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 th mean sense is obtained.
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 where .
Using this and Lemma 4.10 we see that
Using (4.21) combined with Lemma 4.10 and Fatou's
lemma, we show that the third term decays exponentially quickly:Combining (4.27), (4.28), and
(4.29) the inequality (4.26) becomesUsing Lyapunov's inequality, the
inequality (4.30) implieswhere and .
Proof of Proposition 4.2. In order
to prove this proposition we show thatFor each there exists such that .
Define .
Integrating (1.4a)-(1.4b) over ,
then adding and subtracting on both sides we getBy applying Theorem 4.8, we see
that (4.7) and (4.8) hold so Lemma 4.7 may be applied to obtainTaking norms on both sides of
(4.34), squaring both sides, taking suprema, before finally taking expectations
yields:We now consider each term on the
right hand side of (4.35). From Lemma 4.3 we see that the first term
satisfiesIn 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:where .
Take expectations and examine the two terms within the integral. Using Lemma
4.3 we obtainIn order to obtain an
exponential upper bound for the second term within the integral we apply Lemma
4.11 with , and :Next, we obtain an exponential
upper bound on the third term. Using (4.21) and the Burkholder-Davis-Gundy
inequality, there exists a constant such thatNow consider the last term on
the right hand side of (4.35). Using (3.4) we see thatUsing this and the fact that (see (4.16)) we obtainCombining (4.36), (4.38),
(4.39), (4.40), and (4.42) we obtainwhere and .
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 across by the function ,
where ,
integrate over ,
use integration by parts, add and subtract from both sides to obtainwhere , is defined by (4.12) and is defined by
Consider the reformulation of (1.2a)-(1.2b) given by (4.44).
It is well known that can be expressed aswhere the function satisfies and .
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 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 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 th 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 th mean case.
Proposition 5.1. Let satisfy (2.4) and (4.4) and satisfy (2.6). If there exists a constant such that (3.4) holds, and if for all there is a constant vector such that the solution of (1.4a)-(1.4b) satisfies statement (iii) of Theorem
3.2, then there exists a constant ,
independent of ,
such that (3.5) holds.
Proposition 5.2. Let satisfy (2.4) and (4.4) and satisfy (2.6). If there exists a constant such that (3.4) holds, and if for all there is a constant vector such that the solution of (1.4a)-(1.4b) satisfies statement (ii) of Theorem
3.2, then there exists a constant ,
independent of ,
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 satisfy (2.4) and (4.4). Suppose that for all
initial conditions there is a -measurable and almost surely finite random
variable with such that the solution of (1.4a)-(1.4b) satisfies Then obeys
Lemma 5.4 may be extracted from [4]; it is required in the proof
of Proposition 5.2.
Lemma 5.4. Let where for .
Then there exists a -independent constant such that
Proof of Proposition 5.1. In order to prove this result we follow the argument
used in [4, Theorem 4.1]. Let .
By defining the process and the matrix we can rewrite (1.4a)-(1.4b) asthe integral form of which
isUsing and rearranging this becomesAdding and subtracting from the right hand side and applying Lemma
4.7 we obtain:Consider each term on the right
hand side of (5.7). We see that the first term tends to zero as
(3.10) holds
and .
The second term is finite by hypothesis. Again, using the fact that and that assumption (3.10)
holds we see that ,
so the third term tends to a limit as .
Now consider the fourth term. Since ,
we can choose such that .
So the functions and are both integrable. The convolution of these
two integrable functions is itself an integrable function, soThus, it is clear that the
fourth term has a finite limit as .
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 ,
so thereforeThe 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 ,
take norms and expectations across (5.7) to obtainThere exists such thatthus the first, second and third
terms on the right hand side of (5.10) are uniformly bounded on .
Now consider the fourth term. Since ,
we can choose such that so that the functions and are both integrable. The convolution of two
integrable functions is itself an integrable function, soso it is clear that the fourth
term is uniformly bounded on .
Finally, we consider the final term on the right hand side of (5.10). Using
(4.41) we obtainsince .
Thus there is a constant such thatThe proof now follows the line
of reasoning found in [4, Theorem 4.3]: observe thatwhereIt is clear that is normally distributed with zero mean and
variance given byLemma 5.4 and (5.14) may now be
applied to obtain:Allowing 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 th mean sense while Proposition 6.2 deals with
the almost sure case.
Proposition 6.1. Let satisfy (2.4) and (3.1), let satisfy (2.6), let satisfy (2.5), 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), (3.6) and (3.5)
hold, then there exist constants ,
independent of ,
and ,
such that statement (ii) of Theorem 3.1 holds.
Proposition 6.2. Let satisfy (2.4) and (3.1), let satisfy (2.6), let satisfy (2.5), 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), (3.6) and (3.5)
hold, then there exists constant ,
independent of such that statement (iii) of Theorem 3.1
holds.
As in the case where we require an explicit formulation for .
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 satisfy (2.4) and (4.4), let satisfy (2.6) and (4.5), and let f satisfy
(2.5). Suppose that the resolvent of (1.2a)-(1.2b) satisfies (3.3), then the solution of (1.1a)-(1.1b) satisfies almost surely, where and is almost surely finite.
Proof of Proposition 6.1. We
begin by showing that is finite. Clearly, we see
thatNow, consider the difference
between the solution of (1.1a)-(1.1b) and its limit given by (6.1):Using integration by parts this
expression becomesTaking norms on both sides of
equation (6.4), raising the power to on both sides, and taking expectations across
we obtainNow 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 decays exponentially; thus the final term must
decay exponentially. Combining these arguments we see that (3.7) holds, where .
It is now shown that decays exponentially. It is clear from the
resolvent equation (1.2a)-(1.2b) thatConsider each term on the right
hand side of (6.6). We can apply Theorem 4.5 to obtain that decays exponentially quickly to .
In order to show that the second term decays exponentially we proceed as
follows: since decays exponentially and (3.4) holds it is
possible to choose such that the functions and are both in .
The convolution of two integrable functions is itself an integrable function,
soTo 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 under this hypothesis and (3.1). Combining the
above we see that decays exponentially quickly to .
Proof of Proposition 6.2.
Take norms across (6.4) to obtainUsing 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 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 .
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 th mean senses is shown. Proposition 7.1 concerns
the necessity of the conditions in the th mean case while Proposition 7.2 deals with the
almost sure case.
Proposition 7.1. Let satisfy (2.4) and (4.4), let satisfy (2.6), and let satisfy (2.5). If there exists constant such that (3.4) holds, and if for all there is constant vector such that the solution of (1.1a)-(1.1b) satisfies statement (ii) of
Theorem 3.1, then there
exist constants and ,
independent of ,
such that (3.6) and (3.5) hold.
Proposition 7.2. Let satisfy (2.4) and (4.4), let satisfy (2.6), and let satisfy (2.5). If there exists constant such that (3.4) holds, and if for all there is a constant vector such that the solution of (1.1a)-(1.1b) satisfies statement (iii) of
Theorem 3.1, then
there exist constants and ,
independent of ,
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 is an almost surely finite random variable
and where , , and the functions and are defined by (2.8) and respectively. 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 be a sequence of independent Gaussian random
variables with and .
Then
Lemma 7.5. If there is a such that and then where 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 satisfy (2.4) and (4.4). Suppose that for all
initial conditions there is an -measurable and almost surely finite random
variable with such that the solution of (1.1a)-(1.1b) satisfies Then obeys
Lemma 7.7. Let satisfy (2.4) and (4.4). Suppose that for all
initial conditions there is an -measurable and almost surely finite random
variable such that the solution of (1.1a)-(1.1b) satisfies Then obeys (7.7).
Proof of Proposition 7.1. Since (3.7) holds for every initial condition we can choose :
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 from both sides and applying Lemma 7.6 we
obtainwhere ,
the function is defined byand .
Taking expectations across (7.9) and allowing we obtainwhere .
Using this expression for we obtainThe first term on the right-hand
side of (7.12) decays exponentially due to (3.7). Assumptions (3.4) and (3.7)
imply that 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, decays exponentially.
Proving that (3.5) holds breaks into two steps. We
begin by showing thatwhere .
By choosing we can obtain the following reformulation of
(1.1a)-(1.1b) using
methods applied in [15, Proposition 5.1]
Rearranging (7.14), taking
expectations and then norms on both sides we can obtainSince (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 ,
we can choose such that .
It can easily be shown thatFinally, 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
obtainWe 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,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 ,
we obtainwhere is defined by (7.10). Using this expression for ,
(7.9) becomeswhere .
Rearranging the equation and taking norms yieldsThe 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 decays exponentially due to (3.8). In order to
show that 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 such thatwhere .
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 such that (3.5) holds. Using the equivalence
of norms we see that for all and assumption (3.5) implies thatApplying Lemma 7.5 we
obtainChoose any .
For each we can choose a constant such thatNow, summing over we see thatwhere , and .
Now, sincewe see thatSo for we see that .
This giveswhere is finite and .
Now summing over we obtain (3.6), by picking out any .
This concludes the case when (3.5) holds.
Now, consider the case where assumption (3.5) fails to
hold. We choose a constant such that and define the function asand the vector martingale asWe let denote the th component of and denote the quadratic variation of given byWe show at the end of this proof
thatand therefore assume it for the
time being.
Since (3.5) fails to hold there exists an entry ,
of the martingale such thatIt follows that and a.s. Consider the corresponding th entry of ,
denoted ;
it is either bounded or unbounded. If is bounded then is bounded and so, by the Martingale
Convergence Theorem, is bounded: this contradicts (7.34). So, we
suppose the latter, that is unbounded, and proceed to show this is also
contradictory. Since ,
for it is clear that .
Taking the limit superior on both sides of the inequality
yieldsAs is deterministic, there exists an increasing
sequence of deterministic times with such that as .
Consequently, as .
We choose a subsequence of these times with such thatDefine .
ObviouslywhereIt is clear that is an indepenendent normally distributed
sequence with the variance of each given by so we may apply Lemma 7.4.
We now show that assumption (7.33) holds. By changing
the order of integration we can show thatThus, as ,
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 satisfy (2.4) and (3.1). Suppose that there
exists a constant matrix and constants and such that the solution of (1.2a)-(1.2b) satisfies (4.3). If the kernel satisfies (3.2) then there exists a constant such that 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 solutions of (1.4a)-(1.4b); ,
where .
Since (3.9) holds we obtainfor each .
Thus, the resolvent of (1.2a)-(1.2b) decays exponentially to .
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 we can consider the following solutions of (1.4a)-(1.4b); whereWe know that approaches exponentially quickly in the almost sure
sense. Introduceand notice .
Let .
ThenIf we define then 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
thatConsider the solutions of (1.1a)-(1.1b) with initial conditions for and .
Since we see thatwhere is an almost surely finite constant. As both
terms on the left hand side of this expression are decaying exponentially to
zero, must decay exponentially to as .
Thus 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 solutions with initial conditions for and .
Since for ,
we can writewhere 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).