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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

In this paper, we study the exponential convergence of
the solution of

The solution of
(

The case where (

Two papers by Appleby et al. [

This paper examines (

In this
section, we introduce some standard notation as well as giving a precise
definition of (

Let

Let

The set of complex numbers is denoted by

If

We now make our problem precise. We assume that the
function

Under the hypothesis (

The notion of convergence and integrability in

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 (

Let

There exists a constant matrix

For all initial conditions

For all initial conditions

The proof of Theorem

Let

There exists a constant matrix

For all initial conditions

For all initial conditions

This result is interesting in its own right as it
generalises a result in [

It is interesting to note the relationship between the
behaviour of the solutions of (

Theorem

The remainder of this paper deals with the proofs of
Theorems

In this section, sufficient conditions for exponential
convergence of solutions of (

Let

Let

In [

Let

From [

Let

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 [

Let

In the proof of Propositions

Let

Lemma

Let

It is possible to apply this lemma using our

Let

Moreover, if the function

Lemma

Let

Lemma

Suppose the function

The following lemma is used in Proposition

Suppose that

The proofs of Propositions

From Theorem

We can show that the third term on the right hand side
of (

Combining these facts we see that

Consider
the case where

Now consider the case where

Now consider

Using (

In order
to prove this proposition we show that

We can now apply the line of reasoning used in
[

We use a reformulation of (

Consider the reformulation of (

In this section, the necessity of condition (

Let

Let

In order to prove these propositions the integral
version of (

Some supporting results are now stated. Lemma

Let

Lemma

Let

In order to prove this result we follow the argument
used in [

Each term on the right hand side of the inequality has
a finite limit as

By
Lemma

In this section, sufficient conditions for exponential
convergence of solutions of (

Let

Let

As in the case where

Let

We
begin by showing that

It is now shown that

Take norms across (

In this section, the necessity of (

Let

Let

The following lemma is used in the proof of
Proposition

Suppose

In order to prove Lemma

Let

If there is a

Lemmas

Let

Let

Since (

Proving that (

We now return to (

Since Lemma

We
suppose that there exists a constant

Now, since

Now, consider the case where assumption (

Since (

We now show that assumption (

We now combine
the results from Sections

We showed the necessity of (

Let

We begin by proving the equivalence between (i) and (ii).
The implication (i) implies (ii) is the subject of Proposition

We now prove the equivalence between (i) and (iii).
The implication (i) implies (iii) is the subject of Proposition

We begin by proving the equivalence between (i) and (ii).
The implication that (i) implies (ii) is the subject of Proposition

We now prove the equivalence between (i) and (iii).
The implication (i) implies (iii) is the subject of Proposition

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).