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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 516853, 14 pages
Abstract Functional Stochastic Evolution Equations Driven by Fractional Brownian Motion
1Department of Mathematics, West Chester University of Pennsylvania, 25 University Avenue, West Chester, PA 19383, USA
2Department of Mathematics and Computer Science, Goucher College, 1021 Dulaney Valley Road, Baltimore, MD 21204, USA
Received 21 October 2013; Revised 15 December 2013; Accepted 20 December 2013; Published 12 February 2014
Academic Editor: T. Diagana
Copyright © 2014 Mark A. McKibben and Micah Webster. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownian motion in a real separable Hilbert space. Global existence results concerning mild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory.
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 on U; is a given mapping; is a bounded, strongly measurable mapping (where is a real separable Hilbert space and denotes the space of Hilbert-Schmidt operators from into with norm equipped with the strong topology); is a U-valued fBm with Hurst parameter ; and .
Stochastic partial functional differential equations naturally arise in the mathematical modeling of phenomena in the natural sciences (see [1–6]). 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 and have the same law (see [4, 7]). Concrete data from a variety of applications have exhibited behavior that differs from standard Brownian motion (), 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 , 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 [8–11] and differential/evolution equations driven by fBm [12–14] 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, 15–18] 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 , 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, 18–22] and communication networks .
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.
For further background of this section, we refer the reader to [6, 8, 9, 12, 23–28] and the references therein. Throughout this paper, U is a real separable Hilbert space with norm and inner product equipped with a complete orthonormal basis , and is a complete probability space. We suppress the dependence of all random variables on throughout the manuscript and write instead of .
We make use of several different function spaces throughout this paper. The space of all bounded linear operators on is denoted by , while stands for the space of all U-valued random variables for which , where the expectation, E, is defined by . An important subspace is given by where is the family of -algebras generated by and is the Borel class on . The space of continuous U-valued random variables such that is denoted by .
Theorem 1. Let X be a Banach space, and let be a completely continuous map. Then, either has a fixed point, or the set is unbounded.
The probability measure induced by a U-valued random variable , denoted by , is defined by . A sequence is said to be weakly convergent to if , for every bounded, continuous function ; in such case, we write . A family is tight if for each , there exists a compact set such that , for all . Kunita  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.
Definition 2. Let and . Define by . The probability measures induced by are the finite dimensional joint distributions of .
Proposition 3. If a sequence of U-valued random variables converges weakly to a U-valued random variable in the mean-square sense, then the sequence of finite dimensional joint distributions corresponding to converges weakly to the finite dimensional joint distribution of .
The next theorem, in conjunction with Proposition 3, is the main tool used to prove one of the convergence results in this paper.
Theorem 4. Let . If the sequence of finite dimensional joint distributions corresponding to converges weakly to the finite dimensional joint distribution of and is relatively compact, then .
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]. Let be a sequence of independent, one-dimensional fBms with Hurst parameter such that, for all ,(i), (ii),(iii),(iv).In such case, , so that the following definition is meaningful.
Definition 5. For every , is a U-valued fBm, where the convergence is understood to be in the mean-square sense.
It has been shown in [12, 30] that the covariance operator of is a positive nuclear operator such that Next, we outline the discussion leading to the definition of the stochastic integral associated with for bounded, strongly measurable functions . To begin, assume that such a function is simple, meaning that there exists such that where and .
Definition 6. The U-valued stochastic integral is defined by As argued in Lemma 2.2 of , this integral is well defined since Since the set of simple functions is dense in the space of bounded, strongly measurable -valued functions, a standard density argument can be used to extend Definition 6 to the case of a general bounded, strongly measurable integrand.
3. Existence Results
We consider mild solutions of (1) in the following sense.
Definition 7. A stochastic process is a mild solution of (1) if For our first result, we impose the following conditions on (1):(H1) is the infinitesimal generator of a strongly continuous semigroup on such that , for all , for some and ;(H2) is such that there exists a positive constant for which (H3) is a bounded, strongly measurable mapping;(H4) is a U-valued fBm;(H5).(Henceforth, we write , which can be shown to be finite by using (H1) and the Uniform Boundedness Principle.)
The following technical properties involving the stochastic integral , under assumptions (H1), (H3), and (H4), are used in the proofs of the main results in this paper.
Proof. Property (i) can be established as in Lemma 6 in . To verify property (ii), let 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 . To argue the second term goes to zero, we first assume that is a simple function as defined in (5). Arguing as in  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 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 defined by 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 satisfies the following properties.
Lemma 9. Assume that (H1)–(H5) hold. Then, is a well-defined, continuous map.
Proof. Using the discussion in Section 2 and the properties of , one can see that for any , is a well-defined stochastic process, for each . In order to verify the continuity of on , let and consider and 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 . Next, using the Hölder inequality with (H2) yields
which clearly goes to 0 as . Also, the strong continuity of with (H2) enables us to conclude, with the help of the dominated convergence theorem, that
as . Consequently, since is dominated by the expressions in (16) and (17), both of which go to 0 as , it follows that as .
It remains to show that as . Observe that and that Using the property with and enables us to conclude that the right side of (19) goes to 0 as . The second term on the right side of (18) goes to 0 as by Lemma 8(ii). Thus, as when is a simple function. Since the set of all such simple functions is dense in , a standard density argument can be used to extend this conclusion to a general bounded, measurable function . This establishes the continuity of .
Finally, we assert that . 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 (20) to conclude that Taking the supremum over in (21) then implies that , for any . The other estimates can be established as above, and when used in conjunction with Lemma 8, one can readily verify that , for any . Thus, we conclude that is well defined, and the proof of Lemma 9 is complete.
Our first existence result is as follows.
Theorem 10. Assume that (H1)–(H5) hold. Then, (1) has a unique mild solution on .
Proof. We know that is well defined and continuous from Lemma 9. Let . We prove that has a unique fixed point in . To this end, let . 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 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, is a strict contraction on and so has a unique fixed point which coincides with a mild solution of (1) on . Performing this same argument on , , and so on enables us to construct in finitely many steps a unique piecewise-defined function in which is a unique mild solution of (1) on the original interval . This completes the proof.
Next, we consider the following initial-value problem: where and satisfy the following conditions:(H6) is a collection of bounded linear operators for which there exists a positive constant such that (H7) is such that there exists a positive constant for which
Corollary 11. If (H1), (H4), (H5), (H6a), and (H6) hold, then (22) has a unique mild solution on .
Proof. Define by Standard computations involving properties of expectation and Hölder’s inequality imply, with the help of (H6) and (H7), that, for all , Thus, if we let in (H2), we can conclude from Theorem 10 that (25) has a unique mild solution on .
We now develop existence results for (1) in which the Lipschitz condition on 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:(H8) generates a compact - semigroup on U;(H9) is a continuous map such that there exists positive constants and such that for all .
We begin by establishing certain compactness properties of the mapping defined by The well definedness of this mapping is essentially a stochastic analog of Lemma 3.1 in  (where plays the role of the resolvent operator) and its proof follows similarly by making the natural modifications.
Lemma 12. Assume that is a compact semigroup on . Then, is a compact map from into .
Theorem 13. Assume that (H3), (H4), (H5), (H8), and (H9) hold. Then, (1) has at least one mild solution on .
Proof. We use Schaefer’s theorem to prove that (as defined in (13)) has a fixed point. The well definedness of under (H3), (H4), (H5), (H8), and (H9) can be established using reasoning similar to that employed in the proof of Theorem 10. To verify the continuity of , let be a sequence in such that as . Standard computations yield
The continuity of ensures that the right side of (32) goes to 0 as , thereby verifying the continuity of .
Next, let . We will show that the set , as defined in Theorem 1 with in place of , is bounded. Let 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 and , Taking into account that and the choice of , we conclude from (33) that , where is a constant independent of and . So, is bounded.
In order to apply Schaefer’s theorem, it remains to show that is compact. To this end, let and define . Using the notation of (13) and (31), we have We assert that is precompact in . Indeed, the fact that is a bounded subset of (cf. (H9)), it follows from Lemma 12 that the set is precompact in . Since the set is trivially precompact, we conclude that is precompact in . So, Schaefer’s theorem implies that has a fixed point which is a mild solution to (1) on . Performing this same argument on , and so on enables us to construct in finitely many steps a piecewise-defined function in , that is, a mild solution of (1) on the original interval . This completes the proof.
Next, we state a corollary regarding (25) under the following assumptions on :(H10) satisfies the following:(i) is continuous, for almost all ;(ii) is strongly -measurable, for all ;(iii)There exist positive constants and such that for almost all and for all .
Corollary 14. If (H3), (H4), (H5), (H8), and (H10) hold, then (25) has at least one mild solution on .
Proof. An argument similar to the one used in [32, Chapter 26, pg. 561] shows that (H10) guarantees the mapping defined in (28) is well defined and continuous. Routine calculations show that satisfies (H9) with and . Consequently, (25) has at least one mild solution by Theorem 13.
We can formulate a stronger version of Corollary 14 by replacing assumption (H10) by the following:(H11) satisfies (H10) (i) and (ii), and(i)for each , there exists such that for almost all , (ii)
Proposition 15. Assume that (H3), (H4), (H5), (H8), and (H11) hold. Then, (25) has at least one mild solution on .
Proof. We use Schauder’s fixed-point theorem to argue that (as defined in (13) with given by (28)) has a fixed point. The continuity and compactness follow by making slight changes to the proof of Theorem 13. Choose such that For , define the set . It remains to show that there exists an such that . Suppose, by way of contradiction, that, for each , there exists such that . Then, Observe that Note that for each , and hence, , for all . So, by (H11), there exists , such that, for almost all , Using (43) in (42) yields and subsequently, contradicting (41). Consequently, there is an such that . Thus, Schauder’s fixed point theorem guarantees the existence of such that , which is a mild solution of (25) on . Performing this same argument on , , and so on enables us to construct in finitely many steps a piecewise-defined function in , that is, a mild solution of (1) on the original interval . This completes the proof.
4. Convergence and Approximation Results
Throughout this section we assume that , , and satisfy (H1)–(H5).
For each , consider a linear operator and mappings , and satisfying the following conditions:(H12) generates a -semigroup such that , for some (independent of ), for each , and as , for each ;(H13)(i), for all ;(ii) as , for all ;(H14) is a bounded, strongly measurable mapping and as , for all . (Here, the constant is the same one appearing in (H2) and so is independent of .)
Assume that (H1)–(H5) hold. Then, by Theorem 10, (1) has a unique mild solution . By virtue of (H4), (H12)–(H14), Theorem 10 implies that, for each , the initial-value problem has a unique mild solution .
Consider the following initial-value problem: Since is a fixed element of , a standard argument guarantees the existence of a unique mild solution of (47). We need the following lemma.
Lemma 16. If (H12)–(H14) hold, then as .
Proof. Using (H12) in conjunction with Theorem 4.1 in [24, pg. 46], we infer that as , for all , uniformly in . Observe that A standard argument invoking (H12) and (H13), involving the Trotter-Kato Theorem , 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 The uniform boundedness of (cf. (H12)) with (H14) guarantees that the supremum (over ) 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 ) 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.
Theorem 17. If (H1)–(H5) and (H12)–(H14) hold and , where , then as .
Proof. Let be the mild solution of (47). Observe that Taking the expectation, followed by taking square roots in (50), yields the following estimate after some computation Observe that (H12) yields, with the help of Hölder’s inequality, Using (51) and (52) in (50) yields, after taking supremum over , In view of (H12)–(H14) and the fact that , we can apply Lemma 16 to conclude from (53) that as . This completes the proof.
Now, let and denote the probability measures on induced by the mild solutions and of (1) and (46), respectively. Using Theorem 17, we will prove that as , for a special subclass of initial-value problems. Precisely, we have the following.
Theorem 18. Assume that (H1), (H3), (H4), and (H5) hold, in addition to the following:(H15);(H16) (where ) is such that there exists a positive constant for which (H17) (where ) is such that(i), for all ,(ii) as , for all , where is the constant defined in (H15);(H18)The operators are bounded and linear.If (where ), then as .
Proof. We begin by showing that is relatively compact in by appealing to the Arzelá-Ascoli theorem. To this end, we will first show that there exists such that
Note that is given by
Likewise, (H16) guarantees the existence of a positive constant such that , for all , so that a standard argument now yields
Also, is uniformly bounded because of (H12) and (H14) and the uniform boundedness of in . Combining the estimates (57) and (58) and rearranging terms enable us to conclude from (56) that (55) holds because and all constants in (56)–(58) are independent of .
Next, we establish the equicontinuity by showing as , for all , uniformly for all . We estimate each term of the expression for separately. The boundedness of (as guaranteed by (H18)) ensures that Employing Theorem in  and taking into account (H12), (H14), and (59), we conclude that Next, observe that Using (59), (H12), and (H13) when estimating each of the two integrals on the right side of (61) separately yields Next, Using the uniform boundedness of in , one can argue as in Lemma 8 to show that the right side of (63) goes to 0 as . We conclude from the above estimates that as , uniformly for and , as desired. Thus, the family is relatively compact in and hence tight (by Prokorhov’s theorem ).
To finish the proof, we remark that Theorem 17 implies that the finite-dimensional joint distributions of converge weakly to those of (cf. Proposition 3). Hence, Theorem 4 ensures that as . This completes the proof.
5. Extension to the Second-Order Case
Consider the abstract second-order stochastic Cauchy problem in a real separable Hilbert space . Here, is a bounded linear operator; is a linear (possibly unbounded) operator for which exists; and are mappings that satisfy (H2) and (H3), respectively; is a -valued fBm with Hurst parameter ; and .