Characterisation of Exponential Convergence to Nonequilibrium Limits for Stochastic Volterra Equations

Abstract

This paper considers necessary and sufficient conditions for the solution of a stochastically and deterministically perturbed Volterra equation to converge exponentially to a nonequilibrium and nontrivial limit. Convergence in an almost sure and 𝑝th mean sense is obtained.

1. Introduction

In this paper, we study the exponential convergence of the solution of 𝑑𝑋(𝑡)=𝐴𝑋(𝑡)+^𝑡0𝐾(𝑡−𝑠𝑋)𝑋(0()𝑠)=𝑑𝑠𝑋+₀𝑓(𝑡)𝑑𝑡+Σ(𝑡)𝑑𝐵(𝑡),𝑡>0,(1.1a),(1.1b)𝑋 to a nontrivial random variable. Here the solution 𝑛 is an [0,∞)-dimensional vector-valued function on 𝐴, 𝑛×𝑛 is a real 𝐾-dimensional matrix, 𝑛×𝑛 is a continuous and integrable [0,∞)-dimensional matrix-valued function on 𝑓, 𝑛 is a continuous [0,∞)-dimensional vector-valued function on Σ, 𝑛×𝑑 is a continuous [0,∞)-dimensional matrix-valued function on 𝐵(𝑡)=(𝐵₁(𝑡),𝐵₂(𝑡),…,𝐵_𝑑(𝑡)), and 𝑋₀ where each component of the Brownian motion is independent. The initial condition 𝑅^(𝑡)=𝐴𝑅(𝑡)+𝑅^𝑡0(𝐾0)(𝑡=−𝐼𝑠,)𝑅(𝑠)𝑑𝑠,𝑡>0,((11..22ba)) is a deterministic constant vector.

The solution of (1.1a)-(1.1b) can be written in terms of the solution of the resolvent equation 𝑅𝑅 where the matrix-valued function 𝑅_∞ is known as the resolvent or fundamental solution. In [1], the authors studied the asymptotic convergence of the solution 𝑅−𝑅_∞ of (1.2a)-(1.2b) to a nontrivial limit 𝑅−𝑅_∞. It was found that 𝑥^(𝑡)=𝐴𝑥(𝑡)+^𝑡0𝐾𝑥((𝑡0−)=𝑠)𝑥𝑥₀(𝑠)𝑑𝑠+𝑓(𝑡),𝑡>0,(1.3a),(1.3b) being integrable and the kernel being exponentially integrable were necessary and sufficient for exponential convergence. This built upon a result of Murakami [2] who considered the exponential convergence of the solution to a trivial limit and a result of Krisztin and Terjéki [3] who obtained necessary and sufficient conditions for the integrability of 𝑓. A deterministically perturbed version of (1.2a)-(1.2b), 𝑅−𝑅_∞𝑑𝑋(𝑡)=𝐴𝑋(𝑡)+^𝑡0𝐾(𝑡𝑋−(0𝑠))𝑋=(𝑠𝑋)𝑑₀𝑠𝑑𝑡+Σ(𝑡)𝑑𝐵(𝑡),𝑡>0,(1.4a),(1.4b)was also studied in [1]. It was shown that the exponential decay of the tail of the perturbation [0,∞) combined with the integrability of 𝑝 and the exponential integrability of the kernel were necessary and sufficient conditions for convergence to a nontrivial limit.

The case where (1.2a)-(1.2b) is stochastically perturbed 𝑅−𝑅_∞𝑡↦∫^∞𝑡𝑓(𝑠)𝑑𝑠 has been considered. Various authors including Appleby and Freeman [4], Appleby and Riedle [5], Mao [6], and Mao and Riedle [7] have studied convergence to equilibrium. In particular the paper by Appleby and Freeman [4] considered the speed of convergence of solutions of (1.4a)-(1.4b) to equilibrium. It was shown that under the condition that the kernel does not change sign on ℝ then (i) the almost sure exponential convergence of the solution to zero, (ii) the ℝ^𝑛th mean exponential convergence of the solution to zero, and (iii) the exponential integrability of the kernel and the exponential square integrability of the noise are equivalent.

Two papers by Appleby et al. [8, 9] considered the convergence of solutions of (1.4a)-(1.4b) to a nonequilibrium limit in the mean square and almost sure senses, respectively. Conditions on the resolvent, kernel, and noise for the convergence of solutions to an explicit limiting random variable were found. A natural progression from this work is the analysis of the speed of convergence.

This paper examines (1.1a)-(1.1b) and builds on the results in [1, 8, 9]. The analysis of (1.1a)-(1.1b) is complicated, particularly in the almost sure case, due to presence of both a deterministic and stochastic perturbation. Nonetheless, the set of conditions which characterise the exponential convergence of the solution of (1.1a)-(1.1b) to a nontrivial random variable is found. It can be shown that the integrability of 𝑛, the exponential integrability of the kernel, the exponential square integrability of the noise combined with the exponential decay of the tail of the deterministic perturbation, ℝ, are necessary and sufficient conditions for exponential convergence of the solution to a nontrivial random limit.

2. Mathematical Preliminaries

In this section, we introduce some standard notation as well as giving a precise definition of (1.1a)-(1.1b) and its solution.

Let 𝐞_𝑖 denote the set of real numbers and let 𝑖 denote the set of ℝ^𝑛-dimensional vectors with entries in ‖𝐴‖. Denote by 𝐴=(𝑎₁,…,𝑎_𝑛) the ‖‖𝐴‖‖²=^𝑛_𝑖=1𝑎^2𝑖=tr𝐴𝐴^𝑇,(2.1)th standard basis vector in tr. Denote by ℝ^𝑛×𝑛 the standard Euclidean norm for a vector 𝑛×𝑛 given by𝐼where diag(𝑎₁,𝑎₂,…,𝑎_𝑛) denotes the trace of a square matrix.

Let 𝑛×𝑛 be the space of 𝑎₁,𝑎₂,…,𝑎_𝑛 matrices with real entries where 0 is the identity matrix. Let 𝐴=(𝑎_𝑖𝑗)∈ℝ^𝑛×𝑑 denote the ‖⋅‖ matrix with the scalar entries ‖‖𝐴‖‖²=^𝑛_𝑑𝑖=1_𝑗=1||𝑎_𝑖𝑗||².(2.2) on the diagonal and ℂ elsewhere. For 𝑧 the norm denoted by ℂ is defined byRe𝑧

The set of complex numbers is denoted by 𝐴∶[0,∞)→ℝ^𝑛×𝑑; the real part of 𝐴(𝑧)=^∞0𝐴(𝑡)𝑒^−𝑧𝑡𝑑𝑡.(2.3) in 𝜖∈ℝ being denoted by ∫^∞0‖𝐴(𝑡)‖𝑒^−𝜖𝑡𝑑𝑡<∞. The Laplace transform of the function 𝐴(𝑧) is defined asRe𝑧≥𝜖If 𝑧↦𝐴(𝑧) and Re𝑧>𝜖 then 𝐽 exists for ℝ and 𝑉 is analytic for ‖⋅‖.

If 𝐶(𝐽,𝑉) is an interval in 𝜙∶𝐽→𝑉 and 𝜙∶[0,∞)→𝑉 a finite-dimensional normed space with norm 𝐿¹([0,∞),𝑉) then ∫^∞0‖𝜙(𝑡)‖𝑑𝑡<∞ denotes the family of continuous functions 𝜙∶[0,∞)→𝑉. The space of Lebesgue integrable functions 𝐿²([0,∞),𝑉) will be denoted by ∫^∞0‖𝜙(𝑡)‖²𝑑𝑡<∞ where 𝑉. The space of Lebesgue square-integrable functions 𝐾∶[0,∞)→ℝ^𝑛×𝑛 will be denoted by 𝐾∈𝐶([0,∞),ℝ^𝑛×𝑛)∩𝐿¹([0,∞),ℝ^𝑛×𝑛),(2.4) where 𝑓∶[0,∞)→ℝ^𝑛. When 𝑓∈𝐶([0,∞),ℝ^𝑛)∩𝐿¹([0,∞),ℝ^𝑛),(2.5) is clear from the context, it is omitted it from the notation.

We now make our problem precise. We assume that the function Σ∶[0,∞)→ℝ^𝑛×𝑑 satisfiesΣ∈𝐶([0,∞),ℝ^𝑛×𝑑).(2.6)the function 𝐾₁ satisfies𝐾₁∈𝐶([0,∞),ℝ^𝑛×𝑛)and the function 𝐾₁(𝑡)=^∞𝑡𝐾(𝑠)𝑑𝑠,𝑡≥0,(2.7) satisfies𝐾Due to (2.4) we may define 𝑓₁ to be a function 𝑓₁∈𝐶([0,∞),ℝ^𝑛) with𝑓₁(𝑡)=^∞𝑡𝑓(𝑠)𝑑𝑠,𝑡≥0.(2.8)where this function defines the tail of the kernel {𝐵(𝑡)}_𝑡≥0. Similarly, due to (2.5), we may define 𝑑 to be a function (Ω,ℱ,{ℱ^𝐵(𝑡)}_𝑡≥0,ℙ) given byℱ^𝐵(𝑡)=𝜎{𝐵(𝑠)∶0≤𝑠≤𝑡}We let 𝑅 denote 𝑡↦𝑋(𝑡;𝑋₀,𝑓,Σ)-dimensional Brownian motion on a complete probability space Σ where the filtration is the natural one 𝑓.

Under the hypothesis (2.4), it is well known that (1.2a)-(1.2b) has a unique continuous solution 𝑋₀, which is continuously differentiable. We define the function ℱ^𝐵 to be the unique solution of the initial value problem (1.1a)-(1.1b). If 𝑋(𝑡;𝑋₀,Σ,𝑓)=𝑅(𝑡)𝑋₀+^𝑡0𝑅(𝑡−𝑠)𝑓(𝑠)𝑑𝑠+^𝑡0𝑅(𝑡−𝑠)Σ(𝑠)𝑑𝐵(𝑠),𝑡≥0.(2.9) and 𝑋₀,𝑓 are continuous then for any deterministic initial condition Σ there exists an almost surely unique continuous and 𝑝-adapted solution to (1.1a)-(1.1b) given byℝ^𝑛When {𝑋(𝑡)}_𝑡≥0, and 𝑝 are clear from the context, we omit them from the notation.

The notion of convergence and integrability in 𝑋_∞th mean and almost sure senses are now defined: the lim_𝑡→∞𝔼‖𝑋(𝑡)−𝑋_∞‖^𝑝=0-valued stochastic process 𝑝converges in 𝑋_∞th mean to 𝛽_𝑝>0 if limsup_𝑡→∞1𝑡log(𝔼‖‖𝑋(𝑡)−𝑋^∞‖‖^𝑝)≤−𝛽_𝑝;(2.10); the process is {𝑋(𝑡)}_𝑡≥0th mean exponentially convergent to 𝑋_∞ if there exists a deterministic 𝑝 such that^∞0𝔼‖‖𝑋(𝑡)−𝑋_∞‖‖^𝑝𝑑𝑡<∞.(2.11)we say that the difference between the stochastic process ℙ and Ω₀ is integrable in the 𝜔∉Ω₀th mean sense iflim_𝑡→∞𝑋(𝑡,𝜔)=𝑋_∞(𝜔)If there exists a 𝑋-null set 𝑋_∞ such that for every 𝑋, the following holds: 𝑋_∞, then 𝛽₀>0converges almost surely to limsup_𝑡→∞1𝑡log‖‖𝑋(𝑡,𝜔)−𝑋^∞(𝜔)‖‖≤−𝛽₀,a.s.(2.12); we say {𝑋(𝑡)}_𝑡≥0 is almost surely exponentially convergent to 𝑋_∞ if there exists a deterministic ^∞0‖‖𝑋(𝑡,𝜔)−𝑋_∞(𝜔)‖‖²𝑑𝑡<∞.(2.13) such that𝔼[𝑋^𝑝]Finally, the difference between the stochastic process 𝔼𝑋^𝑝 and 𝑝,𝑞∈(0,∞) is square integrable in the almost sure sense if𝑋Henceforth, 𝑌 will be denoted by 𝔼‖𝑋‖^𝑝<∞ except in cases where the meaning may be ambiguous. A number of inequalities are used repeatedly in the sequel; they are stated here for clarity. If, for 𝔼‖𝑌‖^𝑞<∞, the finite-dimensional random variables 𝑝 and 𝑝>0 satisfy 𝔼‖‖𝑋‖‖^𝑝^1/𝑝≤𝔼‖‖𝑋‖‖^𝑞^1/𝑞,0<𝑝≤𝑞.(2.14) and (^𝑛_𝑖=1||𝑥_𝑖||)^𝑘≤𝑛^𝑛𝑘−1_𝑖=1||𝑥_𝑖||^𝑘,𝑛,𝑘∈ℕ.(2.15), respectively, then the Lyapunov inequality is useful when considering the 𝐾th mean behaviour of random variables as any exponent ^∞0𝑡²‖‖𝐾(𝑡)‖‖𝑑𝑡<∞,(3.1) may be considered:ΣThe following proves useful in manipulating norms:𝑓

3. Discussion of Results

We begin by stating the main result of this paper. That is, we state the necessary and sufficient conditions required on the resolvent, kernel, deterministic perturbation, and noise terms for the solution of (1.1a)-(1.1b) to converge exponentially to a limiting random variable. In this paper, we are particularly interested in the case when the limiting random variable is nontrivial, although the result is still true for the case when the limiting value is zero.

Theorem 3.1. Let 𝐾 satisfy (2.4) and eachentryof𝐾doesnotchangesignon[0,∞),(3.2)let 𝑅_∞ satisfy (2.6), and let 𝑅 satisfy (2.5). If 𝑅−𝑅_∞∈𝐿²([0,∞),ℝ^𝑛×𝑛),(3.3) satisfies 𝛼>0,𝛾>0,𝜌>0then the following are equivalent.
(i)There exists a constant matrix 𝑐₁>0 such that the solution 𝐾 of (1.2a)-(1.2b) satisfies^∞0‖‖𝐾(𝑠)‖‖𝑒^𝛼𝑠𝑑𝑠<∞,(3.4)and there exist constants Σ, and ^∞0‖‖Σ(𝑠)‖‖²𝑒^2𝛾𝑠𝑑𝑠<∞,(3.5) such that 𝑓₁ satisfies𝑓‖‖𝑓₁(𝑡)‖‖≤𝑐₁𝑒^−𝛾𝑡,𝑡≥0.(3.6) satisfies𝑋₀and 𝑝>0, the tail of ℱ^𝐵(∞), defined by (2.8) satisfies𝑋_∞(𝑋₀,Σ,𝑓)(ii)For all initial conditions 𝔼‖𝑋_∞‖^𝑝<∞ and constants 𝑋(⋅;𝑋₀,Σ,𝑓) there exists an a.s. finite 𝔼‖‖𝑋(𝑡)−𝑋_∞‖‖^𝑝≤𝑚^∗𝑝𝑒^{−𝛽∗𝑝𝑡},𝑡≥0,(3.7)-measurable random variable 𝛽^∗𝑝 with 𝑚^∗𝑝=𝑚^∗𝑝(𝑋₀) such that the unique continuous adapted process 𝑋₀ which obeys (1.1a)-(1.1b) satisfiesℱ^𝐵(∞)where 𝑋_∞(𝑋₀,Σ,𝑓) and 𝑋(⋅;𝑋₀,Σ,𝑓) are positive constants.(iii)For all initial conditions limsup_𝑡→∞1𝑡log‖‖𝑋(𝑡)−𝑋^∞‖‖≤−𝛽^∗0a.s.,(3.8) there exists an a.s. finite 𝛽^∗0-measurable random variable 𝑓∶=0 such that the unique continuous adapted process 𝐾 which obeys (1.1a)-(1.1b) satisfiesΣwhere 𝐾 is a positive constant.

The proof of Theorem 3.1 is complicated by the presence of two perturbations so as an initial step the case when 𝑅_∞ is considered. That is we consider the conditions required for exponential convergence of (1.4a)-(1.4b) to a limiting random variable.

Theorem 3.2. Let 𝑅 satisfy (2.4) and (3.1) and let 𝛼>0 satisfy (2.6). If 𝛾>0 satisfies (3.2) then the following are equivalent.
(i) There exists a constant matrix 𝐾 such that the solution Σ of (1.2a)-(1.2b) satisfies (3.3) and there exist constants 𝑋₀ and 𝑝>0 such that ℱ^𝐵(∞) and 𝑋_∞(𝑋₀,Σ) satisfy (3.4) and (3.5), respectively.(ii) For all initial conditions 𝔼‖𝑋_∞‖^𝑝<∞ and constants 𝑋(⋅;𝑋₀,Σ) there exists an a.s. finite 𝔼‖‖𝑋(𝑡)−𝑋_∞‖‖^𝑝≤𝑚_𝑝𝑒^{−𝛽𝑝𝑡},𝑡≥0,(3.9)-measurable random variable 𝛽_𝑝 with 𝑚_𝑝=𝑚_𝑝(𝑋₀) such that the unique continuous adapted process 𝑋₀ which obeys (1.4a)-(1.4b) satisfiesℱ^𝐵(∞)where 𝑋_∞(𝑋₀,Σ) and 𝑋(⋅;𝑋₀,Σ) are positive constants.(iii) For all initial conditions limsup_𝑡→∞1𝑡log‖‖𝑋(𝑡)−𝑋^∞‖‖≤−𝛽₀a.s.,(3.10) there exists an a.s. finite 𝛽₀-measurable random variable 𝑅_∞=0 such that the unique continuous adapted process 𝐾,𝑓 which obeys (1.4a)-(1.4b) satisfiesΣwhere 𝐾 is a positive constant.

This result is interesting in its own right as it generalises a result in [4] where necessary and sufficient conditions for exponential convergence to zero are found. Theorem 3.2 collapses to this case if 𝐾.

It is interesting to note the relationship between the behaviour of the solutions of (1.1a)-(1.1b), (1.2a)-(1.2b), (1.3a)-(1.3b), and (1.4a)-(1.4b) and the behaviour of the inputs 𝑓, and 𝐾. It is seen in [1] that Σ being exponentially integrable is the crucial condition for exponential convergence when we consider the resolvent equation. Each perturbed equation then builds on this resolvent case: for the deterministically perturbed equation we require the exponential integrability of 𝑅,𝐾 and the exponential decay of the tail of the perturbation Σ (see [1]); for the stochastically perturbed case we require the exponential integrability of 𝑅,𝐾,Σ and the exponential square integrability of 𝑓. In the stochastically and deterministically perturbed case it is seen that the perturbations do not interact in a way that exacerbates or diminishes the influence of the perturbations on the system: we can isolate the behaviours of the perturbations and show that the same conditions on the perturbations are still necessary and sufficient.

Theorem 3.1 has application in the analysis of initial history problems. In particular this theoretical result could be used to interpret the equation as an epidemiological model. Conditions under which a disease becomes endemic (which is the interpretation that is given when solutions settle down to a nontrivial limit) were studied in [9]. The theoretical results obtained in this paper could be exploited to highlight the speed at which this can occur within a population.

The remainder of this paper deals with the proofs of Theorems 3.1 and 3.2. In Section 4 we prove the sufficiency of conditions on Σ, and 𝑝 for the exponential convergence of the solution of (1.4a)-(1.4b) while in Section 5 we prove the necessity of these conditions. In Section 6 we prove the sufficiency of conditions on 𝐾, and Σ for the exponential convergence of the solution of (1.1a)-(1.1b), while Section 7 deals with the necessity of the condition on 𝑅_∞. In Section 8 we combine our results to prove the main theorems, namely, Theorems 3.1 and 3.2.

4. Sufficient Conditions for Exponential Convergence of Solutions of (1.4a)-(1.4b)

In this section, sufficient conditions for exponential convergence of solutions of (1.4a)-(1.4b) to a nontrivial limit are obtained. Proposition 4.1 concerns convergence in the 𝑅th mean sense while Proposition 4.2 deals with the almost sure case.

Proposition 4.1. Let 𝛼>0 satisfy (2.4) and (3.1), let 𝛾>0 satisfy (2.6) and 𝛽_𝑝>0 be a constant matrix such that the solution 𝑋₀ of (1.2a)-(1.2b) satisfies (3.3). If there exist constants 𝑚_𝑝=𝑚_𝑝(𝑋₀)>0 and 𝐾 such that (3.4) and (3.5) hold, then there exist constants Σ, independent of 𝑅_∞, and 𝑅, such that statement (ii) of Theorem 3.2 holds.

Proposition 4.2. Let 𝛼>0 satisfy (2.4) and (3.1), let 𝛾>0 satisfy (2.6) and 𝛽₀>0 be a constant matrix such that the solution 𝑋₀ of (1.2a)-(1.2b) satisfies (3.3). If there exist constants 𝑝=2 and 𝑝>0 such that (3.4) and (3.5) hold, then there exists a constant 𝐾, independent of Σ such that statement (iii) of Theorem 3.2 holds.

In [8], the conditions which give mean square convergence to a nontrivial limit were considered. So a natural progression in this paper is the examination of the speed of convergence in the mean square case. Lemma 4.3 examines the case when 𝑅_∞ in order to highlight this important case. This lemma may be then used when generalising the result to all 𝑅.

Lemma 4.3. Let 𝛼>0 satisfy (2.4) and (3.1), let 𝛾>0 satisfy (2.6), and let 𝜆>0 be a constant matrix such that the solution 𝑋₀ of (1.2a)-(1.2b) satisfies (3.3). If there exist constants 𝑚=𝑚(𝑋₀)>0 and 𝔼‖‖𝑋(𝑡)−𝑋_∞‖‖²≤𝑚(𝑋₀)𝑒^−2𝜆𝑡,𝑡≥0.(4.1) such that (3.4) and (3.5) hold, then there exist constants 𝑅−𝑅_∞∈𝐿²([0,∞),ℝ^𝑛×𝑛), independent of 𝑅−𝑅_∞∈𝐿¹([0,∞),ℝ^𝑛×𝑛), and 𝑅−𝑅_∞∈𝐿¹([0,∞),ℝ^𝑛×𝑛), such that 𝑅−𝑅_∞

From [8, 9] it is evident that 𝑅−𝑅_∞ is a more natural condition on the resolvent than 𝐾 when studying convergence of solutions of (1.4a)-(1.4b). However, the deterministic results obtained in [1] are based on the assumption that 𝑅_∞. Lemma 4.4 is required in order to make use of these results in this paper; this result isolates conditions that ensure the integrability of 𝑅 once 𝑅 is square integrable.

Lemma 4.4. Let 𝑅−𝑅_∞∈𝐿¹([0,∞),ℝ^𝑛×𝑛).(4.2) satisfy (2.4) and (3.1) and let 𝑅 be a constant matrix such that the solution 𝑅_∞ of (1.2a)-(1.2b) satisfies (3.3). Then the solution 𝐾 of (1.2a)-(1.2b) satisfies 𝑅_∞

We now state some supporting results. It is well known that the behaviour of the resolvent Volterra equation influences the behaviour of the perturbed equation. It is unsurprising therefore that an earlier result found in [1] concerning exponential convergence of the resolvent 𝑅 to a limit 𝛼>0 in needed in the proof of Theorems 3.1 and 3.2.

Theorem 4.5. Let 𝐾 satisfy (2.4) and (3.1). Suppose there exists a constant matrix 𝛽>0 such that the solution 𝑐>0 of (1.2a)-(1.2b) satisfies (4.2). If there exists a constant ‖‖𝑅(𝑡)−𝑅_∞‖‖≤𝑐𝑒^−𝛽𝑡,𝑡≥0.(4.3) such that 𝑋_∞ satisfies (3.4) then there exist constants 𝑋_∞ and 𝐾 such that ^∞0𝑡‖‖𝐾(𝑡)‖‖𝑑𝑡<∞,(4.4)

In the proof of Propositions 4.1 and 4.2, an explicit representation of Σ is required. In [8, 9] the asymptotic convergence of the solution of (1.4a)-(1.4b) was considered. Sufficient conditions for convergence were obtained and an explicit representation of ^∞0‖‖Σ(𝑡)‖‖²𝑑𝑡<∞.(4.5) was found:

Theorem 4.6. Let 𝑅 satisfy (2.4) and 𝑋and let lim_𝑡→∞𝑋(𝑡)=𝑋_∞ satisfy (2.6) and 𝑋_∞Suppose that the resolvent ℱ^𝐵(∞) of (1.2a)-(1.2b) satisfies (3.3). Then the solution 𝑋_∞=𝑅_∞𝑋⁰+^∞0Σ(𝑡)𝑑𝐵(𝑡)a.s.(4.6) of (1.4a)-(1.4b) satisfies 𝑋_∞ almost surely, where 𝐾 is an almost surely finite and 𝑋₀-measurable random variable given by 𝑋_∞(𝑋₀,Σ)

Lemma 4.7 concerns the structure of 𝑡↦𝑋(𝑡;𝑋₀,Σ) in the almost sure case. It was proved in [9].

Lemma 4.7. Let 𝑋(⋅;𝑋l₀im_𝑡→∞𝑋(𝑡;𝑋₀,Σ)=𝑋_∞(𝑋₀,Σ)a.s.,(4.7),Σ)−𝑋_∞(𝑋₀,Σ)∈𝐿²([0,∞),ℝ^𝑛)a.s.(4.8) satisfy (2.4) and (4.4). Suppose that for all initial conditions 𝐴+^∞0𝐾(𝑠)𝑑𝑠𝑋_∞=0a.s.(4.9) there is an almost surely finite random variable 𝐾 such that the solution Σ of (1.4a)-(1.4b) satisfies Σ𝑅_∞Then 𝑅

It is possible to apply this lemma using our a priori assumptions due to Theorem 4.8, which was proved in [9].

Theorem 4.8. Let 𝑋₀ satisfy (2.4) and (4.4) and let ℱ^𝐵(∞) satisfy (2.6). If 𝑋_∞(𝑋₀,Σ) satisfies (4.5) and there exists a constant matrix 𝑋(⋅;𝑋₀,Σ) such that the solution Σ of (1.2a)-(1.2b) satisfies (3.3), then for all initial conditions ^∞0𝑡‖‖𝑅_∞Σ(𝑡)‖‖²𝑑𝑡<∞,(4.10) there is an almost surely finite 𝑀=𝐴+∫^∞0𝐾(𝑡)𝑑𝑡-measurable random variable 𝑇, such that the unique continuous adapted process 𝐽=𝑇⁻¹𝑀𝑇 which obeys (1.4a)-(1.4b) satisfies (4.7).
Moreover, if the function 𝑒_𝑖=1 also satisfies 𝑖then (4.8) holds.

Lemma 4.9 below is required in the proof of Lemma 4.4. It is proved in [8]. Before citing this result some notation is introduced. Let 𝐽 and 𝑒_𝑖=0 be an invertible matrix such that 𝐷_𝑝=diag(𝑒₁,𝑒₂,…,𝑒_𝑛) has Jordan canonical form. Let 𝑃=𝑇𝐷_𝑝𝑇⁻¹ if all the elements of the 𝑄=𝐼−𝑃th row of 𝐾 are zero, and 𝑅_∞ otherwise. Let 𝑅 and put det[𝐼+𝐹(𝑧)]≠0,Re𝑧≥0,(4.11) and 𝐹.

Lemma 4.9. Let 𝐹(𝑡)=−𝑒^−𝑡(𝑄+𝑄𝐴)−(𝑒∗𝑄𝐾)(𝑡)+𝑃^∞𝑡𝐾(𝑢)𝑑𝑢,𝑡≥0.(4.12) satisfy (2.4) and (4.4). If there exists a constant matrix 𝜎∈𝐶([0,∞)×[0,∞),ℝ^𝑝×𝑟) such that the resolvent 𝔼‖‖‖^𝑏𝑎𝜎(𝑠,𝑡)𝑑𝐵(𝑠)‖‖‖^2𝑚≤𝑑_𝑚(𝑝,𝑟)^𝑏𝑎‖‖𝜎(𝑠,𝑡)‖‖²𝑑𝑠^𝑚,(4.13) of (1.2a)-(1.2b) satisfies (3.3), then 𝑑_𝑚(𝑝,𝑟)=𝑝^𝑚+1𝑟^2𝑚+1(2𝑚)!(𝑚!2^𝑚)⁻¹𝑐₂(𝑝,𝑟)^𝑚where 𝐾∈𝐶([0,∞),ℝ^𝑛×𝑛)∩L¹([0,∞),ℝ^𝑛×𝑛) is defined by ^∞0‖‖𝐾(𝑠)‖‖𝑒^𝛼𝑠𝑑𝑠<∞.(4.14)

Lemma 4.10 concerns the moments of a normally distributed random variable. It can be extracted from [4, Theorem 3.3] and it is used in Proposition 4.1.

Lemma 4.10. Suppose the function 𝜆̃>0 then 𝜂̃=2𝜆̃⋀𝛼where ^𝑡0𝑒^{̃−2𝜆(𝑡−𝑠)}𝑒^{−𝛼𝑠}‖‖𝐾(𝑠)‖‖𝑑𝑠≤𝑐𝑒^{−̃𝜂𝑡},(4.15).

The following lemma is used in Proposition 4.2. A similar result is proved in [4].

Lemma 4.11. Suppose that 𝑐 and 𝑋(𝑡)→𝑋_∞If 𝑋_∞ and 𝔼‖‖𝑋_∞‖‖²=𝔼[tr(𝑋_∞𝑋^𝑇∞)]=‖‖𝑅_∞𝑋₀‖‖²+^∞0‖‖𝑅_∞Σ(𝑠)‖‖²𝑑𝑠<∞.(4.16) then 𝔼[‖‖𝑋(𝑡)−𝑋_∞‖‖²]=𝔼[tr(𝑋(𝑡)−𝑋_∞)(𝑋(𝑡)−𝑋_∞)^𝑇],(4.17)where 𝔼[‖‖𝑋(𝑡)−𝑋_∞‖‖²]=‖‖(𝑅(𝑡)−𝑅_∞)𝑋₀‖‖²+^𝑡0‖‖(𝑅(𝑡−𝑠)−𝑅_∞)Σ(𝑠)‖‖²𝑑𝑠+^∞𝑡‖‖𝑅_∞Σ(𝑠)‖‖²𝑑𝑠.(4.18) is a positive constant.

The proofs of Propositions 4.1 and 4.2 and Lemmas 4.3 and 4.4 are now given.

Proof of Lemma 4.3. From Theorem 4.6 we see that ‖‖(𝑅(𝑡)−𝑅_∞)𝑋₀‖‖²≤𝑐₁‖‖𝑋₀‖‖²𝑒^−2𝛽𝑡.(4.19) almost surely where 𝑅 is given by (4.6), so we see that𝑅_∞Since0<𝜆<min(𝛽,𝛾)we use (2.9) and (4.6) to expand the right hand side of (4.17) to obtain𝑒_𝜆ΣIn order to obtain an exponential upper bound on (4.18) each term is considered individually. We begin by considering the first term on the right-hand side of (4.18). Using (3.1) and (3.3) we can apply Lemma 4.4 to obtain (4.2). Then using (3.1), (4.2), and (3.4) we see from Theorem 4.5 that𝑒_𝜆(𝑅−𝑅_∞)∈𝐿²[0,∞)We provide an argument to show that the second term decays exponentially. Using (3.5) and the fact that 𝑒_𝜆 decays exponentially quickly to 𝑒_𝜆(𝑡)=𝑒^𝜆𝑡 we can choose 𝐿²[0,∞) such that 𝐿²[0,∞) and 𝐿²[0,∞) where the function 𝑒^2𝜆𝑡^𝑡0‖‖(𝑅(𝑡−𝑠)−𝑅_∞)Σ(𝑠)‖‖²𝑑𝑠≤^𝑡0𝑒^{2𝜆(𝑡−𝑠)}‖‖𝑅(𝑡−𝑠)−𝑅_∞‖‖²𝑒^2𝜆𝑠‖‖Σ(𝑠)‖‖²𝑑𝑠≤𝑐₂,(4.20) is defined by Σ∶=^∞0‖‖Σ(𝑠)‖‖²𝑒^2𝛾𝑠𝑑𝑠≥^∞𝑡‖‖Σ(𝑠)‖‖²𝑒^2𝛾𝑠𝑑𝑠≥𝑒^2𝛾𝑡^∞𝑡‖‖Σ(𝑠)‖‖²𝑑𝑠.(4.21). Since the convolution of an 𝔼[‖‖𝑋(𝑡)−𝑋_∞‖‖²]≤𝑚(𝑋₀)𝑒^−2𝜆𝑡,(4.22) function with an 𝑚(𝑋₀)=𝑐₁‖𝑋₀‖²+𝑐₂+Σ‖𝑅_∞‖² function is itself an 𝜆<min(𝛽,𝛾) function we get0<𝑝≤2and so the second term of (4.18) decays exponentially quickly.
We can show that the third term on the right hand side of (4.18) decays exponentially using (3.5) and the following argument:𝑝>2
Combining these facts we see that0<𝑝≤2where 𝔼[‖𝑋_∞‖²]<∞ and 𝔼‖‖𝑋_∞‖‖^𝑝≤𝔼‖‖𝑋_∞‖‖²^𝑝/2<∞.(4.23).

Proof of Proposition 4.1. Consider the case where 0≤𝑝≤2 and 𝔼‖‖𝑋(𝑡)−𝑋_∞‖‖^𝑝≤𝔼[‖‖𝑋(𝑡)−𝑋_∞‖‖²]^𝑝/2≤𝑚_𝑝(𝑋₀)𝑒^{−𝛽𝑝𝑡},𝑡≥0,(4.24) separately. We begin with the case where 𝑚_𝑝(𝑋₀)=𝑚(𝑋₀)^𝑝/2. The argument given by (4.16) shows that 𝛽_𝑝=𝜆𝑝. Now applying Lyapunov's inequality we see that𝑝>2We now show that (3.9) holds for 𝑚∈ℕ. Lyapunov's inequality and Lemma 4.3 can be applied as follows:2(𝑚−1)<𝑝≤2𝑚where 𝔼‖𝑋_∞‖^2𝑚 and 𝔼[‖𝑋(𝑡)−𝑋_∞‖^2𝑚].
Now consider the case where 𝔼‖𝑋_∞‖^𝑝. In this case, there exists a constant 𝔼[‖𝑋(𝑡)−𝑋_∞‖^𝑝] such that 𝔼‖‖𝑋_∞‖‖^2𝑚≤𝑐‖‖𝑅_∞𝑋₀‖‖^2𝑚+𝑐^∞0‖‖𝑅_∞Σ(𝑠)‖‖²𝑑𝑠^𝑚<∞,(4.25). We now seek an upper bound on 𝑐 and 𝔼‖𝑋_∞‖^𝑝≤𝔼[‖𝑋_∞‖^2𝑚]^𝑝/2𝑚<∞, which will in turn give an upper bound on 𝔼[‖𝑋(𝑡)−𝑋_∞‖^2𝑚] and 𝑋_∞ by using Lyapunov's inequality. By applying Lemma 4.10 we see that2𝑚where 𝔼[‖‖𝑋≤(𝑡3)−𝑋_∞2𝑚−1‖‖^2𝑚]+‖‖𝔼(𝑅‖‖‖(𝑡)_∞𝑡−𝑅^∞)𝑋₀‖‖^2𝑚+𝔼‖‖‖^𝑡0(𝑅(𝑡−𝑠)−𝑅_∞)Σ(𝑠)𝑑𝐵(𝑠)‖‖‖^2𝑚𝑅_∞Σ(𝑠)𝑑𝐵(𝑠)‖‖‖^2𝑚.(4.26) is a positive constant, so ‖‖(𝑅(𝑡)−𝑅_∞)𝑋₀‖‖^2𝑚≤𝑐₁‖‖𝑋₀‖‖^2𝑚𝑒^{−2𝑚𝛽𝑡}.(4.27).
Now consider ∫^𝑡0‖(𝑅(𝑡−𝑠)−𝑅_∞)Σ(𝑠)‖²𝑑𝑠≤𝑐₂𝑒^−2𝜆𝑡. Using the variation of parameters representation of the solution and the expression obtained for 𝜆<min(𝛽,𝛾), taking norms, raising both sides of the equation to the 𝔼‖‖‖^𝑡0(𝑅(𝑡−𝑠)−𝑅_∞)Σ(𝑠)𝑑𝐵(𝑠)‖‖‖^2𝑚≤𝑑^𝑚(𝑛,𝑑)^𝑡0‖‖𝑅(𝑡−𝑠)−𝑅_∞‖‖²‖‖Σ(𝑠)‖‖²𝑑𝑠^𝑚≤𝑑_𝑚(𝑛,𝑑)𝑐^𝑚2𝑒^{−2𝑚𝜆𝑡}.(4.28)th power, then taking expectations across the inequality, we arrive at𝔼‖‖‖^∞𝑡Σ(𝑠)𝑑𝐵(𝑠)‖‖‖^2𝑚≤𝑑^𝑚(𝑛,𝑑)^∞𝑡‖‖Σ(𝑠)‖‖²𝑑𝑠^𝑚≤𝑑_𝑚(𝑛,𝑑)Σ^𝑚𝑒^{−2𝑚𝛾𝑡}.(4.29)We consider each term on the right hand side of (4.26). By Theorem 4.5 we have𝔼[‖‖𝑋≤(𝑡3)−𝑋_∞2𝑚−1‖‖^2𝑚]𝑐₁‖‖𝑋₀‖‖^2𝑚𝑒^{−2𝑚𝛽𝑡}+𝑑_𝑚(𝑛,𝑑)𝑐^𝑚2𝑒^{−2𝑚𝜆𝑡}+𝑑_𝑚(𝑛,𝑑)‖‖𝑅_∞‖‖^2𝑚Σ^𝑚𝑒^{−2𝑚𝛾𝑡}.(4.30)Now, consider the second term on the right-hand side of (4.26). By (4.20) we see that 𝔼[‖‖𝑋(𝑡)−𝑋_∞‖‖^𝑝]≤𝑚_𝑝(𝑋₀)𝑒^{−𝛽𝑝𝑡},(4.31) where 𝑚_𝑝(𝑋₀)=3^{𝑝((2𝑚−1)/2𝑚)}(𝑐₁‖𝑋₀‖^2𝑚+𝑑_𝑚(𝑛,𝑑)𝑐^𝑚2+𝑑_𝑚(𝑛,𝑑)‖𝑅_∞‖^2𝑚Σ^𝑚)^𝑝/2𝑚. Using this and Lemma 4.10 we see that𝛽_𝑝=𝜆𝑝
Using (4.21) combined with Lemma 4.10 and Fatou's lemma, we show that the third term decays exponentially quickly:𝔼sup_{𝑛−1≤𝑡≤𝑛}‖‖𝑋(𝑡)−𝑋_∞‖‖²≤𝑚(𝑋⁰)𝑒^{−2𝜂(𝑛−1)},𝜂>0.(4.32)Combining (4.27), (4.28), and (4.29) the inequality (4.26) becomes𝑡>0Using Lyapunov's inequality, the inequality (4.30) implies𝑛∈ℕwhere 𝑛−1≤𝑡<𝑛 and Δ(𝑡)=𝑋(𝑡)−𝑋_∞.

Proof of Proposition 4.2. In order to prove this proposition we show that[𝑛−1,𝑡]For each 𝑋_∞ there exists 𝑋(𝑡)−𝑋_∞=+𝑋(𝑛−1)−𝑋_∞+^𝑡𝑛−1𝐴(𝑋(𝑠)−𝑋_∞)𝑑𝑠_𝑡𝑛−1^𝑠0+𝐾(𝑠−𝑢)(𝑋(𝑢)−𝑋_∞)𝑑𝑢𝑑𝑠+^𝑡𝑛−1Σ(𝑠)𝑑𝐵(𝑠)_𝑡𝑛−1𝐴+^∞0𝐾(𝑢)𝑑𝑢𝑋_∞𝑑𝑠−^𝑡𝑛−1^∞𝑠𝐾(𝑣)𝑋_∞𝑑𝑣𝑑𝑠.(4.33) such that Δ(𝑡)=Δ(𝑛−1)+^𝑡𝑛−1(𝐴Δ(𝑠)+(𝐾∗Δ)(𝑠))𝑑𝑠+^𝑡𝑛−1Σ(𝑠)𝑑𝐵(𝑠)−^𝑡𝑛−1𝐾₁(𝑠)𝑑𝑠𝑋_∞.(4.34). Define 𝔼sup_{𝑛−1≤𝑡≤𝑛}‖‖Δ(𝑡)‖‖²≤4+𝔼𝔼‖‖Δ(𝑛su−p1_{𝑛−1≤𝑡≤𝑛})‖‖²+𝔼^𝑛𝑛−1‖‖𝐴‖‖‖‖Δ(𝑠)‖‖+(‖‖𝐾‖‖∗‖‖Δ‖‖)(𝑠)𝑑𝑠²‖‖‖^𝑡𝑛−1Σ(𝑠)𝑑𝐵(𝑠)‖‖‖²+^𝑛𝑛−1‖‖𝐾₁(𝑠)‖‖𝑑𝑠²𝔼‖‖𝑋_∞‖‖².(4.35). Integrating (1.4a)-(1.4b) over 𝔼[‖‖Δ(𝑛−1)‖‖²]≤𝑚(𝑋₀)𝑒^{−2𝜆(𝑛−1)}.(4.36), then adding and subtracting ^𝑛𝑛−1‖‖𝐴‖‖‖‖Δ(𝑠)‖‖+(‖‖𝐾‖‖∗‖‖Δ‖‖)(𝑠)𝑑𝑠²≤≤22^𝑛𝑛−1_𝑛𝑛−1‖‖𝐴‖‖²‖‖Δ(𝑠)‖‖²+(‖‖𝐾‖‖∗‖‖Δ‖‖)²(𝑠)𝑑𝑠≤2_𝑛𝑛−1‖‖𝐴‖‖²‖‖Δ(𝑠)‖‖²+^𝑠0𝑒^{𝛼(𝑠−𝑢)/2}‖‖𝐾(𝑠−𝑢)‖‖^1/2𝑒^{−𝛼(𝑠−𝑢)/2}‖‖𝐾(𝑠−𝑢)‖‖^1/2‖‖Δ(𝑠)‖‖𝑑𝑢²𝑑𝑠‖‖𝐴‖‖²‖‖Δ(𝑠)‖‖²+𝐾_𝛼^𝑠0𝑒^{−𝛼(𝑠−𝑢)}‖‖𝐾(𝑠−𝑢)‖‖‖‖Δ(𝑠)‖‖²𝑑𝑢𝑑𝑠,(4.37) on both sides we get𝐾_𝛼=∫^∞0𝑒^𝛼𝑡‖𝐾(𝑡)‖𝑑𝑡By applying Theorem 4.8, we see that (4.7) and (4.8) hold so Lemma 4.7 may be applied to obtain𝔼^𝑛𝑛−1‖‖𝐴‖‖²‖‖Δ(𝑠)‖‖²𝑑𝑠≤‖‖𝐴‖‖²𝑚(𝑋₀)^𝑛𝑛−1𝑒^−2𝜆𝑠𝑑𝑠≤𝑐₁(𝑋₀)𝑒^{−2𝜆(𝑛−1)}.(4.38)Taking norms on both sides of (4.34), squaring both sides, taking suprema, before finally taking expectations yields:𝐾=𝐾We now consider each term on the right hand side of (4.35). From Lemma 4.3 we see that the first term satisfies𝛼=𝛼,𝜆=𝜆̃In order to obtain an exponential bound on the second term on the right hand side of (4.26) we make use of the Cauchy-Schwarz inequality as follows:𝜂=𝜂̃where 𝔼^𝑛𝑛−1𝐾_𝛼^𝑠0𝑒^{−𝛼(𝑠−𝑢)}‖‖𝐾(𝑠−𝑢)‖‖‖‖Δ(𝑠)‖‖²𝑑𝑢𝑑𝑠≤𝑚(𝑋₀)𝐾_𝛼^𝑛𝑛−1^𝑠0≤𝑐₂𝑒^−𝛼𝑢‖‖𝐾(𝑢)‖‖𝑒^{−2𝜆(𝑠−𝑢)}𝑑𝑢𝑑𝑠(𝑋₀)𝑒^{−𝜂(𝑛−1)}.(4.39). Take expectations and examine the two terms within the integral. Using Lemma 4.3 we obtain𝑐₃>0In order to obtain an exponential upper bound for the second term within the integral we apply Lemma 4.11 with 𝔼[sup_{𝑛−1≤𝑡≤𝑛}‖‖‖^𝑡𝑛−1Σ(𝑠)𝑑𝐵(𝑠)‖‖‖²]≤𝑐₃Σ𝑒^{−2𝛾(𝑛−1)}.(4.40), 𝐾_𝛼∶=^∞0‖‖𝐾(𝑠)‖‖𝑒^𝛼𝑠𝑑𝑠≥𝑒^𝛼𝑡^∞𝑡‖‖𝐾(𝑠)‖‖𝑑𝑠≥𝑒^𝛼𝑡‖‖𝐾₁(𝑡)‖‖.(4.41) and 𝔼‖𝑋_∞‖²<∞:^𝑛𝑛−1‖‖𝐾₁(𝑠)‖‖𝑑𝑠²𝔼‖‖𝑋_∞‖‖²≤𝔼‖‖𝑋_∞‖‖²^𝑛𝑛−1𝐾_𝛼𝑒^−𝛼𝑠𝑑𝑠²≤𝑐₄𝑒^{−2𝛼(𝑛−1)}.(4.42)Next, we obtain an exponential upper bound on the third term. Using (4.21) and the Burkholder-Davis-Gundy inequality, there exists a constant 𝔼sup_{𝑛−1≤𝑡≤𝑛}‖‖𝑋(𝑡)−𝑋_∞‖‖²≤𝑚(𝑋⁰)𝑒^{−2𝜂(𝑛−1)},(4.43) such that𝑚(𝑋₀)=4(𝑚(𝑋₀)+𝑐₁(𝑋₀)+𝑐₂(𝑋₀)+𝑐₃Σ+𝑐₄)Now consider the last term on the right hand side of (4.35). Using (3.4) we see that𝜂<min(2𝜆,𝛼)Using this and the fact that 𝑅^(𝑠)=𝐴𝑅(𝑠)+(𝐾∗𝑅)(𝑠) (see (4.16)) we obtainΦ(𝑡−𝑠)Combining (4.36), (4.38), (4.39), (4.40), and (4.42) we obtainΦ(𝑡)=𝑃+𝑒^−𝑡𝑄where [0,𝑡] and 𝑅_∞.
We can now apply the line of reasoning used in [10, Theorem 4.4.2] to obtain (3.10).

Proof of Lemma 4.4. We use a reformulation of (1.2a)-(1.2b) in the proof of this result. It is obtained as follows: multiply both sides of 𝑌(𝑡)+(𝐹∗𝑌)(𝑡)=𝐺(𝑡),𝑡≥0,(4.44) across by the function 𝑌=𝑅−𝑅_∞, where 𝐹, integrate over 𝐺, use integration by parts, add and subtract 𝐺(𝑡)=𝑒^−𝑡𝑄−𝑒^−𝑡(𝑄𝑅_∞+𝑄𝐴𝑅_∞)+^∞𝑡^∞𝑠−𝑃𝐾(𝑢)𝑅_∞𝑑𝑢𝑑𝑠_∞𝑡𝑄𝐾(𝑢)𝑅_∞𝑑𝑢−(𝑒∗𝑄𝐾𝑅_∞)(𝑡),𝑡≥0.(4.45) from both sides to obtain𝑌where 𝑌(𝑡)=𝐺(𝑡)−^𝑡0𝑟(𝑡−𝑠)𝐺(𝑠)𝑑𝑠,(4.46), 𝑟 is defined by (4.12) and 𝑟+𝐹∗𝑟=𝐹 is defined by𝑟+𝑟∗𝐹=𝐹
Consider the reformulation of (1.2a)-(1.2b) given by (4.44). It is well known that 𝐺 can be expressed as𝑟where the function 𝑝 satisfies 𝑝 and 𝐾. We refer the reader to [11] for details. Consider the first term on the right hand side of (4.46). As (3.1) holds it is clear that the function Σ is integrable. Now consider the second term. Since (3.3) and (4.4) hold we may apply Lemma 4.9 to obtain (4.11). Now we may apply a result of Paley and Wiener (see [11]) to see that 𝛼>0 is integrable. The convolution of an integrable function with an integrable function is itself integrable. Now combining the arguments for the first and second terms we see that (4.2) must hold.

5. On the Necessity of (3.5) for Exponential Convergence of Solutions of (1.4a)-(1.4b)

In this section, the necessity of condition (3.5) for exponential convergence in the almost sure and 𝑋₀th mean senses is shown. Proposition 5.1 concerns the necessity of the condition in the almost sure case while Proposition 5.2 deals with the 𝑋_∞(𝑋₀,Σ)th mean case.

Proposition 5.1. Let 𝑡↦𝑋(𝑡;𝑋₀,Σ) satisfy (2.4) and (4.4) and 𝛾>0 satisfy (2.6). If there exists a constant 𝑋₀ such that (3.4) holds, and if for all 𝐾 there is a constant vector Σ such that the solution 𝛼>0 of (1.4a)-(1.4b) satisfies statement (iii) of Theorem 3.2, then there exists a constant 𝑋₀, independent of 𝑋_∞(𝑋₀,Σ), such that (3.5) holds.

Proposition 5.2. Let 𝑡↦𝑋(𝑡;𝑋₀,Σ) satisfy (2.4) and (4.4) and 𝛾>0 satisfy (2.6). If there exists a constant 𝑋₀ such that (3.4) holds, and if for all 𝐾 there is a constant vector 𝑋₀ such that the solution ℱ^𝐵(∞) of (1.4a)-(1.4b) satisfies statement (ii) of Theorem 3.2, then there exists a constant 𝑋_∞(𝑋₀,Σ), independent of 𝔼‖𝑋_∞‖²<∞, such that (3.5) holds.

In order to prove these propositions the integral version of (1.4a)-(1.4b) is considered. By reformulating this version of the equation an expression for a term related to the exponential integrability of the perturbation is found. Using various arguments, including the Martingale Convergence Theorem in the almost sure case, this term is used to show that (3.5) holds.

Some supporting results are now stated. Lemma 5.3 is the analogue of Lemma 4.7 in the mean square case. It was proved in [8].

Lemma 5.3. Let 𝑡↦𝑋(𝑡;𝑋₀,Σ) satisfy (2.4) and (4.4). Suppose that for all initial conditions 𝔼‖‖𝑋li(m⋅;_𝑡→∞𝑋𝔼₀‖‖𝑋(𝑡;𝑋₀,Σ)−𝑋_∞(𝑋₀,Σ)‖‖²=0,,Σ)−𝑋_∞(𝑋₀,Σ)‖‖²∈𝐿¹([0,∞),ℝ).(5.1) there is a 𝑋_∞-measurable and almost surely finite random variable 𝐴+^∞0𝐾(𝑠)𝑑𝑠𝑋_∞=0a.s.(5.2) with 𝑁=(𝑁₁,…,𝑁_𝑛) such that the solution 𝑁_𝑖∼𝒩(0,𝑣^2𝑖) of (1.4a)-(1.4b) satisfies 𝑖=1,…,𝑛Then {𝑣_𝑖}^𝑛𝑖=1 obeys 𝑑₁>0

Lemma 5.4 may be extracted from [4]; it is required in the proof of Proposition 5.2.

Lemma 5.4. Let 𝔼‖‖𝑁‖‖²≤𝑑₁𝔼‖‖𝑁‖‖]².(5.3) where 0<𝛾<𝛼⋀𝛽₀ for 𝑍(𝑡)=𝑒^𝛾𝑡𝑋(𝑡). Then there exists a 𝜅(𝑡)=𝑒^𝛾𝑡𝐾(𝑡)-independent constant 𝑑𝑍(𝑡)=(𝛾𝐼+𝐴)𝑍(𝑡)+^𝑡0𝜅(𝑡−𝑠)𝑍(𝑠)𝑑𝑠𝑑𝑡+𝑒^𝛾𝑡Σ(𝑡)𝑑𝐵(𝑡),(5.4) such that 𝑍(𝑡)−𝑍(0)=(𝛾𝐼+𝐴)^𝑡0𝑍(𝑠)𝑑𝑠+^𝑡0^𝑠0𝜅(𝑠−𝑢)𝑍(𝑢)𝑑𝑢𝑑𝑠+^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠).(5.5)

Proof of Proposition 5.1. In order to prove this result we follow the argument used in [4, Theorem 4.1]. Let 𝑍(𝑡)=𝑒^𝛾𝑡𝑋(𝑡). By defining the process ^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)=𝑒^𝛾𝑡𝑋(𝑡)−𝑋₀−(𝛾𝐼+𝐴)^𝑡0𝑒^𝛾𝑠𝑋(𝑠)𝑑𝑠−^𝑡0𝑒^𝛾𝑠^𝑠0𝐾(𝑠−𝑢)𝑋(𝑢)𝑑𝑢𝑑𝑠.(5.6) and the matrix 𝑋_∞ we can rewrite (1.4a)-(1.4b) as^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)=−𝑒^𝛾𝑡(𝑋(𝑡)−𝑋_∞)−(𝑋₀−𝑋_∞)−(𝛾𝐼+𝐴)^𝑡0𝑒^𝛾𝑠(𝑋(𝑠)−𝑋_∞)𝑑𝑠_𝑡0𝑒^𝛾𝑠^𝑠0𝐾(𝑠−𝑢)(𝑋(𝑢)−𝑋_∞)𝑑𝑢𝑑𝑠+^𝑡0𝑒^𝛾𝑠𝐾₁(𝑠)𝑑𝑠𝑋_∞.(5.7)the integral form of which is𝛾<𝛽₀Using 𝛾<𝛽₀ and rearranging this becomes𝑒_𝛾(𝑋−𝑋_∞)∈𝐿¹[0,∞)Adding and subtracting 𝑡→∞ from the right hand side and applying Lemma 4.7 we obtain:0<𝛾<𝛼⋀𝛽₀Consider each term on the right hand side of (5.7). We see that the first term tends to zero as (3.10) holds and 𝛾₁>0. The second term is finite by hypothesis. Again, using the fact that 𝛾<𝛾₁<𝛼⋀𝛽₀ and that assumption (3.10) holds we see that 𝑡↦𝑒^𝛾1𝑡𝐾(𝑡), so the third term tends to a limit as 𝑡↦𝑒^𝛾1𝑡(𝑋(𝑡)−𝑋_∞). Now consider the fourth term. Since ‖‖‖^𝑠0𝐾(𝑠−𝑢)(𝑋(𝑢)−𝑋_∞)𝑑𝑢‖‖‖≤𝑐𝑒^−𝛾1𝑠.(5.8), we can choose 𝑡→∞ such that 𝑡→∞. So the functions lim_𝑡→∞^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)existsandisalmostsurelyfinite.(5.9) and 𝛾<𝛼⋀𝛽₁ are both integrable. The convolution of these two integrable functions is itself an integrable function, so𝔼‖‖‖^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)‖‖‖≤+𝔼[‖‖𝑒𝛾𝐼^𝛾𝑡‖‖+𝑋𝐴(𝑡)‖‖−_𝑡0𝑋_∞‖‖]+𝔼[‖‖𝑋₀−𝑋_∞‖‖]+𝔼[𝑒^𝛾𝑠‖‖𝑋(𝑠)−𝑋_∞‖‖]𝑑𝑠_𝑡0𝑒^𝛾𝑠^𝑠0+‖‖𝐾(𝑢)‖‖𝔼[‖‖𝑋(𝑠−𝑢)−𝑋_∞‖‖]𝑑𝑢𝑑𝑠_𝑡0𝑒^𝛾𝑠‖‖𝐾₁(𝑠)‖‖𝑑𝑠𝔼‖‖𝑋_∞‖‖.(5.10)Thus, it is clear that the fourth term has a finite limit as 𝑚₁. Finally, the fifth term on the right hand side of (5.7) has a finite limit at infinity, using (4.41).
Each term on the right hand side of the inequality has a finite limit as 𝔼[𝑒^𝛾𝑡‖‖𝑋(𝑡)−𝑋_∞‖‖]≤𝑚₁𝑒^{−(𝛽1−𝛾)𝑡},(5.11), so therefore[0,∞)The Martingale Convergence Theorem [12, Proposition 5.1.8] may now be applied component by component to obtain (3.5).

Proof of Proposition 5.2. By Lemma 5.3, (5.7) still holds. Define 0<𝛾<𝛼⋀𝛽₁, take norms and expectations across (5.7) to obtain𝛾₁>0There exists 𝛾<𝛾₁<𝛼⋀𝛽₁ such that𝑡↦𝑒^𝛾1𝑡𝐾(𝑡)thus the first, second and third terms on the right hand side of (5.10) are uniformly bounded on 𝑡↦𝑒^𝛾1𝑡𝔼‖𝑋(𝑡)−𝑋_∞‖. Now consider the fourth term. Since ^𝑠0‖‖𝐾(𝑠−𝑢)‖‖𝔼‖‖𝑋(𝑢)−𝑋_∞‖‖𝑑𝑢≤𝑐𝑒^−𝛾1𝑠,(5.12), we can choose [0,∞) such that ^𝑡0𝑒^𝛾𝑠‖‖𝐾₁(𝑠)‖‖𝑑𝑠𝔼‖‖𝑋_∞‖‖≤𝐾_𝛼𝔼‖‖𝑋_∞‖‖^𝑡0𝑒^{−(𝛼−𝛾)𝑠}𝑑𝑠<∞,(5.13) so that the functions 𝛾<𝛼 and 𝑐>0 are both integrable. The convolution of two integrable functions is itself an integrable function, so𝔼‖‖‖^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)‖‖‖≤𝑐.(5.14)so it is clear that the fourth term is uniformly bounded on ‖‖‖^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)‖‖‖²=^𝑛_𝑖=1𝑁_𝑖(𝑡)²,(5.15). Finally, we consider the final term on the right hand side of (5.10). Using (4.41) we obtain𝑁_𝑖(𝑡)=^𝑑_𝑗=1^𝑡0𝑒^𝛾𝑠Σ_𝑖𝑗(𝑠)𝑑𝐵_𝑗(𝑠).(5.16)since 𝑁_𝑖(𝑡). Thus there is a constant 𝑣_𝑖(𝑡)²=^𝑑_𝑗=1^𝑡0𝑒^2𝛾𝑠Σ_𝑖𝑗(𝑠)²𝑑𝑠.(5.17) such that^𝑡0𝑒^2𝛾𝑠‖‖Σ(𝑠)‖‖²𝑑𝑠=^𝑛_𝑑𝑖=1_𝑗=1^𝑡0=𝑒^2𝛾𝑠||Σ_𝑖𝑗(𝑠)||²𝑑𝑠_𝑛_𝑖=1𝑣_𝑖(𝑡)²=≤𝑑𝔼₁‖‖‖^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)‖‖‖²𝔼‖‖‖^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)‖‖‖²≤𝑑₁𝑐².(5.18)The proof now follows the line of reasoning found in [4, Theorem 4.3]: observe that𝑡→∞where𝑝It is clear that 𝐾 is normally distributed with zero mean and variance given byΣLemma 5.4 and (5.14) may now be applied to obtain:𝑓Allowing 𝑅_∞ on both sides of this inequality yields the desired result.

6. Sufficient Conditions for Exponential Convergence of Solutions of (1.1a)-(1.1b)

In this section, sufficient conditions for exponential convergence of solutions of (1.1a)-(1.1b) to a nontrivial limit are found. Proposition 6.1 concerns the 𝑅th mean sense while Proposition 6.2 deals with the almost sure case.

Proposition 6.1. Let 𝛼>0 satisfy (2.4) and (3.1), let 𝜌>0 satisfy (2.6), let 𝛾>0 satisfy (2.5), and let 𝛽^∗𝑝>0 be a constant matrix such that the solution 𝑋₀ of (1.2a)-(1.2b) satisfies (3.3). If there exist constants 𝑚^∗𝑝=𝑚^∗𝑝(𝑋₀)>0, 𝐾 and Σ such that (3.4), (3.6) and (3.5) hold, then there exist constants 𝑓, independent of 𝑅_∞, and 𝑅, such that statement (ii) of Theorem 3.1 holds.

Proposition 6.2. Let 𝛼>0,𝜌>0 satisfy (2.4) and (3.1), let 𝛾>0 satisfy (2.6), let 𝛽^∗0>0 satisfy (2.5), and let 𝑋₀ be a constant matrix such that the solution 𝑓∶=0 of (1.2a)-(1.2b) satisfies (3.3). If there exist constants 𝑋_∞ and 𝐾 such that (3.4), (3.6) and (3.5) hold, then there exists constant Σ, independent of 𝑅 such that statement (iii) of Theorem 3.1 holds.

As in the case where 𝑋 we require an explicit formulation for 𝑋→𝑋_∞(𝑋₀,Σ,𝑓). The proof of this result follows the line of reasoning used in the proof of Theorem 4.6 and is therefore omitted.

Theorem 6.3. Let 𝑋_∞(𝑋₀,Σ,𝑓)=𝑋_∞(𝑋₀,Σ)+𝑅_∞^∞0𝑓(𝑡)𝑑𝑡,a.s.(6.1) satisfy (2.4) and (4.4), let 𝑋_∞(𝑋₀,Σ,𝑓) satisfy (2.6) and (4.5), and let f satisfy (2.5). Suppose that the resolvent 𝔼‖𝑋_∞(𝑋₀,Σ,𝑓)‖^𝑝 of (1.2a)-(1.2b) satisfies (3.3), then the solution 𝔼‖‖𝑋_∞(𝑋₀,Σ,𝑓)‖‖^𝑝≤2^𝑝−1𝔼‖‖𝑋^∞(𝑋₀,Σ)‖‖^𝑝+‖‖‖^∞0𝑅_∞𝑓(𝑠)𝑑𝑠‖‖‖^𝑝<∞.(6.2) of (1.1a)-(1.1b) satisfies 𝑋(⋅;𝑋₀,Σ,𝑓) almost surely, where 𝑋_∞(𝑋₀,Σ,𝑓)and 𝑋(𝑡;=𝑋₀(,𝑋Σ(,𝑡𝑓;𝑋)−₀𝑋_∞(𝑋₀,Σ,𝑓),Σ)−𝑋_∞(𝑋₀,Σ))+^𝑡0(𝑅(𝑡−𝑠)−𝑅_∞)𝑓(𝑠)𝑑𝑠−^∞𝑡𝑅_∞𝑓(𝑠)𝑑𝑠.(6.3) is almost surely finite.

Proof of Proposition 6.1. We begin by showing that 𝑋(=𝑡;𝑋(𝑋₀,(Σ𝑡;,𝑋𝑓₀)−𝑋_∞(𝑋₀,Σ,𝑓),Σ)−𝑋_∞(𝑋₀,Σ))−𝑓₁(𝑡)+(𝑅(𝑡)−𝑅_∞)𝑓₁(0)−^𝑡0𝑅^(𝑡−𝑠)𝑓₁(𝑠)𝑑𝑠.(6.4) is finite. Clearly, we see that𝑝Now, consider the difference between the solution 𝔼‖‖𝑋(𝑡;𝑋₀,Σ,𝑓)−𝑋_∞(𝑋₀,Σ,𝑓)‖‖^𝑝≤4^𝑝−1𝔼‖‖𝑋(𝑡;𝑋⁰,Σ)−𝑋_∞(𝑋₀,Σ)‖‖^𝑝+‖‖𝑓₁(𝑡)‖‖^𝑝+‖‖𝑅(𝑡)−𝑅_∞‖‖^𝑝‖‖𝑓₁(0)‖‖^𝑝+^𝑡0‖‖𝑅^(𝑡−𝑠)‖‖‖‖𝑓₁(𝑠)‖‖𝑑𝑠^𝑝.(6.5) of (1.1a)-(1.1b) and its limit 𝑅^ given by (6.1):𝛽^∗𝑝<min(𝛽_𝑝,𝛽,𝜌)Using integration by parts this expression becomes𝑅^Taking norms on both sides of equation (6.4), raising the power to 𝑅^(𝑡)=𝐴(𝑅(𝑡)−𝑅_∞)+^𝑡0𝐾(𝑡−𝑠)(𝑅(𝑠)−𝑅_∞)𝑑𝑠−𝐾₁(𝑡)𝑅_∞+𝐴+^∞0𝐾(𝑠)𝑑𝑠𝑅_∞.(6.6) on both sides, and taking expectations across we obtain𝑅Now consider the right hand side of (6.5). The first term decays exponentially quickly due to Theorem 3.2. The second term decays exponentially quickly due to assumption (3.6). By applying Lemma 4.4 we see that (4.2) holds so we can apply Theorem 4.5 to show that the third term must decay exponentially. In the sequel, an argument is provided to show that 𝑅_∞ decays exponentially; thus the final term must decay exponentially. Combining these arguments we see that (3.7) holds, where 𝑅−𝑅_∞.
It is now shown that 𝜇 decays exponentially. It is clear from the resolvent equation (1.2a)-(1.2b) that𝑡↦𝑒^𝜇𝑡𝐾(𝑡)Consider each term on the right hand side of (6.6). We can apply Theorem 4.5 to obtain that 𝑡↦𝑒^𝜇𝑡(𝑅(𝑡)−𝑅_∞) decays exponentially quickly to 𝐿¹([0,∞),ℝ^𝑛×𝑛). In order to show that the second term decays exponentially we proceed as follows: since 𝑒^𝜇𝑡‖‖‖^𝑡0𝐾(𝑡−𝑠)(𝑅(𝑠)−𝑅_∞)𝑑𝑠‖‖‖=‖‖‖^𝑡0𝑒^{𝜇(𝑡−𝑠)}𝐾(𝑡−𝑠)𝑒^𝜇𝑠(𝑅(𝑠)−𝑅_∞)𝑑𝑠‖‖‖≤𝑐.(6.7) decays exponentially and (3.4) holds it is possible to choose (𝐴+∫^∞0𝐾(𝑠)𝑑𝑠)𝑅_∞=0 such that the functions 𝑅^ and 0 are both in ‖‖𝑋≤(𝑡;‖‖𝑋𝑋₀(,𝑡;Σ𝑋,𝑓₀)−𝑋_∞(𝑋₀,Σ,𝑓)‖‖,Σ)−𝑋_∞(𝑋₀,Σ)‖‖+‖‖𝑓₁(𝑡)‖‖+‖‖𝑅(𝑡)−𝑅_∞‖‖‖‖𝑓₁(0)‖‖+‖‖‖^𝑡0𝑅^(𝑡−𝑠)𝑓₁(𝑠)𝑑𝑠‖‖‖.(6.8). The convolution of two integrable functions is itself an integrable function, so𝑅^To see that the third term decays exponentially we use (4.41). Finally, we consider the fourth term. By Lemma 4.4 and (3.3) we have that (4.2) holds. In [1, Theorem 6.1] it was shown that 𝛽^∗≤min(𝛽₀,𝛽,𝜌) under this hypothesis and (3.1). Combining the above we see that 𝑝 decays exponentially quickly to 𝑝.

Proof of Proposition 6.2. Take norms across (6.4) to obtain𝐾Using Theorem 3.2, we see that the first term on the right hand side of (6.8) decays exponentially. The second term on the right hand side decays exponentially as (3.6) holds. We can apply Theorem 4.5 to show that the third term must decay exponentially. An argument was provided in Proposition 6.1 to show that Σ decays exponentially. Combining this with (3.6) enables us to show that the fourth term decays exponentially. Using the above arguments we obtain (3.8), where 𝑓.

7. On the Necessity of (3.6) and (3.5) for Exponential Convergence of Solutions of (1.1a)-(1.1b)

In this section, the necessity of (3.6) and (3.5) for exponential convergence of solutions of (1.1a)-(1.1b) in the almost sure and 𝛼>0th mean senses is shown. Proposition 7.1 concerns the necessity of the conditions in the 𝑋₀th mean case while Proposition 7.2 deals with the almost sure case.

Proposition 7.1. Let 𝑋_∞(𝑋₀,Σ,𝑓) satisfy (2.4) and (4.4), let 𝑡↦𝑋(𝑡;𝑋₀,Σ,𝑓) satisfy (2.6), and let 𝜌>0 satisfy (2.5). If there exists constant 𝛾>0 such that (3.4) holds, and if for all 𝑋₀ there is constant vector 𝐾 such that the solution Σ of (1.1a)-(1.1b) satisfies statement (ii) of Theorem 3.1, then there exist constants 𝑓 and 𝛼>0, independent of 𝑋₀, such that (3.6) and (3.5) hold.

Proposition 7.2. Let 𝑋_∞(𝑋₀,Σ,𝑓) satisfy (2.4) and (4.4), let 𝑡↦𝑋(𝑡;𝑋₀,Σ,𝑓) satisfy (2.6), and let 𝜌>0 satisfy (2.5). If there exists constant 𝛾>0 such that (3.4) holds, and if for all 𝑋₀ there is a constant vector 𝑐>0 such that the solution ‖‖𝑓₁(𝑡)+𝜇₁(𝑡,𝜔)‖‖≤𝑐(𝜔)𝑒^−𝜆𝑡,(7.1) of (1.1a)-(1.1b) satisfies statement (iii) of Theorem 3.1, then there exist constants 𝜆>0 and 𝜔∈Ω^∗, independent of ℙ[Ω^∗]=1, such that (3.6) and (3.5) hold.

The following lemma is used in the proof of Proposition 7.2. This lemma allows us to separate the behavior of the deterministic perturbation from the stochastic perturbation in the almost sure case. It is interesting to note that we can prove this lemma without any reference to the integro-differential equation.

Lemma 7.3. Suppose 𝑓₁ is an almost surely finite random variable and 𝜇₁where 𝜇₁(𝑡)=^∞𝑡Σ(𝑠)𝑑𝐵(𝑠),𝑡≥0,(7.2), {𝜉_𝑖}^∞𝑖=1, 𝔼[𝜉_𝑖]=0 and the functions 𝔼[𝜉^2𝑖]=𝑣^2𝑖≥1 and limsup_{𝑚𝑚→∞}_𝑖=1𝜉_𝑖=∞,liminf_{𝑚𝑚→∞}_𝑖=1𝜉_𝑖=−∞,a.s.(7.3) are defined by (2.8) and 𝛾>0respectively. Then (3.5) and (3.6) hold.

In order to prove Lemma 7.3 we require Lemmas 7.4 and 7.5 below. Lemma 7.5 was proved in [13]. The proof of Lemma 7.4 makes use of Kolmogorov's Zero-One Law. It follows the proof of Theorem 2 in [14, Chapter IV, Section 1] and so is omitted.

Lemma 7.4. Let 𝜎∈𝐶([0,∞),ℝ) be a sequence of independent Gaussian random variables with ^∞0𝜎(𝑠)²𝑒^2𝛾𝑠𝑑𝑠<∞,(7.4) and limsup_𝑡→∞1𝑡log|||^∞𝑡𝜎(𝑠)𝑑𝐵(𝑠)|||≤−𝛾,a.s.(7.5). Then {𝐵(𝑡)}_𝑡≥0

Lemma 7.5. If there is a 𝐾 such that 𝑋₀ and ℱ^𝐵(∞)then 𝑋_∞(𝑋₀,Σ)where 𝔼‖𝑋_∞‖²<∞ is a one-dimensional standard Brownian motion.

Lemmas 7.6 and 7.7 are used in the proofs of Propositions 7.1 and 7.2, respectively and are the analogues of Lemmas 5.3 and 4.7. Their proofs are identical in all important aspects and so are omitted.

Lemma 7.6. Let 𝑡↦𝑋(𝑡;𝑋₀,Σ) satisfy (2.4) and (4.4). Suppose that for all initial conditions 𝔼‖‖𝑋li(m⋅;_𝑡→∞𝑋𝔼₀‖‖𝑋(𝑡;𝑋₀,Σ,𝑓)−𝑋_∞(𝑋₀,Σ,𝑓)‖‖²=0,,Σ,𝑓)−𝑋_∞(𝑋₀,Σ,𝑓)‖‖²∈𝐿¹([0,∞),ℝ).(7.6) there is an 𝑋_∞-measurable and almost surely finite random variable 𝐴+^∞0𝐾(𝑠)𝑑𝑠𝑋_∞=0a.s.(7.7) with 𝐾 such that the solution 𝑋₀ of (1.1a)-(1.1b) satisfies ℱ^𝐵(∞)Then 𝑋_∞(𝑋₀,Σ,𝑓) obeys 𝑡↦𝑋(𝑡;𝑋₀,Σ,𝑓)

Lemma 7.7. Let 𝑋(⋅;𝑋l₀im_𝑡→∞𝑋(𝑡;𝑋₀,Σ,𝑓)=𝑋_∞(𝑋₀,Σ,𝑓)a.s.,,Σ,𝑓)−𝑋_∞(𝑋₀,Σ,𝑓)∈𝐿²([0,∞),ℝ^𝑛)a.s.(7.8) satisfy (2.4) and (4.4). Suppose that for all initial conditions 𝑋_∞ there is an 𝑋₀=0-measurable and almost surely finite random variable 𝑋_∞ such that the solution Δ(𝑡)=−𝑋_∞+^𝑡0𝛿(𝑠)𝑑𝑠+^𝑡0𝑓(𝑠)𝑑𝑠+𝜇(𝑡)−^𝑡0𝐾₁(𝑠)𝑑𝑠𝑋_∞,(7.9) of (1.1a)-(1.1b) satisfies Δ(𝑡)=𝑋(𝑡)−𝑋_∞Then 𝛿 obeys (7.7).

Proof of Proposition 7.1. Since (3.7) holds for every initial condition we can choose 𝛿(𝑡)=𝐴Δ(𝑡)+(𝐾∗Δ)(𝑡),(7.10): this simplifies calculations. Moreover using (3.7) in Lemma 7.6 it is clear that assumption (7.7) holds. Consider the integral form of (1.1a)-(1.1b). Adding and subtracting 𝜇(𝑡)=∫^𝑡0Σ(𝑠)𝑑𝐵(𝑠) from both sides and applying Lemma 7.6 we obtain𝑡→∞where −𝔼𝑋_∞=−^∞0𝔼[𝛿(𝑠)]𝑑𝑠−^∞0𝑓(𝑠)𝑑𝑠+^∞0𝐾₁(𝑠)𝑑𝑠𝔼𝑋_∞,(7.11), the function 𝔼[𝛿(𝑡)]=𝐴𝔼[Δ(𝑡)]+(𝐾∗𝔼[Δ])(𝑡) is defined by𝔼[𝑋_∞]and 𝑓₁(𝑡)=−𝔼Δ(𝑡)−^∞𝑡𝔼[𝛿(𝑠)]𝑑𝑠+^∞𝑡𝐾₁(𝑠)𝑑𝑠𝔼𝑋_∞,(7.12). Taking expectations across (7.9) and allowing 𝔼[𝛿(⋅)] we obtain𝑓₁where ‖‖‖^∞0𝑒^𝛾𝑡𝑓(𝑡)𝑑𝑡‖‖‖<∞,(7.13). Using this expression for 𝛾>0 we obtain𝛾<𝛼⋀𝛽₁The first term on the right-hand side of (7.12) decays exponentially due to (3.7). Assumptions (3.4) and (3.7) imply that 𝑒^𝛾𝑡Δ(𝑡)=+Δ(0)+(𝛾𝐼+𝐴)^𝑡0𝑒^𝛾𝑠Δ(𝑠)𝑑𝑠_𝑡0𝑒^𝛾𝑠^𝑠0𝐾(𝑠−𝑢)Δ(𝑢)𝑑𝑢𝑑𝑠−^𝑡0𝑒^𝛾𝑠𝐾₁(𝑠)𝑑𝑢𝑑𝑠𝑋_∞+^𝑡0𝑒^𝛾𝑠𝑓(𝑠)𝑑𝑠+^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠).(7.14) decays exponentially so the second term decays exponentially. The third term on the right-hand side of (7.12) decays exponentially due to the argument given by (4.41). Hence, ‖‖‖^𝑡0𝑒^𝛾𝑠𝑓(𝑠)𝑑𝑠‖‖‖≤+𝔼‖‖‖‖𝑒𝛾𝐼^𝛾𝑡Δ+(𝑡𝐴)‖‖‖‖+_𝑡0𝔼‖‖Δ(0)‖‖+𝑒^𝛾𝑠𝔼‖‖Δ(𝑠)‖‖𝑑𝑠_𝑡0𝑒^𝛾𝑠^𝑠0‖‖𝐾(𝑠−𝑢)‖‖𝔼‖‖Δ(𝑢)‖‖𝑑𝑢𝑑𝑠+^𝑡0𝑒^𝛾𝑠‖‖𝐾₁(𝑠)‖‖𝑑𝑠𝔼‖‖𝑋_∞‖‖.(7.15) decays exponentially.
Proving that (3.5) holds breaks into two steps. We begin by showing that0<𝛾<𝛼⋀𝛽₁where 𝛾₁>0. By choosing 𝛾<𝛾₁<𝛼⋀𝛽₁ we can obtain the following reformulation of (1.1a)-(1.1b) using methods applied in [15, Proposition 5.1] ^𝑠0‖‖𝐾(𝑠−𝑢)‖‖𝔼‖‖Δ(𝑢)‖‖𝑑𝑢≤𝑐𝑒^−𝛾1𝑠.(7.16)Rearranging (7.14), taking expectations and then norms on both sides we can obtain𝔼‖‖‖^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)‖‖‖≤+𝔼‖‖‖‖𝑒𝛾𝐼^𝛾𝑡Δ+(𝑡𝐴)‖‖‖‖+_𝑡0𝔼‖‖Δ(0)‖‖𝑒^𝛾𝑠𝔼‖‖Δ(𝑠)‖‖𝑑𝑠+^𝑡0𝑒^𝛾𝑠^𝑠0+‖‖𝐾(𝑠−𝑢)‖‖𝔼‖‖Δ(𝑢)‖‖𝑑𝑢𝑑𝑠_𝑡0𝑒^𝛾𝑠‖‖𝐾₁(𝑢)‖‖𝑑𝑢𝑑𝑠𝔼‖‖𝑋_∞‖‖+‖‖‖^𝑡0𝑒^𝛾𝑠𝑓(𝑠)𝑑𝑠‖‖‖.(7.17)Since (3.7) holds this implies that both the first and third terms on the right-hand side of (7.15) are bounded. The second term is bounded due to our assumptions. Since 𝔼‖‖‖^𝑡0𝑒^𝛾𝑠Σ(𝑠)𝑑𝐵(𝑠)‖‖‖≤𝐶.(7.18), we can choose 𝑡→∞ such that −𝑋_∞=−^∞0𝛿(𝑠)𝑑𝑠−^∞0𝑓(𝑠)𝑑𝑠−𝜇(∞)+^∞0𝐾₁(𝑠)𝑑𝑠𝑋_∞,(7.19). It can easily be shown that𝛿Finally, we see that the fifth term is bounded using (4.41). So, (7.13) holds.
We now return to (7.14). Again rearranging the equation and taking norms and then expectations across both sides, we obtain𝑋_∞We already provided an argument above to show that the first five terms on the right hand side of this expression are bounded. Also, we know that (7.13) holds. Thus,Δ(𝑡)=−^∞𝑡𝛿(𝑠)𝑑𝑠−𝑓₁(𝑡)−𝜇₁(𝑡)+^∞𝑡𝐾₁(𝑠)𝑑𝑠𝑋_∞,(7.20)The proof is now identical to Proposition 5.2.

Proof of Proposition 7.2. Since Lemma 7.7 holds we can obtain (7.9). Thus, as 𝜇₁(𝑡)=∫^∞𝑡Σ(𝑠)𝑑𝐵(𝑠), we obtain‖‖𝑓₁(𝑡)+𝜇₁(𝑡)‖‖≤‖‖Δ(𝑡)‖‖+‖‖‖^∞𝑡𝛿(𝑠)𝑑𝑠‖‖‖+‖‖‖^∞𝑡𝐾₁(𝑠)𝑑𝑠𝑋_∞‖‖‖.(7.21)where ∫^∞𝑡𝐴Δ(𝑠)𝑑𝑠 is defined by (7.10). Using this expression for ∫^∞𝑡(𝐾∗Δ)(𝑠)𝑑𝑠, (7.9) becomes𝑐>0where ‖‖𝑓₁(𝑡)+𝜇₁(𝑡)‖‖≤𝑐𝑒^−𝜆𝑡∀𝑡≥0,a.s.,(7.22). Rearranging the equation and taking norms yields𝜆<min(𝛽^∗0,𝛼)The first term on the right hand side of (7.21) decays exponentially due to (3.8). Using the argument given in (4.41) we see that the third term on the right hand side of (7.21) decays exponentially. Finally, we consider the second term. Clearly 𝛾 decays exponentially due to (3.8). In order to show that 1≤𝑖≤𝑛 decays exponentially we use an argument similar to that applied in the proof of Proposition 7.1. So there is an almost surely finite random variable 1≤𝑗≤𝑑 such that^∞0Σ_𝑖𝑗(𝑠)²𝑒^2𝛾𝑠𝑑𝑠<∞.(7.23)where limsup_𝑡→∞1𝑡log|||^∞𝑡Σ_𝑖𝑗(𝑠)𝑑𝐵_𝑗(𝑠)|||≤−𝛾,𝜔∈Ω_𝑖𝑗,ℙ[Ω_𝑖𝑗]=1.(7.24). We can now apply Lemma 7.3 to obtain (3.6) and (3.5).

Proof of Lemma 7.3. We suppose that there exists a constant 𝜖∈(0,𝛾) such that (3.5) holds. Using the equivalence of norms we see that for all 𝜔∈Ω_𝑖𝑗 and 𝑐_𝑖𝑗(𝜔,𝜖)≥1 assumption (3.5) implies that|||^∞𝑡Σ_𝑖𝑗(𝑠)𝑑𝐵_𝑗(𝑠)|||≤𝑐_𝑖𝑗(𝜔,𝜖)𝑒^{−(𝛾−𝜖)𝑡}.(7.25)Applying Lemma 7.5 we obtain𝑗Choose any ||𝜇^𝑖1(𝑡)||≤𝑐_𝑖(𝜔,𝜖)𝑒^{−(𝛾−𝜖)𝑡},(7.26). For each 𝜔∈Ω_𝑖=∩^𝑑𝑗=1Ω_𝑖𝑗 we can choose a constant 𝑐_𝑖=∑^𝑑𝑗=1𝑐_𝑖𝑗 such that𝜇^𝑖1(𝑡)=∑^𝑑𝑗=1∫^∞𝑡Σ_𝑖𝑗(𝑠)𝑑𝐵_𝑗(𝑠)Now, summing over ||𝑓^𝑖1(𝑡)+𝜇^𝑖1(𝑡)||²≤^𝑛_𝑖=1||𝑓^𝑖1(𝑡)+𝜇^𝑖1(𝑡)||²=‖‖𝑓₁(𝑡)+𝜇₁(𝑡)‖‖²(7.27) we see that||𝑓^𝑖1(𝑡)+𝜇^𝑖1(𝑡)||≤𝑐(𝜔)𝑒^−𝜆𝑡,𝜔∈Ω^∗.(7.28)where 𝜔∈Ω_𝑖∩Ω^∗, |𝑓^𝑖1(𝑡)|≤|𝑓^𝑖1(𝑡)+𝜇^𝑖1(𝑡)|+|𝜇^𝑖1(𝑡)|≤𝑐(𝜔)𝑒^−𝜆𝑡+𝑐_𝑖(𝜔,𝜖)𝑒^{−(𝛾−𝜖)𝑡} and ||𝑓^𝑖1(𝑡)||≤𝑐_𝑖(𝜔)𝑒^−𝜌𝑡,𝜔∈Ω^∗∩Ω_𝑖,(7.29).
Now, since𝑐_𝑖>0we see that𝜌≤max(𝜆,𝛾−𝜖)So for 𝑖 we see that 𝜔∈∩_𝑖(Ω^∗∩Ω_𝑖). This gives𝛾where 0<𝛾<𝜆 is finite and 𝑑. Now summing over 𝑑(𝑡)=^𝑡01𝛾(𝑒^𝛾𝑠−1)𝑓(𝑠)𝑑𝑠,(7.30) we obtain (3.6), by picking out any 𝑀. This concludes the case when (3.5) holds.
Now, consider the case where assumption (3.5) fails to hold. We choose a constant 𝑀(𝑡)=^𝑡01𝛾(𝑒^𝛾𝑠−1)Σ(𝑠)𝑑𝐵(𝑠).(7.31) such that 𝑀_𝑖 and define the function 𝑖 as𝑀and the vector martingale ⟨𝑀_𝑖⟩ as𝑀_𝑖We let ⟨𝑀_𝑖⟩(𝑡)=^𝑑_𝑗=1^𝑡01𝛾²(𝑒^𝛾𝑠−1)²Σ_𝑖𝑗(𝑠)²𝑑𝑠.(7.32) denote the ‖‖𝑑(𝑡)+𝑀(𝑡)‖‖<𝑐^∗(𝜔),𝜔∈Ω^∗,ℙ[Ω^∗]=1,(7.33)th component of 𝑖,1≤𝑖≤𝑛 and 𝑀 denote the quadratic variation of lim_𝑡→∞⟨𝑀_𝑖⟩(𝑡)=∞.(7.34) given byliminf_𝑡→∞𝑀_𝑖(𝑡)=−∞We show at the end of this proof thatlimsup_𝑡→∞𝑀_𝑖(𝑡)=∞and therefore assume it for the time being.
Since (3.5) fails to hold there exists an entry 𝑖, of the martingale 𝑑 such that𝑑_𝑖It follows that 𝑑_𝑖 and 𝑀_𝑖 a.s. Consider the corresponding ⟨𝑀_𝑖⟩(𝑡)th entry of 𝑑_𝑖, denoted |𝑑_𝑖(𝑡)+𝑀_𝑖(𝑡)|<𝑐^∗(𝜔); it is either bounded or unbounded. If 𝜔∈Ω^∗ is bounded then −𝑐^∗−𝑀_𝑖(𝑡)<𝑑_𝑖(𝑡) is bounded and so, by the Martingale Convergence Theorem, ∞=−𝑐^∗−liminf_𝑡→∞𝑀_𝑖(𝑡)≤limsup_𝑡→∞𝑑_𝑖(𝑡).(7.35) is bounded: this contradicts (7.34). So, we suppose the latter, that 𝑑 is unbounded, and proceed to show this is also contradictory. Since {𝑡_𝑚}^∞𝑚=0, for 𝑡₀=0 it is clear that 𝑑_𝑖(𝑡_𝑚)→∞. Taking the limit superior on both sides of the inequality yields𝑚→∞As 𝑀_𝑖(𝑡_𝑚)→−∞ is deterministic, there exists an increasing sequence of deterministic times 𝑚→∞ with {𝜏_𝑚}^∞𝑚=0 such that 𝜏₀=𝑡₀ as 𝑣^2𝑙∶=^𝑑_𝑗=1^𝜏𝑙_𝜏𝑙−11𝛾²(𝑒^𝛾𝑠−1)²Σ^2𝑖𝑗(𝑠)𝑑𝑠≥1.(7.36). Consequently, 𝑆_𝑖(𝑚)=𝑀_𝑖(𝜏_𝑚) as 𝑆_𝑖(𝑚)=^𝑚_𝑙=1𝜉^𝑙(𝑖)(7.37). We choose a subsequence of these times 𝜉^𝑙(𝑖)=^𝑑_𝑗=1^𝜏𝑙_𝜏𝑙−11𝛾(𝑒^𝛾𝑠−1)Σ_𝑖𝑗(𝑠)𝑑𝐵_𝑗(𝑠).(7.38) with {𝜉^𝑙(𝑖)}^∞𝑙=1 such that𝜉^𝑙(𝑖)Define 𝑣^2𝑙≥1. Obviously𝑑(𝑡)=^𝑡0𝑒^𝛾𝑠(𝑓₁(𝑠)−𝑓₁(𝑡))𝑑𝑠,𝑀(𝑡)=^𝑡0𝑒^𝛾𝑠(𝜇₁(𝑠)−𝜇₁(𝑡))𝑑𝑠.(7.39)where0<𝛾<𝜆It is clear that ‖‖𝑑(𝑡)+𝑀(𝑡)‖‖≤≤𝑐(^𝑡0𝜔𝑒)^𝛾𝑠(_𝑡0‖‖𝑓₁(𝑡)+𝜇₁(𝑡)‖‖+‖‖𝑓₁(𝑠)+𝜇₁(𝑠)‖‖)𝑑𝑠𝑒^𝛾𝑠(𝑒^−𝜆𝑡+𝑒^−𝜆𝑠)𝑑𝑠<𝑐₁(𝜔),𝜔∈Ω^∗.(7.40) is an indepenendent normally distributed sequence with the variance of each 𝐾 given by 𝑅_∞ so we may apply Lemma 7.4.
We now show that assumption (7.33) holds. By changing the order of integration we can show that𝛽>0Thus, as 𝑐>0,𝑅

8. On the Necessary and Sufficient Conditions for Exponential Convergence of Solutions of (1.1a)-(1.1b) and (1.4a)-(1.4b)

We now combine the results from Sections 4 and 5 to prove Theorem 3.2 and combine the results from Sections 6 and 7 to prove Theorem 3.1.

We showed the necessity of (3.5) for the exponential convergence of the solution of (1.4a)-(1.4b) in Section 5. In order to prove the necessity of the exponential integrability of the kernel we require the following result which was extracted from [1].

Theorem 8.1. Let 𝐾 satisfy (2.4) and (3.1). Suppose that there exists a constant matrix 𝛼>0 and constants 𝐾 and 𝑛 such that the solution {𝑋_𝑗(𝑡)}_{𝑗=1,…,𝑛} of (1.2a)-(1.2b) satisfies (4.3). If the kernel 𝑋_𝑗(0)=𝐞_𝑗 satisfies (3.2) then there exists a constant 𝑚₁(𝐞_𝑗)𝑒^−𝛽1𝑡≥𝔼‖‖𝑋_𝑗(𝑡)−𝑋_𝑗(∞)‖‖≥‖‖𝔼𝑋_𝑗(𝑡)−𝑋_𝑗(∞)‖‖=‖‖(𝑅(𝑡)−𝑅(∞))𝐞_𝑗‖‖.(8.1) such that 𝑗=1,…,𝑛 satisfies (3.4).

Proof of Theorem 3.2. We begin by proving the equivalence between (i) and (ii). The implication (i) implies (ii) is the subject of Proposition 4.1. We can demonstrate that (ii) implies (i) as follows. We begin by proving that (3.9) implies (3.4). We consider the following 𝑅 solutions of (1.4a)-(1.4b); 𝑅_∞, where 𝑋₀. Since (3.9) holds we obtain𝑛+1for each 𝑋_𝑗(𝑡)_{𝑗=1,…,𝑛+1}. Thus, the resolvent 𝑋_𝑗(0)=𝐞_𝑗for𝑗=1,…,𝑛,𝑋_𝑛+1(0)=0.(8.2) of (1.2a)-(1.2b) decays exponentially to 𝑋_𝑗(𝑡). We can apply Theorem 8.1 to obtain (3.4) after which Proposition 5.2 can be applied to obtain (3.5). As (8.1) holds it is clear that (3.3) holds.
We now prove the equivalence between (i) and (iii). The implication (i) implies (iii) is the subject of Proposition 4.2. We now demonstrate that (iii) implies (i). We begin by proving that (3.10) implies (3.4). As (3.10) holds for all 𝑋_𝑗(∞) we can consider the following 𝑆_𝑗(𝑡)=𝑋_𝑗(𝑡)−𝑋_𝑛+1(𝑡),(8.3) solutions of (1.4a)-(1.4b); 𝑆_𝑗(0)=𝐞_𝑗 where𝑆=[𝑆₁,…,𝑆_𝑛]∈𝑅^𝑛×𝑛We know that 𝑆^(𝑡)=𝐴𝑆(𝑡𝑆)+(0()𝐾=∗𝐼𝑆.)(𝑡),𝑡>0,(8.4) approaches 𝑆_𝑗(∞)=𝑋_𝑗(∞)−𝑋_𝑛+1(∞) exponentially quickly in the almost sure sense. Introduce𝑆(𝑡)→𝑆_∞and notice ‖‖𝔼[𝑋(𝑡)−𝑋_∞]‖‖≤𝔼‖‖𝑋(𝑡)−𝑋_∞‖‖≤𝑚^∗𝑒^{−𝛽∗1𝑡}.(8.5). Let 𝑛+1. Then𝑋_𝑗If we define 𝑋_𝑗(0)=𝐞_𝑗 then 𝑗=1,…,𝑛 exponentially quickly so we can apply Theorem 8.1 to obtain (3.4). As (3.4) and (3.10) hold we can apply Proposition 5.1 to obtain (3.5). Also evident from this argument is that (3.3) holds. This proves that (iii) implies (i).

Proof of Theorem 3.1. We begin by proving the equivalence between (i) and (ii). The implication that (i) implies (ii) is the subject of Proposition 6.1. Now consider the implication (ii) implies (i). Using (3.7) we see that𝑋_𝑛+1(0)=0Consider the 𝑅(𝑡)𝐞_𝑗=𝑋_𝑗(𝑡)−𝑋_𝑛+1(𝑡) solutions 𝔼[𝑋_𝑗(𝑡)−𝑋_𝑗(∞)]+𝔼[𝑋_𝑛+1(𝑡)−𝑋_𝑛+1(∞)]=𝑅(𝑡)𝐞_𝑗−𝔼[𝑐_𝑗],(8.6) of (1.1a)-(1.1b) with initial conditions 𝑐_𝑗=𝑋_𝑗(∞)−𝑋_𝑛+1(∞) for 𝑡↦𝑅(𝑡)𝐞_𝑗 and 𝔼[𝑐_𝑗]. Since 𝑡→∞ we see that𝑅where 𝑛+1 is an almost surely finite constant. As both terms on the left hand side of this expression are decaying exponentially to zero, 𝑋_𝑗(𝑡) must decay exponentially to 𝑋_𝑗(0)=𝐞_𝑗 as 𝑗=1,…,𝑛. Thus 𝑋_𝑛+1(0)=0 must satisfy (4.3). Now, apply Theorem 8.1 to obtain (3.4) and Proposition 7.1 to obtain (3.6) and (3.5).
We now prove the equivalence between (i) and (iii). The implication (i) implies (iii) is the subject of Proposition 6.2. Once again, consider the 𝑅(𝑡)𝐞_𝑗=𝑋_𝑗(𝑡)−𝑋_𝑛+1(𝑡) solutions 𝑗=1,…,𝑛 with initial conditions (𝑋_𝑗(𝑡)−𝑋_𝑗(∞))−(𝑋_𝑛+1(𝑡)−𝑋_𝑛+1(∞))=𝑅(𝑡)𝐞_𝑗−𝑐_𝑗,(8.7) for 𝑐_𝑗=𝑋_𝑗(∞)−𝑋_𝑛+1(∞) and 𝑋_𝑗. Since 𝑋_𝑗(∞) for 𝑋_𝑛+1, we can write𝑋_𝑛+1(∞)where 𝑅 is an almost surely finite random variable. From (3.8) we know that decays exponentially quickly to , similarly decays exponentially quickly to . Thus, decays exponentially to a limit. As a result (4.3) must hold. Now apply Theorem 8.1 to obtain (3.4) and Proposition 7.2 to obtain (3.6) and (3.5).

Acknowledgments

The authors are pleased to acknowledge the referees for their careful scrutiny of and suggested corrections to the manuscript. The first author was partially funded by an Albert College Fellowship, awarded by Dublin City University’s Research Advisory Panel. The second author was funded by The Embark Initiative operated by the Irish Research Council for Science, Engineering and Technology (IRCSET).