FUNCTIONAL INTEGRO-DIFFERENTIAL STOCHASTIC EVOLUTION EQUATIONS IN HILBERT SPACE

We investigate a class of abstract functional integro-differential stochastic evolution equations in a real separable Hilbert space. Global existence results concerning mild and periodic solutions are formulated under various growth and compactness conditions. Also, related convergence results are established and an example arising in the mathematical modeling of heat conduction is discussed to illustrate the abstract theory.


Introduction
The purpose of this paper is to study the global existence and convergence properties of mild solutions to a class of abstract semi-linear functional stochastic integro-differential equations of the general form x(0) = h(x) + x 0 , in a real separable Hilbert space H. Here, A : D(A) ⊂ H → H is a linear (possibly unbounded) operator, G : C([0, T ]; H) → C([0, T ]; L 2 (Ω; BL(K; H))) (where K is a real separable Hilbert space), F : C([0, T ]; H) → L p (0, T ; L 2 (Ω; H)) (1 ≤ p < ∞), W is a K-valued Wiener process with incremental covariance described by the nuclear operator Q, x 0 is an F 0 -measurable H-valued random variable independent of W , and h : C([0, T ]; H) → L 2 0 (Ω; H).The present work may be regarded as a direct attempt to extend recent results developed in [7,10,16,18,20] to a broader class of functional stochastic equations.The equations considered in the aforementioned papers can be viewed as special cases of (1.1) by making the appropriate identifications of F , G, and h.Moreover, we further extend these results by incorporating more general initial conditions.In particular, mild periodic solutions are obtained.To the authors' knowledge the results in this paper are new even in the case of a classical initial condition (i.e., when h = 0).
The deterministic version of (1.1) (and related equations) coupled with a classical initial condition has been studied extensively both when A is linear and when A is nonlinear.We refer the reader to [8,30] and the references therein.Byszewski [13] introduced nonlocal initial conditions into such abstract initial-value problems and argued that the corresponding models more accurately describe the phenomena since more information was taken into account at the onset of the experiment, thereby reducing the ill effects incurred by a single (possibly erroneous) initial measurement.Since then, many authors have continued this work in several directions and established existence theories for first-order nonlinear evolution equations [2,4,29], second-order equations [7], delay equations [7,28], Volterra integral and integro-differential equations [5,25], and differential inclusions [1].Concrete nonlocal parabolic and elliptic partial (integro-) differential equations arising in the mathematical modeling of various physical, biological, and ecological phenomena, as well as a discussion of the advantages of replacing the classical initial condition with a more general functional, can be found in [13,21] and the references contained therein.
Stochastic differential equations (SDEs) in both finite and infinite dimensions have also received considerable attention.We refer the reader to [10,32] for a thorough discussion in the finite dimensional setting, and [14,19] for the infinite dimensional setting.A semi-group-theoretic development of a theory for the stochastic analogues of deterministic evolution equations is both powerful and beneficial since it enables one to investigate a broad class of stochastic partial differential equations within a unified context.SDEs are important from the viewpoint of applications since they incorporate (natural) randomness into the mathematical description of the phenomena, and, therefore, provide a more accurate description of it.Moreover, coupling such equations with a nonlocal initial condition strengthens the model even further.
The basic tools used in this paper include fixed-point techniques, the theory of (compact) linear semi-groups, results for probability measures, and methods and results for infinite dimensional SDEs.The results are important from the viewpoint of applications since they cover nonlocal generalizations of integro-differential SDEs arising in fields such as electromagnetic theory, population dynamics, and heat conduction in materials with memory [10,17,19,32].
The outline of the paper is as follows.We review some basic facts about linear semi-groups, the theory of SDEs, and probability measures in Section 2.Then, Sections 3 and 4 are devoted to the development of our main existence results, while a discussion of various convergence results immediately follows in Section 5. Finally, the paper concludes with a discussion of a concrete nonlocal integro-partial SDE in Section 6.

Preliminaries
For further background of this section, we refer the reader to [9,11,12,14,15,19,23,30,32].Throughout this manuscript, H and K denote real separable Hilbert spaces equipped with norms • H and • K , respectively, and the space of bounded linear operators from K to H is denoted by BL(K; H) (or simply BL(H) if K = H).Also, for Banach spaces X and Y , the space of continuous functions from X into Y (equipped with the usual sup-norm) shall be denoted by C(X; Y ), while L p (0, T ; X) shall represent the space of X-valued functions that are p-integrable on [0, T ].
Let (Ω, F, P ) be a complete probability space equipped with a normal filtration {F t : 0 ≤ t ≤ T } (i.e., a right-continuous, increasing family of sub σ-algebras of F).An H-valued random variable is an F-measurable function X : Ω → H and a collection of random variables S = {X(t; ω) : Ω → H|0 ≤ t ≤ T } is called a stochastic process.Henceforth, we shall suppress the dependence on ω ∈ Ω and write X(t) instead of X(t; ω) and X : [0, T ] → H in place of S.
The collection of all strongly-measurable, square-integrable H-valued random variables, denoted by L 2 (Ω; H), is a Banach space equipped with norm , where the expectation, E, is defined by One can prove that this is a Banach space when equipped with the norm Definition 2.1: A stochastic process {W (t) : t ≥ 0} in a real separable Hilbert space H is a Wiener process if for each t ≥ 0, (i) W (t) has continuous sample paths and independent increments, Consider the initial-value problem where H ds < ∞ a.s.[P ], and for all 0 ≤ t ≤ T .
(The second integral in (2.3) is taken in the sense of Itó.A complete discussion of the construction of the Itó integral can be found in [14].)It is well-known that (2.2) has a unique mild solution x ∈ C([0, T ]; H), and if stronger regularity restrictions are imposed on the data, this solution is a strong solution (see [19,20]).
The following alternative of the Leray-Schauder principle [24] plays a key role in Section 4.
We conclude this section with some comments regarding probability measures.We refer the reader to [9,11] for a more detailed discussion.
Let X be an H-valued random variable and let P(H) denote the set of all probability measures on H.The probability measure P induced by X, denoted P X , is defined by

Existence Results -Lipschitz Case
Consider the initial-value problem (1.1) in a real separable Hilbert space H under the following assumptions: (H1) The linear operator A : ) The continuity of J is easily verified.Successive applications of Hölder's inequality yields Subsequently, an application of (H2), together with Minkowski's inequality, enables us to continue the string of inequalities in (3.4) to conclude that Taking the supremum over [0, T ] in (3.5) then implies that (Ω; H) (by (H4) and (H5)).Consequently, one can argue as in [20] to conclude that J is a well-defined.
Next, we show that J is a strict contraction.Observe that for x, y ∈ C([0, T ]; H), we infer from (3.3) that For convenience, let I 1 , I 2 , and I 3 represent the first, second, and third terms, respectively, on the right-side of (3.6).Squaring both sides and taking the expectation in (3.6) yields, with the help of Young's inequality, and subsequently, Using reasoning similar to that which led to (3.4), one can show that Also, one can modify the argument of Proposition 1.9 in [20] to conclude that there exists a constant C G (depending only on p, T r(Q), and T ) such that Using (3.8) and (3.9) in (3.7) enables us to conclude that J is a strict contraction, provided that (3.2) is satisfied and thus, has a unique fixed point which coincides with a mild solution of (1.1).This completes the proof.Next, we consider the following initial-value problem studied in [16]. where ) are given mappings satisfying the following conditions: We recover Theorem 2.1 in [16] as the following corollary of Theorem 3. Proof: The Uniform Boundedness Principle guarantees the existence of positive constants M B and M C such that B(t, s) BL ≤ M B and C(t, s) BL ≤ M C , for all 0 ≤ t ≤ s ≤ T .Standard computations involving properties of expectation and Hölder's inequality imply, with the help of (H6), that for all x, y ∈ C([0, T ]; H), Similarly, (H7) enables us to infer that for all x, y ∈ C([0, T ]; H), (i) We also recover Theorem 3.3 in [22] as a corollary to Theorem 3.2 if we replace F and G in (3.11), respectively, by where C is a convolution-type kernel satisfying Assumptions 3.2 on page 361 in [22].The result then follows from Corollary 3.3.
(ii) A result analogous to Corollary 3.3 regarding a delay version of (3.10) (obtained by replacing g(s, x(s)) by g(s, x(s), x(σ(s))), where σ : [0, T ] → [0, T ] is a continuous, nondecreasing function) can be established by making slight modifications to the above argument.A related delay equation is discussed in [7] using compactness methods.
We conclude this section with a comment on a special case of (3.10), namely where x 0 = 0 and h is given by Clearly, h, as given by (3.14), satisfies (H7) with M g = 1.Since M S ≥ 1, condition (3.2) does not hold for such h.To incorporate (3.14) into our theory, we consider that the functions f i and g are defined instead on C((0, ∞); H) and satisfy (H6) and (H7), respectively, with [0, T ] replaced by [0, ∞).Also, we take B and C to be convolution kernels in L 1 (0, ∞) of the type described in Remark 3.4(i).And finally, we assume that A generates a semi-group {S(t) : t ≥ 0} on H such that (H8) There exist M S ≥ 1 and ω > 0 such that S(t) BL ≤ M S e −ωt , for all t ≥ 0.
For conditions that ensure that (H8) holds, see [30], pg.116.Using an approach similar to the one employed in [25], we can now prove that the following initial-value problem has a unique mild solution, provided T is sufficiently large.
Theorem 3.5: Suppose (H1) and (H8) hold, and that f i , g, B, and C are as described above.If also Proof: Arguing as in [22], it follows that for each fixed T > 0 and each y ∈ L 2 0 (Ω; H), the initial-value problem (3.15) (with y in place of x(T )) has a unique mild solution x y on [0, T ] given by On account of (H8), and the assumptions imposed on f i , g, B, and C, (3.16) yields (3.17) Now, using a Gronwall-type inequality in (3.17) (cf.[25], Lemma 4.2), we arrive at for all y, z ∈ L 2 0 (Ω; H), and subsequently, Observe that (3.18) and (H9) imply that Q T is a strict contraction on L 2 (Ω; H), for sufficiently large T .Thus, for T chosen such that (H9) is satisfied, Q T has a unique fixed point y 0 .The corresponding function u = u y 0 is the desired mild solution of (3.15).

Existence Results -Compactness Case
We now develop existence results for (1.1) in which the Lipschitz conditions on F , G, and h are replaced by sublinear growth conditions.This is done at the expense of a compactness restriction on the semi-group.Precisely, we use the following assumptions instead: We begin by establishing certain compactness properties of the mappings Φ 1 : The well-definedness of these two mappings follows from an application of Lebesgue's Dominated Convergence Theorem.Lemma 4.1: Assume that {S(t) : 0 ≤ t ≤ T } is a compact semi-group on H.Then, Proof: Part (i) is essentially a stochastic analog of Lemma 3.1 in [3] (where S(t) plays the role of the resolvent operator) and its proof follows similarly by making the natural modifications.We shall only sketch the proof of (ii).
Since the right-side of (4.4) can be made arbitrarily small, uniformly for v ∈ K r , we conclude that Φ 2 (K r )(t) is totally bounded.This, combined with the work above, yields the precompactness, and the proof is complete.Theorem 4.2: Assume that (H5) and (H10) -(H13) are satisfied.Then, (1.1) has at least one mild solution on [0, T ] provided that Proof: We use Schaefer's theorem to prove that J (as defined in (3.3)) has a fixed point.
The well-definedness of J under (H10) -(H13) can be established using reasoning similar to that employed in the proof of Theorem 3.2.To verify the continuity of J , let . The continuity of F , G, and h ensure that the right-side of (4.5) goes to zero as n → ∞, thereby verifying the continuity of J .
Next, we show that the set ξ(J ), as defined in Theorem 2.3 with C([0, T ]; H) in place of X, is bounded.Let v ∈ ξ(J ) and observe that the Hölder and Young inequalities (with (H12)) yield Also, arguing as in (3.4), we obtain (with the help of (H14)) Hence, (4.6) and (4.7), in conjunction with (H13), enable us to conclude that for all v ∈ ξ(J ) and 0 ≤ t ≤ T , we have Taking into account that λ ≥ 1 and (H14), we conclude from (4.7) that v C ≤ η, where η is a constant independent of v and λ.So, ξ(J ) is bounded.
To apply Schaefer's theorem, we must finally show that J is compact.To this end, let r > 0 and define K r = {v ∈ C([0, T ]; H) : v C ≤ r}.Using the notation of (4.1) and (4.2), we can express (3.3) as We shall prove that J (K r ) is precompact in C([0, T ]; H).First, the facts that {F (v) : Bi and hence, L is totally bounded.Thus, L is precompact in C([0, T ]; H).Hence, Schaefer's theorem implies that J has at least one fixed point x ∈ C([0, T ]; H) which is a mild solution to (1.1).

10) has at least one mild solution on [0, T ].
Proof: We use Schauder's fixed-point theorem [24] to argue that J (as defined in Suppose, by way of contradiction, that for each k ∈ IN , there exists H17) and (H18), there exist ) Using (4.12) in (4.11) yields (with the help of (H13)) and subsequently, . Consequently, there is an n 0 ∈ IN such that J (B n 0 ) ⊂ B n 0 .Thus, Schauder's fixed point theorem guarantees the existence of x ∈ B n 0 such that J (x) = x, which is the mild solution that we seek.
Remark: An inspection of the proof shows that (H13) can be weakened slightly in that instead of imposing the sublinear growth restriction on h, we need only assume that lim

Convergence Results
Throughout this section we assume that A, F , G, and h satisfy (H1)-(H4) and that (3.2) holds.For each n ∈ IN , consider a linear operator (Ω; H) satisfying the following conditions: (Here, the constants M S , M F , M G , and M h are the same ones appearing in (H1)-(H4) and so, are independent of n.) Let x be the mild solution to (1.1) as guaranteed by Theorem 3.2.By virtue of (H6), (H20), (H21)(i), (H22)(i), and (H23)(i), Theorem 3.2 implies that, for each n ∈ IN , the problem (5.1) Consider the following initial-value problem: Since h n (x) + x 0 is a fixed element of L 2 0 (Ω; H), a standard argument (see Ch. 7 in [14]) guarantees the existence of a unique mild solution y n of (5.2).We need the following lemma.
We now state the first of our two main convergence results.A comparable theorem for a nonlinear deterministic evolution equation is discussed in [2].
We begin by showing {P x n } ∞ n=1 is relatively compact in C([0, T ]; H) by appealing to the Arzelá-Ascoli theorem.To this end, we shall first show that there exists η > 0 such that sup n∈IN sup 0≤t≤T x n (t) L 2 (Ω;H) = η < ∞. (5.8) Note that x n is given by Using this fact, together with (H20) and (H23)(i), we arrive at Likewise, (H21)(ii) and (H22)(ii) guarantee that there exist for all n, so that a standard argument now yields and Combining the estimates (5.10)-(5.12)and rearranging terms, we can now conclude from (5.9) that (5.8) holds due to (H24) and the fact that all constants in (5.10)-(5.12)are independent of n.Next, we establish the equicontinuity by showing E x n (t)−x n (s) 4 H → 0 as (t−s) → 0, for all 0 ≤ s ≤ t ≤ T , uniformly for all n ∈ IN .We estimate each term of the representation formula for x n (t) − x n (s) (cf.(5.9)) separately.Employing Theorem 2.4(d) in [30] and taking into account (H20), (H23), and the uniform boundedness of S n (•)A n , we conclude that where Estimating each of the two integrals on the right-side of (5.14) separately yields, from the boundedness of S n (•)A n , (H20), and (H21)(i), that and similarly, Regarding the difference of the stochastic integrals, note that Fubini's theorem, together with basic integral properties, enables us to write Arguing as above, we see that  [11]).
To finish the proof, we remark that Theorem 5.2 implies that the finite-dimensional joint distributions of P x n converge weakly to those of P (cf.Proposition 2.5).Hence, Theorem 2.6 ensures that P x n w → P x as n → ∞.Remark: For the classical version of (5.1) (i.e., when h n = 0, for all n), a Gronwalltype argument can be used to establish the uniform boundedness (in C([0, T ]; H)) of {x n } ∞ n=1 and, in such case, condition (H24) can be dropped.
and d 2 = 2M C b 2 T .Consequently, (3.10) has at least one mild solution by Theorem 4.2.
(3.3) with F and G given by (3.11) has a fixed point.The continuity and compactness follow by making slight changes to the proof of Theorem 4.1.For n ∈ IN , let B n = {x ∈ C([0, T ]; H) : x C ≤ n}.It remains to show that there exists an n ∈ IN such that J (B n ) ⊂ B n .

{P n } converges weakly to the finite dimensional joint distribution of P and {P n }is relatively compact, then P n
1], where B(H) is the Borel class on H.A sequence {P n } ⊂ P(H) is said to be weakly convergent to P if Ω fdP n → Ω fdP , for every bounded, continuous function f : H → IR; in such case, we write P n w → P .Next, a family {P n } is tight if for each > 0, there exists a compact set K such that P n(K ) ≥ 1 − , for all n.Let P ∈ P(H) and 0 ≤ t 1 < t 2 < . ..< t k ≤ T .Define π t 1 ,...,t k : C([0, T ]; H) → H k by π t 1 ,...,t k (X) = (X(t 1 ), . .., X(t k )).The probability measures induced by π t 1 ,...,t k are the finite dimensional joint distributions of P .
≤ t ≤ T .(The Uniform Boundedness Principle and the strong continuity of S(t) together guarantee the existence of a positive constant M S such that S(t) BL ≤ M S for all 0 ≤ t ≤ T .)Our first result is: Theorem 3.2: Assume that (H1) -(H5) hold.Then, (1.1) has a unique mild solution on are bounded subsets of L p (0, T ; L 2 (Ω; H)) and C BL , respectively (cf.(H11) and (H12)), it follows from Lemma 4.1 that the set {Φ 1 Now, let P x and P x n denote the probability measures on C([0, T ]; H) induced by the mild solutions x and x n of (1.1) and (5.1), respectively.Using Theorem 5.2, we can prove that P x n Invoking(5.8)in(5.13),(5.15),(5.16),(5.18),and(5.19)enablesus to conclude that, in fact, E x n (t) − x n (s)4H → 0 as (t − s) → 0, uniformly for 0 ≤ s ≤ t ≤ T and n ∈ IN , as desired.Thus, the family {P x n } ∞ n=1 is relatively compact in C([0, T ]; H) and hence, tight (by Prokorhov's theorem n (t − θ)G n (x n )(τ )dθdW (τ )4