 1/q ˙ kε (ζ)|2q ˙ jε (τ )|2q (E|ψjk (τ )|q ) ≤ E|W E|W dζ. τ −2ε

Hence (4.61) follows from (4.2). Estimates for θ343j (ε, t), j = 1, 2, 3: Using (U3) and (G1) we have .

/ 12 E*θ3431 (t, ε)*2Xα,2 + t ∞ 0 1 12 ' dτ 2 2 ≤C qj E*h (τ, u(τ )) − h (τ − 2ε, u(τ − 2ε))* j j X 0,2 α s (t − τ ) j=1 N + t ∞ 0 1 12 O ' dτ 2 ≤C qj2 ε + E*u(τ ) − u(τ − 2ε)* . X0,2 α s (t − τ ) j=1

Therefore (4.36) implies ∞ . /1/2 ' E*θ3431 (t, ε)*2Xα,2 ≤ Cε1/2−δ qk2 ,

0 < α < 1,

4ε ≤ s ≤ t ≤ T,

k=1

with δ small enough. The property (4.61) allows us to obtain that . / 12 E*θ3432 (t, ε)*2Xα,2 + t ∞ ' ≤C qj qk j,k=1

s

0 1 r1 dτ ∗ ∗ r E*h (τ, ε) − h (τ − 2ε, 2ε)* . jk jk X0,2 (t − τ )α

(4.62)

30

CHUESHOV AND VUILLERMOT

Therefore as above using properties (G1) and (4.36) we can conclude E ∞ F2 . /1/2 ' 432 2 1/2−δ E*θ3 (t, ε)*Xα,2 ≤ Cε qk , 0 < α < 1, 4ε ≤ s ≤ t ≤ T,

(4.63)

k=1

where δ < 1/2. Applying arguments similar to the ones given above we also easily obtain that ∞ . /1/2 ' 433 2 1/2−δ E*θ3 (t, ε)*Xα,2 ≤ Cε qk2 , 0 < α < 1, δ < 1/2, 4ε ≤ s ≤ t ≤ T. (4.64) k=1

Estimates for θ343j (ε, t), j = 4, 5: Now we consider θ3434 (t, ε) and apply the same method as for the proof of estimate (4.54) for θ32 (ε, t). It is clear that ' ' *θ3434 (t, ε)*2Xα,2 = qj qk qj ∗ qk∗ JIjkj ∗ k∗ (t, s), j(=k j ∗ (=k∗

where

and

+

JIjkj ∗ k∗ (t, s) =

t



s

+

t

dτ ∗ bjkj ∗ k∗ (τ, τ ∗ ) · ψjk (τ ) · ψj ∗ k∗ (τ ∗ )

s

$ % bjkj ∗ k∗ (τ, τ ∗ ) = U (t, τ )h∗jk (τ − 2ε, ε), U (t, τ ∗ )h∗j ∗ k∗ (τ ∗ − 2ε, ε) X

α,2

.

The symmetry of the integrands gives that ' ' E*θ3434 (t, ε)*2Xα,2 = qj qk qj ∗ qk∗ EJjkj ∗ k∗ (t, s), j(=k j ∗ (=k∗

where J

jkj ∗ k∗

+

(t, s) = 2

t



s

+

τ

dτ ∗ bjkj ∗ k∗ (τ, τ ∗ ) · ψjk (τ ) · ψj ∗ k∗ (τ ∗ ).

s

If t ≥ s + 4ε, we can present Jjkj ∗ k∗ (t, s) in the form (1)

(2)

(3)

Jjkj ∗ k∗ (t, s) = Jjkj ∗ k∗ (t, s) + Jjkj ∗ k∗ (t, s) + Jjkj ∗ k∗ (t, s), where (1) Jjkj ∗ k∗ (t, s) (2)

=2

Jjkj ∗ k∗ (t, s) = 2 (3)

+

Jjkj ∗ k∗ (t, s) = 2



s t

+



s+4ε

+

t



s+4ε

By Lemma 4.10 the function +

+

s+4ε

τ −4ε

+

τ

dτ ∗ bjkj ∗ k∗ (τ, τ ∗ ) · ψjk (τ ) · ψj ∗ k∗ (τ ∗ ),

s τ −4ε

dτ ∗ bjkj ∗ k∗ (τ, τ ∗ ) · ψjk (τ ) · ψj ∗ k∗ (τ ∗ ), s

+

τ

dτ ∗ bjkj ∗ k∗ (τ, τ ∗ ) · ψjk (τ ) · ψj ∗ k∗ (τ ∗ ).

τ −4ε

dτ ∗ bjkj ∗ k∗ (τ, τ ∗ ) · ψj ∗ k∗ (τ ∗ )

s

is Fτ −4ε -measurable. Therefore, since ψjk (τ ) is F[τ −3ε,τ ]-measurable and since Eψjk (τ ) = 0 for j += k, we have C+ τ −4ε D + t (2) EJjkj ∗ k∗ = 2 dτ E dτ ∗ bjkj ∗ k∗ (τ, τ ∗ ) · ψj ∗ k∗ (τ ∗ ) · Eψjk (τ ) = 0. s

s

NON-RANDOM INVARIANT SETS

31

Since gj (x, t, u) and ∇u gj (x, t, u) are uniformly bounded, we have from (U3) that |bjkj ∗ k∗ (τ, τ ∗ )| ≤ C · (t − τ )−α · (t − τ ∗ )−α . Therefore using (4.61) we have (1)

(3)

|EJjkj ∗ k∗ (t, s)| ≤ CI1 (t, s, 2ε) and |EJjkj ∗ k∗ (t, s)| ≤ CI3 (t, s, 2ε) for 4ε ≤ s + 4ε ≤ t ≤ T , where I1 (t, s, ε) ≤ Cε1−α given (4.50) and I3 (t, s, ε) ≤ Cε1−α given (4.52). We can also easily prove that |EJjkj ∗ k∗ (t, s)| ≤ Cε1−α

for 4ε ≤ t ≤ s + 4ε ≤ T.

Therefore as above we have E*θ3434 *2Xα,2

 4 ' ≤C qj  ε1−α ,

0 ≤ α < 1,

4ε ≤ s ≤ t ≤ T.

(4.65)

0 ≤ α < 1,

4ε ≤ s ≤ t ≤ T.

(4.66)

0 ≤ α < 1,

4ε ≤ s ≤ t ≤ T,

(4.67)

j

In a similar way using Lemma 4.10 again we also have 

E*θ3435 *2Xα,2 ≤ C 

' j

2

qj2  ε1−α ,

Thus from (4.62) – (4.66) we obtain 

E*θ343 *2Xα,2 ≤ C 1 +

' j

with some positive γ. 4.2.5

4

qj  εγ ,

An estimate for θ(ε, t)

Using (4.58), (4.59) and (4.67) we obtain 

E*θ34 (ε, t)*2Xα,2 ≤ C 1 +

' j

4

qj  εγ ,

0 ≤ α < 1,

4ε ≤ s ≤ t ≤ T,

with some γ > 0. Therefore by (4.49), (4.54) and (4.55) we have 

E*θ3 (ε, t)*2Xα,2 ≤ C 1 +

' j

4

qj  εδ ,

0 ≤ α < 1,

4ε ≤ s ≤ t ≤ T,

with some δ > 0. Consequently (4.43) and (4.47) imply that there exist ρ > 0 and α > 1/2 such that  4 ' E*θ(ε, t)*2Xα,2 ≤ C 1 + qj  ερ , 4ε ≤ s ≤ t ≤ T. (4.68) j

This inequality implies the conclusion of Theorem 4.8.

32

CHUESHOV AND VUILLERMOT

4.3

Convergence of ε-residual solutions

In this subsection we complete the proof of Theorem 4.3. We start with the following assertion. ∗ Lemma and (G1) are valid. & 24.11 Assume that the properties (A), (A ), (B), (F0), (G0) Let j qj < ∞ and u0 = u0 (x, ω) be F0 -measurable and E*u0 *rV < ∞ for some r > 2. Let fˆ be given by (4.9). Assume that u ˆε (t, ω) is an approximate ε-residual solution (see Definition 4.7) to Itˆ o’s problem (3.5) with fˆ instead of f and u ˆ(t, ω) be a mild solution to ˆ problem (3.5) with f instead of f given by Theorem 3.2 hold. Then we have

+

T s

0 1 E*ˆ u(t)−ˆ uε (t)*2W 1 (O,Rm ) dt ≤ C· ε2- + E*ˆ u(s) − u ˆε (s)*2X0,2 f or every s ≥ k0 ε. (4.69) 2

Proof. As in the proof of Theorem 3.2 for T ∗ ≤ T − s and L > 0 we introduce spaces VTs ∗ of progressively measurable processes v(t) ≡ v(t, ω) on [s, s+T ∗ ] with values in V ≡ W21 (O, Rm ) such that @+ A1/2 s+T ∗ 0 1 2 2 |v|VTs ∗ ≡ E *v(t)*V + L · *v(t)*X0,2 dt < ∞. (4.70) s

Since u ˆε (t, ω) satisfies (4.33) and u ˆ(t, ω) is a mild solution to Itˆo’s problem (3.5) with fˆ instead of f , we have the following representations u ˆε (t) = U (t, s)ˆ uε (s) + B s [ˆ uε ](t) + θ(ε, t)

(4.71)

u ˆ(t) = U (t, s)ˆ u(s) + B s [ˆ u](t),

(4.72)

and where s

B [ˆ v ](t) =

+

t

U (t, τ )fˆ(τ, v(τ )dτ +

0

∞ ' j=1

qj

+

t

U (t, τ )gj (τ, v(τ )dWj (τ ).

0

An argument given in the proof of Theorem 3.2 allows us to choose T ∗ > 0 and L > 0 such that |B s [ˆ uε ] − B s [ˆ u]|VTs ∗ ≤ q|ˆ uε − u ˆ|VTs ∗ , where q < 1. Therefore, from (4.71) and (4.72) we obtain that |ˆ uε − u ˆ|2V s ∗ ≤ C1 · ε2- + C2 · Iε (s, T ∗ ), T

where Iε (s, T ∗ ) =

+

s

s+T ∗

. / E *U (t, s) (ˆ uε (s) − uˆ(s)) *2V + L*U (t, s) (ˆ uε (s) − uˆ(s)) *2X0,2 dt.

Using property (2.11) we have Iε (s, T ∗ ) ≤ C · E*ˆ uε (s) − u ˆ(s)*2X0,2 . Therefore, we obtain (4.69) for T = T ∗ . Dividing the interval [s, T ] into subintervals of length less than T ∗ , one can prove (4.69) in the general case. Now we are in a position to complete the proof of Theorem 4.3. If the hypotheses of Theorem 4.3 are valid, then the mild solution uε (t) satisfies the assumptions of Proposition 4.6 with r > 8. Therefore by Remark 4.9 the hypotheses of

NON-RANDOM INVARIANT SETS

33

Theorem 4.8 are valid. Consequently uε (t) is an approximate ε-residual solution to Itˆo’s problem (4.40) with fˆ given by (4.9). Thus we can apply Lemma 4.11 to conclude that + T 0 1 E*ˆ u(t) − uε (t)*2W 1 (O,Rm ) dt ≤ C · ε2- + E*ˆ u(s) − uε (s)*2X0,2 (4.73) 2

s

for every s ∈ [k0 ε, T ]. It follows from (4.30) that E*uε (k0 ε) − u0 *2X0,2 ≤ C · ε1/2−δ

(4.74)

for δ > 0 arbitrary small. We also have from (3.7) that E*ˆ u(k0 ε) − u0 *2X0,2 ≤ C · εγ

(4.75)

with positive γ. Since E*ˆ u(t) − uε (t)*2X0,2 ≤ 2E*ˆ u(t) − u0 *2X0,2 + 2E*ˆ uε (t) − u0 *2X0,2 , equations (4.73), (4.74) and(4.75) imply that + T ∗ E*ˆ u(t) − uε (t)*2W 1 (O,Rm ) dt ≤ C · εγ with some γ ∗ > 0. 2

k0 ε

(4.76)

In a similar way from (4.30) we have + k0 ε ∗∗ E*uε (t) − u0 *2W 1 (O,Rm ) dt ≤ C · εγ with some γ ∗∗ > 0. 2

0

A direct calculation gives that lim

ε→0

+

0

k0 ε

E*ˆ u(t) − u0 *2W 1 (O,Rm ) dt = 0. 2

The last two relations and also (4.76) lead to equation (4.8). This concludes the proof of Theorem 4.3.

5

Main results

In this section we consider several situations for which we can guarantee an estimate of the type (4.7) for solutions of the predictable smoothing of the problem (3.5) and hence apply the approximation Theorem 4.3.

5.1

Invariant sets

We start with the case when the predictable smoothing (4.6) possesses a bounded invariant set. Our considerations are based on the approach developed by H. Amann [2] in the deterministic case. In the following we denote by D a closed convex bounded subset of Rm and (cf. [2]) we impose the following convexity condition. (C) Let 0 = m0 < m1 < . . . < mk = m be integers such that Ai (x, t, D) = Amj (x, t, D) and B i (x, D) = B mj (x, D) for mj−1 < i ≤ mj and 1 ≤ j ≤ k. Then D = D1 ×. . .×Dm , where Dj is a closed bounded convex subset of Rmj −mj−1 , 1 ≤ j ≤ k, containing the origin. We also assume that al0 += 0 if bl0 (x) = 0 for all x ∈ ∂O.

34

CHUESHOV AND VUILLERMOT

In particular if Ai (x, t, D) = A1 (x, t, D) and B i (x, D) = B 1 (x, D) for 2 ≤ i ≤ m, then D can be an arbitrary closed bounded convex subset of Rm with 0 ∈ D. Now we describe additional hypotheses concerning the nonlinear functions f l (x, t, u, η) and gjl (x, t, u). We assume that (F1) The function f (x, t, u, η) is continuous with respect to (x, t, u, η) and such that f (·, ·, u, η) is µ-H¨ older continuous in x and µ2 -H¨older continuous in t, uniformly with respect to u and η in bounded subsets of Rm and Rnm . (FT) The following tangency condition holds: /ν, f (x, t, ξ, η)0 ≤ 0 for all (x, t) ∈ O × [0, T ], every ξ ∈ ∂D, every ν ∈ N (ξ), and every η = (η1 , . . . , ηn ) ∈ Rm × . . . Rm satisfying /ν, ηi 0 = 0, i = 1, . . . , m, where N (ξ) is the set of all outer normals to ∂D at ξ ∈ ∂D (we recall that ν ∈ Rm \ {0} is called an outer normal to the boundary ∂D at the point ξ0 if /ν, ξ0 0 = max{/ν, ξ0 : ξ ∈ D}). (GT) The zero tangency condition holds: /ν, gj (x, t, ξ)0 = 0 for all (x, t) ∈ O × [0, T ], every ξ ∈ ∂D and every ν from the set N (ξ) of all outer normals to ∂D at ξ ∈ ∂D. We start with the random problem (4.12) and (4.13). Let {sj (t, ω) : j ∈ N, t ∈ [0, T ]} be a family of random processes with 21 µ-H¨older continuous trajectories on the probability space (Ω, F , P). We assume that the functions sj (t, ω) are measurable with respect to (t, ω) and possesses property (4.11) for some r > 2. Under conditions (F0), (F1), (FT), (G0), (G1), (GT) we consider the system (4.12) and (4.13) of semilinear parabolic random PDE’s. We note that our assumptions concerning f and gj imply that the infinite series in (4.12) converges almost surely. Our first result in this section is the following existence and uniqueness theorem for the problem (4.12) and (4.13). Theorem 5.1 Assume that (A), (B), (C), (F0), (F1), (FT), (G0), (G1) and (GT) are 2 valid. Assume also that (4.11) holds for some r > 2. Let u0 (x) ≡ u0 (x, ω) ∈ CB (O, D) 2 almost surely and E*u0 *C (O) < ∞. Then there exists a unique (up to equivalence) function u(t) ≡ u(x, t, ω) with the properties (i) for any s < 2 − 2/r, s∗ < 2 and 1 < p < ∞ we have s s∗ u(t, ω) ∈ C(0, T ; Wp,B (O)) ∩ L∞ (0, T ; Wp,B (O)) f or almost all ω ∈ Ω

and for every α ∈ [0, 1 − 1/r), 1 < p < ∞ there exists a constant C such that N O *u(t) − u(s)*Xα,p E sup *u(t)*Xα,p + sup ≤ C(1 + E*u0 *C 2 (O) + Sr ), (5.1) |t − s|γ 0≤t≤T 0≤s 1 and thus the assumption (S) of Section 4 is true.

NON-RANDOM INVARIANT SETS

35

Remark 5.2 If sj (t, ω) +≡ 0 for finitely many indices, then without any moment restrictions like (4.11) we have from our assumptions that the function f ∗ (x, t, u, η) = f (x, t, u, η) +

∞ '

gjl (x, t, u) · sj (t, ω)

j=1

possesses the properties (F0), (F1) and (FT). Therefore in this case we can apply Theorem 1 from [2] to prove that for almost every ω ∈ Ω problem (4.12) and (4.13) has a unique classical solution u(x, t) with property (5.2) and such that u = u(x, t, ω) ∈ C 2,1 (O × (0, T ], Rm ) ∩ C 1,0 (O × [0, T ], Rm ). Thus in the case when sj (t, ω) +≡ 0 for finitely many indices, a stronger version of Proposition 4.5 is a direct corollary of the result by H.Amann [2]. The proof of Theorem 5.1 relies on the following lemma. Lemma 5.3 Assume that hypotheses of Theorem 5.1 hold. Let uN (t) ≡ uN (x, t, ω) be a classical solution to problem (4.12) and (4.13) which corresponds to the sequences {sj (t) ≡ sj,N (t)} defined by the equations sj,N (t) = sj (t) for j ≤ N and sj,N (t) ≡ 0 for j > N (see Remark 5.2). Then uN (t) possesses property (5.2) and for every α ∈ [0, 1 − 1/r), 1 < p < ∞ there exists a constant C such that N O E sup sup *uN (t)*Xα,p ≤ C(1 + E*u0 *Xα,p + Sr ), (5.3) N 0≤t≤T

and

N E sup

sup

N 0≤s 1. Now we prove (5.4). Using (4.21), (4.22), (2.15) and (5.6) we obtain , $ %*u(t) − u(s)*Xα,p ≤ C1 |t − s|β−α *u(s)*Xβ+θ,p + |t − s|1−α 1 + *u0 *Xα,p   r 1/r + T '  +C2 |t − s|1−α−1/r  |sj (τ )| dτ  0

provided

1 2

j

< α < 1 − 1r . This implies (5.4) for every admissible α and for all p > 1.

Proof of Theorem 5.1. Lemma 5.3 implies that there exists a set Ω∗ ⊂ Ω of full measure and a constant C = C(ω) such that sup *uN (t)*Xα,p +

0≤t≤T

sup 0≤s 1 (see (2.1) and (2.6)), the function u(t) belongs to the space C([0, T ], C 1 (O)). This property makes it possible to prove the uniqueness of limiting points for the sequence {uN (t)}. Indeed, assume that there exists another function v(t) ∈ C([0, T ], C 1 (O)) which satisfies (4.14) and belongs to C([0, T ], Xα,p ) with α ∈ [0, 1 − 1/r) and 1 < p < ∞. Then from (U3), (4.14) and (2.16) we have *u(t) − v(t)*Xα,p ≤ C1

+

0

+C2

'+ j

0

t

t

*u(τ ) − v(τ )*Xα,p dτ (t − τ )α

*gj (τ, u(τ )) − gj (τ, v(τ ))*Xδ,p · |sj (t)| dτ, (t − τ )α−δ

NON-RANDOM INVARIANT SETS

37

where 1/2 < α < 1 and δ ∈ (0, α). It is clear from (2.18) in Proposition 2.4 that *gj (τ, u(τ )) − gj (τ, v(τ ))*Xδ,p ≤ C*u(τ ) − v(τ )*Xα,p provided n/(2p) < α < 1 − 1/r, where δ < 1/2 and the constant C may depend on u(t) and v(t). Consequently, if we denote ψ(t) = *u(τ ) − v(τ )*Xα,p , we obtain the inequality + t + t ' ψ(τ ) ψ(τ ) ψ(t) ≤ C1 dτ + C · |sj (τ )| dτ 2 α α−δ 0 (t − τ ) 0 (t − τ ) j This inequality implies + t0 ψ(τ ) dτ + C ψ(τ ) · a(τ ) dτ 2 α 0 (t − τ ) 0 0& 1α/δ for all t ≤ t0 ≤ T , where a(τ ) = |s (τ )| . Since we can choose α/δ < r, from (5.9) j j ψ(t) ≤ C1

+

t

we have that a(t) ∈ L1 (0, T ) for the case considered. Therefore the generalized Gronwall method gives  α/δ ' + t0 ' ψ(t) ≤ CT ψ(τ ) ·  |sj (τ, ω)| dτ, ω ∈ Ω∗ j

0

j

for all t ≤ t0 ≤ T . In particular, we can choose t = t0 . Applying now the standard Gronwall lemma we find that ψ(t) ≡ 0. Thus we have uniqueness of the limiting point of the sequence {uN (t, ω)} for each ω ∈ Ω∗ and therefore the sequence {uN (t, ω)} converges almost surely in the space C(0, T ; Xα,p ) to the function u(t, ω) which solves (4.14) almost surely. Other conclusions of Theorem 5.1 are easy consequences of Lemma 5.3 and this convergence property. Theorem 5.1 immediately implies the following assertion. Corollary & 5.4 Assume that (A), (B), (C), (F0), (F1), (FT), (G0), (G1) and (GT) are ε valid. Let j qj < ∞ and {Wj (t)}, 0 < ε < 1, be smooth predictable approximations 2 of Wiener processes Wjε (t) (see Definition 4.1). Let u0 (x) ≡ u0 (x, ω) ∈ CB (O, D) almost r surely and E*u0 *C 2 (O) < R with some R > 0 and r > 2. Then the corresponding predictable smoothing (4.6) of Itˆ o’s problem (3.5) has a unique (up to equivalence) mild solution uε (t) = uε (x, t, ω) such that uε (t) ∈ C(0, T ; Lr (Ω; Xα,p )) for all 0 ≤ α < 1 and p > 1. For this solution we have the following invariance property: uε (x, t) ∈ D

for all

(x, t) ∈ O × [0, T ] almost surely.

(5.10)

qj Wjε (t)

Proof. It is clear that the processes sj (t) ≡ possess property (4.11) for every r > 1 & −1/2 with Sr ≤ Cr ε j qj . Thus Theorem 5.1 can be applied here with any r > 2. Therefore there exists a unique mild solution uε (t) to (4.6) such that (5.10) holds and s uε (t, ω) ∈ C([0, T ], Wp,B (Rm )) ⊂ Xα,p ,

0 ≤ 2α < s < 2,

for almost all ω. Using the Gronwall type argument it is easy to find from (4.6) that *uε (t, ω)*Xα,p ≤ C1 *uε (0, ω)*Xα,p + C2 (ω),

t ∈ [0, T ],

almost surely,



where C2 (ω) ∈ Lr∗ (Ω) for every r ∈ [1, +∞). Therefore the Lebesgue convergence theorem implies that uε (t) ∈ C(0, T ; Lr (Ω; Xα,p )). This corollary and also Theorem 4.3 allow us to prove the following version of the Wong– Zaka¨ı approximation theorem.

38

CHUESHOV AND VUILLERMOT

Theorem 5.5&Assume that (A), (A∗ ), (B), (C), (F0), (F1), (FT), (G0), (G1) and (GT) are valid. Let j qj < ∞ and {Wjε (t), t ≥ 0}0 8. Then lim

ε→0

+

0

T

E*ˆ u(t) − uε (t)*2W 1 (O;Rm ) dt = 0, 2

(5.11)

where u ˆ(t) is the solution to problem (3.5) in the sense of Definition 3.1 with fˆ given by (4.9) instead of f . Theorem 5.5 and Corollary 5.4 imply our main result on the invariance of the deterministic set D for the stochastic Itˆ o parabolic system (1.1) and (1.2). Theorem& 5.6 Assume that (A), (A∗ ), (B), (C), (F0), (F1), (G0), (G1) and (GT) are 1 valid. Let j qj < ∞, u0 (x) ≡ u0 (x, ω) ∈ V ≡ W2,B (O, D) almost surely and E*u0 *2V < ∞. Assume that the function 1 ' 2 f˜(x, t, u, η) = f (x, t, u, η) − · q hj (x, t, u), 2 j j

(5.12)

where hj (x, t, u) = (h1j (x, t, u), . . . , hm j (x, t, u)) given by (4.10), satisfies the tangency condition (FT). Then the mild solution u(t) ≡ u(x, t, ω) to problem (1.1) and (1.2) possesses the property u(x, t, ω) ∈ D

for almost all

(x, t, ω) ∈ O × [0, T ] × Ω.

(5.13)

2 Proof. If we assume that u0 (x, ω) ∈ CB (O, D) almost surely and E*u0 *rC 2 (O) < ∞ for r > 8, then (5.13) easily follows from Theorem 5.5 and Corollary 5.4 with f˜ given by (5.12) instead of f . Thus, in order to prove the theorem for general initial data u0 (x) we only need 2 to construct a sequence {uk (x, ω)} ⊂ CB (O, D) with the properties

lim E*uk − u0 *2V = 0

k→∞

and E*uk *rC 2 (O) < ∞ for some r > 8.

We can do it in two steps. We first approximate u0 (x) by elements uδ (x) = u0 (x)(1 + δ*u0 *V )−1 , where δ > 0. It is clear that lim E*uδ − u0 *2V = 0 and E*uk *rC 2 (O) < ∞ for every r ≥ 1.

δ→0

Since 0 ∈ D and u0 (x) ∈ D, we have that uδ (x) ∈ D. Now we consider uλ,δ (x) = e−A(0)λ uδ (x) for λ > 0. The parabolic regularity theory 2 implies that uλ,δ (x) ∈ CB (O, Rm ). The application of the invariance result from [2] allows us to state that uλ,δ (x) ∈ D for every x ∈ O. It is also obvious that lim E*uλ,δ − uδ *2V = 0 and E*uλ,δ *rC 2 (O) < ∞ for every r ≥ 1.

λ→0

Thus by the standard diagonal process we can construct the required sequence {uk }.

NON-RANDOM INVARIANT SETS

5.2

39

Comparison principle

Another corollary of Theorem 5.5 is a stochastic comparison principle for the parabolic system (1.1) of Itˆ o’s PDE’s. We impose the following additional hypotheses concerning the nonlinear terms f and gj in equation (1.1). (F2) Every component f l of the function f ≡ (f 1 , . . . , f m ) depends on x, t, u and on the derivatives Dul only, i.e. it has the form f l (x, t, u, η) = f˜l (x, t, u, η l ), where f˜l is a continuous mapping from O × [0, T ] × Rm × Rn into R. We also assume that the function f is cooperative, i.e. f l (x, t, u, η) ≤ f l (x, t, v, η),

l = 1, . . . , m,

(5.14)

for all (x, t, η) ∈ O × [0, T ] × Rnm and for all u, v ∈ Rd such that ul = v l and uj ≤ v j for j += i. (G2) Every component gjl of the function gj ≡ (gj1 , . . . , gjm ) depends on x, t and ul only, i.e. it has the form gjl (x, t, u) = g˜jl (x, t, ul ), where g˜l is a mapping from O × [0, T ] × R into R. Remark 5.7 We note that hypothesis (F2) is the standard assumption taken from the deterministic theory of monotone systems (see, e.g. [49]). Sometimes this hypothesis is called quasi-monotonicity condition (see, e.g., [41]). In the case of ordinary differential equations property (5.14) is known as Kamke’s conditions (see, e.g., [49]). On the other hand property (G2) assumes a very special structure of the diffusion terms in (1.1). This is the price we have to pay in order to preserve the monotonicity properties of deterministic cooperative systems perturbed by stochastic terms. We also note that deterministic cooperative parabolic systems were studied by many authors (see, e.g., [29], [49], [55] [41] and the references therein). These systems generate order-preserving (or monotone) dynamical systems. We refer to [35] and [49] for the definitions and the basic facts. In random and stochastic cases order-preserving systems were considered in [5] and [20] (see also [18], where a class of cooperative parabolic Itˆ o’s equations was studied). Theorem 5.8 Assume that the hypotheses of Theorem 4.3 and also (F1), (F2), (G2) are 1 m valid. Let u(t) ≡ (u10 (t), . . . , um 0 (t)) and v(t) ≡ (v0 (t), . . . , v0 (t)) be solutions to problem (3.5) in the sense of Definition 3.1 with initial data u0 (x, ω) = (u10 (x, ω), . . . , um 0 (x, ω)) and v0 (x, ω) = (v01 (x, ω), . . . , v0m (x, ω)) respectively and with fˆ given by (4.9) instead of f . Assume that the predictable smoothing (4.6) of Itˆ o’s problem (3.5) has mild solutions uε (t) and vε (t) for these initial data with properties described in Theorem 4.3. Then the property ul0 (x, ω) ≤ v0l (x, ω), l = 1, . . . , m, f or almost all (x, ω) ∈ O × Ω

(5.15)

implies that ul (x, t, ω) ≤ v l (x, t, ω),

l = 1, . . . , m,

t ∈ [0, T ],

(5.16)

for almost all (x, ω) ∈ O × Ω. Proof. We can apply a deterministic comparison principle (see, e.g., [29], [49], [55]) to the solutions uε (t) and vε (t) of parabolic system (4.6) for almost every ω ∈ Ω. It gives that (5.15) implies the inequalities ulε (x, t, ω) ≤ vεl (x, t, ω),

l = 1, . . . , m,

t ∈ [0, T ],

for all ε > 0 and almost all (x, ω) ∈ O × Ω. Therefore (5.16) follows from (4.8).

40

CHUESHOV AND VUILLERMOT

Corollary 5.9 Assume that the hypotheses of Theorem 5.6 and also (F2), (G2) hold 1 m for u, v ∈ D. Let u(t) ≡ (u10 (t), . . . , um 0 (t)) and v(t) ≡ (v0 (t), . . . , v0 (t)) be solutions 1 m to problem (3.5) with initial data u0 (x, ω) = (u0 (x, ω), . . . , u0 (x, ω)) and v0 (x, ω) = 1 (v01 (x, ω), . . . , v0m (x, ω)). We assume that u0 and v0 are F0 -measurable from V = W2,B (O, D) possessing property (5.15). Then (5.16) holds. Proof. We apply Theorem 5.8 with f˜ given by (5.12) instead of f . Since by (G2) the difference f˜l (x, t, u, η) − f l (x, t, u, η) depends on ul only, property (F2) for f˜ follows from (F2) for f . Remark 5.10 Various types of comparison theorems for stochastic parabolic equations and systems were obtained earlier by several authors (see, e.g., [6], [30],[28],[38], [39] and the references therein). However the case of nonautonomous parabolic systems was not studied before. Moreover, we allow a dependence of the drift term f on the first order derivatives Du.

6

Examples

Now we consider several examples. Example 6.1 Let b = (b1 , . . . , bm ) ∈ Rm , bl > 0, l = 1, . . . , m. We consider the set , [0, b] = u = (u1 , . . . , um ) ∈ Rm : 0 ≤ ul ≤ bl , l = 1, . . . , m

which is said to be an interval in Rm with the end-points 0 and b. Assume that f (x, t, u) is a continuously differentiable function from O × [0, T ] × Rm into Rm , gj (x, t, u) satisfy (G0) and (G1). We also assume that for every (x, t) ∈ O × [0, T ] we have f l (x, t, u) ≥ 0, gjl (x, t, u) = 0,

l = 1, . . . , d, j ∈ Z+

(6.1)

for all u ∈ [0, b] of the form u = (u1 , . . . , ul−1 , 0, xl+1 , . . . , xm ) and f l (x, t, u) ≤ 0, gjl (x, t, u) = 0,

l = 1, . . . , d, j ∈ Z+

(6.2)

for all u ∈ [0, b] of the form u = (u1 , . . . , ul−1 , bl , xl+1 , . . . , xm ). As an example of functions f and gj with these properties we can consider f l (x, t, u) = −αl (x, t, u)ul + (bl − ul )

m '

βli (x, t, u)ui

(6.3)

i=1

and gjl (x, t, u) = σjl (x, t, u)ul (bl − ul ), j ∈ Z+ ,

(6.4)

where l = 1, . . . , m and αl > 0, βli ≥ 0 and σjl are smooth functions. The equations (1.1) and (1.2) with such a kind of nonlinear terms often arise in the study of population dynamics of ecological systems. Assumptions (6.1) and (6.2) imply the tangency conditions (FT) and (GT) with D = [0, b]. Therefore all conclusions of Theorems 5.5 and 5.6 are valid for this case. If we assume additionally that the functions f and gj satisfy (F2) and (G2), then the comparison principle given by Theorem 5.8 is applicable here. We note that in case the functions f and gj are of the form (6.3) and (6.4), the hypotheses (F2) and (G2) are true, if we assume that αl > 0, βli ≥ 0 and σjl are independent of u, for instance. We also note that the corresponding stochastic ordinary differential equation with drift and diffusion terms like (6.3) and (6.4) was studied in [20].

NON-RANDOM INVARIANT SETS

41

As a special case of Example 6.1 we consider the case of a single equation. Example 6.2 We consider the following class of real, parabolic, Itˆo initial-boundary value problems driven by the Wiener processes {Wj (t) : t ∈ R+ 0 , j = 1, . . . , r} :  du(x, t)        

   u(x, 0)      ∂u(x,t,ω) ∂n(a)

=

{div ((a(x)∇u(x, t)) + f (u(x, t), ∇u(x, t))} dt +

&r

j=1

gj (u(x, t))dWj (t, ω),

(x, t) ∈ O × R+ , (6.5)

=

ϕ(x) ∈ (u0 , u1 ), x ∈ O,

=

0,

(x, t, ω) ∈ ∂O × R+ 0 × Ω.

In relations (6.5) the third relation stands for the conormal derivative of u relative to a. We assume that the function a(x) is matrix-valued with entries that belong to C 1+µ (O) and which satisfy aij (·) = aji (·) for every i, j ∈ {1, . . . , n}. Moreover, the matrix a generates an uniformly elliptic operator (cf. assumption (A)). We also assume that there exist u0,1 ∈ R with u0 < u1 such that f ∈ C 2 ([u0 , u1 ] × Rd ) and f (u0 , η) ≥ 0, f (u1 , η) ≤ 0 for all η ∈ Rd and that gj ∈ C 2 ([u0 , u1 ]) and gj (u0 ) = gj (u1 ) = 0 for each j ∈ {1, . . . , r}. The results above give us the invariance of the interval [u0 , u1 ] and the comparison theorem for this case. We note that the models described in Examples 6.1 and 6.2 are relevant to the mathematical analysis of certain phenomena related to the long-time effects of recurrent and random selection in population dynamics. We refer to [22, 23] and also to [5], [11], [18], [36] for recent mathematical analyses of related problems and for further bibliography regarding their history. Example 6.3 We consider the following stochastic perturbation of Ginzburg–Landau type equations arising in the theory of superconductivity of liquids (see some discussion and the references in [1] and [50]): r ' , dul (x, t) = ∆ul + ul f (|u|) )dt + ul gj (|u|) dWj (t, ω),

(x, t) ∈ O × R+ ,

(6.6)

j=1

∂ul (x, t, ω) + k l (x)ul = 0, (x, t, ω) ∈ ∂O × R+ (6.7) 0 × Ω. ∂n $ %1/2 Here l = 1, . . . , m, |u| = (u1 )2 + . . . + (um )2 , and f (ρ) and gj (ρ) are smooth functions on R+ such that there exists R > 0 with property f (R) ≤ 0 and gj (R) = 0 (for instance f (ρ) = ρ2 − R1 and gj (ρ) = qj (ρ2 − R) with 0 < R ≤ R1 ). We assume that k l (x) are smooth non-negative functions on ∂O. Corollary 5.4 and Theorems 5.5 and 5.6 are applied here with , D = u = (u1 , . . . , um ) ∈ Rm : |u|2 ≡ u21 + . . . + u2m ≤ R2 . Acknowledgements

The authors’ research was supported in part by the ETH-Forschungsinstitut fuer Mathematik in Zuerich where this work was completed; the authors would like to thank Professor Marc Burger for his invitation to the above institution. They are also indebted to Professor Herbert Amann for a very stimulating discussion concerning non autonomous, semilinear, deterministic, parabolic partial differential equations.

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References [1] Abrahams, E., Tsuneto, T., Time Variation of the Ginzburg-Landau Order Parameter, Phys. Rev. 152 (1) (1966) 416-432. [2] Amann, H., Invariant Sets and Existence Theorems for Semilinear Parabolic and Elliptic Systems, J.Math. Anal Appl. 65 (1978) 432–467. [3] Amann, H., Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems, in: Schmeisser, H.-J., and H.Tribel, H., (eds), Function Spaces, Differential Operators and Nonlinear Analysis, p. 9–126. Teubner Verlag, Stuttgart Leipzig, 1993. [4] Amann, H., Linear and Quasilinear Parabolic Problems (1995), Birkh¨auser, Berlin. [5] Arnold, L., Chueshov, I.D, Order-Preserving Random Dynamical Systems: Equilibria, Attractors, Applications, Dynamics and Stability of Systems 13 (1998) 265–280. [6] Assing, S., Comparison of Systems of Stochastic Partial Differential Equations, Stoch. Processes Appl. 82 (1999) 259-282. [7] Aubin, J.P., Da Prato, G., The Viability Theorem for Stochastic Differential Inclusions, Stoch. Anal. and Appl. 16 (1998) 1-15. [8] Babin, A.V., Vishik, M.I., Attractors of Evolution Equations (1992), North Holland, Amsterdam. [9] Belopolskaya, Ja.I., Dalecky, Yu.L., Stochastic Equations and Differential Geometry (1990), Kluver Academic Publishers, Dordrecht. ´ ´ ´, B., Etude [10] Berge d’une Classe d’Equations aux D´eriv´ees Partielles Stochastiques: Existence, Unicit´e, Comportement Asymptotique, Th`ese 520 (2001) Universit´e HenriPoincar´e, Nancy. ´, B., Chueshov, I.D., Vuillermot, P.-A., On the Behavior of Solutions to [11] Berge Certain Parabolic SPDE’s Driven by Wiener Processes, Stoch. Processes Appl. 92 (2001) 237-263. ¨ fstro ¨ m, J., Interpolation Spaces: an Introduction (1976), Springer, [12] Bergh, J., Lo New York. [13] Bernfeld, S.R., Hu, Y.Y., Vuillermot, P.-A., Large-Time Asymptotic Equivalence for a Class of Non Autonomous Semilinear Parabolic Equations, Bull. Sci. Math. 122 (5) (1998) 337-368. ´ski, M., Flandoli, F., A Convergence Result for Stochastic [14] Brze´ zniak, Z., Capin Partial Differential Equations, Stochastics 24 (1988) 423-445. [15] Cardon-Weber, C., Millet, A., On Strongly Petrovski˘ı’s Parabolic SPDE’s in Arbitrary Dimension, Preprint 685 (2001), Laboratoire de Probabilit´es et des Mod`eles Al´eatoires, Universit´e Paris 6. [16] Cerrai, S., Stochastic Reaction-Diffusion Systems with Multiplicative Noise and NonLipschitz Reaction Term, Probab. Theory Relat. Fields 125 (2003) 271–304. [17] Chueh, K.N., Conley C.C., Smoller J.A., Positively Invariant Regions for Systems of Nonlinear Diffusion Equations, Indiana Univ. Math. J. 26 (1977) 373-392.

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[18] Chueshov, I.D., Order-Preserving Random Dynamical Systems Generated by a Class of Coupled Stochastic Semilinear Parabolic Equations, in: Fiedler, B., Gr¨oger, K., Sprekels, J. (eds.), International Conference on Differential Equations, EQUADIF 99, Berlin, Aug 1–7, 1999, vol 1. World Scientific, Singapore,(2000) 711–716. [19] Chueshov, I.D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics 1 (2002), Acta Scientific Publishing House, Kharkov. [20] Chueshov, I.D., Monotone Random Systems:Theory and Applications, Lecture Notes in Mathematics 1779 (2002), Springer, New York. [21] Chueshov, I.D., Vuillermot, P.-A., Long-Time Behavior of Solutions to a Class of Quasilinear Parabolic Equations with Random Coefficients, Ann. I. H. Poincar´e-AN 15 (2) (1998) 191-232. [22] Chueshov, I.D., Vuillermot, P.-A., Long-Time Behavior of Solutions to a Class of Stochastic Parabolic Equations with Homogeneous White Noise: Stratonovitch’s Case, Probab. Theory Relat. Fields 112 (1998) 149-202. [23] Chueshov, I.D., Vuillermot, P.-A., Long-Time Behavior of Solutions to a Class of Stochastic Parabolic Equations with Homogeneous White Noise: Itˆ o’s Case, Stoch. Anal. and Appl. 18 (4) (2000) 581-615. [24] Chueshov, I.D., Schmalfuß, B., Parabolic Stochastic Partial Differential Equations with Dynamical Boundary Conditions, (2003), submitted [25] Da Prato, G., Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992) Cambridge University Press, Cambridge. [26] Da Prato, G., Zabczyk, J., Ergodicity for Infinite Dimensional Systems (1996), Cambridge University Press, Cambridge. [27] Daners, D., Koch Medina, P., Abstract Evolution Equations, Periodic Problems and Applications (1992) Longman Sci. Tech., Harlow. [28] Donati-Martin, C., Pardoux, E., White Noise Driven SPDEs with Reflection, Prob. Theory Rel. Fields 95 (1993) 1–24. [29] Fife, P.C., Tang M.M., Comparison Principles for Reaction-Diffusion Systems: Irregular Comparison Functions and Applications to Questions of Stability and Speed of Propagation of Disturbances, J. Diff. Eqs. 40 (1981) 168-185. [30] Geiss, C., Manthey, R., Comparison Theorems for Stochastic Differential Equations in Finite and Infinite Dimensions, Stoch. Processes Appl. 53 (1994) 23–35. [31] Grecksch, W., Tudor, C., Stochastic Evolution Equations-a Hilbert Space Approach, Mathematical Research 85 (1995), Akademie Verlag, Berlin. ¨ ngy, I., On the Approximation of Stochastic Partial Differential Equations, Part [32] Gyo I, Stochastics 25 (1988) 59-85. ¨ ngy, I., On the Approximation of Stochastic Partial Differential Equations, Part [33] Gyo II, Stochastics 26 (1989) 129-164. [34] Hale, J.K., Asymptotic Behavior of Dissipative Systems, AMS-Mathematical Surveys and Monographs 25 (1988), American Mathematical Society, Providence.

44

CHUESHOV AND VUILLERMOT

[35] Hess, P., Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series 247 (1991), Longman Sci. Tech., Harlow. [36] G. Hetzer, G., Shen W., Zhu, S., Asymptotic Behavior of Positive Solutions of Random and Stochastic Parabolic Equations of Fisher and Kolmogorov Types, J. Dyn. Diff. Equations 14 (2002) 139–188. [37] Ikeda, N., Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North Holland Mathematical Library 24 (1981), North Holland, Amsterdam. [38] Kotelenez, P., Comparison Methods for a Class of Function Valued Stochastic Partial Differential Equations, Prob. Theory Rel. Fields 93 (1992) 1–19. [39] Manthey, R., Zausinger, T., Stochastic Evolution Equations in L2ν ρ , Stochastics and Stochastics Reports 66 (1999) 37-85. [40] Milian A., Invariance for Stochastic Equations with Regular Coefficients, Stoch. Anal. and Appl. 15 (1997) 91-101. [41] Pao, C.V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press, New York. [42] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer, New York. [43] Protter, P., Approximations of Solutions of Stochastic Differential Equations Driven by Semi-Martingales, The Annals of Probability 13 (3) (1985) 716-743. [44] Redlinger, R., Compactness Results for Time-Dependent Parabolic Systems, J. Diff. Eqs. 64 (1986) 133–153. [45] Redheffer, R., Walter, W., Invariant Sets for Systems of Partial Differential Equations I. Parabolic Equations, Arch. Rat. Mech. Anal. 67 (1978) 41-52. [46] Sanz-Sol´ e, M., Vuillermot, P.-A., H¨ older-Sobolev Regularity of Solutions to a Class of SPDE’s Driven by a Spatially Colored Noise, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 869-874. [47] Sanz-Sol´ e, M., Vuillermot, P.-A., Equivalence and H¨ older-Sobolev Regularity of Solutions for a Class of Non Autonomous Stochastic Partial Differential Equations, Ann. I. H. Poincar´e-PR 39 (2003) 703-742. [48] Seidler, J., A Note on the Core of Flandoli’s Regularity Results, Preprint, 1993. [49] Smith, H.L., Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, AMS-Mathematical Surveys and Monographs 41 (1995), American Mathematical Society, Providence. [50] Temam, R., Infinite–Dimensional Dynamical Systems in Mechanics and Physics, (1988), Springer, New York. [51] Tessitore, G., Zabczyk, J., Wong-Zaka¨ı Approximations of Stochastic Evolution Equations, (2002), submitted [52] Tribel, H., Interpolation Theory, Functional Spaces and Differential Operators (1978), North Holland, Amsterdam.

NON-RANDOM INVARIANT SETS

45

[53] Twardowska, K., Approximation Theorem of Wong – Zakai Type for Nonlinear Stochastic Partial Differential Equations, Stoch. Anal. Appl. 13 (1995) 601–626. [54] Twardowska, K., Wong-Zaka¨ı Approximations for Stochastic Differential Equations, Acta Applic. Math. 43 (1996) 317-359. [55] Vishnevski˘ı, M.P., Stabilization of Solutions of Weakly Coupled Cooperative Parabolic Systems (Russian), Mat. Sb. 183 (10) (1992) 45-62. [56] Vuillermot, P.-A., Global Exponential Attractors for a Class of Nonautonomous Reaction-Diffusion Equations on RN , Proc. Amer. Math. Soc. 116 (3) (1992) 775-782. [57] Wong, E., Stochastic Processes in Information and Dynamical Systems, Mc Graw-Hill Series in Systems Science (1971), Mc Graw-Hill, New York. [58] Wong, E., Zaka¨ı, M., Riemann-Stieltjes Approximations of Stochastic Integrals, Z. Wahrscheinlichkeitstheorie u. verw. Gebiete 12 (1969) 87-97. [59] Zabczyk, J., Stochastic Invariance and Consistency of Financial Models, Rendiconti Mat. Acc. Lincei 9 (2000) 67-80.