On the relationship between solutions of stochastic and random differential inclusions

On the relationship between solutions of stochastic and random differential inclusions T.Caraballo1, J.A.Langa1, J.Valero2 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. e-mails: [email protected] ; [email protected] 2Universidad Cardenal Herrera CEU, Comissari 3, 03203 Elche, Alicante. Spain. e-mail: v[email protected] Abstract Some results on the relationship of the solutions of a stochastic differential inclusion and the corresponding random differential inclusion obtained after a change of variable are proved. As a consequence, we obtain the pullback convergence of the solutions of the stochastic inclusion to a compact random set. The cases of a reaction-diffusion inclusion perturbed by additive and multiplicative noises are considered. 1 Introduction In our previous works Caraballo et al. [4, 5, 6], the concept of multivalued random dynamical system (MRDS) has been introduced and some applications to stochastic differential inclusions have been considered. In fact, a reaction-diffusion inclusion perturbed by additive and multiplicative noise is considered. As in the single-valued case, the construction of the MRDS relies on the possibility of performing a suitable change of variable driving the stochastic inclusion to a deterministic one depending on a random parameter (a random inclusion), so that the deterministic ideas can be properly adapted to this situation. However, there exits a result due to Da Prato and Frankowska [7] which guarantees the existence of stochastic solutions for a more general noise than the ones considered in [4, 5, 6]. In both situations, we do not have uniqueness of solutions. To define the MRDS in the first case, we take the union of all the solutions to the problem. Consequently, one can think about the relationship between the solutions of the stochastic inclusion and the random inclusion obtained after the change of variable. This is our main aim in this paper. Indeed, we shall prove that in our two particular situations (i.e., additive and multiplicative noises) each solution of the stochastic inclusion correspondes to a solution in the random one. The converse is true under some additional conditions on the solutions of the random inclusion. It is shown in [4, 5, 6] that the random inclusion generates a perfect cocycle having a compact global random attractor. As a consequence, all solutions of the stochastic inclusion with the initial condition in a bounded set of the phase space converge uniformly (in the sense of pullback attraction) to this attractor in the Hausdorff semidistance. This justifies the interest of the 1 results of [4, 5, 6], since they give some information on the asymptotic behaviour of the solutions of the stochastic inclusion. Hopefully, it will be possible in the future to give some light on the possibility of considering more general situations than the ones treated up to date. As far as we know, this problem remains interesting even in the single-valued case (see also Capinski and Cutland [3] for another especial case of Navier-Stokes equation, and Imkeller and Schmalfuss [8] for a general change of variables relating stochastic and random differential equations). 2 The additive case Let Hbe a real separable Hilbert space with the norm k·k and the scalar product h·,·i. The linear operator A:D(A)⊂H→His called m-dissipative if < A(y), y >≤0,for all y∈D(A) and Im(A−λI) = H, for any λ > 0. Let Abe a linear, m-dissipative, self-adjoint and densely defined operator (i.e. D(A) = H) . It follows from these properties that −A=∂ϕ for some proper, lower semicontinuous convex function ϕ:H→(−∞,+∞] (see Barbu [2, p.60]). Moreover, Philips-Lumer Theorem implies that Ais the infinitesimal generator of a semigroup of class C0denoted by S(t).Since Ais m-dissipative, it is known that kS(t)k ≤ 1. The operator S(t) is assumed to be compact for any t > 0. Consider the equation    du (t) dt =Au (t) + f(t),0≤t≤T, u(0) = u0∈H. (1) The function u: [0, T]→His called a strong solution to (1) if u(·)∈C([0, T], H), u(0) = u0,uis absolutely continuous on compact sets of (0, T), u is a.e. differentiable in (0, T ), and du (t) dt =Au (t) + f(t) for a.a. t∈(0.T). It is well known that for any f(·)∈L2(0, T ;H) the equation has a unique strong solution (see Barbu [2, p.189]). Moreover, it is well known that u(t) = S(t)u0+Zt 0 S(t−s)f(s)ds, for any t≥0. Consider the stochastic differential inclusion (see Caraballo et al. [4, 6] for more details)    du (t) dt ∈Au (t) + F(u(t)) + Pm i=1 φi dwi(t) dt ,0≤t≤T, u(0) = u0∈H, (2) where F:H→2Hsatisfies: (F1) Fhas closed, bounded, convex, non-empty values. 2 (F2) There exists C > 0 such that distH(F(u), F (v)) ≤Cku−vk,∀u, v ∈H, where ‘distH’ denotes the Hausdorff metric. Let ζ(t) = Pm i=1 φiwi(t). Let us consider the Wiener probability space (Ω,F,P) defined by Ω = {ω= (w1(·), ..., wm(·)) ∈C(R,Rm)|ω(0) = 0}, equipped with the Borel σ−algebra F, the Wiener measure P,and the usual uniform convergence on bounded sets of R. Each ω∈Ω generates a map ζ(·) = Pm i=1 φiwi(·)∈C(R, H) such that ζ(0) = 0. Consider the maps ρs,t : Ω →Ω ρs,t (ω) (τ) =    ω(s),si τ≤s, ω(τ),si s≤τ≤t, ω(t) , si τ≥t, and define the σ-algebras Fs,t =ρ−1 s,t F. It is clear that Fs,t ⊂ Fs0,t0⊂ F,∀s0≤s≤t≤t0. A process u(t, ω) : [0,∞)×Ω→His said to be adapted if u(t, ·) is F0,t measurable for any t≥0. Now, Theorem 2.1 from [7] provides the existence of at least one solution u(·, ω, u0) to (2) for any u0∈H, that is, an adapted process with values in Hsuch that: 1. u(·, ω, u0) is continuous for P-a.a. ω∈Ω. 2. u(0, ω, u0) = u0. 3. For any t∈[0, T ] u(t) = S(t)u0+Zt 0 S(t−s)f(s)ds + m X i=1 Zt 0 S(t−s)φidwi(s), being f(s, ω) an adapted process such that f(s, ω)∈F(u(s, ω)) , for a.a. (s, ω)∈(0, T )×Ω, EZT 0 kf(s)k2ds<∞. Note that, since T > 0 is arbitrary, we can extend any solution to the whole [0,+∞). Consider now the change of variable v(t) = u(t)−ζ(t). Then, inclusion (2) becomes (formally)    dv (t) dt ∈Av (t) + F(v(t) + ζ(t)) + Aζ (t),0≤t≤T, v(0) = u0. (3) For any u0∈Hand ω∈Ω there exists a strong solution v(·, ω, u0) to (3), i.e., a strong solution to (1) for some f(·)∈L2(0, T;H) such that f(t)∈F(v(t) + ζ(t)) + Aζ (t), a.e. in (0, T). Let us denote the solutions to (2) and (3) by u(·) = Is(u0)f(·) and v(·) = Ir(u0)f(·), respectively. Then, we can prove the following results. 3 Proposition 1 If u(·) = Is(u0)f(·), then the function defined as v(·) = u(·)−ζ(·)satisfies v(·) = Ir(u0) (f(·) + Aζ (·)) .In other words, if u(·)is a solution of the stochastic inclusion (2) in the sense of Da Prato and Frankowska, and we define the function v(·)by the equality v(·) = u(·)−ζ(·),then v(·)is a strong solution to (2) but substituting the original f(t)in the right-hand side of (1) by f(t) + Aζ (t). Proof. Since u(·) is a solution to (2) we have u(t) = S(t)u0+Zt 0 S(t−s)f(s)ds + m X i=1 Zt 0 S(t−s)φidwi(s).(4) Let Y(s) = Pm i=1 S(t−s)φiwi(s). Since dS (t)u0=AS (t)u0dt, we easily get dY (s) = S(t−s) m X i=1 φidwi(s)−AS (t−s) m X i=1 φiwi(s)ds. Hence, using the fact that Aand S(t) commute, we have m X i=1 φiwi(t) = m X i=1 Zt 0 S(t−s)φidwi(s)− m X i=1 Zt 0 S(t−s)Aφiwi(s)ds. (5) Now, combining (4) and (5), we obtain v(t) = u(t)−ζ(t) = S(t)u0+Zt 0 S(t−s) (f(s) + Aζ (s)) ds. It follows v(·) = Ir(u0) (f(·) + Aζ (·)) . Proposition 2 If v(·) = Ir(u0)f(·), being f(t)an adapted process such that ERT 0kf(s)k2ds< ∞, then u(·) := v(·) + ζ(·) = Is(u0) (f(·)−Aζ (·)) . Proof. Since v(·) is a strong solution of (1), we have v(t) = S(t)u0+Zt 0 S(t−s)f(s)ds =S(t)u0+Zt 0 S(t−s)g(s)ds + m X i=1 Zt 0 S(t−s)Aφiwi(s)ds, where g(s) = f(s)−Aζ (s)∈F(v(s) + ζ(s)), a.e. on (0, T ). Using (5) we obtain u(t) = v(t) + ζ(t) = S(t)u0+Zt 0 S(t−s)g(s)ds + m X i=1 Zt 0 S(t−s)φidwi(s). Hence, u(·) = Is(u0)g(·).(Observe that u(·) is adapted since fand ζalso are.) Now, notice that the random differential inclusion (3) generates a perfect mutivalued cocycle G:R+×Ω×H→C(H) by means of G(t, ω)u0=[ v(·)∈D(u0,ω) {v(t) + ζ(t)}, 4 where D(u0, ω) = {v(·) : v(·) is a strong solution to (3)}. Moreover, Ghas compact values (see Caraballo et al. [4]). Definition 3 The closed random set ω7→ A(ω)(that is, a measurable map with closed values) is called a global random attractor of Gif: i) G(t, ω)A(ω) = A(θtω),for all t≥0,P−a.s (that is, it is strictly invariant); ii) For all bounded D⊂X, lim t→+∞dist(G(t, θ−tω)D, A(ω)) = 0,P−a.s.; iii) A(ω)is compact P−a.s. Property ii) means that the initial moment of time goes to -∞and the final moment is 0. This is called the pullback convergence in the literature (Kloeden and Schmalfuss [9]) . If we assume the following conditions: (H1) There exist constants δ > 0, M ≥0 such that hy, ui ≤ (−δ+ε)kuk2+M, for all u∈D(A), y ∈F(u),(6) where ε≥0 is the biggest constant such that hAu, ui≤−εkuk2. (H2) The level sets MR={u∈D(ϕ)| kuk ≤ R, ϕ(u)≤R} are compact in Hfor any R > 0. Then, it is proved in [4, Theorem 16] that Ghas the global random attractorA(ω), which is measurable with respect to the σ−algebra F. As a consequence of Proposition 1 we have u(t, ω, u0)∈G(t, ω)u0, for all (t, u0)∈R+× H, ω ∈Ω and u(·, ω, u0)∈ L (u0, ω),where L(u0, ω) = {u(·, ω, u0) : u(·, ω, u0) is a solution to (2)}. We then obtain that all the solutions of inclusion (2) with initial conditions on a bounded set converge uniformly to A(ω) in the sense of Definition 3. Corollary 4 Assuming (H1) −(H2) ,for any bounded set Bit holds lim t→+∞sup u∈L(u0,ω) sup u0∈B dist (u(t, θ−tω, u0),A(ω)) = 0. 5 These results can be applied to the following reaction-diffusion inclusion    ∂u ∂t ∈∆u+f(u) + h+Pm i=1 φidwi(t) dt ,on O × (0, T), u= 0, on ∂O × (0, T), u(x, 0) = u0(x) on O, (7) where O ⊂ Rnis an open bounded subset with smooth boundary ∂O,h(·)∈L2(O) and f:R→ 2Ris a multivalued map with non-empty, compact convex values. Assume that fis Lipschitz, i.e. there exists C≥0 such that distH(f(x), f(z)) ≤Ckx−zk,for all x, z ∈R.(8) Define the operators A:D(A)→H, F :H→2H, H =L2(O), Au = ∆u, F(u) = {y∈H:y(x)∈f(u(x)) + h(x),a.e. on O} , with D(A) = H2(O)∩H1 0(O). It is assumed that φi∈D(A).The map −Ais the subdifferential of a proper, convex, lower semicontinuous function ϕand the map Fsatisfies (F1) −(F2). Moreover, condition (H2) is satisfied and D(ϕ) = H(see Melnik and Valero [10, Section 3.2.2.]). On the other hand, it is well known that the operator Asatisfies all the conditions assumed before. Hence, Propositions 1-2 hold. If we also assume the existence of M≥0, δ > 0 such that zs ≤(λ1−2δ)|s|2+M1,for all s∈R, z ∈f(s),(9) where λ1is the first eigenvalue of −∆ in H1 0(Ω), then (H1) is also satisfied (see [4, Theorem 17]). It follows that Corollary 4 holds. 3 The multiplicative case Let us now consider the multiplicative case from Caraballo et al. [5]. That is, let us consider the following stochastic differential inclusion in the Stratonovich sense    du (t) dt ∈Au (t) + F(u(t)) + σu(t)◦dw (t) dt ,0≤t≤T, u(0) = u0∈H, (10) where σ∈R,F:H→2Hsatisfies (F1)-(F2) and now the Wiener probability space (Ω,F,P) is defined by Ω = {ω=w(·)∈C(R,R)|ω(0) = 0}, equipped with the Borel σ−algebra F, the Wiener measure P,and the usual uniform convergence on bounded sets of R. The change of variable which takes the stochastic inclusion (10) into a random one is different from the one in the previous section. Indeed, set α(t) = α(t, ω) = e−σwt(ω).Using the change v(t) = α(t)u(t),inclusion (10) becomes (formally)    dv (t) dt ∈Av (t) + α(t)Fα−1(t)v(t),0≤t≤T, v(0) = u0∈H. (11) 6 Thus, as in Section 2 and taking into account the relation between the Ito and Stratonovich integral in this special case, Theorem 2.1 in [7] also ensures the existence of at least one solution to (10) satisfying: 1. u(·, ω, u0) is continuous for P-a.a. ω∈Ω. 2. u(0, ω, u0) = u0. 3. For any t∈[0, T ] u(t) = S(t)u0+Zt 0 S(t−s)f(s)ds +Zt 0 S(t−s)u(s)◦dw (s), where f(s, ω) is an adapted process such that f(s, ω)∈F(u(s, ω)) , for a.a. (s, ω)∈(0, T )×Ω, EZT 0 kf(s)k2ds<∞. On the other hand, given u0∈Hand ω∈Ω,there exists a strong solution of (11), i.e. a strong solution of (1) for some f(·)∈L2(0, T;H) such that f(t)∈α(t)Fα−1(t)v(t)a.e. in (0, T). Using again a similar notation to that one in Section 2, i.e. u(·) = Js(u0)f(·) for the solutions of (10), and v(·) = Jr(u0)f(·) for the solutions of (11), we can prove the following results. Proposition 5 If u(·) = Js(u0)f(·), then v(·) = α(·)u(·) = Jr(u0) (α(·)f(·)) .In other words, if u(·)is a solution to the stochastic inclusion (10) in the sense of Da Prato and Frankowska, and we define the function v(·)by v(·) = α(·)u(·),then v(·)is a strong solution to (11) but substituting the original f(t)in the right-hand side of (1) by α(t)f(t). Proof. Let u(·) be a solution to (10). Then, we have u(t) = S(t)u0+Zt 0 S(t−s)f(s)ds +σZt 0 S(t−s)u(s)◦dw (s),(12) where f(s)∈F(u(s)) a.e. in (0, T)×Ω. Let Y(s) = α(s)S(t−s)u(s),0≤s≤t. Then, it follows dY (s) = −σα(s)S(t−s)u(s)◦dw (s) + α(s)d[S(t−s)u(s)] , and, by a direct integration Y(t)−Y(0) = −σZt 0 α(s)S(t−s)u(s)◦dw (s) +Zt 0 α(s)d[S(t−s)u(s)] .(13) Now, we work with the last term in (13). 7 Observe that d[S(t−s)u(s)] =dS(t−s)S(s)u0+Zs 0 S(s−τ)f(τ)dτ +σZs 0 S(s−τ)u(τ)◦dw (τ) =dS(t)u0+Zs 0 S(t−τ)f(τ)dτ +σZs 0 S(t−τ)u(τ)dw (τ) =S(t−s)f(s)ds +σS (t−s)u(s)◦dw (s), and, consequently, (13) turns into Y(t)−Y(0) = −σZt 0 α(s)S(t−s)u(s)◦dw (s) +Zt 0 S(t−s)α(s)f(s)ds +Zt 0 σα(s)S(t−s)u(s)◦dw (s) =Zt 0 S(t−s)α(s)f(s)ds. Taking into account that Y(t) = v(t) and Y(0) = S(t)u0,it holds v(t) = S(t)u0+Zt 0 S(t−s)˜ f(s)ds, where ˜ f(s) = α(s)f(s). Proposition 6 If v(·) = Jr(u0)f(·), being f(t)an adapted process such that ERT 0kf(s)k2ds< ∞, then u(·) = α−1(·)v(·) = Js(u0)˜ f(·),where ˜ f(s) = α−1(s)f(s). Proof. Since v(·) is a strong solution to (11), we have v(t) = S(t)u0+Zt 0 S(t−s)f(s)ds, where f(s)∈α(s)F(α−1(s)v(s)), a.e. on (0, T). Let us denote Y(s) = α−1(s)S(t−s)v(s) = eσws(ω)S(t−s)v(s),0≤s≤t. Then, arguing as in the preceding proof, it follows dY (s) = σeσwsS(t−s)v(s)◦dws+ eσwsd[S(t−s)v(s)] =σeσwsS(t−s)v(s)◦dws+ eσwsS(t−s)f(s)ds, and, after integration, it yields Y(t)−Y(0) = Zt 0 S(t−s)α−1(s)f(s)ds +σZt 0 S(t−s)α−1(s)v(s)◦dws. 8 Setting u(t) = α−1(t)v(t),˜ f(t) = α−1(t)f(t), it holds u(t) = S(t)u0+Zt 0 S(t−s)˜ f(s)ds +σZt 0 S(t−s)u(s)◦dws. Therefore the proof is complete since the adaptedness of uis immediate. Now, as in the additive case, differential inclusion (11) generates a perfect mutivalued cocycle GM:R+×Ω×H→C(H) by setting GM(t, ω)u0=[ v(·)∈D(u0,ω) α−1(t)v(t), where D(u0, ω) = {v(·) : v(·) is a solution to (11)}. Moreover, GMhas compact values (see Caraballo et al. [5] for more details). If we assume (H1)−(H2) ,then it is proved in [5, Theorem 26] that Ghas the global random attractorA(ω), which is measurable with respect to the σ−algebra F. As a consequence of Proposition 5 we have u(t, ω, u0)∈GM(t, ω)u0, for all (t, u0)∈R+× H, ω ∈Ω and u(·, ω, u0)∈ LM(u0, ω),where LM(u0, ω) = {u(·, ω, u0) : u(·, ω, u0) is a solution to (10)}. We obtain then that all the solutions to inclusion (10) with initial conditions on a bounded set converge uniformly to A(ω) in the sense of Definition 3. Corollary 7 Assuming (H1) −(H2) ,for any bounded set Bit holds lim t→+∞sup u∈LM(u0,ω) sup u0∈B dist (u(t, θ−tω, u0),A(ω)) = 0. These results can be applied to the following reaction-diffusion inclusion      ∂u ∂t ∈∆u+f(u) + h+σu(t)◦dw (t) dt ,on O × (0, T), u= 0, on ∂O × (0, T), u(x, 0) = u0(x) on O, (14) where O ⊂ Rn, h,f, F and Aare defined as for (7). Hence, Propositions 5-6 hold. If we assume also the existence of M≥0, δ > 0 such that zs ≤(λ1−2δ)|s|2+M1,for all s∈R, z ∈f(s), where λ1is the first eigenvalue of −∆ in H1 0(Ω), then (H1) is also satisfied (see [5, Theorem 27]). It follows that Corollary 7 holds. 9