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

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.

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 𝜕²𝑊(𝑧,𝑡)𝜕𝑡²+𝐸𝐼𝜌𝜕⁴𝑊(𝑧,𝑡)𝜕𝑧⁴−𝜂𝜌+𝐸𝐴_𝑐2𝜌𝐿^𝐿₀𝜕²𝑊(𝜉,𝑡)𝜕𝜉²𝜕𝑑𝜉²𝑊(𝑧,𝑡)𝜕𝑧²=0,0≤𝑧≤𝐿,0<𝑡, (1) 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 𝐴_𝑐 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: 𝑊()()=𝜕0,𝑡=𝑊𝐿,𝑡²𝑊(0,𝑡)𝜕𝑧²=𝜕²𝑊(𝐿,𝑡)𝜕𝑧²=0. (2) The initial conditions given by 𝑊(𝑧,0)=𝜍₀(𝑧),𝜕𝑊(𝑧,0)𝜕𝑡=𝜍₁(𝑧),0≤𝑧≤𝐿 (3) 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 {𝛼^𝐻𝑥(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 𝐵^𝐻(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: 𝜕𝜕𝑊(𝑧,𝑡;𝜔)+𝜕𝑡𝐸𝐼𝜌𝜕⁴𝑊(𝑧,𝑡;𝜔)𝜕𝑧⁴𝜕+𝛼³𝑊(𝑧,𝑡;𝜔)𝜕𝑡𝜕𝑧²−𝐻𝜕𝑡𝜌+𝐸𝐴_𝑐2𝜌𝐿^𝐿₀𝜕𝑊(𝜉,𝑡;𝜔)𝜕𝜉²×𝜕𝑑𝜉₂𝑊(𝑧,𝑡;𝜔)𝜕𝑧²(𝑊())()𝜕𝑡=ℱ𝑧,𝑡;𝜔,𝑡;𝜔𝜕𝑡+𝑔𝑧,𝑡;𝜔𝑑𝛽^𝐻(),𝑊()()=𝜕𝑡;𝜔0,𝑡;𝜔=𝑊𝐿,𝑡;𝜔²𝑊(0,𝑡;𝜔)𝜕𝑧²=𝜕²𝑊(𝐿,𝑡;𝜔)𝜕𝑧²𝑊()=0,𝑧,0;𝜔=𝜍₀(),()𝑧;𝜔𝜕𝑊𝑧,0;𝜔𝜕𝑡=𝜍₁(𝑧;𝜔), (4)(5)(6) where 0≤𝑧≤𝐿, 0≤𝑡≤𝑇, 𝜔∈Ω (a complete probability space), {𝛽^𝐻(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 ^𝑡₀𝑏(𝑡,𝑠)𝐹₀(𝑠,𝑊(𝑧,𝑠;𝜔))𝑑𝑠+𝐹₁(𝑡,𝑊(𝑧,𝑡;𝜔))_𝑇0,𝑎(𝑡,𝑠)𝐹₂𝑧,𝑠,𝑊(𝑧,𝑠;𝜔),^𝑠₀𝑘((𝐹𝑠,𝜏,𝑊𝑧,𝜏;𝜔))𝑑𝜏𝑑𝑠,₃(𝑧,𝑡,𝑊(𝑧,𝑡;𝜔))+_ℒ2()0,𝐿𝐹₄𝑧,𝑦,𝑡;𝜔𝜇(𝑧,𝑡)𝑑𝑦, (7)(8)(9) where {𝜇(𝑧,𝑡)∶0≤𝑧≤𝐿,0≤𝑡≤𝑇} is a family of probability measures, and 𝑎, 𝑘, and 𝐹_𝑖(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 Section 2. 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 {𝑒_𝑗|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‖²_𝑈<∞. Also, 𝐶([0,𝑇];𝑈) stands for the space of ℒ²-continuous 𝑈-valued random variables 𝑦∶[0,𝑇]→𝑈 such that 𝑦²_{𝐶])([0,𝑇;𝑈}≡sup_{0≤𝑡≤𝑇}𝐸𝑦(𝑡)²_𝑈<∞. (10) 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 𝒞(𝑈)=𝜑∶𝑈⟶𝑈𝜑iscontinuousand𝜑_𝒞=sup_𝑥∈𝑈𝜑(𝑥)_𝑈𝜆²(𝑥)+sup_{𝑥≠𝑦in𝑈}𝜑(𝑥)−𝜑𝑦_𝑈𝑥−𝑦_𝑈<∞}. (11) For 𝑝≥1, we let ℘^𝑠_𝜆^𝑝(𝑈)=‖𝑚‖𝑚∶𝑈⟶ℝ∣𝑚isassignedmeasureon𝑈suchthat_𝜆𝑝=_𝑈𝜆^𝑝(𝑥)|𝑚|(𝑑𝑥)<∞, (12) where 𝑚=𝑚⁺−𝑚⁻ is the Jordan decomposition of 𝑚 and |𝑚|=𝑚⁺+𝑚⁻. Then define the space ℘_𝜆²(𝑈)=℘^𝑠_𝜆2(𝑈)∩℘(𝑈) equipped with the metric 𝜌 given by 𝜌𝜍₁,𝜍₂=sup_𝑈𝜑(𝑥)𝜍₁−𝜍₂(𝑑𝑥)∶𝜑_𝒞≤1. (13) 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 𝒞_𝜆²=𝒞_𝜆²([0,𝑇];(℘_𝜆²(𝑈),𝜌)) is complete when equipped with the metric 𝐷_𝑇𝜍₁,𝜍₂=sup_{[]𝑡∈0,𝑇}𝜌𝜍₁(𝑡),𝜍₂(𝑡),∀𝜍₁,𝜍₂∈𝒞_𝜆². (14) 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 corresponding strongly continuous sine family {𝑆(𝑡)∶𝑡∈ℝ}⊂𝔅𝔏(𝑈) is defined by 𝑆(𝑡)𝑧=∫^𝑡₀𝐶(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 generator 𝐴 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 exist 𝑀_𝐴≥1 and 𝜔≥0 such that ‖𝐶(𝑡)‖≤𝑀_𝐴𝑒^𝜔|𝑡| and hence, ‖𝑆(𝑡)‖≤𝑀_𝐴𝑒^𝜔|𝑡|,(ii)𝐴∫^𝑟_𝑠𝑆(𝑢)𝑧𝑑𝑢=[𝐶(𝑟)−𝐶(𝑠)]𝑧,  for all 0≤𝑠≤𝑟<∞,(iii)There exists 𝑁≥1 such that ‖𝑆(𝑠)−𝑆(𝑟)‖≤𝑁|∫^𝑟_𝑠𝑒^𝑤|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_∗=𝑀_𝐴𝑒^𝜔𝑇.

Next, we 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 [15, 17]. Let {𝛽^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,…(i)𝛽^𝐻_𝑗(0)=0,(ii)𝐸[𝛽^𝐻_𝑗(𝑡)−𝛽^𝐻_𝑗(𝑠)]²=|𝑡−𝑠|^2𝐻𝜈_𝑗,(iii)𝐸[𝛽^𝐻_𝑗(1)]²=𝜈_𝑗>0,(iv)∑^∞_𝑗=1𝜈_𝑗<∞.

In such case, ∑^∞_𝑗=1𝐸‖𝛽^𝐻_𝑗(𝑡)𝑒_𝑗‖²_𝑈=𝑡^2𝐻∑^∞_𝑗=1𝜈_𝑗<∞, so that the following definition is meaningful.

Definition 3. For every 𝑡≥0, 𝛽^𝐻(𝑡)=∑^∞_𝑗=1𝛽^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 [17] that the covariance operator of {𝛽^𝐻(𝑡)∶𝑡≥0} is a positive nuclear operator 𝑄 such that ()=1tr𝑄𝑡,𝑠2^∞_𝑗=1𝜈_𝑗𝑡^2𝐻+𝑠^2𝐻−|𝑡−𝑠|^2𝐻. (15) We now outline the discussion leading to the definition of the stochastic integral associated with {𝛽^𝐻(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 {𝑔_𝑖∶𝑖=1,…,𝑛}⊆ℝ such that 𝑔(𝑡)=𝑔_𝑖,∀𝑡_𝑖−1≤𝑡≤𝑡_𝑖, (16) where 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛−1<𝑡_𝑛=𝑇 and max_{1≤𝑖≤𝑛}‖𝑔_𝑖‖_{𝔏(𝑉,𝑈)}=𝐾.

Definition 4. The 𝑈-valued stochastic integral ∫^𝑇₀𝑔(𝑡)𝑑𝛽^𝐻(𝑡) is defined by ^𝑇₀𝑔(𝑡)𝑑𝛽^𝐻(𝑡)=^∞_𝑗=1^𝑇₀𝑔(𝑡)𝑑𝛽^𝐻_𝑗(𝑡)𝑒_𝑗=^∞_𝑗=1^𝑛_𝑖=1𝑔_𝑖𝛽^𝐻_𝑗𝑡_𝑖−𝛽^𝐻_𝑗𝑡_𝑖−1𝑒_𝑗. (17) As argued in Lemma 2.2 of [17], this integral is well-defined since 𝐸^𝑇₀𝑔(𝑡)𝑑𝛽^𝐻(𝑡)²_𝑈≤𝐾²𝑇^∞2𝐻_𝑗=1𝜈_𝑗<∞. (18) 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 (∑^𝑛_𝑖=1𝑎_𝑖)^𝑚≤𝑛^𝑚−1∑^𝑛_𝑖=1𝑎^𝑚_𝑖, where 𝑎_𝑖 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 Section 1 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 exists 𝑀_𝐹3>0 for which |𝐹₃(𝑦,𝑡,𝑤)|≤𝑀_𝐹3(1+|𝑤|), for all 𝑦∈(0,𝐿), 𝑡∈[0,𝑇], and 𝑤∈ℝ;(ii)there exists 𝑀_𝐹3>0 for which |𝐹₃(𝑦,𝑡,𝑤₁)−𝐹₃(𝑦,𝑡,𝑤₂)|≤𝑀_𝐹3|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 exists 𝑀_𝐹4>0 for which ‖𝐹₄(𝑦,𝑡,𝑤)‖_ℒ²_(0,𝐿)≤𝑀_𝐹4(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 𝒞_𝜆², 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.

Let 𝑈=𝑉=ℒ²(0,𝐿) (note they are real separable Hilbert spaces) and identify the solution process 𝑥∶[0,𝑇]→𝑈 by 𝑥(𝑡)(𝑧)=𝑊(𝑧,𝑡),∀𝑧∈(0,𝐿),𝑡∈[0,𝑇]. (19) Define the operator 𝐴∶𝐷(𝐴)⊂𝑈→𝑈 by ()=𝜕𝐴𝑥𝑡,⋅⁴𝑥(𝑡,⋅)𝜕𝑧⁴,𝐷(𝐴)=𝑥∈𝐻⁴(0,𝐿)∶𝑥(0)=𝑥(𝐿)=𝑥^′′(0)=𝑥^′′(𝐿)=0. (20) 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 {𝜆_𝑛=(𝑛𝜋)⁴∶𝑛=1,2,3,…} with corresponding eigenvalues 𝑧_𝑛(𝑠)=√2𝐿sin(𝑛𝜋𝑠)∶𝑛=1,2,3,…,𝑠∈[0,𝐿]. (21) Then 𝐴 has spectral representation 𝐴𝑦=^∞_𝑛=1𝜆_𝑛𝑦,𝑧_𝑛𝑧_𝑛. (22) The following operators are well defined because the fractional powers of 𝐴 are positive and self-adjoint: 𝐴^1/2𝑦=^∞_𝑛=1𝜆_𝑛^1/2𝑦,𝑧_𝑛𝑧_𝑛𝜕=−²𝑦𝜕𝑧²,𝐴^1/4𝑦=^∞_𝑛=1𝜆_𝑛^1/4𝑦,𝑧_𝑛𝑧_𝑛=𝜕𝑦.𝜕𝑧 (23) Also, observe that 𝐴^1/4𝑦²_𝑈=𝐴^1/4𝑦,𝐴^1/4𝑦=^∞_𝑛=1𝜆_𝑛^1/2𝑦,𝑧_𝑛²=^𝐿₀𝜕𝑦(𝜕𝑧𝑤,𝑡)²_𝑈𝑑𝑤. (24) It can be shown using standard computations involving the inner product and properties of fractional powers of 𝐴 that ‖𝐴^1/4𝑥(𝑡)‖²_𝑈  and ‖𝐴^1/2𝑥(𝑡)‖²_𝑈 are uniformly bounded on [0,𝑇].

Define the mappings 𝑓∶[0,𝑇]×𝑈×℘_𝜆²(𝑈)→𝑈, 𝑔∶[0,𝑇]→𝔏(𝑉,𝑈), 𝑥₀(⋅), and 𝑥₁(⋅), and the operator 𝐵∶𝑈→𝑈 as follows: 𝑓𝑡,𝑥(𝑡),𝜇(𝑡)(𝑧)=𝐹₃((+𝑧,𝑡,𝑊𝑧,𝑡))_ℒ2()0,𝐿𝐹₄𝜇()−𝐻𝑧,𝑦,𝑡𝑧,𝑡𝑑𝑦𝜌+𝐸𝐴_𝑐2𝜌𝐿^𝐿₀𝜕𝑊(𝜉,𝑡)𝜕𝜉²𝜕𝑑𝜉²𝑊(𝑧,𝑡)𝜕𝑧²=𝐹₃(𝑧,𝑡,𝑊(𝑧,𝑡))+_ℒ2()0,𝐿𝐹₄𝜇()−𝐻𝑧,𝑦,𝑡𝑧,𝑡𝑑𝑦𝜌+𝐸𝐴_𝑐𝐴2𝜌𝐿^1/4𝑥(𝑡)²_𝑈𝐴^1/2𝑥(𝑡),𝑔(𝑡𝑧)(),𝐵𝑥)(=𝑔𝑧,𝑡_′(𝑡)(𝑧)𝜕=𝛼³𝑊(𝑧,𝑡)𝜕𝑡𝜕𝑧²=𝛼𝐴^1/2𝜕𝑊(𝑧,𝑡)𝜕𝑡²,𝑥₀(0)(𝑧)=𝜍₀(𝑧),𝑥₁(0)(𝑧)=𝜍₁(𝑧), (25) 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: 𝑑𝑥^′(𝑡)=𝐵𝑥^′(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)+𝐴𝑥+𝑓𝑡,𝑥,𝜇𝑑𝑡+𝑔𝑑𝛽^𝐻(𝑡)𝑥(0),0≤𝑡≤𝑇,=𝑥₀,𝑥^′(0)=𝑥₁,𝜇(𝑡)=probabilitydistributionof𝑥(𝑡) (26) 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; {𝛽^𝐻(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: 𝑑𝑥^′(𝑡)=𝐵𝑥^′(𝑡)(𝑡)(𝑥𝑡)(𝑡)+𝐴𝑥+ℱ)(𝑑𝑡+𝑔𝑑𝛽^𝐻(𝑡)𝑥(0),0≤𝑡≤𝑇,=𝑥₀,𝑥^′(0)=𝑥₁, (27) 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)𝐹_𝑖∶(0,𝐿)×[0,𝑇]×ℝ→ℝ(𝑖=0,1) are mappings that satisfy the Caratheodory conditions and are such that there exists 𝑀_𝐹𝑖>0 such that 𝐹_𝑖𝑦,𝑡,𝑤₁−𝐹_𝑖𝑦,𝑡,𝑤₂≤𝑀_𝐹𝑖𝑤₁−𝑤₂, (28) for all 𝑦∈(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 𝑀_𝐹2>0 such that 𝐹₂𝑦,𝑡,𝑤₁,𝑧₁−𝐹₂𝑦,𝑡,𝑤₂,𝑧₂≤𝑀_𝐹2𝑤₁−𝑤₂+𝑧₁−𝑧₂, (29) for all 𝑦∈(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 𝑘𝑡,𝑠,𝑥₁−𝑘𝑡,𝑠,𝑥₂≤𝑀_𝑘𝑥₁−𝑥₂, (30) for all 𝑥₁,𝑥₂∈ℝ  and (𝑠,𝑡)∈Δ. Identify the solution process 𝑥∶[0,𝑇]→𝑈 by 𝑥(𝑡)(𝑧)=𝑊(𝑧,𝑡),∀𝑧∈(0,𝐿),𝑡∈[0,𝑇]. (31)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)): ℱ(𝑥)(𝑡)=^𝑡₀𝑎(𝑡,𝑠)×𝐹₂⋅,𝑠,𝑊(⋅,𝑠;𝜔),^𝑠₀𝑘((ℱ(𝑥𝑡)=𝑠,𝜏,𝑊⋅,𝜏;𝜔))𝑑𝜏𝑑𝑠,)(_𝑡0𝑏(𝑡,𝑠)𝐹₀(⋅,𝑠,𝑊(⋅,𝑠;𝜔))𝑑𝑠+𝐹₁(⋅,𝑡,𝑊(⋅,𝑡;𝜔)). (32)(33) 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 process 𝑥∈𝐶([0,𝑇];𝑈) is a mild solution of (26) if(i)𝑥(𝑡)=𝑆(𝑡)𝑥₁+(𝐶(𝑡)−𝑆(𝑡)𝐵)𝑥₀+^𝑡₀𝐶()(𝑠)+𝑡−𝑠𝐵𝑥𝑑𝑠_𝑡0𝑆()𝑓(𝑠)(𝑠)+𝑡−𝑠𝑠,𝑥,𝜇𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑔(𝑠)𝑑𝛽^𝐻(𝑠),0≤𝑡≤𝑇, (34)(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 constant 𝑀_𝑓 such that 𝐸𝑓𝑡,𝑧₁,𝜍₁−𝑓𝑡,𝑧₂,𝜍₂²_𝑈≤𝑀_𝑓𝐸𝑧₁−𝑧₂²_𝑈+𝜌²𝜍₁,𝜍₂, (35) globally on [0,𝑇]×𝑈×℘_𝜆²(𝑈),(ii)there exists a positive constant 𝑀_𝑓 such that 𝐸𝑓(𝑡,𝑧,𝜍)²_𝑈≤𝑀_𝑓1+𝐸‖𝑧‖²_𝑈+‖𝜍‖²_𝜆2; (36) 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){𝛽^𝐻(𝑡)∶0≤𝑡≤𝑇} is a 𝑈-valued fBm;(A16)𝑥₀,𝑥₁∈ℒ²(Ω;𝑈) are (ℱ₀,ℬ(𝑈))-measurable, where {ℱ_𝑡∶0≤𝑡≤𝑇} is the family of 𝜎-algebras ℱ_𝑡 generated by {𝛽^𝐻(𝑠)∶0≤𝑠≤𝑡}.
Henceforth, we write 𝑀^∗=max‖𝑆(𝑡)‖_{𝔅𝔏(𝑈)}∪𝐶(𝑡∶0≤𝑡≤𝑇‖)‖_{𝔅𝔏(𝑈)}∶0≤𝑡≤𝑇, (37) 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 that 𝑔∶[0,𝑇]→𝔏(𝑉,𝑈) satisfies (A13). Then, for all0≤𝑡≤𝑇, 𝐸^𝑡₀𝑆(𝑡−𝑠)𝑔(𝑠)𝑑𝐵^𝐻(𝑠)²_𝑈≤𝐶_𝑡^∞_𝑗=1𝜈_𝑗, (38) 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 4.

Let 𝜇∈𝒞_𝜆² be fixed and define the solution map Φ∶𝐶([0,𝑇];𝑈)→𝐶([0,𝑇];𝑈) by Φ(𝑥)(𝑡)=𝑆(𝑡)𝑥₁+(𝐶(𝑡)−𝑆(𝑡)𝐵)𝑥₀+^𝑡₀𝐶()(𝑠)+𝑡−𝑠𝐵𝑥𝑑𝑠_𝑡0𝑆()𝑓(𝑠)(𝑠)+𝑡−𝑠𝑠,𝑥,𝜇𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑔(𝑠)𝑑𝛽^𝐻(𝑠)=,0≤𝑡≤𝑇,_5𝑖=1𝐼^𝑥_𝑖(𝑡),0≤𝑡≤𝑇. (39) 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, then Φ 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∈𝒞_𝜆². Further, for every 𝑝≥1, there exists a positive constant 𝐶_𝑝 (depending on 𝑝, 𝑇, and 𝜈_𝑗) such that sup_{0≤𝑡≤𝑇}𝐸‖𝑥(𝑡)‖_𝑈^2𝑝≤𝐶_𝑝1+𝐸𝑥_0𝑈^2𝑝+𝐸𝑥_1𝑈^2𝑝. (40)

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 let 𝑥 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 𝜇_𝑥 and 𝜇_𝑦. Then there exists a positive constant 𝑀^′ such that 𝐸𝑥(𝑡)−𝑦(𝑡)²_𝑈≤𝑀^′𝑥₀−𝑦₀²_{ℒ2()Ω;𝑈}+𝑥₁−𝑦₁²_{ℒ2()Ω;𝑈}+𝐷²_𝑇𝜇_𝑥,𝜇_𝑦. (41)

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 𝑧^′′(𝑡)=𝐴𝑧(𝑡)+𝐵𝑧^′(𝑡)((𝑡𝑧(0)+𝐹𝑡,𝑧)),0≤𝑡≤𝑇,=𝑧₀,𝑧^′(0)=𝑧₁, (42) 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 constant 𝑀_𝐹 such that ‖𝐹(𝑡,𝑧₁)−𝐹(𝑡,𝑧₂)‖_𝑈≤𝑀_𝐹‖𝑧₁−𝑧₂‖_𝑈 globally on [0,𝑇]×𝑈;(ii)there exists a positive constant 𝑀_𝐹 such that ‖𝐹(𝑡,𝑧)‖_𝑈≤𝑀_𝐹‖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 𝑧(𝑡)=𝑆(𝑡)𝑥₁+(𝐶(𝑡)−𝑆(𝑡)𝐵)𝑥₀+^𝑡₀𝐶()(𝑠)+𝑡−𝑠𝐵𝑥𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝐹(𝑠,𝑥(𝑠))𝑑𝑠,0≤𝑡≤𝑇. (43) For each 𝜀>0, consider the stochastic initial-value problem 𝑑𝑥_𝜀(𝑡)=𝐴_𝜀𝑥_𝜀(𝑡)+𝐵𝑥^′_𝜀(𝑡)+_𝑈𝐹_1𝜀(𝑡,𝑧)𝜇_𝜀(𝑡)(𝑑𝑧)+𝐹_2𝜀𝑡,𝑥_𝜀(𝑡)𝑑𝑡+𝑔_𝜀(𝑡)𝑑𝛽^𝐻(𝑡)𝑥,0≤𝑡≤𝑇,_𝜀(0)=𝑧₀,𝑥^′_𝜀(0)=𝑧₁,𝜇_𝜀(𝑡)=probabilitydistributionof𝑥_𝜀(𝑡), (44) in 𝑈. Here, we assume that 𝑧₀∈𝐷(𝐴_𝜀)=𝐷(𝐴) and 𝑧₁∈𝐸_𝜀=𝐸, for all 𝜀>0 and that 𝐹_𝑖𝜀∶[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)𝐴_𝜀∶𝐷(𝐴_𝜀)=𝐷(𝐴)→𝐷(𝐴_𝜀) generates a strongly continuous cosine family {𝐶_𝜀(𝑡)∶𝑡≥0} satisfying 𝐶_𝜀(𝑡)→𝐶(𝑡) strongly as 𝜀→0⁺, uniformly in 𝑡∈[0,𝑇]. Also, the corresponding sine family {𝑆_𝜀(𝑡)∶𝑡≥0} satisfying 𝑆_𝜀(𝑡)→𝑆(𝑡) strongly as 𝜀→0⁺, uniformly in 𝑡∈[0,𝑇]. Moreover, max𝐶_𝜀(𝑡)_{𝔅𝔏(𝑈)}+𝑆_𝜀(𝑡)_{𝔅𝔏(𝑈)}∶0≤𝑡≤𝑇≤𝑀^∗, (45) for all 𝜀>0 and 𝑀^∗ is the same bound used for the cosine and sine families generated by 𝐴.(A19)𝐵_𝜀∶𝑈→𝑈 is a bounded linear operator such that 𝐵_𝜀→𝐵 in 𝔅𝔏(𝑈) and ‖𝐵_𝜀‖_{𝔅𝔏(𝑈)}≤‖𝐵‖_{𝔅𝔏(𝑈)}, for all 𝜀>0.(A20)𝐹_2𝜀∶[0,𝑇]×𝑈→𝑈 is Lipschitz in the second variable (with the same Lipschitz constant 𝑀_𝐹 used for 𝐹 in (A17)), and 𝐹_2𝜀(𝑡,𝑢)→𝐹(𝑡,𝑢) as 𝜀→0⁺, for all 𝑢∈𝑈, uniformly in 𝑡∈[0,𝑇].(A21)𝐹_1𝜀∶[0,𝑇]×𝑈→𝑈 is a continuous mapping such that∫_𝑈𝐹_1𝜀(𝑡,𝑧)𝜇_𝜀(𝑡)(𝑑𝑧)→0 uniformly in 𝑡 as 𝜀→0⁺.(A22)𝑔_𝜀∶[0,𝑇]→𝔏(𝑉,𝑈) is a bounded, measurable function such that 𝑔_𝜀(𝑡)→0 as 𝜀→0⁺, uniformly in 𝑡∈[0,𝑇].

Under these assumptions, the following result holds.

Theorem 10. Let 𝑧 and 𝑥_𝜀 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‖𝑥_𝜀(𝑡)−𝑧(𝑡)‖²_𝑈≤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 process 𝑥∈𝐶([0,𝑇];𝑈) is a mild solution of (27) if 𝑥(𝑡)=𝑆(𝑡)𝑥₁+(𝐶(𝑡)−𝑆(𝑡)𝐵)𝑥₀+^𝑡₀𝐶(𝑡−𝑠)𝐵𝑥(𝑠)𝑑𝑠+^𝑡₀𝑆()ℱ(𝑥𝑠)+𝑡−𝑠)(𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑔(𝑠)𝑑𝛽^𝐻(𝑠), (46) 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 𝑀_ℱ>0 for which ℱ(𝑥)−ℱ𝑦_ℒ2≤𝑀_ℱ𝑥−𝑦_𝒞,∀𝑥,𝑦∈𝐶([0,𝑇];𝑈). (47)Define the solution map Γ∶𝐶([0,𝑇];𝑈)→𝐶([0,𝑇];𝑈) by Γ(𝑥)(𝑡)=𝑆(𝑡)𝑥₁+(𝐶(𝑡)−𝑆(𝑡)𝐵)𝑥₀+^𝑡₀𝐶()(𝑠)+𝑡−𝑠𝐵𝑥𝑑𝑠_𝑡0𝑆()ℱ(𝑥𝑠)+𝑡−𝑠)(𝑑𝑠_𝑡0𝑔(𝑠)𝑑𝛽^𝐻(𝑠),0≤𝑡≤𝑇,=𝐼₁(𝑡)+𝐼₂(𝑡)+𝐼₃(𝑡)+𝐼₅(𝑡)+^𝑡₀𝑆(𝑡−𝑠)ℱ(𝑥)(𝑠)𝑑𝑠,0≤𝑡≤𝑇. (48) (cf. (39)). The operator Γ satisfies the following properties.

Lemma 12. If (A11), (A13)–(A16), and (A23) hold, then Γ 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(𝑀²_ℱ+𝑀²_𝐵)<1. Further, for every 𝑝≥1, there exists a positive constant 𝐶_𝑝 (depending on 𝑝, 𝑇, and 𝜈_𝑗) such that

sup_{0≤𝑡≤𝑇}𝐸‖𝑥(𝑡)‖_𝑈^2𝑝≤𝐶_𝑝1+𝐸𝑥_0𝑈^2𝑝+𝐸𝑥_1𝑈^2𝑝. (49a)

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 √2𝑀_∗2𝑇𝑇𝑀_𝐹0𝑀_𝐵+𝑀_𝐹1²+𝑀²_𝐵^1/2𝑇^1/2<1. (49b) 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 √2𝑀_∗2𝑇𝑀_𝐹2|𝑎|_ℒ²1+𝑇³𝑀_𝑘^1/22+𝑀²_𝐵^1/2𝑇^1/2<1. (49c) Moreover, for every 𝑝≥1, estimate (49a) holds.

Finally, we remark that an approximation scheme in the spirit of Theorem 10 can be established for (27) by making the natural modifications to (42) and hypotheses (A20) and (A21).

5. Proofs of Main Results

Proof of Lemma 7. Let 𝜇∈𝒞_𝜆² be fixed and consider the solution map Φ defined in (39).
One can see from the discussion in Section 2 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 𝐸Φ(𝑧)𝑡^∗+ℎ−Φ(𝑧)𝑡^∗2_𝑈≤5⁵_𝑖=1𝐼_𝑖𝑡^∗+ℎ−𝐼_𝑖𝑡^∗2_𝑈. (50) The strong continuity of 𝐶(𝑡) and 𝑆(𝑡) implies that ²_𝑖=1𝐸𝐼_𝑖𝑡^∗+ℎ−𝐼_𝑖𝑡^∗2_𝑈⟶0as|ℎ|⟶0,foreach0≤𝑡≤𝑇. (51) Using (A11) and (A14), together with Hölder’s inequality and properties of cosine operators, yields 𝐸𝐼₃𝑡^∗+ℎ−𝐼₃𝑡^∗2_𝑈=𝐸^𝑡∗₀𝐶𝑡^∗+ℎ−𝑠−𝐶𝑡^∗−𝑠𝐵𝑥(𝑠)𝑑𝑠+^𝑡∗_𝑡^+ℎ_∗𝐶𝑡^∗+ℎ−𝑠𝐵𝑥(𝑠)𝑑𝑠²_𝑈𝑑𝑠≤2𝑡^∗𝑡∗₀𝐸𝐶𝑡^∗+ℎ−𝑠−𝐶𝑡^∗−𝑠𝐵𝑥(𝑠)²_𝑈𝑑𝑠+2^𝑡∗_𝑡^+ℎ_∗𝐸𝐶𝑡^∗+ℎ−𝑠𝐵𝑥(𝑠)²_𝑈𝑑𝑠≤2𝑡^∗𝑡∗₀𝐸𝐶𝑡^∗+ℎ−𝑠−𝐶𝑡^∗−𝑠𝐵𝑥(𝑠)²_𝑈𝑀𝑑𝑠+2ℎ^∗2𝑀²_𝐵‖𝑥‖²_𝐶. (52) 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 𝐸𝐼₄𝑡^∗+ℎ−𝐼₄𝑡^∗2_𝑈≤2𝑇^𝑡∗₀𝐸𝑆𝑡^∗+ℎ−𝑠−𝑆𝑡^∗(𝑠)(𝑠)−𝑠×𝑓𝑠,𝑥,𝜇²_𝑈𝑑𝑠+2ℎ²𝑀^∗2𝐸𝑓𝑠,𝑥(𝑠),𝜇(𝑠)²_𝑈. (53) Since 𝐸𝑓𝑠,𝑥(𝑠),𝜇(𝑠)²_𝑈≤𝑀²_𝑓1+‖𝑥‖²_𝐶+sup_{0≤𝑠≤𝑇}𝜇(𝑠)²_𝜆² (54) 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 that 𝐼₃(𝑡^∗+ℎ)−𝐼₃(𝑡^∗)→0 as |ℎ|→0. Observe that 𝐼₅𝑡^∗+ℎ−𝐼₅𝑡^∗=𝐸^𝑡∗₀^+ℎ𝑆𝑡^∗+ℎ−𝑠𝑔(𝑠)𝑑𝛽^𝐻(𝑠)−^𝑡∗₀𝑆𝑡^∗−𝑠𝑔(𝑠)𝑑𝛽^𝐻(𝑠)²_𝑈=𝐸^∞_𝑗=1^𝑡∗₀^+ℎ𝑆𝑡^∗+ℎ−𝑠𝑔(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)−^∞_𝑗=1^𝑡∗₀𝑆𝑡^∗−𝑠𝑔(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈=𝐸^∞_𝑗=1^𝑡∗_𝑡^+ℎ_∗𝑆𝑡^∗+ℎ−𝑠𝑔(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)+^∞_𝑗=1^𝑡∗₀𝑆𝑡^∗+ℎ−𝑠−𝑆𝑡^∗(𝑠)𝑒−𝑠×𝑔_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈≤2𝐸^∞_𝑗=1^𝑡∗_𝑡^+ℎ_∗𝑆𝑡^∗+ℎ−𝑠𝑔(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈+^∞_𝑗=1^𝑡∗₀𝑆𝑡^∗+ℎ−𝑠−𝑆𝑡^∗(𝑠)𝑒−𝑠×𝑔_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈. (55) 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 𝐸^𝑚_𝑗=1^𝑡∗₀𝑆𝑡^∗+ℎ−𝑠−𝑆𝑡^∗−𝑠𝑔(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈=𝐸^𝑚_𝑗=1^𝑛−1_𝑘=0𝑆_ℎ𝑔_𝑘𝑒_𝑗𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘²_𝑈=𝐸^𝑚_𝑗=1^𝑛−1_𝑘=0𝑆_ℎ𝑔_𝑘𝑒_𝑗𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘,_{𝑚𝑗=1𝑛−1𝑘=0}𝑆_ℎ𝑔_𝑘𝑒_𝑗𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘𝑈≤^𝑚_𝑗=1^𝑛−1_𝑘=0𝑆_ℎ²_{(𝑈)𝔅𝔏}𝐾²𝐸×𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘,𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘ℝ≤sup_{0≤𝑠≤𝑡∗}𝑆_ℎ²_{(𝑈)𝔅𝔏}𝐾²𝐸𝛽^𝐻_𝑗²=sup_{0≤𝑠≤𝑡∗}𝑆_ℎ²_{(𝑈)𝔅𝔏}𝐾²𝑡^∗𝑚2𝐻_𝑗=1𝜈_𝑗, (56) where 𝑆_ℎ=𝑆(𝑡^∗−𝑡_𝑘+ℎ)−𝑆(𝑡^∗−𝑡_𝑘). Hence, 𝐸^∞_𝑗=1^𝑡∗₀𝑆𝑡^∗+ℎ−𝑠−𝑆𝑡^∗−𝑠𝑔(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈≤lim_𝑚→∞𝐸^𝑚_𝑗=1^𝑡∗₀𝑆𝑡^∗+ℎ−𝑠−𝑆𝑡^∗(𝑠)𝑒−𝑠×𝑔_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈≤lim_𝑚→∞sup_{0≤𝑠≤𝑡∗}𝑆_ℎ²_{(𝑈)𝔅𝔏}𝐾²𝑡^∗𝑚2𝐻_𝑗=1𝜈_𝑗=sup_{0≤𝑠≤𝑡∗}𝑆_ℎ²_{(𝑈)𝔅𝔏}𝐾²𝑡^∗∞2𝐻_𝑗=1𝜈_𝑗, (57) and the right side of (57) goes to 0 as |ℎ|→0. Next, observe that 𝐸^∞_𝑗=1^𝑡∗_𝑡^+ℎ_∗𝑆𝑡^∗+ℎ−𝑠𝑔(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈=𝐸^∞_𝑗=1^ℎ₀𝑆(𝑢)𝑔𝑡^∗+ℎ−𝑢𝑒_𝑗𝑑𝛽^𝐻_𝑗𝑡^∗+ℎ−𝑢²_𝑈. (58) Using the property 𝐸(𝛽^𝐻_𝑗(𝑠)−𝛽^𝐻_𝑗(𝑡))²=|𝑡−𝑠|^2𝐻𝜈_𝑗 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 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 6, 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 thatΦis well defined, and the proof of Lemma 7 is complete.

Proof of Theorem 8. Let 𝜇∈𝒞_𝜆² 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 𝐸(Φ𝑥)(𝑡)−Φ𝑦(𝑡)²_𝑈≤2𝑇𝑀^∗2𝑀²_𝐵+𝑀²_𝑓^𝑡₀𝐸𝑥(𝑠)−𝑦(𝑠)²_𝑈𝑑𝑠=𝛿^𝑡₀𝐸𝑥(𝑠)−𝑦(𝑠)²_𝑈𝑑𝑠, (59) where 𝛿=2𝑇(𝑀^∗)²(𝑀²_𝐵+𝑀²_𝑓). For any natural number 𝑛, it follows from successive iteration of (59) that Φ^𝑛𝑥−Φ^𝑛𝑦²_𝐶≤𝑇𝛿^𝑛𝑛!𝑥−𝑦²_𝐶. (60) Since (𝑇𝛿^𝑛/𝑛!)<1 for large enough values of 𝑛, we can conclude from (60) that Φ^𝑛 is a strict contraction on 𝐶 and so for a given 𝑇>0, Φ has a unique fixed point 𝑥_𝜇∈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 that 𝜇 is, in fact, the probability law of 𝑥_𝜇. To this end, let ℒ(𝑥_𝜇)={ℒ(𝑥_𝜇(𝑡))∶𝑡∈[0,𝑇]} represent the probability law of 𝑥_𝜇 and define the map Ψ∶𝒞_𝜆²→𝒞_𝜆² by Ψ(𝜇)=ℒ(𝑥_𝜇). It is not difficult to see that ℒ(𝑥_𝜇(𝑡))∈℘_𝜆²(𝑈), for all 𝑡∈[0,𝑇] since 𝑥_𝜇∈𝐶([0,𝑇];𝑈). In order to verify the continuity of the map 𝑡↦ℒ(𝑥_𝜇(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|>0lim_ℎ→0𝐸𝑥_𝜇(𝑡+ℎ)−𝑥_𝜇(𝑡)²_𝑈=0,∀0≤𝑡≤𝑇. (61) Consequently, since for all 𝑡∈[0,𝑇] and 𝜑∈𝒞_𝜆², it is the case that _𝑈𝜑(𝑥)ℒ𝑥_𝜇(𝑡+ℎ)−ℒ𝑥_𝜇(𝑡)(=𝐸𝜑𝑥𝑑𝑥)_𝜇(𝑡+ℎ)−𝜑𝑥_𝜇(𝑡)≤𝜑_𝜆²𝐸𝑥_𝜇(𝑡+ℎ)−𝑥_𝜇(𝑡)_𝑈, (62) and hence 𝜌ℒ𝑥_𝜇(𝑡+ℎ),ℒ𝑥_𝜇(𝑡)=sup_{𝜑𝜆2≤1𝑈}𝜑(𝑥)ℒ𝑥_𝜇(𝑡+ℎ)−ℒ𝑥_𝜇(𝑡)()|ℎ|𝑑𝑥⟶0as⟶0, (63) for any 0≤𝑡≤𝑇. Thus, 𝑡↦ℒ(𝑥_𝜇(𝑡)) is a continuous map, so that ℒ(𝑥_𝜇)∈𝒞_𝜆², thereby showing that Ψ is well defined. In order to show that Ψ has a unique fixed point in 𝒞_𝜆², let 𝜇,𝜈∈𝒞_𝜆² and let 𝑥_𝜇,𝑥_𝜈 be the corresponding mild solutions of (26). A standard computation yields 𝐸𝑥_𝜇(𝑡)−𝑥_𝜈(𝑡)²_𝑈≤2𝑇𝑀^∗2𝑀²_𝐵+𝑀²_𝑓^𝑡₀𝐸𝑥_𝜇(𝑠)−𝑥_𝜈(𝑠)²_𝑈𝑀𝑑𝑠+2𝑇^∗2𝑀²_𝑓^𝑡₀𝜌²𝜇(𝑠),𝜈(𝑠)𝑑𝑠,0≤𝑡≤𝑇. (64) Observe that 𝜌²(𝜇(𝑠),𝜈(𝑠))≤𝐷²_𝑇(𝜇,𝜈). Thus, continuing the inequality in (64) yields 𝐸𝑥_𝜇(𝑡)−𝑥_𝜈(𝑡)²_𝑈≤2𝑇𝑀^∗2𝑀²_𝐵+𝑀²_𝑓^𝑡₀𝐸𝑥_𝜇(𝑠)−𝑥_𝜈(𝑠)²_𝑈𝑑𝑠+5𝑇²𝑀^∗2𝑀²_𝑓𝐷²_𝑇𝜇,𝜈,0≤𝑡≤𝑇. (65) Applying Gronwall’s lemma now yields 𝐸𝑥_𝜇(𝑡)−𝑥_𝜈(𝑡)²_𝑈≤5𝑇²𝑀^∗2𝑀²_𝑓exp2𝑇𝑀^∗2𝑀²_𝐵+𝑀²_𝑓𝐷²_𝑇(𝑇)𝐷𝜇,𝜈=𝜍²_𝑇𝜇,𝜈,0≤𝑡≤𝑇. (66) We can choose 0≤𝑇≤𝑇 to ensure 𝜍(𝑇)<1, so that taking the supremum above yields 𝑥_𝜇−𝑥_𝜈²_{𝐶])([0,𝑇;𝑈}≤𝜍𝑇𝐷²_𝑇𝜇,𝜈. (67) Since 𝜌ℒ𝑥_𝜇(𝑡),ℒ𝑥_𝜈(𝑡)≤𝐸𝑥_𝜇(𝑡)−𝑥_𝜈(𝑡)_𝑈,∀0≤𝑡≤𝑇, (68) (which follows directly from (62) and (63)), we have Ψ𝜇−Ψ(𝜈)²_𝜆²=𝐷²_𝑇Ψ𝜇,Ψ(𝜈)≤sup_{𝑡∈0,𝑇}𝐸𝑥_𝜇(𝑡)−𝑥_𝜈(𝑡)²_𝑈=𝑥_𝜇−𝑥_𝜈²_{𝐶0,𝑇;𝑈}<𝜍𝑇𝐷²_𝑇𝜇,𝜈, (69) so that Ψ is a strict contraction on 𝒞_𝜆²([0,𝑇];(℘_𝜆²(𝑈),𝜌)). Thus, (26) has a unique mild solution on [0,𝑇] with probability distribution 𝜇∈𝒞_𝜆²([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 of 𝑝th moments of the mild solutions of (26). Let 𝑝≥1 and observe that the standard computations yield, for all 0≤𝑡≤𝑇, 𝐸𝑥_𝜇(𝑡)_𝑈^2𝑝≤5^𝑝𝐸𝑆(𝑡)𝑥_1𝑈^2𝑝+𝐸(𝐶(𝑡)−𝑆(𝑡)𝐵)𝑥_0𝑈^2𝑝+𝐸^𝑡₀𝐶(𝑡−𝑠)𝐵𝑥(𝑠)𝑑𝑠_𝑈^2𝑝+𝐸^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,𝑥(𝑠),𝜇(𝑠)𝑑𝑠_𝑈^2𝑝+𝐸^𝑡₀𝑆(𝑡−𝑠)𝑔(𝑠)𝑑𝛽^𝐻(𝑠)_𝑈^2𝑝≤10^𝑝𝑀^∗2𝑝𝐸𝑥_1𝑈^2𝑝+𝐸𝑥_0𝑈^2𝑝1+𝑀_𝐵^2𝑝+𝑇𝑀_𝐵^𝑡₀𝐸𝑥_𝜇(𝑠)²_𝑈^𝑝+𝑇^𝑡₀𝐸𝑥_𝜇(𝑠)²_𝑈+𝜇(𝑠)²_𝜆²𝑑𝑠^𝑝+𝐶_𝑡^∞_𝑗=1𝜈_𝑗^𝑝. (70)
Applying the Jensen and Hölder inequalities enables us to continue the string of inequalities above as follows: ≤10^𝑝𝑀^∗2𝑝𝐸𝑥_1𝑈^2𝑝+𝐸𝑥_0𝑈^2𝑝1+𝑀_𝐵^2𝑝+𝑇^𝑝𝑀^𝑝_𝐵^𝑡₀𝐸𝑥_𝜇(𝑠)_𝑈^2𝑝𝑑𝑠+𝐶^∗_𝑇2^𝑝𝑡₀𝐸𝑥_𝜇(𝑠)_𝑈^2𝑝+𝜇(𝑠)_𝜆^2𝑝₂𝑑𝑠+𝐶^𝑝_𝑇^∞_𝑗=1𝜈_𝑗^𝑝≤10^𝑝𝑀^∗2𝑝𝐸𝑥_1𝑈^2𝑝+𝐸𝑥_0𝑈^2𝑝1+𝑀_𝐵^2𝑝+𝐶^∗_𝑇2^𝑝𝑇sup_{0≤𝑠≤𝑇}𝜇(𝑠)_𝜆^2𝑝₂+𝐶^𝑝_𝑇^∞_𝑗=1𝜈_𝑗^𝑝+10^𝑝𝑀^∗2𝑝𝑇^𝑝𝑀^𝑝_𝐵+𝐶^∗_𝑇2^𝑝𝑡₀𝐸𝑥_𝜇(𝑠)_𝑈^2𝑝𝑑𝑠. (71)
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 Proposition 9. Computations similar to those used leading to the contractivity of the solution map in Theorem 8 can be used, along with Gronwall’s lemma, to establish this result. The details are omitted.

Proof of Theorem 10. We estimate each term of the representation formula for 𝐸‖𝑥_𝜀(𝑡)−𝑧(𝑡)‖²_𝑈 separately. We begin by observing that (A18) and (A19), together, guarantee the existence of positive constants 𝐾_𝑖 and positive functions 𝛼_𝑖(𝜀) (which decrease to zero as 𝜀→0⁺)(𝑖=1,2,3) such that, for sufficiently small 𝜀, 𝑆_𝜀(𝑡)𝐵𝑧₀−𝑆(𝑡)𝐵𝑧_0𝑈≤𝐾₁𝛼₁(𝜀),𝑆_𝜀(𝑡)𝑧₀−𝑆(𝑡)𝑧_0𝑈≤𝐾₂𝛼₂(𝜀),𝐶_𝜀(𝑡)𝑧₀−𝐶(𝑡)𝑧_0𝑈≤𝐾₃𝛼₃(𝜀), (72) for all 0≤𝑡≤𝑇.
Next, we estimate 𝐸^𝑡₀𝐶_𝜀(𝑡−𝑠)𝐵_𝜀𝑥_𝜀(𝑠)−𝐶(𝑡−𝑠)𝐵𝑧(𝑠)𝑑𝑠²_𝑈. (73) To this end, note that ^𝑡₀𝐸𝐶_𝜀(𝑡−𝑠)𝐵_𝜀𝑥_𝜀(𝑠)−𝐶_𝜀(𝑡−𝑠)𝐵_𝜀𝑧(𝑠)²_𝑈≤𝑀𝑑𝑠_∗2𝑀²_𝐵^𝑡₀𝐸𝑥_𝜀(𝑠)−𝑧(𝑠)²_𝑈𝑑𝑠, (74) and that ^𝑡₀𝐸𝐶_𝜀(𝑡−𝑠)𝐵_𝜀𝑧(𝑠)−𝐶_𝜀(𝑡−𝑠)𝐵𝑧(𝑠)²_𝑈≤𝑀𝑑𝑠_∗2𝑡0𝐸𝐵_𝜀𝑧(𝑠)−𝐵𝑧(𝑠)²_𝑈𝑑𝑠. (75) 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, ^𝑡₀𝐸𝐶_𝜀(𝑡−𝑠)𝐵_𝜀𝑧(𝑠)−𝐶_𝜀(𝑡−𝑠)𝐵𝑧(𝑠)²_𝑈𝑑𝑠≤𝐾₄𝛼₄(𝜀), (76) for all 0≤𝑡≤𝑇. Finally, (A18) and (A19) guarantee that the right side of ^𝑡₀𝐸𝐶_𝜀(𝑡−𝑠)𝐵𝑧(𝑠)−𝐶(𝑡−𝑠)𝐵𝑧(𝑠)²_𝑈≤𝑑𝑠_𝑡0𝐶_𝜀(𝑡−𝑠)−𝐶(𝑡−𝑠)²_𝑈𝐸‖𝐵𝑧(𝑠)‖²_𝑈𝑑𝑠 (77) 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, ^𝑡₀𝐸𝐶_𝜀(𝑡−𝑠)𝐵𝑧(𝑠)−𝐶(𝑡−𝑠)𝐵𝑧(𝑠)²_𝑈𝑑𝑠≤𝐾₅𝛼₅(𝜀), (78) for all 0≤𝑡≤𝑇. Thus, using (75)–(78), together with standard inequalities, in (74) yields 𝐸^𝑡₀𝐶_𝜀(𝑡−𝑠)𝐵_𝜀𝑥_𝜀(𝑠)−𝐶(𝑡−𝑠)𝐵𝑧(𝑠)𝑑𝑠²_𝑈≤16𝑇^1/2𝑀^∗2𝑀²_𝐵^𝑡₀𝐸𝑥_𝜀(𝑠)−𝑧(𝑠)²_𝑈𝑑𝑠+16𝑇^1/2𝐾₄𝛼₄(𝜀)+𝐾₅𝛼₅(𝜀). (79)
Next, we estimate 𝐸^𝑡₀𝑆_𝜀(𝑡−𝑠)𝐹_2𝜀𝑠,𝑥_𝜀(𝑠)−𝑆(𝑡−𝑠)𝐹(𝑠,𝑧(𝑠))𝑑𝑠²_𝑈. (80) 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, ^𝑡₀𝐸𝑆_𝜀(𝑡−𝑠)−𝑆(𝑡−𝑠)𝐹(𝑠,𝑧(𝑠))²_𝑈𝑑𝑠≤𝐾₆𝛼₆(𝜀), (81) for all 0≤𝑡≤𝑇. Also, observe that Young’s inequality and (A20), together, imply ^𝑡₀𝐸𝑆_𝜀(𝑡−𝑠)𝐹_2𝜀𝑠,𝑥_𝜀(𝑠)−𝐹(𝑠,𝑧(𝑠))²_𝑈≤𝑀𝑑𝑠_∗2𝑡0𝐸𝐹_2𝜀𝑠,𝑥_𝜀(𝑠)−𝐹_2𝜀(𝑠,𝑧(𝑠))++𝐹_2𝜀(𝑠,𝑧(𝑠))−𝐹(𝑠,𝑧(𝑠))²_𝑈𝑀𝑑𝑠≤4^∗2𝑡₀𝑀²_𝐹𝐸𝑥_𝜀(𝑠)−𝑧(𝑠)²_𝑈+𝐸𝐹_2𝜀(𝑠,𝑧(𝑠))−𝐹(𝑠,𝑧(𝑠))²_𝑈𝑑𝑠. (82) Note that (A20) guarantees the existence of 𝐾₇>0 and 𝛼₇(𝜀) (as above) such that, for sufficiently small 𝜀>0, 𝐸𝐹_2𝜀(𝑠,𝑧(𝑠))−𝐹(𝑠,𝑧(𝑠))²_𝑈≤𝐾₇𝛼₇(𝜀), (83) for all 0≤𝑡≤𝑇.
So, we can continue the inequality (82) to conclude that ^𝑡₀𝐸𝑆_𝜀(𝑡−𝑠)𝐹_2𝜀𝑠,𝑥_𝜀(𝑠)−𝐹(𝑠,𝑧(𝑠))²_𝑈𝑑𝑠≤4𝑀²_𝐹^𝑡₀𝐸𝑥_𝜀(𝑠)−𝑧(𝑠)²_𝑈𝑑𝑠+4𝑇𝐾₇𝛼₇(𝜀). (84)
Using (82) and (84), together with the Hölder, Minkowski, and Young inequalities, yields 𝐸^𝑡₀𝑆_𝜀(𝑡−𝑠)𝐹_2𝜀𝑠,𝑥_𝜀(𝑠)−𝑆(𝑡−𝑠)𝐹(𝑠,𝑧(𝑠))𝑑𝑠²_𝑈≤4𝑇^1/2𝐾₂𝛼₂(𝜀)+4𝑇𝐾₃𝛼₃(𝜀)+4𝑀^∗2𝑀²_𝐹^𝑡₀𝐸𝑥_𝜀(𝑠)−𝑧(𝑠)²_𝑈𝑑𝑠. (85) Next, (A21) guarantees the existence of 𝐾₈>0 and 𝛼₈(𝜀) (as above) such that, for sufficiently small 𝜀>0, 𝐸_𝑈𝐹_1𝜀(𝑠,𝑧)𝜇_𝜀(𝑠)(𝑑𝑧)²_𝑈≤𝐾₈𝛼₈(𝜀),∀0≤𝑠≤𝑇. (86) As such, we have 𝐸^𝑡₀𝑆_𝜀(𝑡−𝑠)_𝑈𝐹_1𝜀(𝑠,𝑧)𝜇_𝜀(𝑠)(𝑑𝑧)𝑑𝑠²_𝑈≤𝑀𝑑𝑠_∗2𝑇^𝑡₀𝐸_𝑈𝐹_1𝜀(𝑠,𝑧)𝜇_𝜀(𝑠)(𝑑𝑧)²_𝑈≤𝑀^∗2𝑇²𝐾₈𝛼₈(𝜀), (87) for all 0≤𝑡≤𝑇.
It remains to estimate 𝐸‖∫^𝑡₀𝑆_𝜀(𝑡−𝑠)𝑔_𝜀(𝑠)𝑑𝐵^𝐻(𝑠)‖²_𝑈. Observe that (A22) implies the existence of 𝐾₉>0 and 𝛼₉(𝜀) (as above) such that, for sufficiently small 𝜀>0, 𝐸𝑔_𝜀(𝑡)²_{𝔏()𝑉;𝑈}≤𝐾₉𝛼₉(𝜀),0≤𝑡≤𝑇. (88)
First, assume that 𝑔_𝜀 is a simple function. Observe that for 𝑚∈ℕ, 𝐸^𝑚_𝑗=1^𝑡₀𝑆_𝜀(𝑡−𝑠)𝑔_𝜀(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈=𝐸^𝑚_𝑗=1^𝑛−1_𝑘=0𝑆_𝜀𝑡−𝑡_𝑘𝑔_𝑘𝑒_𝑗𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘²_𝑈=𝐸^𝑚_𝑗=1^𝑛−1_𝑘=0𝑆_𝜀𝑡−𝑡_𝑘𝑔_𝑘𝑒_𝑗𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘,_{𝑚𝑗=1𝑛−1𝑘=0}𝑆_𝜀𝑡−𝑡_𝑘𝑔_𝑘𝑒_𝑗𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘𝑈≤^𝑚_𝑗=1^𝑛−1_𝑘=0𝑆_𝜀𝑡−𝑡_𝑘²_{(𝑈)𝔅𝔏}𝐾₉𝛼₉(𝜀)𝐸×𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘,𝛽^𝐻_𝑗𝑡_𝑘+1−𝛽^𝐻_𝑗𝑡_𝑘ℝ≤sup_{0≤𝑠≤𝑡}𝑆_𝜀𝑡−𝑡_𝑘²_{(𝑈)𝔅𝔏}𝐾₉𝛼₉(𝜀)𝐸𝛽^𝐻_𝑗²=sup_{0≤𝑠≤𝑡}𝑆_𝜀𝑡−𝑡_𝑘²_{(𝑈)𝔅𝔏}𝐾₉𝛼₉(𝜀)𝑡^𝑚2𝐻_𝑗=1𝜈_𝑗. (89) Hence, 𝐸^𝑡₀𝑆_𝜀(𝑡−𝑠)𝑔_𝜀(𝑠)𝑑𝛽^𝐻(𝑠)²_𝑈=𝐸^∞_𝑗=1^𝑡₀𝑆_𝜀(𝑡−𝑠)𝑔_𝜀(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈≤lim_𝑚→∞𝐸^𝑚_𝑗=1^𝑡₀𝑆_𝜀(𝑡−𝑠)𝑔_𝜀(𝑠)𝑒_𝑗𝑑𝛽^𝐻_𝑗(𝑠)²_𝑈≤lim_𝑚→∞sup_{0≤𝑠≤𝑡}𝑆_𝜀𝑡−𝑡_𝑘²_{(𝑈)𝔅𝔏}𝐾₉𝛼₉(𝜀)𝑡^𝑚2𝐻_𝑗=1𝜈_𝑗≤𝑀²_𝑆𝑡^∞2𝐻_𝑗=1𝜈_𝑗𝐾₉𝛼₉(𝜀),0≤𝑡≤𝑇. (90) 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 function𝑔_𝜀.
Now, using (72)–(90), we conclude that there exist positive constants 𝐾_𝑖(𝑖=1,…,9) and positive functions 𝛼_𝑖(𝜀)  (𝑖=1,…,9) such that 𝐸𝑥_𝜀(𝑡)−𝑧(𝑡)²_𝑈≤⁹_𝑖=1𝐾_𝑖𝛼_𝑖(𝜀)+4𝑀^∗2𝑀²_𝐹×^𝑡₀𝐸𝑥_𝜀(𝑠)−𝑧(𝑠)²_𝑈𝑑𝑠,0≤𝑡≤𝑇, (91) so that an application of Gronwall’s lemma implies 𝐸𝑥_𝜀(𝑡)−𝑧(𝑡)²_𝑈≤Ψ(𝜀)exp(𝜍𝑡),0≤𝑡≤𝑇, (92) where 𝜍=4(𝑀^∗)²𝑀²_𝐹 and Ψ(𝜀)=∑⁹_𝑖=1𝐾_𝑖𝛼_𝑖(𝜀). This completes the proof.

Proof of Lemma 12. Using the discussion in Section 2 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 𝐸Γ(𝑧)𝑡^∗+ℎ−Γ(𝑧)𝑡^∗2_𝑈≤5⁵_{𝑖=1,𝑖≠4}𝐸𝐼_𝑖𝑡^∗+ℎ−𝐼_𝑖𝑡^∗2_𝑈+𝐸^𝑡∗₀^+ℎ𝑆𝑡^∗ℱ(𝑥𝑠)−+ℎ−𝑠)(𝑑𝑠_𝑡∗0𝑆𝑡^∗−𝑠ℱ(𝑥)(𝑠)𝑑𝑠²_𝑈. (93)
Using Hölder’s inequality with (A23) yields 𝐸^𝑡∗_𝑡^+ℎ_∗𝑆𝑡^∗+ℎ−𝑠ℱ(𝑥)(𝑠)𝑑𝑠²_𝑈≤2𝑀²_𝐹𝑀^∗2ℎ²×1+‖𝑥‖²_{𝐶])([0,𝑇;𝑈}+‖ℱ(0)‖²_{𝐶])([0,𝑇;𝑈}, (94) which goes to 0 as |ℎ|→0. Also, 𝐸^𝑡∗₀[𝑆(ℎ)−𝐼]𝑆𝑡^∗−𝑠ℱ(𝑥)(𝑠)𝑑𝑠²_𝑈≤𝑇^𝑡∗₀[𝑆(ℎ)−𝐼]𝑆𝑡^∗−𝑠²_{(𝑈)𝔅𝔏}×𝐸‖ℱ(𝑥)(𝑠)‖²_𝑈𝑑𝑠≤𝑇sup_{0≤𝑠≤𝑇}[𝑆(ℎ)−𝐼]𝑆𝑡^∗−𝑠²_{(𝑈)𝔅𝔏}‖ℱ(𝑥)‖_ℒ², (95) and the right side goes to 0 as |ℎ|→0 here as well. Successive applications of Hölder’s inequality yields 𝐸^𝑡₀𝑆(𝑡−𝑠)ℱ(𝑥)(𝑠)𝑑𝑠²_𝑈^1/2≤𝑇^1/2𝑀^∗𝑇₀‖ℱ(𝑥)(𝑠)‖²_{ℒ2()Ω;𝑈}𝑑𝑠^1/2≤𝑇^1/2𝑀^∗‖ℱ(𝑥)‖_ℒ². (96) Subsequently, an application of (A23), together with Minkowski’s inequality, enables us to continue the string of inequalities in (96) to conclude that 𝐸^𝑡₀𝑆(𝑡−𝑠)ℱ(𝑥)(𝑠)𝑑𝑠²_𝑈^1/2≤𝑇^1/2𝑀^∗𝑀_ℱ‖𝑥‖_𝒞+‖𝐹(0)‖_ℒ². (97) Taking the supremum over [0,𝑇] in (97) then implies that ∫^𝑡₀𝑆(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 Lemma 6, they can be used to verify that sup_{0≤𝑡≤𝑇}𝐸‖Γ(𝑥)(𝑡)‖²_𝑈<∞, 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 Theorem 13. We know that Γ 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])(𝑡)−(𝑡)=Γ𝑥)(Γ𝑦_𝑡0𝐶()𝐵𝑥(𝑠)(𝑠)+𝑡−𝑠−𝑦𝑑𝑠_𝑡0𝑆(𝑡−𝑠)ℱ(𝑥)(𝑠)−ℱ𝑦(𝑠)𝑑𝑠. (98) Squaring both sides, taking the expectation in (98), and applying Young’s inequality yields 𝐸(Γ𝑥)(𝑡)−Γ𝑦(𝑡)²_𝑈≤2𝐸^𝑡₀𝑆(𝑡−𝑠)ℱ(𝑥)(𝑠)−ℱ𝑦(𝑠)𝑑𝑠²_𝑈+2𝐸^𝑡₀𝐶(𝑡−𝑠)𝐵𝑥(𝑠)−𝑦(𝑠)𝑑𝑠²_𝑈≤2𝑀^∗2𝑇ℱ(𝑥)−ℱ𝑦²_ℒ2+𝑀²_𝐵𝑥−𝑦²_𝐶≤2𝑀^∗2𝑇𝑀²_ℱ+𝑀²_𝐵𝑥−𝑦²_𝐶, (99) and so Γ𝑥−Γ𝑦²_𝐶≤2𝑀^∗2𝑇𝑀²_ℱ+𝑀²_𝐵𝑥−𝑦²_𝐶. (100) Hence, Φ is a strict contraction, provided that 𝑀^∗2𝑇(𝑀²_ℱ+𝑀²_𝐵)<1 and so has a unique fixed point which coincides with a mild solution of (27). This completes the proof.

Proof of Corollary 14. For (i), define ℱ∶𝐶([0,𝑇];𝑈)→ℒ²(0,𝑇;ℒ²(Ω;𝑈)) by ℱ(𝑥)(𝑡)=^𝑡₀𝑏(𝑡,𝑠)𝐹₀(𝑠,𝑥(𝑠))𝑑𝑠+𝐹₁(𝑡,𝑥(𝑡)),∀0≤𝑡≤𝑇. (101) The Uniform Boundedness Principle with (A6) guarantees the existence of a positive constant 𝑀_𝑏 such that ‖𝐵(𝑡,𝑠)‖_𝔅𝔏≤𝑀_𝑏, 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), ℱ(𝑥)−ℱ𝑦²_ℒ2≤2^𝑇₀𝑇𝑀²_𝑏^𝑠₀𝐸𝐹₀(𝜏,𝑥(𝜏))−𝐹₀𝜏,𝑦(𝜏)²_𝑈𝐹𝑑𝜏+𝐸₁(𝑠,𝑥(𝑠))−𝐹₁𝑠,𝑦(𝑠)²_𝑈≤2𝑇𝑇𝑀_𝑏𝑀_𝐹0+𝑀_𝐹1𝑥−𝑦_𝐶. (102) Thus, if we let 𝑀_ℱ=2𝑇[𝑇𝑀_𝑏𝑀_𝐹0+𝑀_𝐹1] 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 𝑀_ℱ=2𝑀_𝐹2𝑇|𝑎|_ℒ²_((0,𝑇)²₎1+𝑀_𝑘𝑇^31/2 (103) 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 Section 5 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 solution 𝑊∈𝐶([0,𝑇];ℒ²(0,𝑇)) with probability distribution 𝜇. Moreover, for every 𝑝≥1, there exists a constant 𝐶_𝑝 for which sup_{0≤𝑡≤𝑇}𝐸‖𝑊(⋅,𝑡)‖_ℒ^2𝑝_2()0,𝐿≤𝐶_𝑝1+𝐸𝜉_0ℒ^2𝑝_2()0,𝐿+𝐸𝜉_1ℒ^2𝑝_2()0,𝐿. (104)

Proof. We have illustrated in Section 3 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 Section 3. To see that (A12) is satisfied, observe that (A1)(i) implies that 𝐹₃(𝑡,⋅,𝑥(𝑡,⋅))_ℒ²_(0,𝐿)≤𝑀_𝐹3𝒟[1+|𝑥(𝑡,𝑧)|]²𝑑𝑧^1/2≤2𝑀_𝐹3𝐿+‖𝑥(𝑡,⋅)‖²_ℒ2()0,𝐿^1/2≤2𝑀_𝐹3√𝐿+‖𝑥‖_{𝐶([0,𝑇];ℒ}²_(0,𝐿))≤𝑀^∗_𝐹31+‖𝑥‖_{𝐶([0,𝑇];ℒ}²_(0,𝐿)), (105) for all 0≤𝑡≤𝑇 and 𝑥∈𝐶([0,𝑇];ℒ²(0,𝐿)), where 𝑀^∗_𝐹3=2𝑀_𝐹3√2𝐿,if𝐿>1,𝑀_𝐹3,if𝐿≤1. (106) Also, from (A1)(ii), we obtain 𝐹₃(𝑡,⋅,𝑥(𝑡,⋅))−𝐹₃𝑡,⋅,𝑦(𝑡,⋅)_ℒ2(0,𝐿)≤𝑀_𝐹3^𝐿₀𝑥(𝑡,𝑧)−𝑦(𝑡,𝑧)²𝑑𝑧^1/2≤𝑀_𝐹3𝑥−𝑦_{𝐶([0,𝑇];ℒ2(0,𝐿))}. (107)
Next, using (A2)(i) together with the Hölder inequality, we observe that _ℒ2()0,𝐿𝐹₄𝑡,⋅,𝑦𝜇(𝑡,⋅)𝑑𝑦_ℒ2(0,𝐿)=^𝐿_{0ℒ2()0,𝐿}𝐹₄𝑡,𝑧,𝑦𝜇(𝑡,𝑧)𝑑𝑦²𝑑𝑧^1/2≤^𝐿_{0ℒ2()0,𝐿}𝐹₄𝑡,𝑧,𝑦²_ℒ2()0,𝐿𝜇(𝑡,𝑧)𝑑𝑦𝑑𝑧^1/2≤𝑀_𝐹4^𝐿_{0ℒ2()0,𝐿}1+𝑦_ℒ2(0,𝐿)²𝜇(𝑡,𝑧)𝑑𝑦𝑑𝑧^1/2≤𝑀_𝐹4√𝐿𝜇(𝑡)_𝜆²≤𝑀_𝐹4√𝐿1+𝜇(𝑡)_𝜆²,∀0≤𝑡≤𝑇,𝜇∈℘_𝜆²(𝑈). (108) Also, invoking (A2)(ii) enables us to see that for all 𝜇,𝜈∈℘_𝜆²(𝑈), _ℒ2()0,𝐿𝐹₄𝜇()−𝑡,⋅,𝑦𝑡,⋅𝑑𝑦_ℒ2()0,𝐿𝐹₄𝑡,⋅,𝑦𝜈(𝑡,⋅)𝑑𝑦_ℒ2(0,𝐿)=_ℒ2()0,𝐿𝐹₄𝑡,⋅,𝑦𝜇(𝑡,⋅)−𝜈(𝑡,⋅)𝑑𝑦_ℒ2(0,𝐿)≤𝜌𝜇(𝑡),𝜈(𝑡)_ℒ2(0,𝐿)≤√𝐿𝜌𝜇(𝑡),𝜈(𝑡),∀0≤𝑡≤𝑇. (109) Combining (105) and (108), we see that 𝑓 satisfies (A12)(i) with 𝑀_𝑓1=2⋅max𝑀_𝐹4√𝐿,𝑀^∗_𝐹3, (110) and combining (107) and (109), we see that𝑓 satisfies (A12)(ii) with 𝑀_𝑓=max𝑀_𝐹3,√𝐿. (111) This verifies (A12). Next, (A13), (A15), and (A16) hold by assumption, and (A14) follows because of the uniform boundedness of {‖𝐴^1/2𝑥(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 Section 1 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.