On a Class of Measure-Dependent Stochastic Evolution Equations Driven by fBm

We investigate a class of abstract stochastic evolution equations driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space. We establish the existence and uniqueness of a mild solution, a continuous dependence estimate, and various convergence and approximation results. Finally, the analysis of three examples is provided to illustrate the applicability of the general theory.


Introduction
The focus of this investigation is the class of abstract measure-dependent stochastic evolution equations driven by fractional Brownian motion (fBm) of the general form dx(t) = Ax(t) + f t,x(t),μ(t) dt + g(t)dB H (t), 0 ≤ t ≤ T, x(0) = x 0 , μ(t) = probability distribution of x(t) (1.1) in a real separable Hilbert space U. (By the probability distribution of x(t), we mean μ(t)(A) = P({ω ∈ Ω : x(t,ω) ∈ A}) for each A ∈ B(U), where B(U) stands for the Borel class on U.) Here, A : D(A) ⊂ U → U is a linear (possibly unbounded) operator which generates a strongly continuous semigroup {S(t) : t ≥ 0} on U; f : [0,T]×U ×℘ λ 2 (U)→U (where ℘ λ 2 (U) denotes a particular subset of probability measures on U) is a given mapping; g : [0,T] → L(V ,U) is a bounded, measurable mapping (where V is a real separable Hilbert space and L(V ,U) denotes the space of Hilbert-Schmidt operators from V into U with norm • L(V ,U) ); {B H (t) : t ≥ 0} is a V -valued fBm with Hurst parameter H ∈ (1/2,1); and x 0 ∈ L 2 (Ω;U).
Stochastic partial functional differential equations naturally arise in the mathematical modeling of phenomena in the natural sciences (see [1][2][3][4][5][6][7][8]). 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 {x(αt) : 0 ≤ t ≤ T} and {α H x(t) : 0 ≤ t ≤ T} have the same law (see [4,6]).Indeed, while the case when H = 1/2 generates a standard Brownian motion, concrete data from a variety of applications have exhibited other values of H, and it seems that this difference enters in a non-negligible way in the modeling of this phenomena.In fact, since B H (t) is not a semimartingale unless H = 1/2, the standard stochastic calculus involving the It ó 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 [9][10][11] and differential/evolution equations driven by fBm [12][13][14][15] 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 in Section 2.
Often times, a more accurate model of such phenomena can be formulated by allowing the nonlinear perturbations to depend, in addition, on the probability distribution of the state process at time t.A prototypical example in the finite-dimensional setting would be an interacting N-particle system in which (1.1) describes the dynamics of the particles x 1 ,...,x N moving in a space U 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 (see [16][17][18]).Related infinite-dimensional problems in a Hilbert space setting have recently been examined (see [19][20][21]).
The purpose of this work is to study the class of abstract stochastic evolution equations obtained by accounting for more general nonlinear perturbations (in the sense of McKean-Vlasov equations, as described in [19]) in the mathematical description of phenomena involving an fBm.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,[16][17][18][19]22], communication networks [4], Sobolev-type equations arising in the study of consolidation of clay [8], shear in second-order fluids [23], and fluid flow through fissured rocks [24].As a part of our general discussion, we establish an approximation result concerning the effect of the dependence of the nonlinearity on the probability law of the state process, as well as the noise arising from the stochastic integral, for a special case of (1.1) arising often in applications.
The remainder of the paper is organized as follows.First, we make precise the necessary notation and function spaces, and gather certain preliminary results in Section 2. The main results are stated in Section 3 while their proofs form the contents of Section 4. Finally, we conclude the paper with a discussion of three concrete examples in Section 5.

Preliminaries
For details of this section, we refer the reader to [12,[25][26][27][28][29] and the references therein.Throughout this paper, U is a real separable Hilbert space with norm • U and inner product •, • U equipped with a complete orthonormal basis {e j | j = 1,2,...}.Also, (Ω,Ᏺ,P) is a complete probability space.For brevity, we suppress the dependence of all random variables on ω throughout the manuscript.
We begin by making precise the definition of a U-valued fBm and related stochastic integral used in this manuscript.The approach we use coincides with the one formulated and analyzed in [10,12].Let {B H j (t)|t ≥ 0} ∞ j=1 be a sequence of independent, one-dimensional fBms with Hurst parameter H ∈ (1/2,1) such that for all j = 1,2,... the following hold: j=1 B H j (t)e j is a U-valued fBm, where the convergence is understood to be in the mean-square sense.
It has been shown in [12] that the covariance operator of {B H (t) : Next, we outline the discussion leading to the definition of the stochastic integral associated with {B H (t) : t ≥ 0} for bounded, measurable functions, as presented in [10,12].To begin, assume that g : [0,T] → L(V ,U) is a simple function, that is, there exists where

.3)
As argued in [12, Lemma 2.2], this integral is well defined since Since the set of simple functions is dense in the space of bounded, measurable L(V ,U)valued functions, a standard density argument can be used to extend Definition 2.2 to the case of a general bounded, measurable integrand.
We make use of several different function spaces throughout this paper.For one, BL(U) is the space of all bounded linear operators on U while L 2 (Ω;U) stands for the space of all U-valued random variables y for which E y 2 U < ∞.Also, C([0,T];U) stands for the space of L 2 -continuous U-valued random variables y : [0,T] → U such that (2.5) The remaining function spaces coincide with those used in [19]; we recall them here for convenience.First, B(U) stands for the Borel class on U and ℘(U) represents the space of all probability measures defined on B(U) equipped with the weak convergence topology.Define λ : U → R + by λ(x) = 1 + x U , x ∈ U, and consider the space ( For p ≥ 1, we let where m = m + − m − is the Jordan decomposition of m, and |m| = m + + m − .Then, define the space ℘ λ 2 (U) = ℘ s λ 2 (U) ∩ ℘(U) equipped with the metric ρ given by It is known that (℘ λ 2 (U),ρ) is a complete metric space.The space of all continuous ℘ λ 2 (U)-valued functions defined on [0,T], denoted by , is complete when equipped with the metric (2.9) In addition to the familiar Young, Hölder, and Minkowski inequalities, the inequality of the form ( n i=1 a i ) m ≤ m n−1 n i=1 a m i , where a i is a nonnegative constant (i = 1,...,n) and m,n ∈ N, will be used to establish various estimates.
Proposition 2.4 [28, page 37].If a sequence {X n } of U-valued random variables converges weakly to a U-valued random variable X in the mean-square sense, then the sequence of finite dimensional joint distributions corresponding to {P Xn } converges weakly to the finite dimensional joint distribution of P X .
The next theorem, in conjunction with Proposition 2.4, is the main tool used to prove one of the convergence results in this paper.
Theorem 2.5.Let {P n } ⊂ ℘(U).If the sequence of finite dimensional joint distributions corresponding to {P n } converges weakly to the finite dimensional joint distribution of P and {P n } is relatively compact, then P n w − → P.

Statement of results
We consider mild solutions of (1.1) in the following sense.The following conditions on (1.1) are assumed throughout the manuscript unless otherwise specified.
(A1) A : D(A) ⊂ U → U is the infinitesimal generator of a strongly continuous semigroup {S(t) : t ≥ 0} on U such that S(t) BL(U) ≤ M exp(αt) for all 0 ≤ t ≤ T, for some (Henceforth, we write M S = max 0≤t≤T S(t) BL(U) , which is finite by (A1).) The following more general version of [12,Lemma 6], stated without proof, is critical in establishing several estimates.
where C t is a positive constant depending on t, M S , and the growth bound on g, and {ν j : j ∈ N} is defined as in the discussion leading to Definition 2.1.
The first integral on the right-hand side of (3.4) is taken in the Bochner sense while the second is defined in Section 2. The operator Φ satisfies the following properties.
The main existence-uniqueness result is as follows.Mild solutions of (1.1) depend continuously on the initial data and probability distribution of the state process in the following sense.Proposition 3.5.Assume that (A1)-(A5) hold, and let x and y be the mild solutions of (1.1) (as guaranteed to exist by Theorem 3.4) corresponding to initial data x 0 and y 0 with respective probability distributions μ x and μ y .Then, there exists a positive constant M * such that We now formulate various convergence and approximation results.For the first such result, let n ≥ 1 and consider the Yosida approximation of (1.1) given by where μ n (t) is the probability law of x n (t), and R(n;A) = (I − nA) −1 is the resolvent operator of A. Assuming that (A1)-(A5) hold, one can invoke Theorem 3.4 to deduce that Eduardo Hernandez et al. 7 (3.6) has a unique mild solution x n ∈ C([0,T];U) with probability law μ n ∈ Ꮿ λ 2 .The following convergence result holds.
Theorem 3.6.Let x denote the unique mild solution of (1.1) on [0,T] as guaranteed by Theorem 3.4.Then, the sequence of solutions of (3.6) The following corollary is needed to establish the weak convergence of probability measures.
Corollary 3.7.The sequence of probability laws μ n corresponding to the mild solutions x n of (3.6) converges in Ꮿ λ 2 to the probability law μ corresponding to the mild solution x of (1.1) as n → ∞.
Remark 3.8.We observe for later purposes that Corollary 3.7 implies that In view of Theorem 3.6 and Corollary 3.7, the following continuity-type result can be established as in [19].The details are omitted.
Proposition 3.9.Assume that E x 0 4 U < ∞.Then, for any function F : [0,T] × U → R satisfying the following: (i) for each N ∈ N, there exists a positive continuous function k N (t) such that (ii) there exists a positive continuous function l(t) such that it is the case that We now consider the weak convergence of the probability measures induced by the mild solutions of (3.6).Let P x denote the probability measure generated by the mild solution x of (1.1) and P xn the probability measure generated by x n as in (3.6).We have the following.
Next, we present a generalization of [12,Theorem 2] which allows for measure dependence in the nonlinearity.Specifically, let m ∈ N and t ∈ [0,T] be given, and partition the interval [0,t] using the points {t m j = (t/m)( j) : j = 0,1,...,m}.For each j ∈ {1, ...,m}, consider the following recursively defined sequence: where where μ m (s Using this lemma, together with a standard Gronwall-type argument, yields the following approximation result.Theorem 3.12.Let {x(t) : 0 ≤ t ≤ T} be the (unique) mild solution process of (1.1) with probability law {μ(t) : 0 ≤ t ≤ T}.Then, for each Next, we formulate a result in which a deterministic initial-value problem is approximated by a sequence of stochastic equations of a particular form of (1.1) arising frequently in applications.Specifically, consider the deterministic initial-value problem and for each ε > 0, consider the stochastic initial-value problem ) are given mappings.Regarding (3.13), we assume that A satisfies (A1) and that the following hold that.
Eduardo Hernandez et al. 9 (ii) there exists a positive constant M F such that F(t,z) U ≤ M F z U globally on [0, T] × U, (A1) and (A6) guarantee the existence of a unique mild solution z of (3.13) on [0,T] given by As for (3.14), we impose the following conditions on the data for each ε > 0: . Under these assumptions, the following result holds.Theorem 3.13.Let z and x ε be the mild solutions of (3.13) and (3.14) on [0,T], respectively.Then, there exist a positive constant σ and a positive function ψ(ε) (which decreases to 0 as

Proofs
Proof of Lemma 3.3.Let μ ∈ Ꮿ λ 2 be fixed and consider the solution map Φ defined in (3.4).Using the discussion in Section 2 and the properties of x, one can see that for any x ∈ C([0,T];U), Φ(x)(t) is a well defined stochastic process for each 0 ≤ t ≤ T. In order to verify the L 2 -continuity of Φ on [0, T], let z ∈ C([0,T];U) and consider 0 ≤ t * ≤ T and |h| sufficiently small.Observe that the strong continuity of S(t) implies that the right-hand side of (4.2) goes to 0 as |h| → 0. Next, using the Hölder inequality with (A2) yields which clearly goes to 0 as |h| → 0. Also, where and the right-hand side of (4.7) goes to 0 as |h| → 0. Next, observe that Using the property E(B H j (s) − B H j (t)) 2 = |t − s| 2H ν j with s = t * + h and t = t * , we can argue as above to conclude that the right-hand side of (4.8) goes to 0 as |h| → 0. Consequently, I 3 (t * + h) − I 3 (t * ) → 0 as |h| → 0 when g is a simple function.Since the set of all such simple functions is dense in L(V ,U), a standard density argument can be used to extend this conclusion to a general bounded, measurable function g.This establishes the L 2 -continuity of Φ.
Finally, we assert that Φ(C([0,T];U)) ⊂ C([0,T];U).Indeed, the necessary estimates can be established as above, and when used in conjunction with Lemma 3.2, one can readily verify that sup 0≤t≤T E Φ(x)(t) 2  U < ∞ for any x ∈ C([0,T];U).Thus, we conclude that Φ is well defined, and the proof of Lemma 3.3 is complete.
Proof of Theorem 3.4.Let μ ∈ Ꮿ λ 2 be fixed and consider the operator as Φ defined in (3.4).We know that Φ is well defined and L 2 -continuous from Lemma 3.3.We now prove that Φ has a unique fixed point in C([0,T;U]).Indeed, for any x, y ∈ C([0,T;U]), (3.4) implies that Consequently, for a given μ ∈ Ꮿ λ 2 and T > 0, Φ has a unique fixed point x μ ∈ C([0,T];U), provided that TM f M S < 1.In such case, we conclude that x μ is a mild solution of (1.1).
Proof of Proposition 3.5.Computations similar to those used leading to the contractivity of the solution map in Theorem 3.4 can be used, along with Gronwall's lemma, to establish this result.The details are omitted.
Proof of Theorem 3.6.Observe that (4.17) 14 Journal of Applied Mathematics and Stochastic Analysis Standard computations imply that Further, the triangle inequality and (A2), together, imply The boundedness of E f (s,x(s),μ(s) 2 U independent of n, together with the strong convergence of nR(n;A) − I to 0, enables us to infer that the right-hand side of (4.19) goes to 0 as n → ∞.Next, using Lemma 3.2 yields (4.20) Using (4.18)-(4.20) in (4.17) gives rise to an inequality of the form where β i (i = 1,2) are constant multiples of the quantity nR(n;A) − I 2 BL(U) .Consequently, applying Gronwall's lemma and then taking the supremum over 0 ≤ t ≤ T yields Since the right-hand side of (4.22) goes to 0 as n → ∞, the conclusion of the theorem follows.
Proof of Corollary 3.7.This follows from the fact that Proof of Theorem 3.10.Throughout the proof, C i denotes a suitable positive constant independent of n.We will show that {P xn } ∞ n=1 is relatively compact using the Arzela-Ascoli theorem.
First, we can show that {x n } is uniformly bounded in C([0,T];U), that is, sup n∈N sup 0≤t≤T E x n (t) 2  U < ∞.The mild solution x n of (3.6) is given by the variation Eduardo Hernandez et al. 15 of parameters formula Let t ∈ [0,T].We consider each of the three terms on the right-hand side of (4.24) separately.First, since nR(n;A) is contractive for each n, it follows that Routine arguments involving (A2) and Remark 3.8 yield Arguing as in Lemma 3.3, using the contractivity of nR(n;A), yields Combining (4.25)-(4.27),we obtain that Applying Gronwall's lemma now yields the uniform boundedness of {x n } in C([0,T];U).Next, we establish the equicontinuity of {x n }.We will show that for every n ∈ N and for fixed 0 ≤ s ≤ t ≤ T, E x n (t) − x n (s) 4  U → 0 (independently of n) as t − s → 0. Indeed, for 0 ≤ s ≤ t ≤ T, using the semigroup properties yields Also, Eduardo Hernandez et al. 17 One can argue as in Lemma 3.3 to show that the first term on the right-hand side of (4.31) goes to 0 as (t − s) → 0. Likewise, an application of Lemma 3.2 shows that the second term also goes to 0 as (t − s) → 0. Thus, the estimates (4.29)-(4.31)then yield the equicontinuity of {x n }.Therefore, we conclude that the family {P xn } ∞ n=1 is relatively compact by Arzela-Ascoli, and therefore tight (cf.Section 2).Hence, by Proposition 2.4, the finite dimensional joint distributions of P xn converge weakly to P and so, by Theorem 2.5, Proof of Lemma 3.11.A standard Gronwall argument involving (A2)(ii) and Lemma 3.2 can be used to establish this result.
Proof of Theorem 3.12.Since for all 0 ≤ t ≤ T, using (A2)(i), Lemma 3.2, and the observations in the proof of Corollary 3.7 yields (4.33) So, an application of Gronwall's lemma yields (4.34) Observe that as m → ∞ the right-hand side of (4.34) goes to 0 since t − t m m−1 → 0 as m → ∞.This completes the proof.
Proof of Theorem 3.13.We estimate each term of the representation formula for E x ε (t) − z(t) 2  U separately.First, (A7) guarantees the existence of a positive constant K 1 and a positive function α 1 (ε) (which decreases to 0 as ε → 0) such that for sufficiently small ε > 0, To this end, note that the continuity of F, together with (A6), enables us to infer the existence of K 2 > 0 and α 2 (ε) (as above) such that for sufficiently small ε > 0, for all 0 ≤ t ≤ T. Also, observe that Young's inequality and (A8), together, imply (4.37) Note that (A8) guarantees the existence of K 3 > 0 and α 3 (ε) (as above) such that for sufficiently small ε > 0, E F 2ε (s,z(s)) − F(s,z(s)) 2 U ≤ K 3 α 3 (ε) for all 0 ≤ t ≤ T. So, we can continue the inequality (4.37) to conclude that Next, (A9) guarantees the existence of K 4 > 0 and α 4 (ε) (as above) such that for sufficiently small ε > 0, As such, we have for all 0 ≤ t ≤ T. It remains to estimate E t 0 S ε (t − s)g ε (s)dB H (s) 2 U .Note that (A10) implies the existence of K 5 > 0 and α 5 (ε) (as above) such that for sufficiently small ε > 0, Hence, Since the set of all such simple functions is dense in L(V ,U), a standard density argument can be used to establish estimate (4.44) for a general bounded, measurable function g ε .Now, using (4.35)-(4.44),we conclude that there exist positive constants K i (i = 1,...,5) such that so that an application of Gronwall's lemma implies where σ = 4M 2 S M 2 F and Ψ(ε) = 5 i=1 K i α i (ε).This completes the proof.

( 4 . 1 )
10 Journal of Applied Mathematics and Stochastic Analysis Since the semigroup property enables us to write