# Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion

Authors: Mark A. McKibben
Journal: International Journal of Stochastic Analysis
 (2013)

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## Abstract

We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model.


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## Body

## 1. Introduction

The mathematical description of the dynamic buckling has been the subject of investigation for decades and is of interest to the engineering world. Dynamic buckling arises in various ways including vibrations, single load pulses of large amplitude, occurrence of a suddenly applied load, and flutter enhanced bending (see [1]). For the purpose of our study we restrict our attention to the dynamic buckling of a hinged extensible beam which is stretched or compressed by an axial force.Dickey [2] initiated an investigation of the hyperbolic partial differential equation
(1)∂2W(z,t)∂t2+EIρ∂4W(z,t)∂z4ccccccc-(ηρ+EAc2ρL∫0L∂2W(ξ,t)∂ξ2dξ)∂2W(z,t)∂z2=0,ccccccccccccccccccccccccccccccccccc0≤z≤L,0<t,
where E is Young’s modulus, I is the cross-sectional moment of inertia, ρ is the density, η is an axial force,  L is the natural length of the beam, and Ac is the cross-sectional area; these are all positive parameters. Here, W(z,t) describes the transverse deflection of an extensible beam at point z at time t. The nonlinear term in (1) accounts for the change in tension of the beam due to extensibility. The ends of the beam are held at a fixed distance apart and the ends are hinged; this translates to the following boundary conditions:
(2)W(0,t)=W(L,t)=∂2W(0,t)∂z2=∂2W(L,t)∂z2=0.
The initial conditions given by
(3)W(z,0)=ς0(z),∂W(z,0)∂t=ς1(z),0≤z≤L
describe the initial deflection and initial velocity at each pointzof the beam.Fitzgibbon [3] and Ball [4] established the general existence theory for IBVP (1)–(3). Patcheu [5] subsequently incorporated a nonlinear friction force term into the mathematical model to account for dissipation; this was done by replacing the right side of (1) by the term ζ|(∂W(z,t))/∂t|, where ζ is a bounded linear operator. More recently, Balachandran and Park [6] further introduced the term -λ(∂4W(z,t)/∂t2∂z2) into (1) to account for the fact that during vibration, the elements of the beam not only undergo translator motion but also rotate.The above results were established for the deterministic case and did not account for environmental noise. As Kannan and Bharucha-Reid [7] points out, the variability in measurement obtained experimentally suggests that studying a stochastic version of the model is advantageous. Indeed, doing so enables us to understand the effects of noise on the behavior of the phenomenon. To this end, Mahmudov and McKibben [8] studied a stochastic version of (1) (with damping), assuming that noise was incorporated into the model via a standard one-dimensional Brownian motion process.The purpose of the present work is to generalize those results in two ways. One of the directions for this generalization is that we broaden the class of nonlinear forcing terms incorporated into the model. We do so in two ways, one of which is to consider more general functional-type forcing terms that capture general integral terms, semilinear terms, and others under the same abstract form. For the second way, we point out that accounting for certain types of nonlinearities in the mathematical modeling of some phenomena—for instance, nonlinear waves and the dynamic buckling of a hinged extensible beam—requires that the nonlinearities depend on the probability distribution of the solution process. This notion was first studied in the finite-dimensional setting [9, 10], was initiated in the infinite-dimensional setting by Ahmed and Ding [11], and subsequently was studied by various authors [12, 13].The second direction of the generalization is to incorporate noise into the model via a more general fractional Brownian motion process. Regarding the mathematical description of the noise term, it is often the case that the standard Brownian motion is insufficient. Indeed, it has been shown that certain processes arising in a broad array of applications—such as communication networks, self-similar protein dynamics, certain financial models, and nonlinear waves—exhibit a self-similarity property in the sense that the processes{x(αt):0≤t≤T} and {αHx(t):0≤t≤T} have the same law (see [14–16]). 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 nonnegligible way in the modeling of this phenomena. In fact, since BH(t) is not a semimartingale unless H=1/2, the standard stochastic calculus involving the Itó integral cannot be used in the analysis of related stochastic evolution equations. Several authors have studied the formulation of stochastic calculus for fBm and differential/evolution equations driven by fBm in the past decade [14–18].Specifically, we consider the following stochastic IBVP:(4)∂(∂W(z,t;ω)∂t)+[EIρ∂4W(z,t;ω)∂z4+α∂3W(z,t;ω)∂t∂z2]∂t-[(Hρ+EAc2ρL∫0L|∂W(ξ,t;ω)∂ξ|2dξ)×∂2W(z,t;ω)∂z2]∂t=ℱ(W(z,t;ω),t;ω)∂t+g(z,t;ω)dβH(t;ω),(5)W(0,t;ω)=W(L,t;ω)=∂2W(0,t;ω)∂z2=∂2W(L,t;ω)∂z2=0,(6)W(z,0;ω)=ς0(z;ω),∂W(z,0;ω)∂t=ς1(z;ω),
where 0≤z≤L, 0≤t≤T, ω∈Ω (a complete probability space), {βH(t;ω):0≤t≤T} is a fractional Brownian motion, and the nonlinear forcing term ℱ(W(z,t;ω),t;ω) will take on various forms including
(7)∫0tb(t,s)F0(s,W(z,s;ω))ds+F1(t,W(z,t;ω)),(8)∫0Ta(t,s)F2(z,s,W(z,s;ω),∫0sk(s,τ,W(z,τ;ω))dτ)ds,(9)F3(z,t,W(z,t;ω))+∫ℒ2(0,L)F4(z,y,t;ω)μ(z,t)(dy),
where {μ(z,t):0≤z≤L,0≤t≤T} is a family of probability measures, and a, k, and Fi(i=1,2,3,4) are appropriate mappings.The plan of the paper is as follows. We will collect some preliminary information on cosine function theory, special function spaces, probability measures, and fractional Brownian motion in Section2. Then we present abstract formulations of the IBVPs under consideration in Section 3. The main theoretical results together with corollaries illustrating the applicability to the IBVPs introduced above are gathered in Section 4. The proofs of the main results are provided in Section 5, followed by an interpretation of these results for the specific nonlinear beam model in Section 6. An extensive reference list then follows.


## 2. Preliminaries

For details of this section, we refer the reader to [4, 16, 17, 19, 20] and the references therein. Throughout this paper,U and V are real separable Hilbert spaces with norms ∥·∥U and ∥·∥V, and inner products 〈·,·〉U and 〈·,·〉V equipped with a complete orthonormal basis {ej|j=1,2,…}. Also,(Ω,𝔉,P) is a complete probability space. Henceforth, for brevity, we will suppress the dependence of random variables on ω.We make use of several different function spaces throughout this paper.𝔅𝔏(U) is the space of all bounded linear operators onU, while ℒ2(U)=ℒ2((0,T);U) stands for the space of all U-valued random variables y for which E∥y∥U2<∞. Also, C([0,T];U) stands for the space of ℒ2-continuous U-valued random variables y:[0,T]→U such that
(10)∥y∥C([0,T];U)2≡sup0≤t≤TE∥y(t)∥U2<∞.
The space of Hilbert Schmidt operators from V into U is denoted by 𝔏(V,U) and is equipped with the norm ∥·∥𝔏(V,U).The remaining function spaces coincide with those used in [11]; 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→(0,∞) by λ(x)=1+∥x∥U, x∈U, and consider the space
(11)𝒞(U)={φ:U⟶U|φiscontinuousand∥φ∥𝒞=supx∈U∥φ(x)∥Uλ2(x)+supx≠yinU∥φ(x)-φ(y)∥U∥x-y∥U<∞}.
For p≥1, we let
(12)℘λps(U)={∫m:U⟶ℝ∣misassignedmeasureonUsuchthat∥m∥λp=∫Uλp(x)|m|(dx)<∞},
where m=m+-m- is the Jordan decomposition of m and |m|=m++m-. Then define the space ℘λ2(U)=℘λ2s(U)∩℘(U) equipped with the metric ρ given by
(13)ρ(ς1,ς2)=sup{∫Uφ(x)(ς1-ς2)(dx):∥φ∥𝒞≤1}.
It is known that (℘λ2(U),ρ) is a complete metric space. The space of all continuous ℘λ2(U)-valued functions defined on [0,T] is denoted by 𝒞λ2=𝒞λ2([0,T];(℘λ2(U),ρ)) is complete when equipped with the metric
(14)DT(ς1,ς2)=supt∈[0,T]ρ(ς1(t),ς2(t)),∀ς1,ς2∈𝒞λ2.
We recall some facts about cosine families of operators defined as follows.Definition 1.
(i) The one-parameter family{C(t):t∈ℝ}⊂𝔅𝔏(U), satisfying(a)
C(0)=I,(b)
C(t)z is continuous in t on ℝ, for all z∈U,(c)
C(t+s)+C(t-s)=2C(t)C(s), for all t,s∈ℝ, is called a strongly continuous cosine family.
(ii) The correspondingstrongly continuous sine family{S(t):t∈ℝ}⊂𝔅𝔏(U) is defined by S(t)z=∫0tC(s)zds, for all t∈ℝ, for all z∈U.
(iii) The(infinitesimal) generatorA:U→U of {C(t):t∈ℝ} is given by Az=(d2/dt2)C(t)z|t=0, for all z∈D(A)={z∈U:C(·)z∈C2(ℝ;U)}.
It is known that the infinitesimal generatorA is a closed, densely-defined operator on U (see [21]). Such a cosine and (corresponding sine) family and its generator satisfy the following properties.Proposition 2.
Suppose thatAis the infinitesimal generator of a cosine family of operators {C(t):t∈ℝ} (cf. Definition 1). Then the following hold.(i)
There existMA≥1 and ω≥0 such that ∥C(t)∥≤MAeω|t| and hence, ∥S(t)∥≤MAeω|t|,(ii)
A∫srS(u)zdu=[C(r)-C(s)]z,  for all 0≤s≤r<∞,(iii)
There existsN≥1 such that ∥S(s)-S(r)∥≤N|∫srew|s|ds|, for all 0≤s≤r<∞.The Uniform Boundedness Principle, together with (i) above, implies that both{C(t):t∈[0,T]} and{S(t):t∈[0,T]} are uniformly bounded by M*=MAeωT.Next, we make precise the definition of aU-valued fBm and related stochastic integral used in this paper. The approach we use coincides with the one formulated and analyzed in [15, 17]. Let {βjH(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,…(i)
βjH(0)=0,(ii)
E[βjH(t)-βjH(s)]2=|t-s|2Hνj,(iii)
E[βjH(1)]2=νj>0,(iv)
∑j=1∞νj<∞.In such case,∑j=1∞E∥βjH(t)ej∥U2=t2H∑j=1∞νj<∞, so that the following definition is meaningful.Definition 3.
For everyt≥0, βH(t)=∑j=1∞βjH(t)ej is a U-valued fBm, where the convergence is understood to be in the mean-square sense.
It has been shown in [17] that the covariance operator of {βH(t):t≥0} is a positive nuclear operator Q such that
(15)trQ(t,s)=12∑j=1∞νj[t2H+s2H-|t-s|2H].
We now outline the discussion leading to the definition of the stochastic integral associated with {βH(t):t≥0} for bounded, measurable functions. To begin, assume that g:[0,T]→𝔏(V,U) is a simple function; that is, there exists {gi:i=1,…,n}⊆ℝ such that
(16)g(t)=gi,∀ti-1≤t≤ti,
where 0=t0<t1<⋯<tn-1<tn=T and max1≤i≤n∥gi∥𝔏(V,U)=K.Definition 4.
TheU-valued stochastic integral ∫0Tg(t)dβH(t) is defined by
(17)∫0Tg(t)dβH(t)=∑j=1∞(∫0Tg(t)dβjH(t))ej=∑j=1∞(∑i=1ngi[βjH(ti)-βjH(ti-1)])ej.
As argued in Lemma 2.2 of [17], this integral is well-defined since
(18)E∥∫0Tg(t)dβH(t)∥U2≤K2T2H∑j=1∞νj<∞.
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 4 to the case of a general bounded measurable integrand.
Finally, in addition to the familiar Young, Hölder, and Minkowski inequalities, the inequality of the form(∑i=1nai)m≤nm-1∑i=1naim, where ai is a nonnegative constant (i=1,…,n) and m,n∈ℕ, will be used to establish various estimates.


## 3. Abstract Formulation

The goal of this section is to reformulate the stochastic IBVPs introduced in Section1 as abstract stochastic evolution equations. There will be two related abstract formulations depending on the nature of the nonlinearity. We begin with the reformulation (4)–(6) assuming that the forcing term ℱ(W(z,t;ω),t;ω) is given by (9).We impose the following conditions:(A1)
F3:(0,L)×[0,T]×ℝ→ℝ satisfies the Caratheodory conditions (i.e., measurable in (x,t) and continuous in the third variable) such that
(i)
there existsMF3¯>0 for which |F3(y,t,w)|≤MF3¯(1+|w|), for all y∈(0,L), t∈[0,T], and w∈ℝ;(ii)
there existsMF3>0 for which |F3(y,t,w1)-F3(y,t,w2)|≤MF3|w1-w2|, for all y∈(0,L), t∈[0,T], and w1,w2∈ℝ;(A2)
F4:(0,L)×[0,T]×ℒ2(0,L)→ℒ2(0,L) satisfies the Caratheodory conditions such that
(i)
there existsMF4¯>0 for which ∥F4(y,t,w)∥ℒ2(0,L)≤MF4¯(1+∥w∥ℒ2(0,L)), for all y∈(0,L), t∈[0,T], and w∈ℒ2(0,L);(ii)
F4(y,t,·):ℒ2(0,L)→ℒ2(0,L) belongs to 𝒞λ2, for every y∈(0,L), t∈[0,T];(A3)
g:(0,L)×[0,T]→𝔏(ℒ2(0,L),ℒ2(0,L)) is a bounded measurable function;(A4)
μ(·,t)∈℘λ2(ℒ2(0,L)) is the probability law of W(·,t), for each t∈[0,T];(A5)
ς0(·), ς1(·)∈ℒ2(Ω;ℒ2(0,L)).We reformulate the associated IBVP by making the following identifications as in [8]; we recall the highlights of that formulation here for completeness of the discussion.LetU=V=ℒ2(0,L) (note they are real separable Hilbert spaces) and identify the solution process x:[0,T]→U by
(19)x(t)(z)=W(z,t),∀z∈(0,L),t∈[0,T].
Define the operator A:D(A)⊂U→U by
(20)Ax(t,·)=∂4x(t,·)∂z4,D(A)={x∈H4(0,L):x(0)=x(L)=x′′(0)=x′′(L)=0}.
It is known that A is a positive, self-adjoint operator on U (cf. [21, 22]). As such, A generates a strongly continuous cosine family of operators on U. We will express the other terms on the left side of (4) using fractional powers of A, as in [22]. To this end, the eigenvalues of A are {λn=(nπ)4:n=1,2,3,…} with corresponding eigenvalues
(21){zn(s)=2Lsin(nπs):n=1,2,3,…,s∈[0,L]}.
Then A has spectral representation
(22)Ay=∑n=1∞λn〈y,zn〉zn.
The following operators are well defined because the fractional powers of A are positive and self-adjoint:
(23)A1/2y=∑n=1∞λn1/2〈y,zn〉zn=-∂2y∂z2,A1/4y=∑n=1∞λn1/4〈y,zn〉zn=∂y∂z.
Also, observe that
(24)∥A1/4y∥U2=〈A1/4y,A1/4y〉=∑n=1∞λn1/2〈y,zn〉2=∫0L∥∂y∂z(w,t)∥U2dw.
It can be shown using standard computations involving the inner product and properties of fractional powers of A that ∥A1/4x(t)∥U2  and ∥A1/2x(t)∥U2 are uniformly bounded on [0,T].Define the mappingsf:[0,T]×U×℘λ2(U)→U, g:[0,T]→𝔏(V,U), x0(·), and x1(·), and the operator B:U→U as follows:
(25)f(t,x(t),μ(t))(z)=F3(z,t,W(z,t))+∫ℒ2(0,L)F4(z,y,t)μ(z,t)(dy)-(Hρ+EAc2ρL∫0L|∂W(ξ,t)∂ξ|2dξ)∂2W(z,t)∂z2=F3(z,t,W(z,t))+∫ℒ2(0,L)F4(z,y,t)μ(z,t)(dy)-(Hρ+EAc2ρL∥A1/4x(t)∥U2)A1/2x(t),g(t)(z)=g(z,t),B(x′(t))(z)=α∂3W(z,t)∂t∂z2=αA1/2(∂W(z,t)∂t2),x0(0)(z)=ς0(z),x1(0)(z)=ς1(z),
for all z∈(0,L) and t∈[0,T].These identifications can be used to formulate (4)–(6), with nonlinear forcing term given by (9) as the following abstract second-order stochastic evolution equation in a real separable Hilbert space U:
(26)dx′(t)=(Bx′(t)+Ax(t)+f(t,x(t),μ(t)))dt+g(t)dβH(t),0≤t≤T,x(0)=x0,x′(0)=x1,μ(t)=probabilitydistributionofx(t)
in a real separable Hilbert space U. (By the probability distribution of x(t), we mean μ(t)(z)=P({ω∈Ω:x(t,ω)∈z}), for each z∈ℬ(U).) Here, A:D(A)⊂U→U is a linear (possibly unbounded) operator which generates a strongly continuous cosine family {C(t):t≥0} on U with associated sine family {S(t):t≥0}; f:[0,T]×U×℘λ2(U)→U; g:[0,T]→𝔏(V,U) is a bounded, measurable mapping; B:U→U is a bounded linear operator; {βH(t):t≥0} is a V-valued fBm with Hurst parameter H∈(1/2,1); andx0,x1∈ℒ2(Ω;U).We next consider the same IBVP, with the exception that the forcing term is given by a mapping of the form (7) or (8). While all other identifications remain identical, we will describe such nonlinear forcing terms more generally as a so-called functional. Precisely, we consider the following abstract second-order functional stochastic evolution equation in a real separable Hilbert space U:
(27)dx′(t)=(Bx′(t)+Ax(t)+ℱ(x)(t))dt+g(t)dβH(t),0≤t≤T,x(0)=x0,x′(0)=x1,
where ℱ:C([0,T];U)→ℒ2(0,T;ℒ2(Ω;U)) is a given functional and all other identifications are the same as for (26). We impose the following conditions for (7) and (8):(A6)
{b(t,s):0≤t≤s≤T}⊂𝔅𝔏(U);(A7)
Fi:(0,L)×[0,T]×ℝ→ℝ(i=0,1) are mappings that satisfy the Caratheodory conditions and are such that there exists MFi>0 such that
(28)|Fi(y,t,w1)-Fi(y,t,w2)|≤MFi|w1-w2|,for ally∈(0,L), t∈[0,T], and w1,w2∈ℝ;(A8)
F2:(0,L)×[0,T]×ℝ×ℝ→ℝ satisfies the Caratheodory conditions and is such that there exists MF2>0 such that
(29)|F2(y,t,w1,z1)-F2(y,t,w2,z2)|≤MF2[|w1-w2|+|z1-z2|],for ally∈(0,L), t∈[0,T], and w1,w2,z1,z2∈ℝ;(A9)
a∈ℒ2((0,T)2;ℝ);(A10)
k:Δ×ℝ→ℝ, where Δ={(s,t):0<s<t<T} satisfies
(30)|k(t,s,x1)-k(t,s,x2)|≤Mk|x1-x2|,
for all x1,x2∈ℝ  and (s,t)∈Δ. Identify the solution process x:[0,T]→U by
(31)x(t)(z)=W(z,t),∀z∈(0,L),t∈[0,T].It can be shown that under these assumptions, the following are well-defined mappings from C([0,T];U) into ℒ2(0,T;ℒ2(Ω;U)):
(32)ℱ(x)(t)=∫0ta(t,s)ccccccc×F2(·,s,W(·,s;ω),∫0sk(s,τ,W(·,τ;ω))dτ)ds,(33)ℱ(x)(t)=∫0tb(t,s)F0(·,s,W(·,s;ω))ds+F1(·,t,W(·,t;ω)).
The strategy is to formulate a general theory for these two abstract second-order problems and then to apply the results to the specific IBVPs formulated in Section 1.


## 4. Statement of Results

We consider mild solutions of (26) in the following sense.Definition 5.
A stochastic processx∈C([0,T];U) is a mild solution of (26) if(i)
(34)x(t)=S(t)x1+(C(t)-S(t)B)x0+∫0tC(t-s)Bx(s)ds+∫0tS(t-s)f(s,x(s),μ(s))ds+∫0tS(t-s)g(s)dβH(s),0≤t≤T,(ii)
μ(t) is the probability distribution of x(t), for all 0≤t≤T.
The following conditions on (26) are imposed on the data and mappings in (26):(A11)
A:D(A)⊂U→U is the infinitesimal generator of a strongly continuous cosine family {C(t):t≥0} on U with associated sine family {S(t):t≥0} on U;(A12)
f:[0,T]×U×℘λ2(U)→U satisfies the following.
(i)
There exists a positive constantMf such that
(35)E∥f(t,z1,ς1)-f(t,z2,ς2)∥U2≤Mf[E∥z1-z2∥U2+ρ2(ς1,ς2)],globally on[0,T]×U×℘λ2(U),(ii)
there exists a positive constantMf¯ such that
(36)E∥f(t,z,ς)∥U2≤Mf¯[1+E∥z∥U2+∥ς∥λ22];globally on[0,T]×U×℘λ2(U);(A13)
g:[0,T]→𝔏(V,U) is a bounded, measurable mapping;(A14)
B:U→U is a bounded linear operator;(A15)
{βH(t):0≤t≤T} is a U-valued fBm;(A16)
x0,x1∈ℒ2(Ω;U) are (ℱ0,ℬ(U))-measurable, where {ℱt:0≤t≤T} is the family of σ-algebras ℱt generated by {βH(s):0≤s≤t}.
Henceforth, we write(37)M*=max({∥S(t)∥𝔅𝔏(U):0≤t≤T}ccccc∪{∥C(t)∥𝔅𝔏(U):0≤t≤T}),
which is finite by (A11).
The following lemma, introduced in (me and dave) and stated here without proof, is critical in establishing several estimates.Lemma 6.
Assume thatg:[0,T]→𝔏(V,U) satisfies (A13). Then, for all0≤t≤T,
(38)E∥∫0tS(t-s)g(s)dBH(s)∥U2≤Ct∑j=1∞νj,
where Ct is a positive constant depending on t, MS, and the growth bound on g, and {νj:j∈ℕ} is defined as in the discussion leading to Definition 4.Letμ∈𝒞λ2 be fixed and define the solution map Φ:C([0,T];U)→C([0,T];U) by
(39)Φ(x)(t)=S(t)x1+(C(t)-S(t)B)x0+∫0tC(t-s)Bx(s)ds+∫0tS(t-s)f(s,x(s),μ(s))ds+∫0tS(t-s)g(s)dβH(s),0≤t≤T,=∑i=15Iix(t),0≤t≤T.
The first two integrals on the right side of (39) are taken in the Bochner sense, while the third is defined in Section 2. The operator Φ satisfies the following properties.Lemma 7.
If (A11)–(A16) hold, thenΦ is a well-defined ℒ2-continuous mapping.Our first main result is as follows.Theorem 8.
If (A11)–(A16) hold, then (26) has a unique mild solution x on[0,T] with corresponding probability law μ∈𝒞λ2. Further, for every p≥1, there exists a positive constant Cp (depending on p, T, and νj) such that
(40)sup0≤t≤TE∥x(t)∥U2p≤Cp(1+E∥x0∥U2p+E∥x1∥U2p).Mild solutions of (26) depend continuously on the initial data and probability distribution of the state process in the following sense.Proposition 9.
Assume that (A11)–(A16) hold, and letx and y be the mild solutions of (26) (as guaranteed to exist by Theorem 8) corresponding to initial data x0, x1 and y0, y1 with respective probability distributions μx and μy. Then there exists a positive constant M′ such that
(41)E∥x(t)-y(t)∥U2≤M′[∥x0-y0∥ℒ2(Ω;U)2ccc+∥x1-y1∥ℒ2(Ω;U)2ccc+DT2(μx,μy)].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 (26) arising frequently in applications. Specifically, consider the deterministic initial-value problem
(42)z′′(t)=Az(t)+Bz′(t)+F(t,z(t)),0≤t≤T,z(0)=z0,z′(0)=z1,
where z0∈D(A), z1∈E={u∈U:C(·)u∈C1(ℝ;U)}, B:U→U is a bounded linear operator, A satisfies (A11), and F:[0,T]×U→U satisfies the following conditions:(A17)(i)
there exists a positive constantMF such that ∥F(t,z1)-F(t,z2)∥U≤MF∥z1-z2∥U globally on [0,T]×U;(ii)
there exists a positive constantMF¯ such that ∥F(t,z)∥U≤MF¯∥z∥U globally on [0,T]×U.It is known that under these conditions, (42) has a unique mild solution z given by the representation formula
(43)z(t)=S(t)x1+(C(t)-S(t)B)x0+∫0tC(t-s)Bx(s)ds+∫0tS(t-s)F(s,x(s))ds,0≤t≤T.
For each ε>0, consider the stochastic initial-value problem
(44)dxε(t)=(∫UF1ε(t,z)με(t)(dz)+F2ε(t,xε(t))Aεxε(t)+Bxε′(t)+∫UF1ε(t,z)με(t)(dz)+F2ε(t,xε(t)))dt+gε(t)dβH(t),0≤t≤T,xε(0)=z0,xε′(0)=z1,με(t)=probabilitydistributionofxε(t),
in U. Here, we assume that z0∈D(Aε)=D(A) and z1∈Eε=E, for all ε>0 and that Fiε:[0,T]×U→U(i=1,2) are given mappings. We impose the following conditions on the data and mappings in (44), for each ε>0.(A18)
Aε:D(Aε)=D(A)→D(Aε) generates a strongly continuous cosine family {Cε(t):t≥0} satisfying Cε(t)→C(t) strongly as ε→0+, uniformly in t∈[0,T]. Also, the corresponding sine family {Sε(t):t≥0} satisfying Sε(t)→S(t) strongly as ε→0+, uniformly in t∈[0,T]. Moreover,
(45)max{∥Cε(t)∥𝔅𝔏(U)+∥Sε(t)∥𝔅𝔏(U):0≤t≤T}≤M*,
for all ε>0 and M* is the same bound used for the cosine and sine families generated by  A.(A19)
Bε:U→U is a bounded linear operator such that Bε→B in 𝔅𝔏(U) and ∥Bε∥𝔅𝔏(U)≤∥B∥𝔅𝔏(U), for all ε>0.(A20)
F2ε:[0,T]×U→U is Lipschitz in the second variable (with the same Lipschitz constant MF used for F in (A17)), and F2ε(t,u)→F(t,u) as ε→0+, for all u∈U, uniformly in t∈[0,T].(A21)
F1ε:[0,T]×U→U is a continuous mapping such that∫UF1ε(t,z)με(t)(dz)→0 uniformly in t as ε→0+.(A22)
gε:[0,T]→𝔏(V,U) is a bounded, measurable function such that gε(t)→0 as ε→0+, uniformly in t∈[0,T].Under these assumptions, the following result holds.Theorem 10.
Letz and xε be the mild solutions of (42) and (44) on[0,T], respectively. Then there exist a positive constant ς and a positive function ψ(ε) (which decreases to 0 as ε→0+) such that E∥xε(t)-z(t)∥U2≤ψ(ε)exp(ςt), for all t∈[0,T].We now turn our attention to (27). By a mild solution of (27), we mean the following.Definition 11.
A stochastic processx∈C([0,T];U) is a mild solution of (27) if
(46)x(t)=S(t)x1+(C(t)-S(t)B)x0+∫0tC(t-s)Bx(s)ds+∫0tS(t-s)ℱ(x)(s)ds+∫0tS(t-s)g(s)dβH(s),
for all 0≤t≤T.We assume that conditions (A11), (A13)–(A16) hold, in addition to the following condition on the functional forcing term:(A23)
ℱ:C([0,T];U)→ℒ2(0,T;ℒ2(Ω;U)) is such that there exists Mℱ>0 for which
(47)∥ℱ(x)-ℱ(y)∥ℒ2≤Mℱ∥x-y∥𝒞,∀x,y∈C([0,T];U).Define the solution map Γ:C([0,T];U)→C([0,T];U) by
(48)Γ(x)(t)=S(t)x1+(C(t)-S(t)B)x0+∫0tC(t-s)Bx(s)ds+∫0tS(t-s)ℱ(x)(s)ds+∫0tg(s)dβH(s),0≤t≤T,=I1(t)+I2(t)+I3(t)+I5(t)+∫0tS(t-s)ℱ(x)(s)ds,0≤t≤T.
(cf. (39)). The operator Γ satisfies the following properties.Lemma 12.
If (A11), (A13)–(A16), and (A23) hold, thenΓ is a well-defined ℒ2-continuous mapping.The proof of the following theorem follows almost immediately from this result.Theorem 13.
If (A11), (A13)–(A16), and (A23) hold, then (27) has a unique mild solution x on [0,T] provided that M*2T(Mℱ2+MB2)<1. Further, for every p≥1, there exists a positive constant Cp (depending on p, T, and νj) such that(49a)
sup
0
≤
t
≤
T
E
∥
x
(
t
)
∥
U
2
p
≤
C
p
(
1
+
E
∥
x
0
∥
U
2
p
+
E
∥
x
1
∥
U
2
p
)
.Corollary 14.
Assume that (A11) and (A13)–(A16) hold.(i)
If (A6) and (A7) are satisfied, then (27) where the forcing term is given by (33) has a unique mild solution on [0,T] provided that
(49b)2M*[2T(TMF0MB+MF1)2+MB2]1/2T1/2<1.
Moreover, for every p≥1, estimate (49a) holds.(ii)
If (A8)–(A10) are satisfied, then (27) where the forcing term is given by (32) has a unique mild solution on [0,T] provided that
(49c)2M*[[2TMF2|a|ℒ2(1+T3Mk)1/2]2+MB2]1/2T1/2<1.
Moreover, for every p≥1, estimate (49a) holds.Finally, we remark that an approximation scheme in the spirit of Theorem10 can be established for (27) by making the natural modifications to (42) and hypotheses (A20) and (A21).


## 5. Proofs of Main Results

Proof of Lemma7.
Letμ∈𝒞λ2 be fixed and consider the solution map Φ defined in (39).
One can see from the discussion in Section2 and the properties of x 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 ℒ2-continuity of Φ on [0,T], let z∈C([0,T];U) and consider 0≤t*≤T and |h| sufficiently small so that all terms are well defined. Observe that
(50)E∥Φ(z)(t*+h)-Φ(z)(t*)∥U2≤5∑i=15∥Ii(t*+h)-Ii(t*)∥U2.
The strong continuity of C(t) and S(t) implies that
(51)∑i=12E∥Ii(t*+h)-Ii(t*)∥U2⟶0as|h|⟶0,cccccccccccccccccccccccforeach0≤t≤T.
Using (A11) and (A14), together with Hölder’s inequality and properties of cosine operators, yields
(52)E∥I3(t*+h)-I3(t*)∥U2=E∥∫0t*[C(t*+h-s)-C(t*-s)]Bx(s)ds+∫t*t*+hC(t*+h-s)Bx(s)ds∥U2ds≤2t*∫0t*E∥[C(t*+h-s)-C(t*-s)]Bx(s)∥U2ds+2∫t*t*+hE∥C(t*+h-s)Bx(s)∥U2ds≤2t*∫0t*E∥[C(t*+h-s)-C(t*-s)]Bx(s)∥U2ds+2h(M*)2MB2∥x∥C2.
The strong continuity of C(t), along with the Lebesgue dominated convergence theorem, implies that the right side of (52) goes to 0 as |h|→0. Regarding the fourth term of (50), similar computations involving (A11) and (A12) yield the estimate
(53)E∥I4(t*+h)-I4(t*)∥U2≤2T∫0t*E∥[S(t*+h-s)-S(t*-s)]ccccccccccc×f(s,x(s),μ(s))∥U2dsccccccccccc+2h2(M*)2E∥f(s,x(s),μ(s))∥U2.
Since
(54)E∥f(s,x(s),μ(s))∥U2≤Mf2[1+∥x∥C2+sup0≤s≤T∥μ(s)∥λ22]
and the right side is independent of h, it follows immediately from (A11) that the right side of (53) goes to 0 as |h|→0.
It remains to show thatI3(t*+h)-I3(t*)→0 as |h|→0. Observe that
(55)I5(t*+h)-I5(t*)=E∥∫0t*+hS(t*+h-s)g(s)dβH(s)-∫0t*S(t*-s)g(s)dβH(s)∥U2=E∥∑j=1∞∫0t*+hS(t*+h-s)g(s)ejdβjH(s)-∑j=1∞∫0t*S(t*-s)g(s)ejdβjH(s)∥U2=E∥∑j=1∞∫t*t*+hS(t*+h-s)g(s)ejdβjH(s)+∑j=1∞∫0t*[S(t*+h-s)-S(t*-s)]×g(s)ejdβjH(s)∫0t*[S(t*+h-s)-S(t*-s)]∥U2≤2E[∥∑j=1∞∫t*t*+hS(t*+h-s)g(s)ejdβjH(s)∥U2+∥∑j=1∞∫0t*[S(t*+h-s)-S(t*-s)]×g(s)ejdβjH(s)∑j=1∞∫0t*[S(t*+h-s)-S(t*-s)]∥U2].
For the moment, assume that g is a simple function as defined in (16). Observe that for m∈ℕ, arguing as in Lemma 6 of [17] yields
(56)E∥∑j=1m∫0t*[S(t*+h-s)-S(t*-s)]g(s)ejdβjH(s)∥U2=E∥∑j=1m∑k=0n-1S¯hgkej(βjH(tk+1)-βjH(tk))∥U2=E〈∑j=1m∑k=0n-1S¯hgkej(βjH(tk+1)-βjH(tk)),∑j=1m∑k=0n-1S¯hgkej(βjH(tk+1)-βjH(tk))〉U≤∑j=1m∑k=0n-1∥S¯h∥𝔅𝔏(U)2K2E×(βjH(tk+1)-βjH(tk),βjH(tk+1)-βjH(tk))ℝ≤sup0≤s≤t*∥S¯h∥𝔅𝔏(U)2K2E(βjH)2=sup0≤s≤t*∥S¯h∥𝔅𝔏(U)2K2(t*)2H∑j=1mνj,
where S¯h=S(t*-tk+h)-S(t*-tk). Hence,
(57)E∥∑j=1∞∫0t*[S(t*+h-s)-S(t*-s)]g(s)ejdβjH(s)∥U2≤limm→∞E∥∑j=1m∫0t*[S(t*+h-s)-S(t*-s)]cccccccccccccc×g(s)ejdβjH(s)∑j=1m∫0t*[S(t*+h-s)-S(t*-s)]∥U2≤limm→∞sup0≤s≤t*∥S¯h∥𝔅𝔏(U)2K2(t*)2H∑j=1mνj=sup0≤s≤t*∥S¯h∥𝔅𝔏(U)2K2(t*)2H∑j=1∞νj,
and the right side of (57) goes to 0 as |h|→0. Next, observe that
(58)E∥∑j=1∞∫t*t*+hS(t*+h-s)g(s)ejdβjH(s)∥U2=E∥∑j=1∞∫0hS(u)g(t*+h-u)ejdβjH(t*+h-u)∥U2.
Using the property E(βjH(s)-βjH(t))2=|t-s|2Hνj with s=t*+h and t=t*, we can argue as above to conclude that the right side of (58) goes to 0 as |h|→0. Consequently, I3(t*+h)-I3(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 ℒ2-continuity of Φ.
Finally, we assert thatΦ(C([0,T];U))⊂C([0,T];U). Indeed, the necessary estimates can be established as above, and when used in conjunction with Lemma 6, one can readily verify that sup0≤t≤TE∥Φ(x)(t)∥U2<∞, for any x∈C([0,T];U). Thus, we conclude thatΦis well defined, and the proof of Lemma 7 is complete.Proof of Theorem8.
Letμ∈𝒞λ2 be fixed and consider the operator as Φ defined in (39). We know that Φ is well defined and ℒ2-continuous from Lemma 7. We now prove that Φ has a unique fixed point in C([0,T];U). Indeed, for any x,y∈C([0,T];U), (39) implies that
(59)E∥(Φx)(t)-(Φy)(t)∥U2≤2T(M*)2(MB2+Mf2)∫0tE∥x(s)-y(s)∥U2ds=δ∫0tE∥x(s)-y(s)∥U2ds,
where δ=2T(M*)2(MB2+Mf2). For any natural number n, it follows from successive iteration of (59) that
(60)∥Φnx-Φny∥C2≤Tδnn!∥x-y∥C2.
Since (Tδn/n!)<1 for large enough values of n, we can conclude from (60) that Φn is a strict contraction on C and so for a given T>0, Φ has a unique fixed point xμ∈C (by the Banach contraction mapping principle), and this coincides with a mild solution of (26) as desired.
To complete the proof, we must show thatμ is, in fact, the probability law of xμ. To this end, let ℒ(xμ)={ℒ(xμ(t)):t∈[0,T]} represent the probability law of  xμ and define the map Ψ:𝒞λ2→𝒞λ2 by Ψ(μ)=ℒ(xμ). It is not difficult to see that ℒ(xμ(t))∈℘λ2(U), for all t∈[0,T] since xμ∈C([0,T];U). In order to verify the continuity of the map t↦ℒ(xμ(t)), we first comment that an argument similar to the one used to establish Lemma 7 can be used to show that for sufficiently small |h|>0(61)limh→0E∥xμ(t+h)-xμ(t)∥U2=0,cccccccccccccccccc∀0≤t≤T.
Consequently, since for all t∈[0,T] and φ∈𝒞λ2, it is the case that
(62)|∫Uφ(x)(ℒ(xμ(t+h))-ℒ(xμ(t)))(dx)|=|E[φ(xμ(t+h))-φ(xμ(t))]|≤∥φ∥λ2E∥xμ(t+h)-xμ(t)∥U,
and hence
(63)ρ(ℒ(xμ(t+h)),ℒ(xμ(t)))=sup∥φ∥λ2≤1∫Uφ(x)(ℒ(xμ(t+h))-ℒ(xμ(t)))(dx)⟶0ccccccccccccccccccccccccccccccccccccccccccas|h|⟶0,
for any 0≤t≤T. Thus, t↦ℒ(xμ(t)) is a continuous map, so that ℒ(xμ)∈𝒞λ2, thereby showing that Ψ is well defined. In order to show that Ψ has a unique fixed point in 𝒞λ2, let μ,ν∈𝒞λ2 and let xμ,xν be the corresponding mild solutions of (26). A standard computation yields
(64)E∥xμ(t)-xν(t)∥U2≤2T(M*)2(MB2+Mf2)∫0tE∥xμ(s)-xν(s)∥U2ds+2T(M*)2Mf2∫0tρ2(μ(s),ν(s))ds,0≤t≤T.
Observe that ρ2(μ(s),ν(s))≤DT2(μ,ν). Thus, continuing the inequality in (64) yields
(65)E∥xμ(t)-xν(t)∥U2≤2T(M*)2(MB2+Mf2)∫0tE∥xμ(s)-xν(s)∥U2ds+5T2(M*)2Mf2DT2(μ,ν),0≤t≤T.
Applying Gronwall’s lemma now yields
(66)E∥xμ(t)-xν(t)∥U2≤5T2(M*)2Mf2exp(2T(M*)2(MB2+Mf2))DT2(μ,ν)=ς(T)DT2(μ,ν),0≤t≤T.
We can choose 0≤T¯≤T to ensure ς(T¯)<1, so that taking the supremum above yields
(67)∥xμ-xν∥C([0,T];U)2≤ς(T¯)DT2(μ,ν).
Since
(68)ρ(ℒ(xμ(t)),ℒ(xν(t)))≤E∥xμ(t)-xν(t)∥U,cccccccccccccccccccccccccccccc∀0≤t≤T,
(which follows directly from (62) and (63)), we have
(69)∥Ψ(μ)-Ψ(ν)∥λ22=DT¯2(Ψ(μ),Ψ(ν))≤supt∈[0,T¯]E∥xμ(t)-xν(t)∥U2=∥xμ-xν∥C([0,T¯];U)2<ς(T¯)DT¯2(μ,ν),
so that Ψ is a strict contraction on 𝒞λ2([0,T¯];(℘λ2(U),ρ)). Thus, (26) has a unique mild solution on [0,T¯] with probability distribution μ∈𝒞λ2([0,T¯];(℘λ2(U),ρ)). The solution can then be extended, by continuity, to the entire interval [0,T] in finitely many steps, thereby completing the proof of the existence-uniqueness portion of the theorem.
Next, we establish the boundedness ofpth moments of the mild solutions of (26). Let p≥1 and observe that the standard computations yield, for all 0≤t≤T,
(70)E∥xμ(t)∥U2p≤5p[∥∫0tS(t-s)g(s)dβH(s)∥U2pE∥S(t)x1∥U2p+E∥(C(t)-S(t)B)x0∥U2pccc+E∥∫0tC(t-s)Bx(s)ds∥U2pccc+E∥∫0tS(t-s)f(s,x(s),μ(s))ds∥U2pccc+E∥∫0tS(t-s)g(s)dβH(s)∥U2p]≤10p(M*)2p[(Ct∑j=1∞νj)pE∥x1∥U2p+E∥x0∥U2p(1+MB2p)ccccccccccc+(TMB∫0tE∥xμ(s)∥U2)pccccccccccc+(T∫0t(E∥xμ(s)∥U2+∥μ(s)∥λ22)ds)pccccccccccc+(Ct∑j=1∞νj)p].
Applying the Jensen and Hölder inequalities enables us to continue the string of inequalities above as follows:(71)≤10p(M*)2p[(∑j=1∞νj)pE∥x1∥U2p+E∥x0∥U2p(1+MB2p)cccccccccccc+TpMBp∫0tE∥xμ(s)∥U2pdsccccccccccc+CT*2p∫0t(E∥xμ(s)∥U2p+∥μ(s)∥λ22p)dsccccccccccc+CTp(∑j=1∞νj)p]≤10p(M*)2p[(sup0≤s≤T∥μ(s)∥λ22p+CTp(∑j=1∞νj)p)E∥x1∥U2p+E∥x0∥U2p(1+MB2p)ccccccccccc+CT*2pT(sup0≤s≤T∥μ(s)∥λ22p+CTp(∑j=1∞νj)p)]+10p(M*)2p(TpMBp+CT*2p)∫0tE∥xμ(s)∥U2pds.
Therefore, an application of Gronwall’s lemma, followed by taking the supremum over[0,T], yields the desired result. This completes the proof of Theorem 8.Proof of Proposition9.
Computations similar to those used leading to the contractivity of the solution map in Theorem8 can be used, along with Gronwall’s lemma, to establish this result. The details are omitted.Proof of Theorem10.
We estimate each term of the representation formula forE∥xε(t)-z(t)∥U2 separately. We begin by observing that (A18) and (A19), together, guarantee the existence of positive constants Ki and positive functions αi(ε) (which decrease to zero as ε→0+)(i=1,2,3) such that, for sufficiently small ε,
(72)∥Sε(t)Bz0-S(t)Bz0∥U≤K1α1(ε),∥Sε(t)z0-S(t)z0∥U≤K2α2(ε),∥Cε(t)z0-C(t)z0∥U≤K3α3(ε),
for all 0≤t≤T.
Next, we estimate(73)E∥∫0t(Cε(t-s)Bεxε(s)-C(t-s)Bz(s))ds∥U2.
To this end, note that
(74)∫0tE∥Cε(t-s)Bεxε(s)-Cε(t-s)Bεz(s)∥U2ds≤(M*)2MB2∫0tE∥xε(s)-z(s)∥U2ds,
and that
(75)∫0tE∥Cε(t-s)Bεz(s)-Cε(t-s)Bz(s)∥U2ds≤(M*)2∫0tE∥Bεz(s)-Bz(s)∥U2ds.
Since (A19) guarantees that the right side of (75) goes to zero as ε→0+ (uniformly in t), we know that there exist K4>0 and positive function α4(ε) as in (72) such that, for sufficiently small ε>0,
(76)∫0tE∥Cε(t-s)Bεz(s)-Cε(t-s)Bz(s)∥U2ds≤K4α4(ε),
for all 0≤t≤T. Finally, (A18) and (A19) guarantee that the right side of
(77)∫0tE∥Cε(t-s)Bz(s)-C(t-s)Bz(s)∥U2ds≤∫0t∥Cε(t-s)-C(t-s)∥U2E∥Bz(s)∥U2ds
tends to zero as ε→0+, uniformly in t, so that again there exist K5>0 and positive function α5(ε) as in (72) such that, for sufficiently small ε>0,
(78)∫0tE∥Cε(t-s)Bz(s)-C(t-s)Bz(s)∥U2ds≤K5α5(ε),
for all 0≤t≤T. Thus, using (75)–(78), together with standard inequalities, in (74) yields
(79)E∥∫0t(Cε(t-s)Bεxε(s)-C(t-s)Bz(s))ds∥U2≤16T1/2[(M*)2MB2∫0tE∥xε(s)-z(s)∥U2ds]+16T1/2(K4α4(ε)+K5α5(ε)).
Next, we estimate(80)E∥∫0t(Sε(t-s)F2ε(s,xε(s))-S(t-s)F(s,z(s)))ds∥U2.
The continuity ofF, together with (A20), enables us to infer the existence of K6>0 andα6(ε) (as above) such that, for sufficiently small ε>0,
(81)∫0tE∥[Sε(t-s)-S(t-s)]F(s,z(s))∥U2ds≤K6α6(ε),
for all 0≤t≤T. Also, observe that Young’s inequality and (A20), together, imply
(82)∫0tE∥Sε(t-s)[F2ε(s,xε(s))-F(s,z(s))]∥U2ds≤(M*)2∫0tE∥F2ε(s,xε(s))-F2ε(s,z(s))+ccccccccccccc+F2ε(s,z(s))-F(s,z(s))∥U2ds≤4(M*)2∫0t[∥F2ε(s,z(s))-F(s,z(s))∥U2MF2E∥xε(s)-z(s)∥U2ccccccccccccc+E∥F2ε(s,z(s))-F(s,z(s))∥U2]ds.
Note that (A20) guarantees the existence of K7>0 and α7(ε) (as above) such that, for sufficiently small ε>0,
(83)E∥F2ε(s,z(s))-F(s,z(s))∥U2≤K7α7(ε),
for all 0≤t≤T.
So, we can continue the inequality (82) to conclude that
(84)∫0tE∥Sε(t-s)[F2ε(s,xε(s))-F(s,z(s))]∥U2ds≤4MF2∫0tE∥xε(s)-z(s)∥U2ds+4TK7α7(ε).
Using (82) and (84), together with the Hölder, Minkowski, and Young inequalities, yields
(85)E∥∫0t(Sε(t-s)F2ε(s,xε(s))-S(t-s)F(s,z(s)))ds∥U2≤4T1/2[∫0tE∥xε(s)-z(s)∥U2dsK2α2(ε)+4TK3α3(ε)cccccccc+4(M*)2MF2∫0tE∥xε(s)-z(s)∥U2ds].
Next, (A21) guarantees the existence of K8>0 and α8(ε) (as above) such that, for sufficiently small ε>0,
(86)E∥∫UF1ε(s,z)με(s)(dz)∥U2≤K8α8(ε),∀0≤s≤T.
As such, we have
(87)E∥∫0tSε(t-s)∫UF1ε(s,z)με(s)(dz)ds∥U2ds≤(M*)2T∫0tE∥∫UF1ε(s,z)με(s)(dz)∥U2≤(M*)2T2K8α8(ε),
for all 0≤t≤T.
It remains to estimateE∥∫0tSε(t-s)gε(s)dBH(s)∥U2. Observe that (A22) implies the existence of K9>0 and α9(ε) (as above) such that, for sufficiently small ε>0,
(88)E∥gε(t)∥𝔏(V;U)2≤K9α9(ε),0≤t≤T.
First, assume thatgε is a simple function. Observe that for m∈ℕ,
(89)E∥∑j=1m∫0tSε(t-s)gε(s)ejdβjH(s)∥U2=E∥∑j=1m∑k=0n-1Sε(t-tk)gkej(βjH(tk+1)-βjH(tk))∥U2=E〈∑j=1m∑k=0n-1Sε(t-tk)gkej(βjH(tk+1)-βjH(tk)),ccccc∑j=1m∑k=0n-1Sε(t-tk)gkej(βjH(tk+1)-βjH(tk))〉U≤∑j=1m∑k=0n-1∥Sε(t-tk)∥𝔅𝔏(U)2K9α9(ε)Eccccccccc×(βjH(tk+1)-βjH(tk),βjH(tk+1)-βjH(tk))ℝ≤sup0≤s≤t∥Sε(t-tk)∥𝔅𝔏(U)2K9α9(ε)E(βjH)2=sup0≤s≤t∥Sε(t-tk)∥𝔅𝔏(U)2K9α9(ε)t2H∑j=1mνj.
Hence,
(90)E∥∫0tSε(t-s)gε(s)dβH(s)∥U2=E∥∑j=1∞∫0tSε(t-s)gε(s)ejdβjH(s)∥U2≤limm→∞E∥∑j=1m∫0tSε(t-s)gε(s)ejdβjH(s)∥U2≤limm→∞(sup0≤s≤t∥Sε(t-tk)∥𝔅𝔏(U)2K9α9(ε)t2H∑j=1mνj)≤(MS2t2H∑j=1∞νj)K9α9(ε),0≤t≤T.
Since the set of all such simple functions is dense in 𝔏(V,U), a standard density argument can be used to establish estimate (90) for a general bounded, measurable functiongε.
Now, using (72)–(90), we conclude that there exist positive constants Ki¯(i=1,…,9) and positive functions αi(ε)  (i=1,…,9) such that
(91)E∥xε(t)-z(t)∥U2≤∑i=19Ki¯αi(ε)+4(M*)2MF2×∫0tE∥xε(s)-z(s)∥U2ds,0≤t≤T,
so that an application of Gronwall’s lemma implies
(92)E∥xε(t)-z(t)∥U2≤Ψ(ε)exp(ςt),0≤t≤T,
where ς=4(M*)2MF2 and Ψ(ε)=∑i=19Ki¯αi(ε). This completes the proof.Proof of Lemma12.
Using the discussion in Section2 and the properties of x, it follows that for any x∈C([0,T];U), Γ(x)(t) is a well-defined stochastic process, for each 0≤t≤T. In order to verify the ℒ2-continuity of Γ on [0,T], let z∈C([0,T];U) and consider 0≤t*≤T and |h| sufficiently small. Observe that
(93)E∥Γ(z)(t*+h)-Γ(z)(t*)∥U2≤5∑i=1,i≠45E∥Ii(t*+h)-Ii(t*)∥U2+E∥∫0t*+hS(t*+h-s)ℱ(x)(s)ds-∫0t*S(t*-s)ℱ(x)(s)ds∥U2.
Using Hölder’s inequality with (A23) yields(94)E∥∫t*t*+hS(t*+h-s)ℱ(x)(s)ds∥U2≤2MF2(M*)2h2×[1+∥x∥C([0,T];U)2+∥ℱ(0)∥C([0,T];U)2],
which goes to 0 as |h|→0. Also,
(95)E∥∫0t*[S(h)-I]S(t*-s)ℱ(x)(s)ds∥U2≤T∫0t*∥[S(h)-I]S(t*-s)∥𝔅𝔏(U)2ccccccccc×E∥ℱ(x)(s)∥U2ds≤Tsup0≤s≤T∥[S(h)-I]S(t*-s)∥𝔅𝔏(U)2∥ℱ(x)∥ℒ2,
and the right side goes to 0 as |h|→0 here as well. Successive applications of Hölder’s inequality yields
(96)[E∥∫0tS(t-s)ℱ(x)(s)ds∥U2]1/2≤T1/2M*[∫0T∥ℱ(x)(s)∥ℒ2(Ω;U)2ds]1/2≤T1/2M*∥ℱ(x)∥ℒ2.
Subsequently, an application of (A23), together with Minkowski’s inequality, enables us to continue the string of inequalities in (96) to conclude that
(97)[E∥∫0tS(t-s)ℱ(x)(s)ds∥U2]1/2≤T1/2M*[Mℱ∥x∥𝒞+∥F(0)∥ℒ2].
Taking the supremum over [0,T] in (97) then implies that ∫0tS(t-s)ℱ(x)(s)ds∈C([0,T];U), for any x∈C([0,T];U).
The other estimates can be established as before, and when used in conjunction with Lemma6, they can be used to verify that sup0≤t≤TE∥Γ(x)(t)∥U2<∞, for any x∈C([0,T];U). Thus, we conclude that Γ is well defined, and the proof of Lemma 12 is complete.Proof of Theorem13.
We know thatΓ is well defined and ℒ2-continuous from Lemma 12. We now prove that Γ has a unique fixed point in C([0,T];U). For any x,y∈C([0,T;U])(98)(Γx)(t)-(Γy)(t)=∫0tC(t-s)B(x(s)-y(s))ds+∫0tS(t-s)[ℱ(x)(s)-ℱ(y)(s)]ds.
Squaring both sides, taking the expectation in (98), and applying Young’s inequality yields
(99)E∥(Γx)(t)-(Γy)(t)∥U2≤2E∥∫0tS(t-s)[ℱ(x)(s)-ℱ(y)(s)]ds∥U2+2E∥∫0tC(t-s)B(x(s)-y(s))ds∥U2≤2(M*)2T[∥ℱ(x)-ℱ(y)∥ℒ22+MB2∥x-y∥C2]≤2(M*)2T(Mℱ2+MB2)∥x-y∥C2,
and so
(100)∥Γx-Γy∥C2≤2(M*)2T(Mℱ2+MB2)∥x-y∥C2.
Hence, Φ is a strict contraction, provided that M*2T(Mℱ2+MB2)<1 and so has a unique fixed point which coincides with a mild solution of (27). This completes the proof.Proof of Corollary14.
For (i), defineℱ:C([0,T];U)→ℒ2(0,T;ℒ2(Ω;U)) by
(101)ℱ(x)(t)=∫0tb(t,s)F0(s,x(s))ds+F1(t,x(t)),cccccccccccccccccccccccccccccc∀0≤t≤T.
The Uniform Boundedness Principle with (A6) guarantees the existence of a positive constant Mb such that ∥B(t,s)∥𝔅𝔏≤Mb, for all 0≤t≤s≤T. Standard computations involving properties of expectation and Hölder’s inequality imply that for all x,y∈C([0,T];U),
(102)∥ℱ(x)-ℱ(y)∥ℒ22≤2∫0T[TMb2∫0sE∥F0(τ,x(τ))-F0(τ,y(τ))∥U2dτcccccccc+E∥F1(s,x(s))-F1(s,y(s))∥U2∫0sE∥F0(τ,x(τ))-F0(τ,y(τ))∥U2]≤2T[TMbMF0+MF1]∥x-y∥C.
Thus, if we let Mℱ=2T[TMbMF0+MF1] in (A23) we can conclude from Theorem 13 that the IBVP has a unique mild solution on [0,T], provided that (49a) holds.
For (ii), using similar reasoning with(103)Mℱ=2MF2T|a|ℒ2((0,T)2)(1+MkT3)1/2
enables us to invoke Theorem 13 to conclude that the IBVP has a unique mild solution on [0,T], provided that (49b) holds.


## 6. Application of Theory to Nonlinear Beam Model

We now apply the results proved in Section5 to the original nonlinear beam model. Consider the IBVP (4)–(6) equipped with the nonlinear forcing term (9).Proposition 15.
Assume (A1)–(A5). Then this IBVP has a unique mild solutionW∈C([0,T];ℒ2(0,T)) with probability distribution μ. Moreover, for every p≥1, there exists a constant Cp for which
(104)sup0≤t≤TE∥W(·,t)∥ℒ2(0,L)2p≤Cp(1+E∥ξ0∥ℒ2(0,L)2p+E∥ξ1∥ℒ2(0,L)2p).Proof.
We have illustrated in Section3 that the IBVP (4)–(6) equipped with (9) can be reformulated as the abstract evolution equation (26). We verify that with those identifications, (A11)–(A16) are satisfied.
We have already addressed (A11) in Section3. To see that (A12) is satisfied, observe that (A1)(i) implies that
(105)∥F3(t,·,x(t,·))∥ℒ2(0,L)≤M-F3[∫𝒟[1+|x(t,z)|]2dz]1/2≤2M-F3[L+∥x(t,·)∥ℒ2(0,L)2]1/2≤2M-F3[L+∥x∥C([0,T];ℒ2(0,L))]≤MF3*[1+∥x∥C([0,T];ℒ2(0,L))],
for all 0≤t≤T and x∈C([0,T];ℒ2(0,L)), where
(106)MF3*={2M-F3L,ifL>1,2M-F3,ifL≤1.
Also, from (A1)(ii), we obtain
(107)∥F3(t,·,x(t,·))-F3(t,·,y(t,·))∥ℒ2(0,L)≤MF3[∫0L|x(t,z)-y(t,z)|2dz]1/2≤MF3∥x-y∥C([0,T];ℒ2(0,L)).
Next, using (A2)(i) together with the Hölder inequality, we observe that(108)∥∫ℒ2(0,L)F4(t,·,y)μ(t,·)(dy)∥ℒ2(0,L)=[∫0L[∫ℒ2(0,L)F4(t,z,y)μ(t,z)(dy)]2dz]1/2≤[∫0L∫ℒ2(0,L)∥F4(t,z,y)∥ℒ2(0,L)2μ(t,z)(dy)dz]1/2≤M-F4[∫0L(∫ℒ2(0,L)(1+∥y∥ℒ2(0,L))2μ(t,z)(dy))dz]1/2≤M-F4L∥μ(t)∥λ2≤M-F4L(1+∥μ(t)∥λ2),∀0≤t≤T,μ∈℘λ2(U).
Also, invoking (A2)(ii) enables us to see that for all μ,ν∈℘λ2(U),
(109)∥∫ℒ2(0,L)F4(t,·,y)μ(t,·)(dy)-∫ℒ2(0,L)F4(t,·,y)ν(t,·)(dy)∥ℒ2(0,L)=∥∫ℒ2(0,L)F4(t,·,y)(μ(t,·)-ν(t,·))(dy)∥ℒ2(0,L)≤∥ρ(μ(t),ν(t))∥ℒ2(0,L)≤Lρ(μ(t),ν(t)),∀0≤t≤T.
Combining (105) and (108), we see that f satisfies (A12)(i) with
(110)M-f1=2·max{M-F4L,MF3*},
and combining (107) and (109), we see thatf satisfies (A12)(ii) with
(111)Mf=max{MF3,L}.
This verifies (A12). Next, (A13), (A15), and (A16) hold by assumption, and (A14) follows because of the uniform boundedness of {∥A1/2x(t)∥ℒ2(0,L):0≤t≤T}. Thus, we can invoke Theorem 8 to conclude that this IBVP has a unique mild solution x∈C([0,T];ℒ2(0,L)) with probability law {μ(t,·):0≤t≤T}.Similar results can be established for the IBVP (4)–(6) equipped with the nonlinear forcing term described by (32) or (33) by using Corollary 14 and Theorem 13. We omit the details.Finally, we remark that the mild solutions for each of the IBVPs introduced in Section1 depend continuously on the initial data and that the approximation scheme in which the impact of the stochastic term can be made sufficiently negligible (cf. Theorem 10) can be applied to each of these IBVPs.



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