Journal of Stochastic Analysis
Volume 1 Number 3 Article 2
August 2020
Continuous Dependence on the Coefficients for Mean-Field Fractional Stochastic Delay Evolution Equations
Brahim Boufoussi Cadi Ayyad University, Marrakesh, Morocco, [email protected]
Salah Hajji Regional Center for the Professions of Education and Training, Marrakesh, Morocco, [email protected]
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Recommended Citation Boufoussi, Brahim and Hajji, Salah (2020) "Continuous Dependence on the Coefficients for Mean-Field Fractional Stochastic Delay Evolution Equations," Journal of Stochastic Analysis: Vol. 1 : No. 3 , Article 2. DOI: 10.31390/josa.1.3.02 Available at: https://digitalcommons.lsu.edu/josa/vol1/iss3/2 Journal of Stochastic Analysis
Vol. 1, No. 3 (2020) Article 2 (14 pages) digitalcommons.lsu.edu/josa DOI: 10.31390/josa.1.3.02
CONTINUOUS DEPENDENCE ON THE COEFFICIENTS FOR MEAN-FIELD FRACTIONAL STOCHASTIC DELAY EVOLUTION EQUATIONS
BRAHIM BOUFOUSSI AND SALAH HAJJI*
Abstract. We prove that the mild solution of a mean-field stochastic func- tional differential equation, driven by a fractional Brownian motion in a Hilbert space, is continuous in a suitable topology, with respect to the initial datum and all coefficients.
1. Introduction In this paper we are concerned by the following stochastic delay differential equation of McKean-Vlasov type: dx(t)=(Ax(t)+f(t, x ,P ))dt + g(t)dBH (t),t [0,T] t x(t) (1.1) x(t)=ϕ(t),t [ r, 0], ∈ ∈ − where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators, (S(t))t 0, in a Hilbert space X,thedrivingprocess H ≥ B is a fractional Brownian motion with Hurst parameter H (1/2, 1), Px(t) is the probability distribution of x(t), x ([ r, 0],X) is the function∈ defined by t ∈C − xt(s)=x(t + s) for all s [ r, 0], and f :[0, + ) ([ r, 0],X) 2(X) 0 ∈ − ∞ ×C − ×P → X, g :[0, + ) 2(Y,X) are appropriate functions. Here 2(X) denotes the space of probability∞ →L measures with finite second order moment.P That is for each 2 0 µ 2(X), x µ(dx) < and 2(Y,X) denotes the space of all Q-Hilbert- Schmidt∈P operators| | from Y into∞ X (seeL section 2 below). The purpose! of this work is twofold, on one hand to prove existence and unique- ness of the mild solution to Eq.(1.1), on the other to provide, in an appropriate topology, sufficient conditions for the continuity of the map (ϕ, A, f, g) x, where x denotes the mild solution to Eq(1.1). → The most important feature of Equation (1.1) is the dependence of the non- linearity on the state distribution. Stochastic equations of type (1.1), in finite or infinite dimension, have recieved, during the past decades, a great deal of attention. Because they appear as a mathematical modeling to various dynamical systems involving interactions of a great number of particles (refering to quantum mechanic or statistical physics). Roughly speaking, Eq (1.1) can be seen as a mathematical
Received 2020-2-19; Accepted 2020-7-24; Communicated by the editors. 2010 Mathematics Subject Classification. Primary 60H10; Secondary 60G22. Key words and phrases. Mild solution, semigroup of bounded linear operators, fractional Brownian motion, Wiener integral; Wasserstein distance. *Correspondingauthor. 1 2 BRAHIM BOUFOUSSI AND SALAH HAJJI model for large population particle systems which possess the path-dependence in microscopic level:
dXk,M (t)=(AXk,M (t)+f(t, Xk,M ,µ (t))dt + g(t)dBH (t),t [0,T] t M (1.2) Xk,M (t)=ϕ(t),t [ r, 0], ∈ ∈ − The coefficient of particle k 1, 2,...,M depends on the state of particle k and ∈{ } also on the current empirical distribution of all particle locations, that is µM (t)= M 1 d M δXk,M (t). In the finite dimensional situations (X = R ), delayed stochastic 1 diff"erential equations (1.1) driven by Wiener process, are obtained naturally as a weak limit of Xk,M by letting the particle number M . We can refer to some relevant works: for example to Budhiraja et al [6] where→∞ the large deviation of stochastic delays mean-field is considered. Delayed neural network with fractional noise is investigated in Touboul [20]. We also refer to Lasry and Lions[14] for applications in game theory. The continuous dependence problem we consider here, is motivated by several considerations, first, it provide a justification for the use of numerical schemes, where necessarily one replaces continuous objects by discretized approximations. Furthermore, approximating the unbounded operator A by a bounded one (such as e.g. the Yosida approximation) is a powerful technique to obtain estimates for mild solutions to SPDEs. We would like to mention that this problem for the stochastic differential equa- tions without delay have been studied intensively. Da Prato and Zabczyk [9] studied the dependence of the solution on the initial datum ξ, Marinelli et al [15, 16] studied the problem for the case of Poisson noise. The dependence of the solution on the coefficients f and g were considered by Peszat and Zabczyk [18] and Seidler [19]. The dependence of the solution on A, f, g and ϕ were considered by Brze´zniak [5], Kunze and Neerven[12, 13]. Govindan and Ahmed [10] studied the Yosida approximations of mild solutions of a semilinear McKean-Vlasov type stochastic evolution equation in a real Hilbert space. However, in the case of delay, As far as we know, there exist only a few papers published in this field. In Ahmed et al [1], the authors discuss the dependence on the initial conditions of the solution of a stochastic functional differential equations with discontinuous data, driven by a Brownian motion, in a finite dimensional space. In [4], the authors studied the dependence on the initial condition for the mild solution of a functional differential equation in Hilbert space driven by a fBm. To the best of our knowledge, the dependence on the coefficients A, f, g and ϕ of the solution of a mean-field SFDEs, driven by a fBm in a Hilbert space, has not been considered before in the literature and the aim of this paper is to close this gap. The rest of this paper is organized as follows, In Section 2 we introduce some notations, concepts, and basic results about fractional Brownian motion, Wiener integral over Hilbert spaces and we recall some preliminary results about Wasser- stein distance. The existence and uniqueness of mild solutions are discussed in Section 3 by using Banach fixed point theorem. In Section 4, we investigate the CONTINUOUS DEPENDENCE ON THE COEFFICIENTS 3 dependence on the coefficients A, f, g and the initial data ϕ for the mild solution of Equation (1.1).
2. Preliminaries In this section we collect some notions, conceptions and lemmas on Wiener integrals with respect to an infinite dimensional fractional Brownian motion and we recall some basic results about Wasserstein distance which will be used throughout the whole of this paper. Let (Ω, , P) be a complete probability space. Consider a time interval [0,T] with arbitraryF fixed horizon T and let βH (t),t [0,T] the one-dimensional fractional Brownian motion with Hurst parameter{ ∈H (1}/2, 1). This means by definition that βH is a centered Gaussian process with∈ covariance function: 1 R (s, t)= (t2H + s2H t s 2H ). H 2 −| − | Moreover βH has the following Wiener integral representation: t H β (t)= KH (t, s)dβ(s) (2.1) #0 where β = β(t),t [0,T] is a Wiener process, and KH (t; s) is the kernel given by { ∈ } t 1 H H 3 H 1 K (t, s)=c s 2 − (u s) − 2 u − 2 du H H − #s H(2H 1) − for t>s,wherecH = β(2 2H,H 1 ) and β(, ) denotes the Beta function. We put − − 2 KH (t, s)=0ift s. $ We will denote by≤ the reproducing kernel Hilbert space of the fBm. In fact is H H the closure of set of indicator functions 1[0;t],t [0,T] with respect to the scalar product { ∈ } 1[0,t], 1[0,s] = RH (t, s). ' (H H The mapping 1[0,t] β (t) can be extended to an isometry between and the → T H H first Wiener chaos and we will denote by 0 ϕ(s)dβ (s) the image of ϕ by the previous isometry. ! 2 Let us consider the operator K∗ from to L ([0,T]) defined by H H T ∂K (K∗ ϕ)(s)= ϕ(r) (r, s)dr H ∂r #s We refer to Nualart.[17] for the proof of the fact that KH∗ is an isometry between and L2([0,T]). Moreover for any ϕ ,wehave H ∈H T T H ϕ(t)dβ (t)= (KH∗ ϕ)(t)dβ(t) (2.2) #0 #0 Let X and Y be two real, separable Hilbert spaces and let (Y,X) be the space of bounded linear operator from Y to X. For the sake of convenience,L we shall use the same notation to denote the norms in X, Y and (Y,X). Let Q (Y,Y ) L ∈L be an operator defined by Qe = λ e with finite trace trQ = ∞ λ < . n n n n=1 n ∞ where λn 0(n =1, 2...) are non-negative real numbers and en (n =1, 2...)is ≥ { }% 4 BRAHIM BOUFOUSSI AND SALAH HAJJI a complete orthonormal basis in Y . We define the infinite dimensional fBm on Y with covariance Q as
H H ∞ H B (t)=BQ (t)= λnenβn (t) n=1 " & H where βn are real, independent fBm’s. In order to define Wiener integrals with respect to the Q-fBm, we introduce the 0 0 space 2 := 2(Y,X) of all Q-Hilbert-Schmidt operators ψ : Y X. We recall that ψL (Y,XL ) is called a Q-Hilbert-Schmidt operator, if → ∈L
2 ∞ 2 ψ 0 := λnψen < * *L2 * * ∞ n=1 " & 0 and that the space 2 equipped with the inner product ϕ,ψ 0 = n∞=1 ϕen,ψen L ' (L2 ' ( is a separable Hilbert space. 0 % Now, let φ(s); s [0,T] be a function with values in 2(Y,X), The Wiener integral of φ with respect∈ to BH is defined by L
t t t H ∞ H ∞ φ(s)dB (s)= λnφ(s)endβn (s)= λn(KH∗ (φen)(s)dβn(s) #0 n=1 #0 n=1 #0 " & " & (2.3) H where βn is the standard Brownian motion used to present βn as in (2.1) and we have the following result (see Boufoussi and Hajji. [3])
0 T 2 Lemma 2.1. If ψ :[0,T] 2(Y,X) satisfies 0 ψ(s) 0 ds < then the above →L * *L2 ∞ sum in (2.3) is well defined as a X-valued random! variable and we have t t H 2 2H 1 2 E ψ(s)dB (s) 2Ht − ψ(s) 0 ds * * ≤ * * 2 #0 #0 L We now introduce a class of metrics which we will use in this paper. For any p 1, we denote by p(X) the subspace of (X) of the probability measures of order≥ p. P P For any p 1 and µ, ν p(X), the p-Wasserstein distance Wp(µ, ν)isdefined by: ≥ ∈P 1/p p Wp(µ, ν)=inf x y π(dx, dy) ; X X | − | '(# × ) π (X X) with marginals µ and ν ∈Pp × * Notice that if x and x& are random variables of order p,then
p 1/p W (P ,P ) (E x x& ) . p x x! ≤ | − | Notice also that H¨older’s inequality implies that: W (µ, ν) W (µ, ν),µ,ν (X), 1 p q p ≤ q ∈Pp ≤ ≤ CONTINUOUS DEPENDENCE ON THE COEFFICIENTS 5
The following characterization of the 1-Wasserstein distance W1 is a direct con- sequence of the Kantorovich duality theorem. See [Carmona and Delarue. [8], Corollary 5. 4 ].
Lemma 2.2. If (E,d) is a complete separable metric space, and µ, ν 1(E), then, ∈P
W1(µ, ν)= sup φ(x)(µ ν)(dx). φ: φ(x) φ(y) d(x,y) E − | − |≤ # where the supremum is taken over all the real valued continuous functions
3. Existence and Uniqueness Result In this section, we study the existence and uniqueness of mild solutions for Equation (1.1). For this equation we make the following assumption on the coef- ficients: ( ) The operator A generates a strongly continuous analytic semigroup A (S(t))t 0, on X. We recall≥ that a closed operator A generates a strongly continuous analytic semigroup on X if and only if A is densely defined and sectorial, i.e., there exist M 1 and ω R such that λ C : eλ>ω is ≥ ∈ { ∈ + } contained in the resolvent set ρ(A) and sup eλ>ω (λ ω)R(λ,A) M. In this context, we say that A is sectorial of( type* (M,−ω). *≤ Concerning f and g, we shall assume: ( ) The function f :[0,T] ([ r, 0],X) 2(X) X satisfies the following F conditions: ×C − ×P → (i) There exists a positive constant Cf > 0 such that, for all t [0,T] x, y ([ r, T ],X) and µ, ν ([0,T], (X)) ∈ ∈C − ∈C P2 t t 2 2 f(s, xs,µ(s)) f(s, ys,ν(s)) ds Cf x(s) y(s) ds 0 * − * ≤ r * − * # +#− t 2 + W2 (µ(s),ν(s))ds #0 , (ii) For arbitrary µ ([0,T], (X)), there exists a positive constant ∈C P2 Cf = Cf (µ) > 0 such that T - - f(s, 0,µ(s)) 2ds C . * * ≤ f #0 ( ) The function g :[0, + ) 0(Y,X) satisfies- G ∞ →L2 T 2 g(s) 0 ds < . * * 2 ∞ #0 L Moreover, we assume that ϕ ([ r, 0], L2(Ω,X)). ∈C − Definition 3.1. A X-valued process x(t),t [ r, T ] , is called a mild solution of equation (1.1) if { ∈ − } i) x(.) ([ r, T ], L2(Ω,X)), ii) x(t)=∈Cϕ(t−), r t 0. − ≤ ≤ 6 BRAHIM BOUFOUSSI AND SALAH HAJJI
iii) For arbitrary t [0,T], we have ∈ t x(t)=S(t)ϕ(0) + S(t s)f(s, xs,µ(s))ds 0 − t # + S(t s)g(s)dB(s) P a.s − − #0 We can now state the main result of this section. Theorem 3.2. Suppose that ( ), ( ) and ( ) are satisfied. Then for every ϕ A F G ∈ ([ r, 0], L2(Ω,X)), Eq.(1.1) has a unique mild solution on [ r, T ]. C − − Proof. Let µ ([0,T], 2(X)) be temporarily fixed. The classical existence result for stochastic∈ functionalC P differential equations (cf. [7]) guarantees the existence and uniqueness of a mild solution of the equation H dx(t)=(Ax(t)+f(t, xt,µ(t)))dt + g(t)dB (t),t [0,T] x(t)=ϕ(t),t [ r, 0], ∈ ∈ − We denote its solution by xµ. Let (x )= (x (t)) : t [0,T] denote the law of x and define an operator L µ {L µ ∈ } µ ψ on ([0,T], 2(X)) by ψ(µ)= (xµ). Then it is clear that to prove the existence of mildC solutionsP to equation (1.1)L is equivalent to find a fixed point for the operator ψ. Next we will show by using Banach fixed point theorem that ψ has a unique fixed point. We divide the subsequent proof into two steps. Step 1. For arbitrary µ ([0,T], 2(X)), let us prove that t ψ(µ)(t) is continuous on the interval [0,T∈].C P → Let 0 Then, Gronwall’s lemma implies that 2 2 sup E xµ(s) xν (s) γ(t)supW2 (µ(s),ν(s)). s [0,t] * − * ≤ s [0,t] ∈ ∈ where 2 2 2 t 2 γ(t)=t M Cf exp M Cf and M =sup S(t) 2 0 t T * * + , ≤ ≤ Hence, it follows that 2 2 sup W2 (ψ(µ)(s),ψ(ν)(s)) γ(t)supW2 (µ(s),ν(s)). s [0,t] ≤ s [0,t] ∈ ∈ Since γ(0) = 0 < 1. Then there exists 0 0 2 (F1(t, s, x(t + s)) F1(t, s, y(t + s)))ds * r − * #− 0 r C2 x(t + s) y(t + s) 2ds ≤ r * − * #− t rC2 x(s) y(s) 2ds (3.3) ≤ r * − * #− By Lemma 2.2, we obtain that 2 2 2 F2(z)(µ(t) ν(t))(dz)F3(t) W1 (µ(t),ν(t)) F3 * X − * ≤ * *∞ # 2 2 W2 (µ(t),ν(t)) F3 (3.4) ≤ * *∞ Inequalities (3.2), (3.3) and (3.4) imply that: t t 2 2 2 f(s, xs,µ(s)) f(s, ys,ν(s)) ds 2rC T x(s) y(s) ds 0 * − * ≤ r * − * # #− t 2 2 +2 F3 W2 (µ(s),ν(s))ds * *∞ #0 The last inequality implies that the function f satisfies the condition (i)inhy- pothesis ( ). The secondF condition (ii) is trivial. Consequently, all the hypotheses in Theo- rem 3.2 are satisfied and we can ensure the existence and uniqueness of the mild solution of the system (3.1) on the interval [ r, T ]. − 4. Continuous Dependence on the Coefficients In this section, we present and prove our main result concerning the dependence on the coefficients A, f, g and the initial datum for the mild solution to equation (1.1). Let us consider for each n N, the equation ∈ dx(t)=(A x(t)+f (t, x ,P ))dt + g (t)dBH (t),t [0,T] n n t x(t) n (4.1) x(t)=ϕ (t),t [ r, 0], ∈ n ∈ − To establish our main result we need to impose the following assumptions: ( ) The operators A and A are densely defined, closed, and uniformly • A1 n sectorial on X in the sense, there exist numbers M 1 and ω R such ≥ ∈ that A and each An is sectorial of type (M,ω). ( ) The operators A converge to A in the strong resolvent sense: • A2 n lim R(λ,An)x = R(λ,A)x for all eλ>ω and x X n →∞ + ∈ Remark 4.1. One of the classical example of An is the Yosida approximation. In fact, let us consider an operator A which generates a strongly continuous analytic semigroup (S(t))t 0, on X. If we take its Yosida approximands defined by An := 2 ≥ n R(n, A) n. Then assumptions ( 1) and ( 2) hold for these operators. See [Haase.[11],− Proposition 2.1.1 ] and [ArendtA etA al [2], Section 3.6]. CONTINUOUS DEPENDENCE ON THE COEFFICIENTS 9 Remark 4.2. Under ( 1), the operators A and An generate strongly continuous an- A ωt alytic semigroups S, Sn satisfying the uniform bounds S(t) , Sn(t) Me ,t 0. * * * *≤ ≥ The following Trotter-Kato type approximation theorem is well known; see [[2], Theorem 3.6.1] Lemma 4.3. Under condition ( ) and ( ), we have: A1 A2 For all t 0 and x X, Sn(t)x S(t)x as n and the convergence is uniform on compact≥ subsets∈ of [0, )→ X. →∞ ∞ × For the nonlinearities f and fn, we will make the following assumptions: ( ) The maps f,f :[0,T] ([ r, 0],X) (X) X satisfy condition • F1 n ×C − ×P2 → ( ) with uniform constants Cf and Cf . (F ) For all x ([ r, T ],X) and µ ([0,T], (X)), • F2 ∈C − ∈C P2 T - 2 lim fn(s, xs,µ(s)) f(s, xs,µ(s)) ds =0 n | − | →∞ #0 For the functions g and gn, we make the following assumptions: ( ) The maps g, g :[0, + ) 0(Y,X) satisfy • G1 n ∞ →L2 T T 2 2 gn(s) 0 ds, g(s) 0 ds < . * * 2 * * 2 ∞ #0 L #0 L ( 2) • G T 2 lim gn(s) g(s) 0 ds =0 n * − * 2 →∞ #0 L The main result of this paper is given in the next theorem. Theorem 4.4. Suppose that the operators A and A satisfy ( ) and ( ),the n A1 A2 nonlinearities f and fn satisfy ( 1) and ( 2), the functions g and gn satisfy ( 1) F F 2 G and ( 2) and the initial data ϕ and ϕn satisfy ϕn ϕ in ([ r, 0], L (Ω,X)) and let x Gand xn be the mild solutions to (1.1) and (4.1),→ respectively.C − Then, xn x → in ([ r, T ], L2(Ω,X)) as n ,thatis: C − →∞ lim sup E xn(t) x(t) 2=0 n r t T | − | →∞ − ≤ ≤ Proof. Let t [0,T] and 0 u t. By the triangle inequality one has ∈ ≤ ≤ E xn(u) x(u) 2 | − | 2 3E Sn(u)ϕn(0) S(u)ϕ(0) ≤ | u − | n 2 +3E Sn(u s)fn(s, xs ,Pxn(s)) S(u s)f(s, xs,Px(s))ds | 0 − − − | # u +3E S (u s)g (s) S(u s)g(s)dB(s) 2 | n − n − − | #0 3(I + I + I ) (4.2) ≤ 1 2 3 We estimate the various terms of the right hand side separately. 10 BRAHIM BOUFOUSSI AND SALAH HAJJI For the first term, by using the fact that the operator norm of Sn(t) is bounded by MeωT we get that 2 2 I1 2E (Sn(u) S(u))ϕ(0) +2E Sn(u)(ϕ(0) ϕn(0)) ≤ | − | 2| 2 2ωT − | 2 2E sup (Sn(u) S(u))ϕ(0) +2M e sup E ϕ(s) ϕn(s) ≤ 0 u T | − | r s 0 | − | ≤ ≤ − ≤ ≤ Moreover, by Trotter-Kato’s theorem, Sn(.)ϕ(0) converges to S(.)ϕ(0) P-a.s. uni- formly on compact sets, i.e. lim sup (Sn(u) S(u))ϕ(0) =0 n 0 u T | − | →∞ ≤ ≤ which implies, together with 2 2 2ωT 2 sup (Sn(u) S(u))ϕ(0) 4M e ϕ(0) 0 u T | − | ≤ | | ≤ ≤ 2 2 and E ϕ(0) < , that lim E sup (Sn(u) S(u))ϕ(0) = 0, thanks to the n | | ∞ →∞ 0 u T | − | dominated convergence theorem. Then,≤ ≤ 2 I1 γn := 2E sup (Sn(u) S(u))ϕ(0) ≤ 0 u T | − | ≤ ≤ 2 2ωT 2 +2M e sup E ϕ(s) ϕn(s) 0 (4.3) r s 0 | − | → − ≤ ≤ For the second term I2, the triangle inequality yields u n 2 I2 3E Sn(u s)(fn(s, xs,Px(s)) fn(s, xs ,Pxn(s)))ds ≤ | 0 − − | # u 2 +3E Sn(u s)(f(s, xs,Px(s)) fn(s, xs,Px(s)))ds | 0 − − | # u +3E (S (u s) S(u s))f(s, x ,P )ds 2 | n − − − s x(s) | #0 3(I + I + I ) (4.4) ≤ 21 22 23 By using Cauchy Schwarz inequality and condition ( ), we obtain that F1 u u 2 2ωT n 2 2 I21 TM e Cf E x(s) x (s) ds + W2 (Px(s),Pxn(s))ds ≤ r | − | 0 .#− # * 2 2ωT 2 TM e Cf r sup E ϕ(s) ϕn(s) ≤ r s 0 | − | . − ≤ ≤ u +2 sup E x(v) xn(v) 2 ds (4.5) 0 r v s | − | # − ≤ ≤ * Cauchy Schwarz inequality implies that T I TM2e2ωT E f(s, x ,P ) f (s, x ,P ) 2 ds (4.6) 22 ≤ | s x(s) − n s x(s) | #0 The condition ( ) assures that F2 T 2 lim f(s, xs,Px(s)) fn(s, xs,Px(s)) ds =0 n | − | →∞ #0 CONTINUOUS DEPENDENCE ON THE COEFFICIENTS 11 The condition ( ) assures that F1 T f(s, x ,P ) f (s, x ,P ) 2 ds | s x(s) − n s x(s) | #0 T T 4 f(s, x ,P ) f(s, 0,P ) 2 ds +4 f(s, 0,P ) 2 ds ≤ | s x(s) − x(s) | | x(s) | #0 #0 T T +4 f (s, x ,P ) f (s, 0,P ) 2 ds +4 f (s, 0,P ) 2 ds | n s x(s) − n x(s) | | n x(s) | #0 #0 T 2 1 8Cf x(s) ds +8Cf L (Ω, P) ≤ r | | ∈ #− Then we conclude by the Lebesgue- dominated theorem that T 2 lim E f(s, xs,Px(s)) fn(s, xs,Px(s)) ds =0 n | − | →∞ #0 In order to estimate the term I , let us denotes S h(t)= t S(t s)h(s)ds,then, 23 ∗ 0 − u (S (u s) S(u s))f(s, x ,P )ds = S f!(u) S f(u) n − − − s x(s) n ∗ − ∗ #0 where f(u)=f(u, xu,Px(u)). - - By Theorem 3.1 in [16], we have - lim sup Sn f(u) S f(u) =0 n u T | ∗ − ∗ | →∞ ≤ and since - - T 2 2 ωT 2 1 sup Sn f(u) S f(u) 4TM e f(s, xs,Px(s)) ds L (Ω, P) u T | ∗ − ∗ | ≤ 0 | | ∈ ≤ # Then we conclude- by the- Lebesgue dominated theorem that 2 lim E sup Sn f(u) S f(u) =0 n u T | ∗ − ∗ | →∞ ≤ which implies - - u 2 lim sup E (Sn(u s) S(u s))f(s, xs,Px(s))ds = 0 (4.7) n u T | 0 − − − | →∞ ≤ # Inequalities (4.4), (4.5), (4.6) and (4.7) together imply that: u 2 2Tω n 2 I2 δn +6TM e Cf sup E x(v) x (v) ds (4.8) ≤ 0 r v s | − | # − ≤ ≤ where 2 2ωT 2 δn := 3rTM e Cf sup E ϕ(s) ϕn(s) r s 0 | − | − ≤ ≤ u 2 +3 sup E (Sn(u s) S(u s))f(s, xs,Px(s))ds u T | 0 − − − | ≤ # T +3TM2e2Tω E f(s, x ,P ) f (s, x ,P ) 2 ds | s x(s) − n s x(s) | #0 12 BRAHIM BOUFOUSSI AND SALAH HAJJI with lim δn =0 n →∞ For the last term I3, by Lemma 2.1, we have u 2H 1 2 I3 2HT − S(u s)g(s) Sn(u s)gn(s) ds ≤ 0 | − − − | # u 2H 1 2 4HT − Sn(u s)(g(s) gn(s)) ds ≤ 0 | − − | .# u + (S(u s) S (u s))g(s) 2 ds | − − n − | #0 * 2H 1 4HT − (I + I ) (4.9) ≤ 31 32 The condition ( ) assures that G2 T I M 2e2Tω g(s) g (s) 2 ds 0 as n (4.10) 31 ≤ | − n | → →∞ #0 u 2 Let us now prove that lim sup (S(u s) Sn(u s))g(s) ds =0. n u T 0 | − − − | →∞ ≤ # In the simplest case g(s)=ξ for some constant ξ X, we have by Lemma 4.3 ∈ that (S(u s) Sn(u s))ξ 0 as n and the convergence is uniform on [0,T], then,− it follow− that− → →∞ u 2 lim sup (S(u s) Sn(u s))g(s) ds =0. n u T 0 | − − − | →∞ ≤ # The same result remain true when g is a simple function. The general case can be easily proved by approximating g by simple functions then, u 2 lim sup (S(u s) Sn(u s))g(s) ds =0 (4.11) n u T 0 | − − − | →∞ ≤ # Inequalities (4.9), (4.10) and (4.11) together imply that: u 2H 1 2 I3 λn := 4HT − sup (S(u s) Sn(u s))g(s) ds ≤ u T 0 | − − − | . ≤ # T +M 2e2Tω g(s) g (s) 2 ds (4.12) | − n | #0 / with lim λn =0 n Thus,→∞ inequalities (4.2),(4.3), (4.8) and (4.12) together imply n 2 sup E x (u) x(u) 3(γn + δn + λn) 0 u t | − | ≤ ≤ ≤ t 2 2Tω n 2 +18TM e Cf sup E x (u) x(u) ds 0 r u s | − | # − ≤ ≤ CONTINUOUS DEPENDENCE ON THE COEFFICIENTS 13 Using the fact that x = ϕ and xn = ϕ on [ r, 0] we get n − n 2 2 sup E x (u) x(u) 3(γn + δn + λn)+ sup E ϕn(u) ϕ(u) r u t | − | ≤ r u 0 | − | − ≤ ≤ − ≤ ≤ t 2 2Tω n 2 +18TM e Cf sup E x (u) x(u) ds 0 r u s | − | # − ≤ ≤ which implies, by Gronwall’s inequality, lim sup E xn(u) x(u) 2=0 n r u T | − | →∞ − ≤ ≤ This completes the proof. ! Acknowledgment. The authors would like to thank the referee for making very useful comments and suggestions which lead to improvement of this work. References 1. Ahmed,T.A., Elsanousi, S.A. and Mohammed, S.E.A.: Existence theorems for stochas- tic functional differential equations with discontinuous initial data,preprint,Universityof Antwerp (UIA), 1998. 2. Arendt, W., Batty, C. J. K., Hieber, M., and Neubrander, F.: Vector-valued Laplace trans- forms and Cauchy problems.vol.96ofMonographsinMathematics,BirkhuserVerlag,Basel. 2001. 3. Boufoussi, B. and Hajji, S.: Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space.Statist. Probab. Lett; 82,(2012), 1549–1558. 4. Boufoussi, B., Hajji, S., and Lakhel, E.: Functional differential equations in Hilbert spaces driven by a fractional Brownian motion.Afr.Mat.DOI10.1007/s13370-011-0028-8.(2011). 5. Brze´zniak, Z.: On stochastic convolution in Banach spaces and applications.Stochastics Stochastics Rep. 61,(1997), 245–295. 6. Budhiraja, A, Dupuis, E, and Fischer, M : Large deviation properties of weakly interacting processes via weak convergence methods.AnnalsofProbability,Vol.40,(2012),74–102. 7. Caraballo, T., Garrido-Atienza, M. J., and Taniguchi, T.: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion.NonlinearAnalysis74,(2011),3671–3684. 8. Carmona, R. and Delarue, F.: Probabilistic theory of mean field games with applications. I. Mean field FBSDEs, control, and games.ProbabilityTheoryandStochasticModelling,83. Springer. 2018. 9. Da Prato, G. and J. Zabczyk, J.: Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications,CambridgeUniversityPress,Cambridge. 1992. 10. Govindan, T. E. and Ahmed, N. U.: On Yosida Approximations of McKean–Vlasov type stochastic evolution equations.StochasticAnalysisandApplications,33(3):(2015),383–398. 11. Haase, M. H. A.: The functional calculus for sectorial operators, vol. 169 of Operator Theory: Advances and Applications,BirkhuserVerlag,Basel.2006. 12. Kunze, M. C. and van Neerven, J. M. A. M.: Approximating the coefficients in semilinear stochastic partial differential equations.J.Evol.Equ.,11(3):(2011),577–604. 13. Kunze, M. C., and van Neerven, J. M. A. M.: Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations.J.DifferentialEquations253, (2012), 1036–1068. 14. Lasry, J. M. and Lions, P. L.: Mean field games.JapanJ.Math.,2,(2007),229–260. 15. Marinelli, C., Pr´evˆot,C., and M. R¨ockner, M.: Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise.J.Funct.Anal.258(2010), 616–649. 14 BRAHIM BOUFOUSSI AND SALAH HAJJI 16. Marinelli, C., Di Persioc, L., and Ziglio, G.: Approximation and convergence of solu- tions to semilinear stochastic evolution equations with jumps.Journal of Functional Analysis 264,(2013), 2784–2816. 17. Nualart, D.: The Malliavin Calculus and Related Topics, second edition, Springer-Verlag, Berlin. 2006. 18. Peszat, S. and Zabczyk, J.: Strong Feller property and irreducibility for diffusions on Hilbert spaces.Ann.Probab.,23,(1995),157–172. 19. Seidler, J.: Weak convergence of infinite-dimensional diffusions.StochasticAnal.Appl.,15, (1997), 399–417. 20. Touboul, J.: Mean-fields eqautions for stochastic firing-rate neural fields with delays: Deriva- tion and noise-induced transitions.PhysicaD:NonlinearPhenomena.2012. Brahim Boufoussi: Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakesh, Morocco E-mail address: [email protected] Salah Hajji: Department of Mathematics, Regional Center for the Professions of Education and Training, Marrakesh, Morocco E-mail address: [email protected]