1. Introduction

In this paper, we study the exponential convergence of the solution of to a nontrivial random variable. Here the solution is an -dimensional vector-valued function on , is a real -dimensional matrix, is a continuous and integrable -dimensional matrix-valued function on , is a continuous -dimensional vector-valued function on , is a continuous -dimensional matrix-valued function on and where each component of the Brownian motion is independent. The initial condition is a deterministic constant vector.

The solution of (1.1a)-(1.1b) can be written in terms of the solution of the resolvent equation where the matrix-valued function is known as the resolvent or fundamental solution. In [1], the authors studied the asymptotic convergence of the solution of (1.2a)-(1.2b) to a nontrivial limit . It was found that 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 . A deterministically perturbed version of (1.2a)-(1.2b), was also studied in [1]. It was shown that the exponential decay of the tail of the perturbation combined with the integrability of 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 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 then (i) the almost sure exponential convergence of the solution to zero, (ii) the th 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 , 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, , 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 denote the set of real numbers and let denote the set of -dimensional vectors with entries in . Denote by the th standard basis vector in . Denote by the standard Euclidean norm for a vector given bywhere denotes the trace of a square matrix.

Let be the space of matrices with real entries where is the identity matrix. Let denote the matrix with the scalar entries on the diagonal and elsewhere. For the norm denoted by is defined by

The set of complex numbers is denoted by ; the real part of in being denoted by . The Laplace transform of the function is defined asIf and then exists for and is analytic for .

If is an interval in and a finite-dimensional normed space with norm then denotes the family of continuous functions . The space of Lebesgue integrable functions will be denoted by where . The space of Lebesgue square-integrable functions will be denoted by where . When is clear from the context, it is omitted it from the notation.

We now make our problem precise. We assume that the function satisfiesthe function satisfiesand the function satisfiesDue to (2.4) we may define to be a function withwhere this function defines the tail of the kernel . Similarly, due to (2.5), we may define to be a function given byWe let denote -dimensional Brownian motion on a complete probability space where the filtration is the natural one .

Under the hypothesis (2.4), it is well known that (1.2a)-(1.2b) has a unique continuous solution , which is continuously differentiable. We define the function to be the unique solution of the initial value problem (1.1a)-(1.1b). If and are continuous then for any deterministic initial condition there exists an almost surely unique continuous and -adapted solution to (1.1a)-(1.1b) given byWhen , and are clear from the context, we omit them from the notation.

The notion of convergence and integrability in th mean and almost sure senses are now defined: the -valued stochastic process converges in th mean to if ; the process is th mean exponentially convergent to if there exists a deterministic such thatwe say that the difference between the stochastic process and is integrable in the th mean sense ifIf there exists a -null set such that for every , the following holds: , then converges almost surely to ; we say is almost surely exponentially convergent to if there exists a deterministic such thatFinally, the difference between the stochastic process and is square integrable in the almost sure sense ifHenceforth, will be denoted by 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 , the finite-dimensional random variables and satisfy and , respectively, then the Lyapunov inequality is useful when considering the th mean behaviour of random variables as any exponent may be considered:The following proves useful in manipulating norms:

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 satisfy (2.4) and let satisfy (2.6), and let satisfy (2.5). If satisfies then the following are equivalent.
(i)There exists a constant matrix such that the solution of (1.2a)-(1.2b) satisfiesand there exist constants , and such that satisfies satisfiesand , the tail of , defined by (2.8) satisfies(ii)For all initial conditions and constants there exists an a.s. finite -measurable random variable with such that the unique continuous adapted process which obeys (1.1a)-(1.1b) satisfieswhere and are positive constants.(iii)For all initial conditions there exists an a.s. finite -measurable random variable such that the unique continuous adapted process which obeys (1.1a)-(1.1b) satisfieswhere 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 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 satisfy (2.4) and (3.1) and let satisfy (2.6). If satisfies (3.2) then the following are equivalent.
(i) There exists a constant matrix such that the solution of (1.2a)-(1.2b) satisfies (3.3) and there exist constants and such that and satisfy (3.4) and (3.5), respectively.(ii) For all initial conditions and constants there exists an a.s. finite -measurable random variable with such that the unique continuous adapted process which obeys (1.4a)-(1.4b) satisfieswhere and are positive constants.(iii) For all initial conditions there exists an a.s. finite -measurable random variable such that the unique continuous adapted process which obeys (1.4a)-(1.4b) satisfieswhere 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 .

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 , and . It is seen in [1] that 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 and the exponential decay of the tail of the perturbation (see [1]); for the stochastically perturbed case we require the exponential integrability of and the exponential square integrability of . 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 , and 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 , and for the exponential convergence of the solution of (1.1a)-(1.1b), while Section 7 deals with the necessity of the condition on . 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 th mean sense while Proposition 4.2 deals with the almost sure case.

Proposition 4.1. Let satisfy (2.4) and (3.1), let satisfy (2.6) and be a constant matrix such that the solution of (1.2a)-(1.2b) satisfies (3.3). If there exist constants and such that (3.4) and (3.5) hold, then there exist constants , independent of , and , such that statement (ii) of Theorem 3.2 holds.

Proposition 4.2. Let satisfy (2.4) and (3.1), let satisfy (2.6) and be a constant matrix such that the solution of (1.2a)-(1.2b) satisfies (3.3). If there exist constants and such that (3.4) and (3.5) hold, then there exists a constant , independent of 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 in order to highlight this important case. This lemma may be then used when generalising the result to all .

Lemma 4.3. Let satisfy (2.4) and (3.1), let satisfy (2.6), and let be a constant matrix such that the solution of (1.2a)-(1.2b) satisfies (3.3). If there exist constants and such that (3.4) and (3.5) hold, then there exist constants , independent of , and , such that

From [8, 9] it is evident that is a more natural condition on the resolvent than when studying convergence of solutions of (1.4a)-(1.4b). However, the deterministic results obtained in [1] are based on the assumption that . 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 once is square integrable.

Lemma 4.4. Let satisfy (2.4) and (3.1) and let be a constant matrix such that the solution of (1.2a)-(1.2b) satisfies (3.3). Then the solution of (1.2a)-(1.2b) satisfies

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 to a limit in needed in the proof of Theorems 3.1 and 3.2.

Theorem 4.5. Let satisfy (2.4) and (3.1). Suppose there exists a constant matrix such that the solution of (1.2a)-(1.2b) satisfies (4.2). If there exists a constant such that satisfies (3.4) then there exist constants and such that

In the proof of Propositions 4.1 and 4.2, an explicit representation of 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 was found:

Theorem 4.6. Let satisfy (2.4) and and let satisfy (2.6) and Suppose that the resolvent of (1.2a)-(1.2b) satisfies (3.3). Then the solution of (1.4a)-(1.4b) satisfies almost surely, where is an almost surely finite and -measurable random variable given by

Lemma 4.7 concerns the structure of in the almost sure case. It was proved in [9].

Lemma 4.7. Let satisfy (2.4) and (4.4). Suppose that for all initial conditions there is an almost surely finite random variable such that the solution of (1.4a)-(1.4b) satisfies Then

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 satisfy (2.4) and (4.4) and let satisfy (2.6). If satisfies (4.5) and there exists a constant matrix such that the solution of (1.2a)-(1.2b) satisfies (3.3), then for all initial conditions there is an almost surely finite -measurable random variable , such that the unique continuous adapted process which obeys (1.4a)-(1.4b) satisfies (4.7).
Moreover, if the function also satisfies then (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 and be an invertible matrix such that has Jordan canonical form. Let if all the elements of the th row of are zero, and otherwise. Let and put and .

Lemma 4.9. Let satisfy (2.4) and (4.4). If there exists a constant matrix such that the resolvent of (1.2a)-(1.2b) satisfies (3.3), then where is defined by

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 then where .

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

Lemma 4.11. Suppose that and If and then where 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 almost surely where is given by (4.6), so we see thatSincewe use (2.9) and (4.6) to expand the right hand side of (4.17) to obtainIn 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 thatWe provide an argument to show that the second term decays exponentially. Using (3.5) and the fact that decays exponentially quickly to we can choose such that and where the function is defined by . Since the convolution of an function with an function is itself an function we getand 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:
Combining these facts we see thatwhere and .

Proof of Proposition 4.1. Consider the case where and separately. We begin with the case where . The argument given by (4.16) shows that . Now applying Lyapunov's inequality we see thatWe now show that (3.9) holds for . Lyapunov's inequality and Lemma 4.3 can be applied as follows:where and .
Now consider the case where . In this case, there exists a constant such that . We now seek an upper bound on and , which will in turn give an upper bound on and by using Lyapunov's inequality. By applying Lemma 4.10 we see thatwhere is a positive constant, so .
Now consider . Using the variation of parameters representation of the solution and the expression obtained for , taking norms, raising both sides of the equation to the th power, then taking expectations across the inequality, we arrive atWe consider each term on the right hand side of (4.26). By Theorem 4.5 we haveNow, consider the second term on the right-hand side of (4.26). By (4.20) we see that