ABSTRACT SEMILINEAR STOCHASTIC ITÓ-VOLTERRA INTEGRODIFFERENTIAL EQUATIONS

SEMILINEAR STOCHASTIC ITÓ-VOLTERRA INTEGRODIFFERENTIAL EQUATIONS DAVID N. KECK AND MARK A. MCKIBBEN Received 31 October 2005; Revised 3 March 2006; Accepted 14 April 2006 We consider a class of abstract semilinear stochastic Volterra integrodifferential equations in a real separable Hilbert space. The global existence and uniqueness of a mild solution, as well as a perturbation result, are established under the so-called Caratheodory growth conditions on the nonlinearities. An approximation result is then established, followed by an analogous result concerning a so-called McKean-Vlasov integrodifferential equation, and then a brief commentary on the extension of the main results to the time-dependent case. The paper ends with a discussion of some concrete examples to illustrate the abstract theory. Copyright © 2006 D. N. Keck and M. A. McKibben. 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.

(1.5) Also, (1.4) with has been studied in [21][22][23], for both the case when A is autonomous and when A is timedependent.For L of either form above, under appropriate conditions such as [29, Theorem 1.4, page 46], (1.4) admits a resolvent family {R(t) : t ≥ 0} in the following sense.
Definition 1.1.A family {R(t) : t ≥ 0} of bounded linear operators on H is a resolvent for (1.4) whenever (i) R(t) is strongly continuous in t, (ii) R(0) = I, (iii) R(t)D(A) ⊂ D(A) and AR(t)z = R(t)Az, for all z ∈ D(A), t ≥ 0, (iv) (dR(t)/dt)z = z + (a * AR)z = z + (R * Aa)z, where * denotes the usual convolution over [0,t].(See [29, page 32].) Assuming the classical Lipschitz condition on f , it has been shown (see [21,29]) that there exists a unique mild solution on [0, T], for any T > 0, that can be represented by the variation of parameters formula involving the resolvent family, namely, (1.7) Certain applications, such as those mentioned in [21][22][23][24][25][26], indicate that a stochastic version of (1.4) warrants study.Indeed, Mīshura [24][25][26] studied the stochastic Volterra integral equation x(t;ω) = L x(t;ω) + f (t;ω), 0 ≤ t ≤ T, ( (with L given by (1.2)) and established conditions under which such an equation could be reduced to one involving Skorokhod integrals (to allow, in particular, for a natural D. N. Keck and M. A. McKibben 3 treatment of the equations used to describe the motion of an incompressible viscoelastic fluid).As pointed out in that paper, the results mentioned above in the deterministic case can be applied for each given fixed ω ∈ Ω to ensure the existence of a resolvent family {R(t; ω) : t ≥ 0}.However, the stochastic version must be F t -adapted, and so, to guarantee this, certain natural conditions (such as those in [25,Theorem 2]) are imposed; these conditions hold for a broad class of operators.
The purpose of the present investigation is to continue the above work by considering the more general It ó-Volterra integrodifferential equation (1.1) which contains an additional stochastic term involving a Wiener process.The main results in the present paper constitute an extension of the results in [13,[21][22][23] to the stochastic setting, and can be viewed as a counterpart to the results in [24,26] under more general growth conditions.Moreover, we consider a so-called McKean-Vlasov variant of (1.1) in which the mappings f and g depend on the probability law μ(t) of the state process x(t) (i.e., μ(t)B = P({ω ∈ Ω : x(t;ω) ∈ B}) for each Borel set B on H).Precisely, we study (1.9) A prototypical example of such a problem in the finite-dimensional setting would be an interacting N-particle system in which (1.9) describes the dynamics of the particles x 1 ,...,x N moving in the space H in which the probability measure μ is taken to be the empirical measure μ N (t) = (1/N) N k=1 δ xk(t) , where δ xk(t) denotes the Dirac measure.Researchers have investigated related models concerning diffusion processes in the finitedimensional case (e.g., see [10,11,27]) and have more recently devoted attention to the study of the infinite-dimensional version (see [1,19]).Our discussion of (1.9) serves as a counterpart to these results for a class of stochastic Volterra equations.
We will be concerned with mild solutions to (1.1) in the following sense.
x(s))dW(s).In the case when (1.1) admits a resolvent family, a mild solution in both cases of Definition 1.2 can be represented by a stochastic version of (1.7), namely, The structure of this paper is as follows.In Section 2 we state some preliminary information regarding function spaces and inequalities.Then, we state the main results concerning existence and uniqueness of mild solutions of (1.1), along with an approximation result, a discussion of a so-called McKean-Vlasov variant of (1.1), and commentary on analogous results for the time-dependent case in Section 3. We provide the proofs in Section 4, and finally present a discussion of some examples in Section 5.

Preliminaries
For details of this section and additional background, we refer the reader to [9,14,17,18,29] and the references therein.Throughout this paper, H and K are real separable Hilbert spaces with respective norms For any Banach space Z, Ꮿ([0,T];Z) stands for the function space which is itself a Banach space when equipped with the norm and L p (0,T;Z) represents the space with the usual norm.We abbreviate these two spaces as Ꮿ(Z) and ᏸ p (Z), respectively.When considering (1.9), we will make use of the following additional function spaces used in [1].First, B(H) stands for the Borel class on H and P(H) represents the space of all probability measures defined on B(H) equipped with the weak convergence topology.Let λ(x) = 1 + x H , x ∈ H, and define the space where (2.7) Then, we can define the space P λ 2 (H) = P s λ 2 (H) ∩ P(H) equipped with the metric ρ given by (2.8) it has been shown that (P λ 2 (H),ρ) is a complete metric space.Finally, the space of all continuous P λ 2 (H)-valued measures defined on [0,T], denoted by Ꮿ λ 2 (T), is complete when equipped with the metric (2.9) The following estimate on the It ó integral established in [16, Proposition 1.9] is an important tool in obtaining certain estimates. (2.10) Finally, in addition to the familiar Young, Hölder, and Minkowski inequalities, the following inequality (which follows from the convexity of x m , m ≥ 1) plays an important role in establishing various estimates, where a i is a nonnegative constant (i = 1,...,m).

Statement of main results
We begin by establishing the existence and uniqueness of a mild solution to (1.1) (in the sense of Definition 1.2) under the so-called Caratheodory growth conditions (see [12]).Precisely, we consider ( ) is continuous in both variables, and nondecreasing and concave in the second variable with N(t,0 for an appropriate positive constant D, then z(t) = 0, for all 0 ≤ t ≤ T; (H6) for any fixed T > 0, β > 0, the initial-value problem has a global solution on [0, T]; (H7) x 0 is an F 0 -measurable random variable in L 2 (Ω;H) independent of W. Examples of functions Nsatisfying (H4)(ii) and (H5) can be found in [12,15].Aside from the mapping that would generate a Lipschitz condition (namely N(t,u) = Mu, for some positive constant M), some other typical examples (see [15]) for the mapping N in (H4)-(H5) include Conditions that ensure (H3) holds are discussed, for instance, in [25].
We have the following theorem.
Furthermore, we assert that uniqueness is guaranteed to be preserved under sufficiently small perturbations.Indeed, consider a perturbation of (1.1) given by An argument in the spirit of [7] can be used to establish the following result.

D. N. Keck and M. A. McKibben 7
Proposition 3.2.Assume that (H1)-(H7) hold, and that f and g satisfy (H4) and (H5) with appropriate mappings K and N.Then, (3.4) has a unique mild solution, provided that (H8) there exist δ ∈ (0,T) and w ∈ C((0,∞);(0,∞)) which is nondecreasing and , for all r ∈ (0,δ).In the case of a Lipschitz growth condition, routine calculations can be used to establish the following estimates.Proposition 3.3.Assume that (H1)-(H7) hold (with N(t,u) = K(t,u) = Mu, for some M > 0) and that x 0 , x 0 satisfy (H7).Denote the corresponding unique mild solutions of (1.1) (as guaranteed to exist by Theorem 3.1) respectively by x, x.Then, (i) there exist (ii) for each p ≥ 2, there exists a positive constant C p,T (depending only on p and T) such that We now formulate a result in which a related deterministic Volterra integrodifferential equation (as considered in [21,29]) is approximated by a sequence of stochastic equations of the form (1.1).Precisely, consider the deterministic initial-value problem For every ε > 0, consider the stochastic initial-value problem Here, L and L ε are given by either (1.2) or (1.3).Also, assume that L, f , and g satisfy (H1)-(H4) (appropriately modified) with N(t,u) = K(t,u) = Mu, for some M > 0 (i.e., f and g satisfy a Lipschitz condition) so that the results in [21,29] guarantee the existence of a unique global mild solution z of (3.7).Regarding (3.8), we impose the following conditions, for every ε > 0: (H9) Under these assumptions, Theorem 3.1 ensures the existence of a unique mild solution of (3.8), for every ε > 0. We have the following convergence result.
Theorem 3.4.Let z and x ε be the mild solutions to (3.7) and (3.8), respectively.Then, there exist ξ > 0 and a positive function Ψ(ε) which decreases to 0 as ε → 0 + such that for any p ≥ 2, Next, we turn our attention to a so-called McKean-Vlasov variant of (1.1) given by (1.9) in which f : [0,T] × H × P λ 2 (H) → H and g : [0,T] × H × P λ 2 (H) → L 2 (K;H) now depend on the probability law μ(•) of the state process x(•).In addition to (H1)-( H3) and (H7), we replace (H4)-(H6) by the following modified hypothesis: (H12) there exist H), and x, y ∈ L 2 (Ω;H), (iii) there exists M N >0 such that N(t,u)≤M N u, for all 0≤t ≤ T and 0≤u<∞.Remark 3.5.We point out that while the existence portion of the argument for (1.9) can be established using essentially the same argument used to prove Theorem 3.1 without strengthening the assumption on N, the dependence of f and g on the probability measure μ creates an additional difficulty when trying to show that μ(t) is the probability law of x(t).Indeed, it seems that the concavity of N in the second variable (which guarantees the existence of positive constants α 1 and α 2 such that N(t,u) ≤ α 1 + α 2 u, for all 0 ≤ t ≤ T and 0 ≤ u < ∞) is not quite strong enough.However, taking α 1 = 0 (i.e., condition (H12)(ii) becomes a Lipschitz-type condition) is sufficient.Since the nonlinearities involved in McKean-Vlasov equations are often Lipschitz continuous (cf.Example 5.5 in Section 5), the following theorem concerning (1.9) constitutes a reasonable result from the viewpoint of applications; the case of a more general nonlinearity remains an interesting open question.
We have the following analog of Theorem 3.1.
Results analogous to Propositions 3.2 and 3.3 can also be established for (1.9) by making the natural modifications to the hypotheses and proofs.
Finally, in all the previous theorems the operator A in the two definitions of L was independent of t.We now briefly comment on the nonautonomous versions of (1.1) and (1.9), where the operator L(x(t)) is defined by either (1.2) or (1.3) with A replaced by {A(t) : 0 ≤ t ≤ T}.In order to proceed in a manner similar to the one currently employed, conditions need to be prescribed under which (i) a resolvent family {R(t, s) : 0≤t ≤ s<∞} is guaranteed to exist, and (ii) it is F t -adapted.Conditions guaranteeing (i) can be found in [13,20], while the approach used in [25] can be modified to establish sufficient conditions that ensure (ii) holds.Once (i) and (ii) hold, each of the results formulated above can be extended to the time-dependent case by making suitable modifications involving the use of the properties of the time-dependent resolvent family (rather than the autonomous one) in the arguments.

Proofs
Proof of Theorem 3.1.Consider the recursively-defined sequence of successive approximations defined as follows: Also, consider the initial-value problem where ) Using (H6), we deduce that there exists 0 < T ≤ T such that (4.2) has a unique solution z : [0,T] → R given by We will divide the proof of Theorem 3.1 into stages, beginning with the following assertion.
for each 0 ≤ t ≤ T * ≤ T and for each n ≥ 1.
Proof.We prove (i) by induction.To begin, for n = 1, observe that standard computations involving the use of (H4)(i) yield for all 0 ≤ t ≤ T. Now, assume that E x n (t) 2 H ≤ z(t), for all 0 ≤ t ≤ T. Similar computations yield K s,z(s) ds (using the inductive hypothesis and monotonicity of K) for all 0 ≤ t ≤ T. Thus, (i) holds by induction.Next, in order to prove (ii), let δ > 0 be fixed and proceed by induction.For n = 1, observe that for all 0 ≤ t ≤ T, we obtain (using (4.3) and the choice of z 0 ) Also, the continuity of z and K guarantees the existence of 0 < T * ≤ T such that D. N. Keck and M. A. McKibben 11 so that, in conjunction with (4.7), we conclude that for all 0 ≤ t ≤ T * ≤ T, as desired.This completes the proof of Claim 1.
Next, we assert the following.
Proof.Let n,m ≥ 1. Routine calculations used in conjunction with the monotonicity of N yield for all 0 ≤ t ≤ T * ≤ T. This completes the proof of Claim 2.
Proof.We establish (i) using induction on n.To begin, we show the string of inequalities holds for n = 2.To this end, observe that using (4.15) and the monotonicity of N yields Now, assume that γ n (t) ≤ γ n−1 (t), for all 0 ≤ t ≤ T * * , and observe that This completes the proof of (i).The argumen t for (ii) is equally as straightforward and will be omitted.
Using Claim 3, we deduce that {γ n (•)} is a decreasing sequence in n.Moreover, for each given n ≥ 1, it is easy to see that γ n (t) is an increasing function of t.Finally, with all of the preliminary work now complete, we can now prove that (1.1) has a mild solution x on [0, T * * ].To this end, define the function γ : [0,T * * ] → R by Observe that γ is nonnegative and continuous, γ(0) = 0, and Since N(t,0) = 0, for all 0 ≤ t ≤ T, we conclude that the left-hand side of (4.22) tends to 0 as n → ∞.Thus, x is indeed a mild solution of (1.1) on [0,T * * ], as desired.A standard argument can now be employed to prove that the above solution can be extended in finitely many steps to the entire interval [0, T].
Proof of Theorem 3.4.We proceed by estimating each term of the representation formula (cf.(1.7) and (1.10)) for E x ε (t) − z(t) p H separately. Throughout the proof, C i are positive constants and β i (ε) are positive functions which decrease to 0 as ε → 0 + .To begin, note that (H9) guarantees the existence of C 1 and β 1 (ε) such that for sufficiently small ε > 0, Next, regarding the term , the continuity of f ε , together with (H9), ensures the existence of C 2 and β 2 (ε) such that for small enough ε > 0, for all 0 ≤ t ≤ T. Also, observe that Young's inequality and (H10) together yield Note that (H10) guarantees the existence of C 3 and β 3 (ε) such that for small enough ε > 0, for all 0 ≤ t ≤ T, thereby enabling us to conclude from (4.27) that for all 0 ≤ t ≤ T. Using (4.26) and (4.29) together with the Hölder, Minkowski, and Young inequalities yields Similarly, in order to estimate p H , we note that computations similar to those leading to (4.30), together with Lemma 2.1, yield (The argument of [16, Proposition 1.9] guarantees the existence of a bound L g (independent of ε > 0) that applies for all mappings g ε under consideration.)Also, (H11) guarantees the existence of C 4 and β 4 (ε) such that for small enough ε > 0, for all 0 ≤ t ≤ T. Substituting (4.32) in (4.31) yields Using (4.33), in conjunction with (4.25)-(4.30),enables us to conclude that for all ε > 0 small enough to ensure that (4.25)-(4.33)hold simultaneously, there exists a constant η > 0 (namely, An application of Gronwall's lemma in (4.34) subsequently yields for all 0 ≤ t ≤ T, where Ψ(ε) = 4 i=1 C i β i (ε).This completes the proof of Theorem 3.4.
Proof of Theorem 3.6.Let μ ∈ Ꮿ λ 2 (T) ≡ C([0,T];(P λ 2 (H),ρ)) be fixed.The existence and uniqueness of a mild solution x μ on [0, T] of (1.9) can be established as in the proof of Theorem 3.1.We must further show that μ is, in fact, the probability law of x μ .Toward this end, following the approach used in [1,19], let L(x μ ) = {L(x μ (t)) : t ∈ [0,T]} denote the probability law of x μ and define the operator As such, we only need to show that t → L(x μ (t)) is continuous.To this end, observe that for sufficiently small |h| > 0, the continuity of x μ , K, and N implies that Further, for all t ∈ [0,T] and ϕ ∈ C ρ (H), the definition of the metric ρ (cf.(2.8)) yields (4.37) 16 Stochastic It ó-Volterra integrodifferential equations Thus, we may conclude that lim thereby showing that L(x μ ) ∈ Ꮿ λ 2 (T).Finally, note that if x is a mild solution of (1.9), then clearly its probability law L(x) = μ is a fixed point of Ψ. Conversely, if μ is a fixed point of Ψ, then the variation of parameters representation formula (parametrized by μ) defines a solution x μ which, in turn, has a probability law μ belonging to the space Ꮿ λ 2 (T).Thus, in order to complete the proof it suffices to show that the operator Ψ has a unique fixed point in Ꮿ λ 2 (T).To this end, let μ, ν be any two elements of Ꮿ λ 2 (T) and let x μ and x ν be the corresponding mild solutions of (1.9).Standard computations employing the use of (H12) yield where M T = 2M 2 R (T + L g ).Using (H12)(iii), we continue the inequality in (4.39) to obtain for all 0 ≤ t ≤ T. Observe that for 0 < T ≤ T chosen sufficiently small, there exists a constant 0 < C < 1 (independent of μ,ν) for which Thus, Ψ is a contraction on Ꮿ λ 2 (T) and therefore, has a unique fixed point.As such, (1.9) has a unique mild solution on [0,T] with probability distribution μ ∈ Ꮿ λ 2 (T).This procedure can be repeated in order to extend the solution, by continuity, to the entire interval [0,T] in finitely many steps, thereby completing the proof.

Examples
Example x(0,z) = x 0 (z), a.e. on D, ( where and β is a standard N-dimensional Brownian motion.We impose the following conditions: (H13) a ∈ L 1 ((0,T);R) is an F t -adapted, positive, nonincreasing, convex kernel; (H14) F satisfies the Caratheodory conditions (i.e., measurable in t and continuous in x) and is such that (i) there exists M F > 0 such that |F(t, x)| ≤ M F [1 + |x|], for all 0 ≤ t ≤ T and x ∈ R, (ii) there exists M F > 0 such that |F(t, x) − F(t, y)| ≤ M F |x − y|, for all 0 ≤ t ≤ T and x, y ∈ R; (H15) G satisfies the Caratheodory conditions and is such that (i) there exists all 0 ≤ t ≤ T and x, y ∈ R; (H16) x 0 is an F 0 -measurable random variable independent of β with finite second moment.The following theorem is a stochastic analog of [3,Theorem 6.2].
It is known that A is a positive definite, self-adjoint operator in H (see [3]).Moreover, using the properties of A and (H13), it follows from [25]  Next, we consider a variant of (5.1) in which the mapping F now depends, in addition, on the probability law of the state process.Precisely, we consider x(t,z) = 0, a.e. on (0, T) × ∂D, x(0,z) = x 0 (z), a.e. on D, (5.4) where , and μ(t,•) ∈ P λ 2 (L 2 (D)) is the probability law of x(t,•).We impose the following modified version of hypotheses (H14)-(H15).
(H17) F 1 satisfies the Caratheodory conditions (i.e., measurable in (t,z) and continuous in the third variable) and is such that (i) there exists H18) F 2 satisfies the Caratheodory conditions and is such that (i) there exists (H19) G satisfies the Caratheodory conditions and is such that (i) there exists We have the following theorem.Proof.Let H, K, and A be defined as in the proof of Theorem 5.2.Define the maps f : F 2 (t,z, y)μ(t,z)(dy), (5.5) for all 0 ≤ t ≤ T, z ∈ D, and x ∈ H.We must show that f and g satisfy (H4).To this end, observe that from (H17)(i), we obtain for all 0 ≤ t ≤ T, x ∈ Ꮿ(L 2 (D)), where 2M F1 , i fm(D) ≤ 1.
(5.8) (Here, m denotes the Lebesgue measure in R n .)Also, from (H17)(ii), we obtain  (ii) A concrete example of (1.1) (with L given by (1.3)) that arises in the study of viscoelasticity is discussed in [22,23].A stochastic version of this example in the spirit of those discussed in this section can be established in a similar manner, assuming that conditions comparable to those in [25] are imposed to ensure the existence of an F tadapted resolvent family.
.6) D. N. Keck and M. A. McKibben 5 Let m = m + − m − be the Jordan decomposition of m, |m| = m + + m − , and for p ≥ 1, let (4.31) D. N. Keck and M. A. McKibben 15 follows by an application of Gronwall's inequality that
• H and • K .Several function spaces are used throughout the paper.As mentioned earlier, L 2 (K;H) denotes the space of all Hilbert-Schmidt operators from K into H with norm denoted as • L 2 (K;H) .The space of all bounded linear operators on H will be denoted by B(H) with norm • B(H) , while the collection of all strongly measurable square integrable H-valued random variables x is denoted by L 2 (Ω;H) equipped with norm and a ε satisfy (H1)-(H2).Also,(3.8)admits an F tadapted resolvent {R ε (t) : t ≥ 0} such that R ε (t) → R(t) strongly as ε → 0 + , uniformly in t ∈ [0,T] and {R ε (t) : 0 ≤ t ≤ T} is uniformly bounded by M R (the same constant as defined in (H3), independent of ε); (H10) f ε : [0,T] × H → H is Lipschitz in the second variable (with the same Lipschitz constant M as for f and g) and f ε (t,z) → f (t,z) as ε → 0 + , for all z ∈ H, uniformly in t ∈ [0,T]; (H11) g ε : [0,T] × H → L 2 (K;H)is Lipschitz in the second variable (with the same Lipschitz constant M as for f and g) and g ε (t,z) → 0 as ε → 0 + , for all z ∈ H, uniformly in t ∈ [0,T].
5.1.Let D be a bounded domain in R N with smooth boundary ∂D.Consider the following initial boundary value problem: and[29, page 38] that condition (H3) is satisfied with M R = 1.Next, F and G, respectively, generate functions f : [0,T] × H → H and g : [0,T] × H → L 2 (K,H) by the following identifications: Mu, where M = max{M F ,M G }).We can now rewrite (5.1) in the form (1.1) (with L given by (1.2)) in H, and apply Theorem 3.1 to conclude that (5.1) has a unique mild solution x ∈ Ꮿ([0,T];L 2 (L 2 (D))).and K given by (3.4), for instance.The existence and uniqueness of a mild solution in such case is still guaranteed by Theorem 3.1.