Stochastic Functional Partial Differential Equations: Existence, Uniqueness and Asymptotic Decay Property

STOCHASTIC FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS: EXISTENCE, UNIQUENESS AND ASYMPTOTIC STABILITY TOM´ AS CARABALLO Dpto. Ecuaciones Diferenciales y An´alisis Num´erico. Universidad de Sevilla. Apartado de Correos 1160. 41080–SEVILLA, SPAIN Email: [email protected] KAI LIU* AUBREY TRUMAN Department of Mathematics University of Wales Swansea Singleton Park, Swansea SA2 8PP, UK Email: [email protected] ABSTRACT Existence and uniqueness of strong solutions for a class of stochastic functional differential equations in Hilbert spaces are established. Sufficient conditions which guarantee the transference of mean square and pathwise exponential stability from stochastic partial differential equations to stochastic functional partial differential equations are studied. The stability results derived are also applied to stochastic ordinary differential equations with hereditary characteristics. In particular, as a direct consequence our main results improve some of those from Mao and Shah [13] in which it is proved that under certain conditions pathwise exponential stability is transferred from nondelay equations to delay ones if the constant time lag appearing in the problem is sufficiently small, while in our treatment the transference actually holds for arbitrary bounded delay variables not only in finite but in infinite dimensions. Keywords: Stochastic partial differential equation; Stochastic functional partial differential equation; Mean square and pathwise exponential stability. AMS 1991 Classifications: 60H15, 34K40. * Author of Correspondence. 1 1. INTRODUCTION The study of stochastic functional differential equations is motivated by the fact that when one wants to model some evolution phenomena arising in Physics, Biology and Engineering etc., some hereditary characteristics such as aftereffect, time lag and time delay can appear in the variables (see, for example, Kolmanovskii and Myshkis [8], Mohammed [15]). On the other hand, one of the most important and interesting problems in the analysis of stochastic functional differential equations is their stability, the theory of which (mainly for finite dimensional systems) has been greatly developed over the last several years. As is well known, in the case without any hereditary features, Lyapunov’s technique is available to obtain sufficient conditions for the stability of solutions of stochastic (partial) differential equations. However, in the case of stochastic differential equations with hereditary properties, for instance, even with constant time delays, Lyapunov’s method becomes difficult to apply effectively as Krasovskii [10] pointed out for the study of stability of ordinary differential equations, and as Kushner [11] and El’sgol’ts and Norkin [5] (among others) did for stochastic differential equations. The main reason is that it is much more difficult (or even impossible in some cases) to construct proper Lyapunov functions (or functionals) for stochastic functional differential equations than for those without any hereditary characteristics. As a consequence, a comparison technique has been developed by various authors such as Krasovskii [10] and Mao and Shah [13] (among others). Let us illustrate this point of our motivation in more detail. Consider the following stochastic functional differential equation x(t) = Zt 0 f(s, x(s), x(s−h1)) ds +Zt 0Z0 −h2 g(s, x(s+r))h(r)drdw(s), t > 0,(1) where h1>0, h2>0, or equivalently, x(t) = Zt 0 f(s, x(s), x(s)) ds +Zt 0 g(s, x(s)) dw(s) +Zt 0hf(s, x(s), x(s−h1)) −f(s, x(s), x(s))ids +Zt 0hZ0 −h2 g(s, x(s+r))h(r)dr −g(s, x(s))idw(s). (2) We can regard (1) as the perturbed system of the corresponding stochastic differential equation without hereditary characteristics x(t) = Zt 0 f(s, x(s), x(s)) ds +Zt 0 g(s, x(s)) dw(s), t > 0.(3) Clearly, if the time lag scales h1>0, h2>0 are sufficiently small, the perturbation term Zt 0hf(s, x(s), x(s−h1)) −f(s, x(s), x(s))ids 2 +Zt 0hZ0 −h2 g(s, x(s+r))h(r)dr −g(s, x(s))idw(s), could be expected to be so small that the perturbed equation (1) would behave asymptotically as Equation (3) does. For instance, we could expect that if Equation (3) is exponentially stable and the time lags h1>0, h2>0 are small enough, then Equation (1) will remain exponentially stable. So, in order to find out whether the functional equation (1) is exponentially stable, one can check the exponential stability of the equation (3) and then compute whether the time lags h1>0, h2>0 are sufficiently small. In other words, the difficult problem of stability for functional equations would have been transferred to an easier one (the stability of equations without hereditary characteristics). Motivated by the intuitive ideas described above, Mao and Shah [13] obtained some sufficient conditions for the p-th moment exponential stability (and also pathwise stability) of stochastic ordinary differential delay equations. For example, consider the following onedimensional stochastic delay differential equation dx(t) = f(t, x(t), x(t−h)) dt +g(t, x(t)) dw(t), t > 0,(4) where h > 0, or equivalently, dx(t) = f(t, x(t), x(t)) dt +g(t, x(t)) dw(t) + [f(t, x(t), x(t−h)) −f(t, x(t), x(t))] dt. (5) It was proved in [13] that under some circumstances pathwise exponential stability is transferred from the nondelay equation (i.e., h= 0 in (5)) to the delay one (4) if the constant time lag h > 0 appearing in the problem is sufficiently small. Nevertheless, it is worth pointing out that the results derived in [13] are somewhat restrictive for many practical applications. In fact, the situation turns out to be rather complicated when one considers the general functional differential equations, even the usual stochastic delay differential systems. For instance, there exist a wide variety of interesting problems in which it is possible to ensure that if nondelay equations are exponentially stable, then delay ones remain exponentially stable whatever the delay interval could be, what is more even if, the delays are not constants. In this work, by a completely different approach from that in [13] we shall carry out a much more delicate investigation. For instance, by applying some general results to be derived in Section 4 to the equation (1), we can prove that under some circumstances mean square and pathwise exponential stability of (1) are transferable from the equation (3) for arbitrary delay constant h1>0, but for the time lag h2>0 which must be sufficiently small. One of the main aims of this paper is to give sufficient conditions which contain as a special case the corresponding results in finite dimension (that is, for stochastic ordinary differential equations) to transfer the exponential stability of stochastic partial differential equations to stochastic functional partial differential equations. The problem we are referring to is devoted to the consideration of an infinite dimensional version of (1) in which f has the following form: f(t, x, y) = A(t, x) + f1(t, y), 3 with the family of (non-linear) operators A(t, ·) satisfying some kinds of coercivity conditions (see Section 2) as well as f1satisfying Lipschitz continuous ones. We would also like to mention that, in some sense, a suitable coercivity condition implies the (exponential) stability of solutions in mean square (and also pathwise exponential stability) in nondelay cases (see Caraballo and Liu [3] and Chow [4]). In addition to this, we will be able to assure exponential stability in mean square (and, as a consequence, pathwise exponential stability) for a great number of finite dimensional stochastic functional differential equations while the results of Mao and Shah [13] only give this kind of stability for certain delay systems in which the delay must be constant and sufficiently small. Here, we shall analyze only the second moment of solutions. Although we should emphasize that this study can be extended to the p-th moment (p≥2), which is important if it permits us to obtain some information about the stability of sample paths. We also remark that, as is well known, mean square exponential stability, energy equality and Borel–Cantelli’s lemma could imply pathwise exponential stability (see, for instance, Caraballo and Liu [3], Mao [12]). In Section 2 we begin with some preliminary results. We have not seen a general treatment on existence and uniqueness of strong solutions of stochastic functional differential equations in infinite dimensions in the literature. In Section 3 we shall first establish a result, which is easy to verify in many situations, of existence and uniqueness of strong solutions for a class of stochastic partial functional differential equations. The results of exponential stability are studied in Section 5. Finally, two examples are given in Section 6 to illustrate the theory derived in the preceding sections. 2. PRELIMINARIES First of all, we introduce the framework in which our analysis is going to be carried out. Let V, H, K be real, separable Hilbert spaces such that V ,→H≡H0,→V0, where V0is the dual of Vand the injections are continuous and dense. In particular, we also assume both Vand V0are uniformly convex. We denote by k · k ,| · | and k · k∗the norms in V,Hand V0respectively; by h·,·i the duality product between V0, V , and by (·,·) the scalar product in H. Let w(t) be a Wiener process defined on a certain complete probability space (Ω,F, P) and take values in the separable Hilbert space K, with incremental covariance operator W. Let (Ft)t≥0be the σ-algebras generated by {w(s),0≤s≤t}, then w(t) is a martingale relative to (Ft)t≥0and we have the following representation of w(t) : w(t) = ∞ X i=1 βi(t)ei, where {ei}i≥1is an orthonormal set of eigenvectors of W,βi(t) are mutually independent real Wiener processes with incremental covariance λi>0, Wei=λieiand trW= P∞ i=1 λi<∞(tr denotes the trace of an operator, see Pardoux [16]). 4 For an operator G∈ L(K, H), the space of all bounded linear operators from Kinto H, we denote by kGk2its Hilbert-Schmidt norm, i.e. kGk2 2= tr(GWG∗). Given h≥0, p≥2 and T > 0, we denote by Ip(−h, T;V) the space of all V–valued processes (x(t))t∈[−h,T ](we will write x(t) for short) measurable (from [−h, T]×Ω into V), and satisfying: (1). x(t) is Ft-measurable almost surely in t(in the sequel, we will write a.e.t.), where we set Ft=F0for t≤0; (2). ERT −hkx(t)kpdt < +∞. It is not difficult to check that the space Ip(−h, T;V) is a closed subspace of Lp(Ω × [−h, T],F ⊗ B([−h, T]), dP ⊗dt;V),where B([−h, T]) denotes the Borel σ–algebra on [−h, T]. We also write L2(Ω; C(−h, T ;H)) instead of L2(Ω,F, dP;C(−h, T;H)), where C(−h, T;H) denotes the space of all continuous functions from [−h, T] into H. Let C=C([−h, 0], H) be the space of all continuous functions from [−h, 0] into Hwith sup-norm kψkC= sup−h≤s≤0|ψ(s)|,ψ∈C,Lp V=Lp([−h, 0]; V) and Lp H= Lp([−h, 0]; H). Given a stochastic process x(t)∈Ip(−h, T;V)∩L2(Ω; C(−h, T;H)), we associate with an Lp V∩C-valued stochastic process xt: Ω →Lp V∩C,t≥0, by setting xt(s)(ω) = x(t+s)(ω), s∈[−h, 0]. The first purpose of this paper is to establish an existence and uniqueness result for a class of nonlinear stochastic partial functional differential equations of the form (dx(t) = (A(t, x(t)) + f(t, xt))dt +g(t, xt)dw(t), t ∈[0, T] x(t) = ψ(t), t ∈[−h, 0],(6) where, in general, the operators are assumed to be nonlinear. In fact, we are interested in the case in which A(t, ·) : V→V0is a family of nonlinear monotone and coercive operators, f(t, ·) : C→Hand g(t, ·) : C→ L(K, H) are Lipschitz continuous. It is worth pointing out that, in many applications, Ausually denotes a partial differential operator (linear or nonlinear), while fand gare first order partial differential ones (cf. [3][16][17]). We will first establish the desired results by a variational type of argument, which is similar to that one carried out by Pardoux’s [16] for a case without delays, but subject to necessary changes to make our scheme go through when f(t, ·) : C→Hand g(t, ·) : C→ L(K, H). Then we will treat the more general case with f(t, ·) : L2 V→Hand g(t, ·) : L2 V→ L(K, H) by using a Galerkin approximation technique. 5 3. EXISTENCE AND UNIQUENESS OF SOLUTIONS Let A(t, ·) : V→V0be a family of (nonlinear) operators defined a.e.t. and p≥2. Assume the following hypotheses: (a.1) Coercivity: ∃α > 0, λ, ν ∈R1such that: −2hA(t, x), xi+λ|x|2+ν≥αkxkp,∀x∈V , a.e.t.; (a.2) Monotonicity: −2hA(t, x)−A(t, y), x −yi+λ|x−y|2≥0,∀x, y ∈V , a.e.t.; (a.3) Boundedness: ∃γ > 0 : kA(t, x)k∗≤γkxkp−1,∀x∈V , a.e.t.; (a.4) Hemicontinuity: θ∈R1→ hA(t, x +θy), zi ∈ R1is continuous ∀x, y, z ∈V , a.e.t.; (a.5) Measurability: t∈(0, T)→A(t, x)∈V0is Lebesgue −measurable ∀x∈V , a.e.t.. Let f(t, ·) : L2 H→Hbe a family of nonlinear operators defined a.e.t., and satisfy: (f.1) f(t, 0) = 0 ; (f.2) Lipschitz condition: ∃k1=k1(h)>0 such that |f(t, η)−f(t, ξ)| ≤ k1kη−ξkC,∀η, ξ ∈C , a.e.t.; (f.3) Measurability: t∈(0, T)→f(t, η)∈His Lebesgue–measurable, ∀η∈L2 H. And let g(t, ·) : L2 H→ L(K, H) be another nonlinear operator family defined a.e.t. and satisfy: (g.1) g(t, 0) = 0 ; (g.2) Lipschitz condition: ∃k2=k2(h)>0 such that kg(t, η)−g(t, ξ)k2≤k2kη−ξkC,∀η, ξ ∈C , a.e.t.; (g.3) Measurability: t∈(0, T)→g(t, η)∈ L(K, H) is Lebesgue–measurable ∀η∈ L2 H. Given an initial value ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) , 6 the objective in this section is that under the conditions described above, we hopefully find a unique process x(t)∈Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) such that              x(t) = ψ(0) + Zt 0 [A(s, x(s)) + f(s, xs)] ds +Zt 0 g(s, xs)dw(s), P −a.s., ∀t∈[0, T], x(t) = ψ(t), P −a.s., ∀t∈[−h, 0]. (∗) Remark. (1) First, it is worth mentioning that although the results can be proved for p > 1, the interesting situations in the applications appear when p≥2. Because of this, we content ourselves with the analysis of the case p≥2. (2). We observe that if x∈L2(0, T;C), then in view of (f.1)–(f.3), f(x)∈L2(0, T;H) where f(x)(t) = f(t, xt).Moreover, the mapping x∈L2(0, T;C)7→ f(x)∈L2(0, T;H) is continuous and so measurable. Since η∈C7→ f(t, η)∈His continuous a.e.t., it follows that if x(t), t ∈[−h, T] is an H–valued and Ft–adapted stochastic process, so is f(t, xt), t ≥0. In addition, if x∈L2(Ω ×(0, T); C),then f(x)∈L2(Ω ×(0, T); H) . Finally, if xnis a bounded sequence in L2(Ω×(0, T); C), f(xn) is bounded in L2(Ω× (0, T); H) once again. Similar results are deduced from (g.1)–(g.3) for g:L2(0, T;C)→L2(0, T;L(K, H)) defined by g(x)(t) = g(t, xt).These remarks imply that the integrals appearing in (∗) are well defined. (3). In order to avoid unnecessary technicalities in the following stability analysis, we content ourselves with the consideration of Equation (∗) instead of a more general one. However, it is worth pointing out that under some similar conditions, it is possible to extend the results derived here to more general stochastic systems involving coefficients such as f(t, x(t), xt) and g(t, x(t), xt) as well as to remove Conditions (f.1), (g.1). 3.1. Uniqueness of solutions Now we shall prove that there exists at most one solution of (∗). This result will be deduced mainly from (a.2) and Itˆo’s formula. Theorem 1. Assume the preceding hypotheses hold. Then, there exists at most one solution of (∗)in Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) . Proof. Suppose that x, y ∈Ip(−h, T ;V)∩L2(Ω; C(−h, T;H)) are two solutions of (∗). Then, applying Itˆo’s formula to (∗) and taking into account (a.2), we obtain |x(t)−y(t)|2= 2 Zt 0 hA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids + 2 Zt 0 (f(s, xs)−f(s, ys), x(s)−y(s)) ds 7 + 2 Zt 0 (x(s)−y(s),(g(s, xs)−g(s, ys)) dw(s)) +Zt 0 kg(s, xs)−g(s, ys)k2 2ds. ≤λZt 0 |x(s)−y(s)|2ds + 2 Zt 0 |x(s)−y(s)||f(s, xs)−f(s, ys)|ds + 2 Zt 0 (x(s)−y(s),(g(s, xs)−g(s, ys)) dw(s)) +Zt 0 kg(s, xs)−g(s, ys)k2 2ds. Now, it follows from (f.2) and (g.2) that for any t∈[0, T ] Esup 0≤s≤t |x(s)−y(s)|2≤(|λ|+ 1) Zt 0 E|x(s)−y(s)|2ds + (k2 1+k2 2)Zt 0 Ekxs−ysk2 Cds + 2Esup 0≤s≤tZs 0 (x(r)−y(r),(g(r, xr)−g(r, yr))dw(r)). (7) However, by Burkholder-Davis-Gundy’s inequality, we have Esup 0≤s≤tZs 0 (x(r)−y(r),(g(r, xr)−g(r, yr))dw(r)) ≤3Ensup 0≤s≤t |x(s)−y(s)|hZt 0 kg(s, xs)−g(s, ys)k2 2dsi1/2o ≤1 4Esup 0≤s≤t |x(s)−y(s)|2+KZt 0 Ekg(s, xs)−g(s, ys)k2 2ds ≤1 4Esup 0≤s≤t |x(s)−y(s)|2+K·k2 2Zt 0 Ekxs−ysk2 Cds (8) for some positive constant K > 0. On the other hand, since x(s) = y(s) for s≤0, we easily get Zt 0 Ekxs−ysk2 Cds =Zt 0 Esup −h≤r≤0 |xs(r)−ys(r)|2ds =Zt 0 Esup −h≤r≤0 |x(s+r)−y(s+r)|2ds ≤Zt 0 Esup 0≤r≤s |x(r)−y(r)|2ds. (9) 8 Thus, it follows from (7)–(9) Esup 0≤s≤t |x(s)−y(s)|2≤2h|λ|+1+k2 1+k2 2+2k2 2KiZt 0 Esup 0≤r≤s |x(r)−y(r)|2ds, ∀t∈[0, T]. Now, Gronwall’s lemma obviously implies uniqueness. Remark. (1) Observe that if we assume the following monotonicity hypothesis (a.2)’ For all ξ, η ∈Lp(−h, T;V) with ξ0=η0such that −2hA(t, ξ(t))+ f(t, ξt)−A(t, η(t)) −f(t, ηt), ξ(t)−η(t)i+λ|ξ(t)−η(t)|2 ≥ kg(t, ξt)−g(t, ηt)k2 2a.e.t ∈[0, T], instead of (a.2), uniqueness would have been easily deduced. Indeed, notice that in this case, Itˆo’s formula and (a.2)’ imply E|x(t)−y(t)|2≤λZt 0 E|x(s)−y(s)|2ds ∀t∈[0, T], for arbitrary two solutions x, y of the problem. Moreover, it is sufficient to assume an integral version of (a.2)’, namely, (a.2)” For all ξ, η ∈Lp(−h, T;V) with ξ0=η0such that −2Zt 0 hA(s, ξ(s))+ f(s, ξs)−A(s, η(s)) −f(s, ηs), ξ(s)−η(s)ids +λZt 0 |ξ(s)−η(s)|2ds ≥Zt 0 kg(s, ξs)−g(s, ηs)k2 2ds a.e.t ∈[0, T]. (2) Conversely, it is not difficult to prove by carrying out similar computations to the ones in (8) that (a.2), (f.2) and (g.2) imply (a.2)” (of course, with different parameter λ from that one in (a.2)). 3.2. Existence of strong solutions First of all, we state a theorem on existence and uniqueness of solutions of stochastic evolution equations. Next, by means of this result we will prove the desired existence of solution of (∗). Theorem 2. Assume (a.1)–(a.5) hold with λ= 0 . Then, there exists a unique process x∈Ip(0, T;V)∩L2(Ω; C(0, T;H)) such that x(t) = ψ(0) + Zt 0 [A(s, x(s)) + f1(s)] ds +M(t), P −a.s. , ∀t∈[0, T], 9 By virtue of (a.2), we get −2EZT 0 hA(xn)−A(z), xn−zids +λE ZT 0 |xn−z|2ds ≥0 (36) for all z∈Lp(Ω ×(0, T); V)∩L2(Ω ×(0, T); H) . Nevertheless, (27), (30) and (31) allow us to take limits in (36) and, it follows −2EZT 0 hv−A(z), x −zids +λE ZT 0 |x−z|2ds ≥0.(37) Now, if we set z=x−θz2(for θ > 0, z2∈Lp(Ω ×(0, T); V)∩L2(Ω ×(0, T); H) ), we get −2EZT 0 hv−A(x−θz2), θz2ids +λθ2EZT 0 |z2|2ds ≥0.(38) In (38), we divide by θ, take limit as θ→0 and then use the hemicontinuity (a.4) to obtain: −EZT 0 hv−A(x), z2ids ≥0,∀z2∈Lp(Ω ×(0, T); V)∩L2(Ω ×(0, T); H),(39) and therefore v=A(x). Since (32) is true with v=A(x), the proof of Theorem 3 is now complete. 4. EXISTENCE AND UNIQUENESS BY A GALERKIN APPROXIMATION First of all, we would like to point out that in many situations, it is convenient to consider another norm k·kL2 Hinstead of k·kCfor initial datum spaces in (∗). The arguments in the last section still carry through when the norm k · kCis replaced by k · kL2 Hin (f.2) and (g.2). In this section we shall investigate an existence and uniqueness result for (∗) in a more general situation. Precisely, let us assume hypotheses (a.1)–(a.5) for the family of operators A(t, ·). Suppose f(t, ·) : L2 H→His a family of nonlinear operators defined a.e.t. and satisfying: (F.1) f(t, 0) = 0 ; (F.2) Lipschitz condition: ∃k1=k1(h)>0 such that |f(t, η)−f(t, ξ)| ≤ k1kη−ξkL2 V,∀η, ξ ∈L2 V,a.e.t.; (F.3) Measurability: t∈(0, T)→f(t, η)∈His Lebesgue–measurable, ∀η∈L2 V. And let g(t, ·) : L2 H→ L(K, H) be another nonlinear operator family defined a.e.t. and satisfying: 16 (G.1) g(t, 0) = 0 ; (G.2) Lipschitz condition: ∃k2=k2(h)>0 such that kg(t, η)−g(t, ξ)k2≤k2kη−ξkL2 V,∀η, ξ ∈L2 V,a.e.t.; (G.3) Measurability: t∈(0, T )→g(t, η)∈ L(K, H) is Lebesgue–measurable ∀η∈ L2 V. Theorem 4. In addition to (a.1)–(a.5), (F.1)–(F.3) and (G.1)–(G.3), suppose the two following hypotheses hold: (C) There exist α > 0, λ, ν, τ ∈R1such that for all ξ∈Lp(−h, T;V) −2hA(t, ξ(t))+ f(t, ξt), ξ(t)i+λ|ξ(t)|2+τkξ0k2 Lp V+ν ≥αkξ(t)kp+kg(t, ξt)k2 2a.e.t ∈[0, T]; (M) For all ξ, η ∈Lp(−h, T;V)with ξ0=η0, −2hA(t, ξ(t)) + f(t, ξt)−A(t, η(t)) −f(t, ηt), ξ(t)−η(t)i+λ|ξ(t)−η(t)|2 ≥ kg(t, ξt)−g(t, ηt)k2 2a.e.t ∈[0, T]. Then, for each ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) there exists a unique solution of the problem (∗)in Ip(−h, T;V)∩L2(Ω; C(−h, T ;H)). Proof. Uniqueness follows immediately from Itˆo’s formula, Assumption (M) and Gronwall’s lemma. Indeed, let x, y ∈Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) be two solutions to (∗). Then, it is easy to obtain E|x(t)−y(t)|2= 2 Zt 0 EhA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids + 2 Zt 0 E(f(s, xs)−f(s, ys), x(s)−y(s)) ds +Zt 0 Ekg(s, xs)−g(s, ys)k2 2ds. ≤λZt 0 E|x(s)−y(s)|2ds, from which uniqueness follows by means of Gronwall’s lemma. As for the existence, we shall split the proof into the following four steps. STEP 1. Finite-dimensional approximation Let {v1, v2, ..., vn, ...}be an orthonormal basis of Hwhere vi∈Vfor all i≥1. Let Vn=Hn=V0 ndenote the vector space generated by {v1, ...vn}. Let Pn∈ L(H, Hn) be 17 the orthogonal projection from Honto Hn. Then, Pncan be extended to an operator ˜ Pn from V0onto V0 nin the following way ˜ Pnu= n X i=1 hu, viivi, u ∈V0. Let {l1, l2, ..., ln, ...}denote an orthonormal basis in K, and let πn∈ L(K, Kn) be the projection from Konto Kn= span{l1, ..., ln}. Now we consider the problem (∗1)      d(xn(t), vi) = hA(t, xn(t)) + f(t, xn t), viidt + (vi, g(t, xn t)d(πnw(t))),1≤i≤n, xn(t) = Pnψ(t), t ∈[−h, 0]. This equation can be rewritten in an equivalent way as follows. Let An(t, ·) denote the family of operators from Vninto V0 ndefined as An(t, x) = ˜ PnA(t, x), x ∈Vn. Assume fn(t, ·) : L2 Hn→Hngiven by fn(t, ξ) = Pnf(t, ξ) for ξ∈L2 Hn,gn(t, ·) : L2 Hn→ L(Kn, Hn) defined by gn(t, ξ) = Png(t, ξ) for ξ∈L2 Hn, and, finally let Wn(t) denote the Kn–valued Wiener process defined by Wn(t) = πnw(t). Then, Eq. (∗1) can be rewritten as (∗2) (dxn(t) = (An(t, xn(t)) + fn(t, xn t)dt +gn(t, xn t)dWn(t) xn(t) = ψn(t) = Pnψ(t), t ∈[−h, 0]. Although Eq. (∗2) can be considered as an Itˆo stochastic differential equation in Rn, we can not apply the classic results on existence and uniqueness of solutions since Andoes not satisfy a Lipschitz type of condition. However, we can apply to this situation the results proved in the preceding section, i.e., Theorem 3 with k · kCreplaced by k · kL2 Hin (f.2) and (g.2). Indeed, it is easy to check that An, fn, gn, Wnand ψnsatisfy the assumptions in Theorem 3 by replacing V,H,V0by Vn,Hn,V0 n. Therefore, for each natural number n≥1, there exists a unique xn∈Ip(−h, T;Vn)∩L2(Ω; C(−h, T;Hn)) which is the solution to (∗2). Owing to the natural injections, we have that, in fact, xn∈Ip(−h, T;V)∩L2(Ω; C(−h, T;H)). STEP 2. A priori computations As in the proof of Theorem 3, we set xn:= xn(s), An(xn) := An(s, xn(s)), fn(xn) := fn(s, xn s) and gn(xn) := gn(s, xn s). The energy equality implies |xn(t)|2=|ψn(0)|2+ 2 Zt 0 hAn(xn) + fn(xn), xnids + 2 Zt 0 (xn, gn(xn)dWn(s)) + trhhZ. 0 gn(xn)dWn(s)iit, (40) 18 and consequently, |xn(t)|2=|ψn(0)|2+ 2 Zt 0 hA(xn) + f(xn), xnids + 2 Zt 0 (xn, g(xn)dWn(s)) + trhhZ. 0 g(xn)dWn(s)iit. (41) On the one hand, since trhhZ. 0 g(xn)dWn(s)iit=Zt 0 kg(xn)πnk2 2ds ≤Zt 0 kg(xn)k2 2ds, we immediately get from Condition (C) that |xn(t)|2≤ |ψn(0)|2+ 2 Zt 0 hA(xn) + f(xn), xnids + 2 Zt 0 (xn, g(xn)dWn(s)) + Zt 0 kg(xn)k2 2ds ≤ |ψn(0)|2+νt +tτkψnk2 L2 Hn +λZt 0 |xn|2ds −αZt 0 kxnkpds + 2 Zt 0 (xn, g(xn)dWn(s)), (42) which, after taking expectations, yields that E|xn(t)|2+αZt 0 Ekxnkpds ≤E|ψ(0)|2+νt +tτEkψk2 L2 H+λZt 0 E|xn|2ds. (43) Consequently, there exist positive constants c1, c2such that sup −h≤t≤T E|xn(t)|2≤c1(44) EZT 0 kxn(t)kpdt ≤c2,(45) and, as p≥2, there exists c3>0 such that EZT 0 kxn(t)k2dt ≤c3.(46) On the other hand, (42) immediately yields that Esup 0≤t≤T |xn(t)|2≤E|ψ(0)|2+νT +TτEkψk2 L2 H+|λ|ZT 0 E|xn(t)|2dt + 2Esup 0≤t≤TZt 0 (xn, g(xn)dWn(s)). (47) 19 Evaluating the last term in (47) by applying Burkholder-Davis-Gundy’s inequality (cf. see [12]), (G.2) and taking into account (46), we have 2Esup 0≤t≤TZt 0 (xn, g(xn)πndw(s))≤6E  ZT 0 |xn|2kg(xn)k2 2ds!1/2  ≤1 3Esup 0≤t≤T |xn(t)|2+c4ZT 0 Ekg(xn)k2 2ds ≤1 3Esup 0≤t≤T |xn(t)|2+c4k2 2ZT 0 Ekxn sk2 L2 Vds ≤1 3Esup 0≤t≤T |xn(t)|2+c5+c6Ekψk2 L2 V. (48) Hence, there exists a positive constant c7such that Esup 0≤t≤T |xn(t)|2≤c7.(49) So, we have finally proved that {xn}n≥1is bounded in Ip(−h, T;V)∩L2(Ω; C(−h, T;H)), {A(xn)}n≥1is bounded in Lp0(Ω ×(0, T); V0), {f(xn)}n≥1is bounded in L2(Ω ×(0, T); H), {g(xn)}n≥1is bounded in L2(Ω ×(0, T); L(K, H)), where A(xn), f(xn), g(xn) are defined in the obvious way and p0denotes the conjugate of p. STEP 3. Taking weak limits Owing to the last assertions in Step 2, we can ensure that there exists a subsequence {xnk}of {xn}such that xnk* x in Ip(−h, T;V) and weakly star in L2(Ω; L∞(−h, T;H)), xnk(T)* ξ in L2(Ω; H), A(xnk)* χ in Lp0(Ω ×(0, T); V0), f(xnk)* σ in L2(Ω ×(0, T); H), g(xnk)* ζ in L2(Ω ×(0, T); L(K, H)). Let θ:R1→R1be defined as θ(t) = 0 if t < 0 1 if t≥0. If ϕis a function from [0, T] into R1, we can define another function ϕ: (−ρ, T +ρ)→R1(where ρis a positive fixed number) in the following way: ϕ(t) = nϕ(t) if t∈[0, T] 0 otherwise. 20 This permits us to rewrite Eq. (∗1) (with n=nk) as follows (xnk(t), vi) = (ψ(0), vi)θ(t)−(xnk(T), vi)θ(t−T) +Zt 0 hA(xnk) + f(xnk), viids +Zt 0 (vi, g(xnk)πnkdw(s)),∀t∈(−ρ, T +ρ), i = 1, ..., nk. (50) Observe that, as the map φ∈L2(Ω×(0, T); L2(K, H)) 7→ R. 0φ(s)dw(s)∈L2(Ω×(0, T); H) is linear and continuous, then it is weakly continuous (where L2(K, H) denotes the space of all Hilbert-Schmidt operators from Kinto H). Now, we shall prove that g(xnk)πnk* ζ, as k→ ∞, in L2(Ω ×(0, T); L2(K, H)). Indeed, this convergence is equivalent to EZT 0 tr(Q∗g(xnk)πnk)dt →EZT 0 tr(Q∗ζ)dt, for all Q∈L2(Ω ×(0, T); L2(K, H)), and also to EZT 0 tr(g(xnk)πnkQ)dt →EZT 0 tr(ζQ)dt. Therefore, it is sufficient to prove that Qπnk→Qin L2(Ω ×(0, T); L2(K, H)). But this is an immediate consequence of Theorem I. 2.3 in Pardoux [16]. Now, we can take weak limits in (50) and obtain: (x(t), vi) = (ψ(0), vi)θ(t)−(ξ, vi)θ(t−T) + Zt 0 hχ+σ, viids +Zt 0 (vi, ζ)dw(s),∀t∈(−ρ, T +ρ),∀i≥1, (51) so it follows that ξ=x(T) dx(t)=(χ(t) + σ(t))dt +ζ(t)dw(t), t ∈[0, T],(52) x(t) = ψ(t), t ∈[−h, 0].(53) Therefore, it remains to prove that χ+σ=A(x) + f(x) and ζ=g(x). This will be done in the next step. STEP 4. Final step: the monotonicity method Consider v∈Lp(Ω ×(−h, T); V)∩L2(Ω ×(−h, T); H) and set unk=−2EZT 0 e−λthA(xnk) + f(xnk)−A(v)−f(v), xnk−vidt +λE ZT 0 e−λt|xnk−v|2dt −EZT 0 e−λtkg(xnk)−g(v)k2 2dt. (54) 21 Note that unk≥0 due to Assumption (M). On the other hand, we can take limits in the terms of (54) except for the following term ynk=−2EZT 0 e−λthA(xnk) + f(xnk), xnkidt +λE ZT 0 e−λt|xnk|2dt −EZT 0 e−λtkg(xnk)k2 2dt. (55) But, (41) immediately yields that E|xnk(t)|2=E|Pnkψ(0)|2+ 2EZt 0 hA(xnk) + f(xnk), xnkids +EtrhhPnkZ. 0 g(xnk)dWnkiit. (56) In particular, (56) proves that the function t7→ E|xnk(t)|2is absolutely continuous and hence de−λtE|xnk(t)|2+λe−λtE|xnk(t)|2=e−λtdE|xnk(t)|2.(57) Now, it can be obtained that e−λT E|xnk(T)|2≤E|Pnkψ(0)|2−λZT 0 e−λtE|xnk(t)|2dt + 2 ZT 0 e−λtEhA(xnk) + f(xnk), xnkidt +ZT 0 e−λtEkg(xnk)k2 2dt, (58) and therefore, ynk≤E|ψ(0)|2−e−λT E|xnk(T)|2.(59) As an immediate consequence, it follows that lim sup k→∞ ynk≤E|ψ(0)|2−e−λT E|x(T)|2.(60) Applying Itˆo’s formula to Eq. (52), we can get e−λT E|x(T)|2=E|ψ(0)|2−λZT 0 e−λtE|x|2dt + 2 ZT 0 e−λtEhχ+σ, xidt +ZT 0 e−λtEkζk2 2dt. (61) So lim sup k→∞ ynk≤ZT 0 e−λtE2hχ+σ, xi+λ|x|2− kζk2 2dt, (62) 22 and finally 0≤lim sup k→∞ unk≤ − 2EZT 0 e−λthχ+σ−A(v)−f(v), x −vidt +λE ZT 0 e−λt|x−v|2dt −EZT 0 e−λtkζ−g(v)k2 2dt. (63) If we take v=xin (63), it follows that ζ=g(x) and, also −2EZT 0 e−λthχ+σ−A(v)−f(v), x −vidt +λE ZT 0 e−λt|x−v|2dt ≥0.(64) In order to finish the proof, we only need to use hemicontinuity (a.4). Indeed, we notice that the function falso satisfies a similar property and it is easy to deduce from (F.2) that the map θ∈R17→ (f(t, η +θξ), x)∈R1is continuous for all η, ξ ∈L2 V, x ∈ Hand a.e.t∈[0, T]. Now, in (64) setting v=x−θu for θ > 0 and u∈Lp(Ω × (−h, T); V)∩L2(Ω ×(−h, T) : H), dividing by θand letting θtend to 0, we then get ∀u∈Lp(Ω ×(−h, T); V)∩L2(Ω ×(−h, T); H) −2EZT 0 e−λthχ+σ−A(x)−f(x), uidt ≥0.(65) Consequently, χ+σ=A(x) + f(x) and the proof of the theorem is complete. 5. STABILITY OF STRONG SOLUTIONS In this section we shall show that under suitable conditions exponential stability can be transferred from equations without time lags to those with time lag ones. Since we are mainly interested in exponential stability problems for the second moment of solutions, we will assume there exists a process x∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)) ,∀T > 0, which is the strong solution of the following problem: (dx(t) = [A(t, x(t)) + f(t, xt)] dt +g(t, xt)dw(t), t ≥0, x(t) = ψ(t), t ∈[−h, 0].(66) In other words, x(t) satisfies the following integral equation (in V0): x(t) = ψ(0) + Zt 0 [A(s, x(s)) + f(s, xs)]ds +Zt 0 g(s, xs)dw(s), P −a.s., t ≥0, (67) 23 and x(t) = ψ(t), t ∈[−h, 0]. In particular, in this section we suppose all the conditions in Section 3 hold so that there exists a unique strong solution for the stochastic functional differential equation (66). For simplicity, we also suppose in the section that the coefficients A,fand gare continuous with respect to time t. First of all, we investigate the case without hereditary characteristics. In other words, consider Eq. (67) with h= 0 and thus k1(h) = k1>0, k2(h) = k2>0 in (f.2), (g.2), then the equation (66) reduces to (dx(t) = [A(t, x(t)) + f(t, x(t))] dt +g(t, x(t)) dw(t), t ≥0, x(0) = x0.(68) If it is possible to know the existence of some Lyapunov function, we could obtain mean square stability of solutions. Indeed, assume there exist v∈C2(H;R+) and positive constants ci,1≤i≤4 , such that v0(x)∈Vfor all x∈Vand c1|x|2≤v(x)≤c2|x|2,Lv(x)≤ −c3v(x),|v0(x)| ≤ c4|x|, for all x∈V, where Lis the associated diffusion operator defined as Lv(x) = hA(t, x) + f(t, x), v0(x)i+1 2tr[v00(x)g(t, x)Wg∗(t, x)],∀x∈V, we can get (applying Itˆo’s formula to function ec3tv(x), x∈Hand Equation (68)) ec3tv(x(t)) = v(x(0)) + c3Zt 0 ec3sv(x(s)) ds +Zt 0 ec3shA(s, x(s)) + f(s, x(s)), v0(x(s))ids +Zt 0 ec3s(v0(x(s)), g(s, x(s)) dw(s)) 1 2Zt 0 ec3str[v00(x(s))g(s, x(s))Wg∗(s, x(s))] ds. Taking expectations and observing that Lv(x)≤ −c3v(x) , we have ec3tEv(x(t)) ≤Ev(x(0)) + c3Zt 0 ec3sEv(x(s)) ds +Zt 0 ec3sELv(x(s)) ds ≤Ev(x(0)), and consequently Ev(x(t)) ≤e−c3tEv(x(0)) ,∀t≥0. ¿From the assumptions on v, we easily deduce that E|x(t)|2≤c2 c1 e−c3tE|x(0)|2,∀t≥0 24 which means mean square exponential stability of the trivial solution of (68). Although, as we have mentioned before, the construction of Lyapunov functions is not, in general, a trivial problem, there exists a condition that makes v(x) = |x|2become a natural Lyapunov function. This is the following hypothesis: (H): there exists a positive constant γ > 0 such that 2hA(t, x) + f(t, x), xi+kg(t, x)k2 2≤ −γ|x|2,∀x∈V. Indeed, on this occasion Lv(x) = 2hA(t, x) + f(t, x), xi+kg(t, x)k2 2 ≤ − γ|x|2, therefore, setting c3=γ, we obtain exponential stability in mean square sense. Remark. Observe that in a variety of practical situations, the following assumption (H)0 (which seems easier to check) implies (H): (H)0: there exists a positive constant α > 0 such that −2hA(t, x), xi ≥ αkxk2,∀x∈Vand −α+ 2k1β2+k2 2β2<0, where k1,k2both are nonnegative constants in (f.2), (g.2) and β > 0 denotes the constant satisfying |x| ≤ βkxk,∀x∈V . Indeed, note that 2hA(t, x) + f(t, x), xi+kg(t, x)k2 2 ≤ − αkxk2+ 2(f(t, x), x) + tr[g(t, x)Wg∗(t, x)] ≤ − αkxk2+ 2|f(t, x)||x|+ tr[g(t, x)Wg∗(t, x)] ≤ − αkxk2+ 2k1β2kxk2+k2 2β2kxk2 ≤[−α+ 2k1β2+k2 2β2]β−2|x|2, and denote γ= [α−2k1β2−k2 2β2]β−2, the assumption (H) holds. In what follows, we shall show that the same hypotheses as above (mainly (f.2),(g.2) and (H)0) imply mean square exponential stability of the trivial solution of the stochastic functional differential equation (66). However, it is particularly worth pointing out that on this occasion the constants k1,k2are generally dependent on the time lag constant h > 0. This fact simply means that in order to obtain exponential stability, the time lag must be sufficiently small. However, as will be shown by Examples 1, 2 below, on some occasions such as the time delay case, the constant k1or k2could be independent on h > 0 so that the stability is true for any h > 0, a result which improves that of Mao and Shah [13] in 25