Abstract Functional Stochastic Evolution Equations Driven by Fractional Brownian Motion

and Applied Analysis 3 for bounded, strongly measurable functions g : [0, T] → L(V; U). To begin, assume that such a function g is simple, meaning that there exists {g i : i = 1, . . . , n} ⊆ L(V; U) such that g (t) = gi, ∀ti−1 ≤ t < ti, (5) where 0 = t 0 < t 1 < ⋅ ⋅ ⋅ < t n−1 < t n = T and max 1≤i≤n ‖g i ‖L(V;U) = K. Definition 6. TheU-valued stochastic integral ∫T 0 g(t)dβ H (t) is defined by


Introduction
The purpose of this paper is to study the global existence and convergence properties of mild solutions to a class of abstract functional stochastic evolution equations of the general form in a real separable Hilbert space . Here, : ( ) ⊂ → is a linear (possibly unbounded) operator which generates a strongly continuous semigroup { ( ) : ≥ 0} on U; F : C ( [0, ]; L 2 (Ω; )) → L 2 ((0, ); L 2 (Ω; )) is a given mapping; : [0, ] → ( ; ) is a bounded, strongly measurable mapping (where is a real separable Hilbert space and L( ; ) denotes the space of Hilbert-Schmidt operators from into with norm ‖ ⋅ ‖ L( ; ) equipped with the strong topology); { ( ) : ≥ 0} is a U-valued fBm with Hurst parameter ∈ (1/2, 1); and 0 ∈ L 2 0 (Ω; ). Stochastic partial functional differential equations naturally arise in the mathematical modeling of phenomena in the natural sciences (see [1][2][3][4][5][6]). It has been shown that some applications, such as communication networks and certain financial models, exhibit a self-similarity property in the sense that the processes { ( ) : 0 ≤ ≤ } and { ( ) : 0 ≤ ≤ } have the same law (see [4,7]). Concrete data from a variety of applications have exhibited behavior that differs from standard Brownian motion ( = 1/2), and it seems that these differences enter in a nonnegligible way in the modeling of this phenomena. In fact, since ( ) is not a semimartingale unless = 1/2, the standard stochastic calculus involving the Itó integral cannot be used in the analysis of related stochastic evolution equations. There have been several papers devoted to the formulation of stochastic calculus for fBm [8][9][10][11] and differential/evolution equations driven by fBm [12][13][14] published in the past decade. We provide an outline of only the necessary concomitant technical details concerning the construction of the stochastic integral driven by an fBm and some of its properties in Section 2.
The present work may be regarded as a direct attempt to extend results developed in [1,12,[15][16][17][18] to a broader class of functional stochastic equations. The equations considered in the aforementioned papers can be viewed as special cases of (1) by appropriately defining the functional F, the correct space U, and the appropriate value of . In particular, the existence and convergence results we present constitute generalizations of the theory governing standard models arising in the mathematical modeling of nonlinear diffusion processes [1,15,[18][19][20][21][22] and communication networks [4].
The outline of the paper is as follows. We collect some preliminary information about certain function spaces, linear semigroups, probability measures, the definition of fBm, and the stochastic integral driven by a fBmin Section 2. The main existence results in the Lipschitz and compactness cases are discussed in Section 3, while convergence results are developed in Section 4. An extension of an existence result of the case of second-order stochastic evolution equations is discussed in Section 5. The paper concludes with a discussion of three different stochastic partial differential equations in Section 6 as an illustration of the abstract theory.
We make use of several different function spaces throughout this paper. The space of all bounded linear operators on is denoted by BL( ), while L 2 (Ω; ) stands for the space of all U-valued random variables for which ‖ ‖ 2 < ∞, where the expectation, E, is defined by ( ) = ∫ Ω ( ) . An important subspace is given by where {G : 0 ≤ ≤ } is the family of -algebras G generated by { ( ) : 0 ≤ ≤ } and B( ) is the Borel class on . The space of L 2 -continuous U-valued random variables is denoted by C([0, ]; L 2 (Ω; )).
The following alternative of the Leray-Schauder principle [29] plays a role in Section 3.  [27] established the equivalence of tightness and relative compactness of a family of probability measures. Consequently, the Arzelá-Ascoli theorem can be used to establish tightness.  We next make precise the definition of a U-valued fBm and related stochastic integral used in this paper. The approach we use coincides with the one formulated and analyzed in [12,30] be a sequence of independent, one-dimensional fBms with Hurst parameter ∈ (1/2, 1) such that, for all = 1, 2, . . ., In such case, ] < ∞, so that the following definition is meaningful.
where the convergence is understood to be in the mean-square sense.
As argued in Lemma 2.2 of [30], this integral is well defined since Since the set of simple functions is dense in the space of bounded, strongly measurable L( ; )-valued functions, a standard density argument can be used to extend Definition 6 to the case of a general bounded, strongly measurable integrand.

Existence Results
We consider mild solutions of (1) in the following sense.
Proof. Property (i) can be established as in Lemma 6 in [12]. To verify property (ii), let 0 ≤ ≤ and observe that The strong continuity of (⋅), together with (H3), guarantees that the first term on the right side of (10) goes to zero as ℎ → 0. To argue the second term goes to zero, we first assume that is a simple function as defined in (5). Arguing as in [12] yields the estimate where is defined as in part (i) of this lemma. Using (11) in the second term on the right side of (10) yields The convergence of ∑ ∞ =1 ] ensures that the right side of (12) goes to zero as → ∞. As such, property (ii) holds for a simple function . It is not difficult to extend the argument to general bounded, strongly measurable functions . This completes the proof.

Consider the solution map
The first integral on the right side of (13) is taken in the Bochner sense, while the second is defined in Section 2. The operator Φ satisfies the following properties.
Proof. Using the discussion in Section 2 and the properties of , one can see that for any ∈ C ([0, ]; L 2 (Ω; )), Φ( )( ) is a well-defined stochastic process, for each 0 ≤ ≤ . In order to verify the continuity of Φ on [0, ], let ∈ C ([0, ]; L 2 (Ω; )) and consider 0 ≤ * ≤ and |ℎ| sufficiently small. Observe that The semigroup property enables us to write So, the strong continuity of (⋅) implies that the right side of (15) goes to 0 as | ℎ | → 0. Next, using the Hölder inequality with (H2) yields which clearly goes to 0 as | ℎ | → 0. Also, the strong continuity of (⋅)with (H2) enables us to conclude, with the help of the dominated convergence theorem, that (16) and (17), both of which go to 0 as |ℎ| → 0, it follows that and that Abstract and Applied Analysis 5 Using the property ( ( ) − ( )) 2 = | − | 2 ] with = * + ℎ and = * enables us to conclude that the right side of (19) goes to 0 as |ℎ| → 0. The second term on the right side of (18) goes to 0 as |ℎ| → 0 by Lemma 8(ii). Thus, ‖ 3 ( * + ℎ) − 3 ( * )‖ → 0 as | ℎ | → 0 when is a simple function. Since the set of all such simple functions is dense in L( ; ), a standard density argument can be used to extend this conclusion to a general bounded, measurable function . This establishes the continuity of Φ.
Our first existence result is as follows. Proof. We know that Φ is well defined and continuous from Lemma 9. Let = 1/( 2 2 F + 1). We prove that Φ has a unique fixed point in C ([0, ]; L 2 (Ω; )). To this end, let , ∈ C ([0, ]; L 2 (Ω; )). Observe that (13) implies that Squaring both sides and taking the expectation in (22) yields, with the help of Young's inequality, Taking the supremum over [0, ] in (23) and applying reasoning similar to that which led to (16) yield where the last inequality in (24) follows from the choice of . Hence, Φ is a strict contraction on [0, ] and so has a unique fixed point which coincides with a mild solution of (1) on [0, ]. Performing this same argument on [ , 2 ], [2 , 3 ], and so on enables us to construct in finitely many steps a unique piecewise-defined function in C ([0, ]; L 2 (Ω; )) which is a unique mild solution of (1) on the original interval [0, ]. This completes the proof.
We now develop existence results for (1) in which the Lipschitz condition on F is replaced by the combination of continuity and a sublinear growth condition. This is done at the expense of a compactness restriction on the semigroup. Precisely, we use the following assumptions instead: is a continuous map such that there exists positive constants 1 and 2 such that for all ∈ C ([0, ]; L 2 (Ω; )).
We begin by establishing certain compactness properties of the mapping Φ 1 : L 2 ((0, ); L 2 (Ω; )) → C ([0, ]; L 2 (Ω; )) defined by The well definedness of this mapping is essentially a stochastic analog of Lemma 3.1 in [31] (where ( ) plays the role of the resolvent operator) and its proof follows similarly by making the natural modifications. Proof. We use Schaefer's theorem to prove that Φ (as defined in (13) The continuity of F ensures that the right side of (32) goes to 0 as → ∞, thereby verifying the continuity of Φ.
Next, let = 1/(3 2 2 1 +1). We will show that the set (Φ), as defined in Theorem 1 with C ([0, ]; L 2 (Ω; )) in place of X, is bounded. Let V ∈ (Φ) and observe that, arguing as in (20), applications of the Hölder and Young inequalities (with (H8)) yield Also, from Lemma 8 we can infer that Thus, we conclude that, for all ∈ (Φ) and 0 ≤ ≤ , Taking into account that ≥ 1 and the choice of , we conclude from (33) that ‖ ‖ C ≤ , where is a constant independent of and . So, (Φ) is bounded.

Convergence and Approximation Results
Throughout this section we assume that A, F, and satisfy (H1)-(H5). , for some > 0 (independent of n), for each ∈ N, and ‖ − ‖ → 0 as → ∞, for each ∈ ( ); Consider the following initial-value problem: Since 0 is a fixed element of L 2 0 (Ω; ), a standard argument guarantees the existence of a unique mild solution of (47). We need the following lemma.
A standard argument invoking (H12) and (H13), involving the Trotter-Kato Theorem [28], can be used to conclude that each of the first three terms on the right side of (48) goes to 0 as → ∞. As for the fourth term, observe that Abstract and Applied Analysis 9 The uniform boundedness of { ( ) : 0 ≤ ≤ , ∈ N} (cf. (H12)) with (H14) guarantees that the supremum (over [0, ]) of the first term on the right side of (49) goes to 0 as → ∞. An argument in the spirit of the one used to verify Lemma 8 (ii) can be used to show the supremum (over [0, ]) of the second term in (49) and also goes to 0 as → ∞, as needed. This completes the proof.
The following is the first of our two main convergence results.
To finish the proof, we remark that Theorem 17 implies that the finite-dimensional joint distributions of converge weakly to those of P (cf. Proposition 3). Hence, Theorem 4 ensures that → as → ∞. This completes the proof.

Extension to the Second-Order Case
Consider the abstract second-order stochastic Cauchy problem ≥ 0} is ã-valued fBm with Hurst parameter ∈ (1/2, 1); and 0 , 1 ∈ L 2 0 (Ω;̃). We will convert (64) to a first-order system that, in turn, can be represented abstractly in the form (1). To this end, let Then, As such, ] . (67) The space = (C 1/2 ) ×̃is a Banach space when equipped with the usual graph norm. Define Since we can use these identifications to rewrite (64) abstractly in the form We assume that the following conditions are satisfied.
Since B ∈ BL( ), it follows that + generates a strongly continuous semigroup on . As such, we can view (70) (and so, (64)) as a special case of (1). Theorem 10 can be applied directly to (70) under the same hypotheses to conclude that (64) has a unique mild solution ∈ C ([0, ]; L 2 (Ω; )).

Applications
Let D be a bounded domain in R with smooth boundary D and consider the initial-boundary value problem for all 1 , 2 , 1 , 2 ∈ R and almost all ∈ (0, ); for all 1 , 2 ∈ R and almost all ( , ) ∈ . Let = = L 2 (D) and set It is well known that generates a 0 -semigroup on ( ) (see [28], Chapter 7). Next, define One can use (H22)-(H25) to verify that satisfies (H2) with and that is strongly measurable. Thus, (71) can be rewritten in the form (1) in U, with A, F, and defined above so that an application of Theorem 10 immediately yields the following result.

(85)
In view of (H26)-(H29), together with the Hölder and Young inequalities, one can verify that F satisfies (H2) with Hence, (82) can be written in the abstract form (64) in and so can be transformed into (1) via the procedure outlined in Section 5. As such, an application of Theorem 10 immediately yields the following result.