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Growth and fluctuation in perturbed nonlinear Volterra equations

We develop precise bounds on the growth rates and fluctuation sizes of unbounded solutions of deterministic and stochastic nonlinear Volterra equations perturbed by external forces. The equation is sublinear for large values of the state, in the sense that the state dependence is negligible relative to linear functions. If an appropriate functional of the forcing term has a limit L at infinity, the solution of the differential equation behaves asymptotically like the underlying unforced equation when L is zero, like the forcing term when L is infinite and inherits properties of both the forcing term and underlying differential equation for finite positive values of L. Our approach carries over in a natural way to stochastic equations with additive noise, and we treat the illustrative cases of Brownian and Levy noise.


Introduction
We analyse the long-run dynamics of solutions to the scalar Volterra integro-differential equation (1.1) In particular, we concentrate on the behaviour of unbounded but non-explosive solutions, i.e. x ∈ C(R + ; R) but lim sup t→∞ |x(t)| = ∞. As suggested in the title we draw a distinction between when solutions of (1.1) grow, lim t→∞ x(t) = ∞, and when solutions can be said to fluctuate asymptotically, lim inf t→∞ x(t) = −∞ and lim sup t→∞ x(t) = +∞. When solutions grow it is natural to ask at what rate they grow and when they fluctuate to ask if the size of these fluctuations can be captured in an appropriate sense; this paper investigates these types of questions for equations such as (1.1). Throughout µ is a measure on (R + , B(R + )) obeying µ(E) ≥ 0 for all E ∈ B(R + ), µ(R + ) = M ∈ (0, ∞).
(A+) When we do not restrict ourselves to positive solutions we ask for a degree of symmetry in the problem to simplify the analysis. In particular, we require "asymptotic oddness" of the nonlinearity in the following sense: f ∈ C(R; R) and lim |x|→∞ |f (x)| φ(|x|) = 1 for some φ ∈ C 1 (R + ; (0, ∞)).
After developing results regarding the asymptotics of unbounded solutions of (1.1) we extend our deterministic analysis to consider the asymptotic behaviour of the related stochastic Volterra equation µ(ds)f (X(t − s)) dt + dZ(t), t > 0, (1.2) where Z is a semimartingale. We establish a simple existence and uniqueness theorem for equation (1.2) and then specialise to the cases of Brownian and Lévy noise in order to prove precise asymptotic results. The differential equations (1.1) and (1.2) can be viewed as perturbations of the underlying deterministic Volterra integro-differential equation y ′ (t) = [0,t] µ(ds)f (y(t − s)), t > 0, y(0) = ψ. (1. 3) When f is positive and sublinear at infinity (in a sense to be made precise shortly), we have shown in earlier work (see Appleby and Patterson [2]) that the solution y(t) of (1.3) obeys y(t) → ∞ as t → ∞ and grows asymptotically like the solution of the ordinary differential equation z ′ (t) = M f (z(t)), t > 0; z(0) = ψ, (1.4) in the sense that lim t→∞ F (y(t)) M t = 1, (1.5) where F is the function defined by du, x > 0. (1.6) This comparison makes sense because (1.4) integrates to give It is natural to ask how large the forcing terms h in (1.1) and Z in (1.2) can become while the solutions x of (1.1) and X of (1.2) continue to grow in the manner described by (1.5). Furthermore, can we identify a new asymptotic regime or growth rate if the forcing terms exceed this critical rate? The main goal of this paper is to identify such critical rates of growth on h and Z, and to determine precise estimates on the growth rate of solutions, or the rate of growth of the partial maxima when solutions fluctuate. Much of our analysis flows from the simple matter of integrating (1.1) to obtain the forced Volterra integral equation (1.7) Since Itô stochastic "differential" equations are rigorously formulated in integral form it is perhaps even more natural to treat (1.2) similarly, which results in The representation (1.7) shows that the solution to (1.1) is a functional of the "aggregate" behaviour of the forcing term h purely through H and hence it is natural to formulate asymptotic results in terms of H. When studying the asymptotic behaviour of many forced differential systems it is frequently the case that the "aggregate" or "average" behaviour of the forcing terms are important, rather than more restrictive pointwise estimates. When studying stochastic equations pointwise estimates become unrealistically restrictive-or indeed impossible-and it is more natural and perhaps necessary to consider average behaviour. Another issue is whether the deterministic or stochastic character of the perturbation matters, or is it simply a question of the "size" of the perturbation. For these reasons we have found it of interest to study deterministic and stochastic equations in parallel, especially because it transpires that the general form of many results in the stochastic case can be conjectured by appealing to corresponding deterministic results.
To help the discussion we make our hypotheses more specific and outline typical results. In order for solutions of (1.3) to behave similarly to those of (1.4), it is important that f be sublinear : for example, we do not expect linear Volterra equations of the form (1.3) to share the exact exponential rate of growth of a linear ordinary differential equation in which all the mass of µ is concentrated at zero (cf. Gripenberg et al. [5,Theorem 7.2.3]). Also, as we are interested in growing solutions, it is quite natural that the function f should be in some sense monotone. In previous work we showed that if f is asymptotic to a C 1 function φ which is increasing and obeys φ ′ (x) → 0 as x → ∞, then the solution of (1.3) obeys (1.5) [2]. We retain this hypothesis and occasionally strengthen it so that φ ′ (x) decays monotonically to 0 as x → ∞; the implications and technical motivations for such hypotheses are discussed in Section 2.
Before stating our main results precisely we give a heuristic argument as to their likely validity. In this discussion we consider the simple (deterministic) case in which both the solution and the perturbation are positive. If the unperturbed equation (1.3) is integrated as above, H ≡ 0. In this case, the solution of the integral equation is, roughly, of order F −1 (M t). This leads to the naive idea that if H is of smaller order than y (i.e., than F −1 (M t)), then H on the right-hand side of (1.7) could be absorbed into x on the left-hand side, without changing the leading order asymptotic behaviour of x. However, if H dominates y, or is of comparable order, such an outcome is improbable and the asymptotic behaviour of x is unlikely to be determined by y. Since the asymptotic behaviour of (1.3) is described well by F (y(t))/M t → 1 as t → ∞, and F −1 is increasing, it is natural to seek to characterise the forcing term as "small" or "large" according as to whether F (H(t))/M t tends to a small or large limit as t → ∞ (if such a limit exists). Define the dimensionless parameter L ∈ [0, ∞] by lim t→∞ F (H(t)) M t = L. (1.9) In some sense L = 1 is critical; for L < 1, H is dominated by the solution of (1. Examples show that the limits in the first part of (1.11) are not, in general, equal to 1 or 1 + L. Further investigation for finite and positive L leads to better estimates, especially when L > 1. The critical character of the case when L = 1 is demonstrated by the following result: if L ∈ (1, ∞) then We notice that this provides sharper estimates for large L than the asymptotic bounds given for L ∈ (0, ∞) above and identifies that x is of order H. We also show by means of examples that when L ∈ (0, 1], the limit can result, so that x can only be expected to be exactly of the order of H for L > 1 (see example 4.2). However, if L ∈ (0, 1], it is not necessarily the case that x(t)/H(t) → ∞ as t → ∞ (see example 4.1). Notice finally that as L → ∞, equation (1.12) correctly anticipates that x(t)/H(t) → 1 as t → ∞, which is what pertains when L = +∞. To generalise the analysis above to stochastic equations, and for notational convenience, we define the following functional for all functions f and γ ∈ C(R + ; (0, ∞)) such that the above limit is well defined.

Discussion of Hypotheses
To begin we define a useful equivalence relation on the space of positive continuous functions; in essence, two functions are equivalent if they have the same leading order asymptotic behaviour.
and lim x→∞ f (x)/x = 0 implies that F , defined by (1.6), obeys lim x→∞ F (x) = ∞. Hence the following convenient lemma can be proven immediately by asymptotic integration. Lemma 2.2. If f, φ ∈ C((0, ∞); (0, ∞)) are asymptotically equivalent and obey We impose the following sublinearity hypothesis on the nonlinear function f : In many cases the following slightly stronger hypothesis is necessary If f is an increasing, sublinear function, then lim inf x→∞ f ′ (x) = 0 but it is still possible that lim sup x→∞ f ′ (x) = ∞ in the "worst" case. In previous work we provided an example of such a pathological f but such nonlinearities are unlikely to arise naturally in applications so condition (A3) is a relatively mild strengthening of sublinearity in this context [2]. Assuming further that φ ′ tends to zero monotonically, as in (A4), one can establish the following lemmata which prove crucial in the asymptotic analysis of (1.1) and (1.2).
The connection between the "natural" size hypothesis on H, (1.9), and the functional condition, (1.13), is supplied by the following result.
The relevant existence and uniqueness theory regarding solutions of (1.1) can be found in Gripenberg at al. [5] and guarantees that (1.1) has a solution x ∈ C(R + ; R) in the framework of this article (see [5, Corollary 12.3.2 and Theorem 13.5.1] and note that sublinearity guarantees a global linear bound and therefore non-explosion of solutions).
Occasionally, we employ the standard Landau "O" and "o" notation. If a and b are in C(R + ; R), we say that b is O(a) if |b(t)| ≤ Ka(t) for some K ∈ (0, ∞) and t sufficiently large, and b is o(a) if b(t)/a(t) → 0 as t → ∞.

Deterministic Volterra Equations
3.1. Growth Results. Throughout this section we suppose that (A+) holds so that 0 < x(t) → ∞ as t → ∞, subject to a positive initial condition. Our first result provides an easy to check sufficient condition on H which guarantees solutions of (1.1) retain the rate of growth of solutions to the ordinary differential equation (1.4). This sufficient condition is of a different character to conditions involving the functional L f (·) and expresses more explicitly the idea that the perturbation term, H, should be small relative to the solution of (1.4).
then solutions of (1.1) obey Now we formulate a sufficient condition for lim t→∞ F (x(t))/M t = 1 to hold in terms of L f (·). Compared to condition (3.1) such functional based conditions have much better scope for generalization. We also prove that when the solution of (1.1) retains the growth rate of solutions of (1.4) it is of a strictly larger order of magnitude than the perturbation term, H.
The rationale is as follows in the case L f (H) ∈ (0, ∞), with the case of L f (H) = ∞ being similar. By hypothesis H(t) > 0 for t > 0 and as f is a positive function, t → t 0 f (H(s))ds is increasing. Therefore, H either tends to ∞ or to a finite limit. In the former case, H(t) → ∞ as t → ∞ automatically. If, to the contrary, the limit is finite, then H(t) tends to a finite positive limit as t → ∞. But this forces t 0 f (H(s)) ds → ∞ as t → ∞, a contradiction. When L f (H) is nonzero but finite we expect the solution of (1.1) to inherit properties of both the underlying ordinary differential equation and the perturbation term. Our next theorem investigates results of the type (1.5) when L f (H) ∈ (0, ∞); we show that the growth of solutions to (1.1) is at least as fast as that of solutions to the underlying ordinary differential equation and we prove an upper bound on the growth rate. The resulting upper bound is linear in L f (H) and this is intuitively appealing as a "larger" H should speed up growth. However, this upper estimate on the growth rate is not sharp in general. Without additional hypotheses this upper bound is hard to improve but can be shown to be suboptimal for specific classes of nonlinearity, for example when f is regularly varying with less than unit index. We will demonstrate this possible improvement in further work.
We note that the asymptotic lower bound on the quantity x(t)/H(t) in the result above agrees with Theorem 3.2 as L f (H) tends to zero, in the sense that it correctly predicts lim t→∞ x(t)/H(t) = ∞ when L f (H) = 0.
The results of this section can all be restated with positivity assumptions on f and H replaced by (A2) and H ∈ C(R + ; R).
In this case one obtains upper bounds on the rate of growth of solutions of (1.1) in terms of the related ODE, i.e. results of the type lim sup t→∞ F (|x(t)|)/M t < ∞.
The main results of this section are all proven by comparison arguments and the careful asymptotic analysis of the resulting differential inequalities. Since we assume positivity of H to ensure asymptotic growth of solutions, it is straightforward to show that lim inf t→∞ F (x(t))/M t ≥ 1; this is proven by a translation argument and appealing to [2,Corollary 2]. The proof of the corresponding upper bound, lim sup t→∞ F (x(t))/M t < ∞, is more involved but can be roughly summarized as follows: Step 1: Use monotonicity and finiteness of the measure to construct the crude upper inequality where H ǫ includes constants and lower order terms, φ is a monotone function asymptotic to f and we define I ǫ (t) = t T φ(x(s)) ds for t ≥ T .
Step 2: Using hypotheses on the size of the perturbation term try to show that H ǫ is o(I ǫ ) (or O(I ǫ ) respectively).
Step 3: Conclude the argument via a variation on Bihari's inequality.

Fluctuation
Results. The existence of the limit L f (H) (even when it takes the value +∞) is too strong a condition if we hope to apply our deterministic arguments to related equations with stochastic perturbations. We seek to weaken the hypothesis L f (H) ∈ (0, ∞) as follows: assume that there exists a function γ such that γ ∈ C((0, ∞); (0, ∞)) is increasing and obeys lim t→∞ γ(t) = ∞ and lim sup t→∞ We now make hypotheses on L f (γ), as opposed to L f (H). We take lim sup t→∞ |H(t)|/γ(t) = 1, rather than positive and finite since we can always normalise this quantity while keeping the properties of γ unchanged. Since L f (γ) ∈ (0, ∞) forces γ to be eventually increasing, we simply suppose that γ is always increasing for ease of exposition but there is strictly no need to make this assumption. Under (F1) we can permit highly irregular behaviour in H as long as we can capture some underlying regularity in the asymptotics of H via a well-behaved auxiliary function, γ. For example, in applications to stochastic equations, H could be a stochastic process whose partial maxima are described in terms of a deterministic function; this is the case for classes of processes obeying so-called iterated logarithm laws for instance. The following result illustrates the immediate utility of the hypothesis (F1) for deterministic equations and furthermore details how this hypothesis carries over to the case when L f (γ) = ∞.
Case (a.) of the result above indicates that when the perturbation is of intermediate size, in the sense that L f (γ) ∈ (1, ∞), solutions of (1.1) are at most the same order of magnitude as H, modulo a multiplier. In case (b.), when the perturbation is so large that L f (γ) = ∞, solutions of (1.1) have partial maxima of exactly the same order as those of H. This conclusion is strongly hinted at in case (a.) of Theorem 3.4 if one lets L f (γ) → ∞ in that result.
The restriction L f (γ) > 1 is in fact crucial to the proof of Theorem 3.6 and cannot be relaxed within the framework of the current argument. We make this comment precise at the relevant moment during the proof itself (see remark 8.2). In fact, L f (γ) > 1 is not a purely technical contrivance but is also essential to the validity of our result. In example 4.2 we demonstrate that when L f (γ) ∈ (0, 1] it is possible to have lim t→∞ |x(t)|/γ(t) = ∞.
If lim sup t→∞ |H(t)|/γ(t) = 0 in (F1) we can use the following hypothesis and the arguments from Theorem 3.4 to extend the scope of the result above.
In the presence of limited information about the behaviour of H, in the sense that (F2) holds, the result above tells us that the solution of (1.1) is roughly the same order of magnitude as H, in the sense that x also obeys (F2), when L f (γ ± ) = ∞. When L f (γ ± ) ∈ (1, ∞] we are still left with a weak conclusion and we are tempted to ask if this is an artifact of the argument used to establish it. Example 4.5 shows that we cannot expect to conclude that lim sup t→∞ |x(t)|/γ + (t) = 0 in general in this case. However, in attempting to apply this result it is likely that the user would actually seek to refine their choice of γ + in order to obtain a γ + obeying L f (γ + ) = ∞ and hence make the stronger conclusion that x is o(γ + ).
Theorem 3.5 could equally well be stated as follows: These two statements are proved independently of one another but we chose to present them as part of a single result as we feel this is the manner in which they would prove most useful in practice; choosing γ + and γ − "close together" can give useful bounds on the size of the solution but using either bound in isolation only gives very crude information (see example 6.1 for an illustration of this comment).
If we consider the case of positive, growing solutions once more we can impose hypotheses regarding the functional L f (·) directly on H in Theorem 3.4 and quickly establish the following result.
Under stronger hypotheses the result above provides more refined conclusions than Theorems 3.4 and 3.5. In particular, case (a.) establishes bounds which demonstrate that x will closely track the asymptotic behaviour of H and case (b.) establishes that when the noise term, H, is sufficiently large x(t) ∼ H(t) as t → ∞. Furthermore, when x(t) ∼ H(t) as t → ∞, x is of a strictly larger order of magnitude than the solution of the corresponding ordinary differential equation. We also note that this result allows us to pick up fluctuations in the solution even when H is nonnegative. Even though the solution grows to infinity it may not do so monotonically and the conclusion of Theorem 3.6 identifies upper and lower rates of growth of the solution (G L H(t) and G U H(t), respectively, when L f (γ) ∈ (1, ∞) and when L f (γ) = ∞ the fluctuations are entirely determined by H).
The results of this section are proven via the usual machinery of comparison and asymptotic analysis but also rely crucially on the construction of a linear differential inequality to achieve sharp results. The key steps in the argument can be understood as follows: Step 1: Using (3.4), derive the nonlinear differential inequality Step 2: Use (A4) to derive the linear differential inequality Since we can solve this inequality directly, there is no additional loss of sharpness here.
Step 3: Careful asymptotic analysis of the solution to the inequality (3.7) using hypotheses on L f (H) yield upper bounds on the size of the solution to (1.1).
Step 4: The upper bounds achieved in Step 3 are recycled and further simple estimation yields the conclusions shown in the results above. The steps outlined above are also very successful in the presence of random forcing, as we will demonstrate presently.

Deterministic Examples
Consider the Volterra integro-differential equation given by In the notation of (1. We construct examples by choosing a solution x, up to asymptotic equivalence, and then using (4.1) to figure out how large the perturbation term, H, must have been to generate a solution of this size. We defer the calculations relevant to this section until Section 11. For simplicity we forego any mention of hypotheses of the form (F1) in this section and concentrate on the special case γ = H with H positive.
Example 4.1. In this example we demonstrate that the limits in Theorem 3.3 are not always equal to 1 or 1+L f (H) and furthermore that L f (H) ∈ (0, 1] does not in general imply that lim t→∞ 1−β so we can choose an advantageous value of L f (H). For the purposes of this example it is sufficient to take L f (H) = 1 and β = 1/2. With these choices and the reader can compare this with the conclusion of Theorem 3.3. Finally, note that This example highlights the potential problems that emerge when one attempts to address the case L f (H) ∈ (0, 1] (resp. L f (γ)) in the context of Theorem 3.4. In particular, one cannot extend the conclusion of Theorem 3.4 to cover L f (H) ∈ (0, 1] without additional hypotheses because when L f (H) ∈ (0, 1] it is possible to have lim t→∞ x(t)/H(t) = ∞. Choose x(t) = exp λ(t) + 2(t + 1) − e = exp(P (t)) − e for t ≥ 0 and let λ(t) = (1 + t) α for some α ∈ (0, 1/2). In this case H(t) ∼ K P (t) 2α−1 exp(P (t)). Furthermore, H obeys L f (H) = 1 and by construction lim t→∞ x(t)/H(t) = ∞. However, we still have lim t→∞ F (x(t))/M t = 1.
We now show that the bounds on the lim t→∞ x(t)/H(t) and lim inf t→∞ F (x(t))/M t obtained in Theorems 3.3 and 3.6 can actually be attained. Once more suppose that , achieving the upper bound predicted by Theorem 3.6. Futhermore, a simple calculation reveals that lim t→∞ F (x(t))/M t = 1, achieving the lower bound from Theorem 3.3.

Stochastic Volterra Equations
We now study the pathwise asymptotic behaviour of solutions to (1.2). Our approach is to treat (1.2) as a perturbed version of (1.1) where the forcing term is now stochastic and hence to leverage our deterministic results as much as possible. After proving a simple general theorem which establishes existence of unique strong solutions to (1.2) we use the pathwise asymptotic theory for continuous Brownian martingales and α-stable Lévy processes to show that the main results from the previous section are sufficiently general that we can extend them to provide asymptotic estimates on the pathwise growth and fluctuation of solutions to (1.2). We also explain how our results provide a programme for establishing similar pathwise bounds for broader classes of admissible stochastic noise.
Assume henceforth that we are working on a given probability space (Ω, F , P, (F t ) t≥0 ) which is complete and has a right continuous filtration. We ask that the nonlinear function f : R → R obeys the following local Lipschitz condition: for each d > 0 there exists K d > 0 such that and that f obeys a global linear bound of the form where K and η are positive constants. In order to leverage the framework of Métivier and Pellaumail [7] we make a slight modification to the formulation of (1.2): consider the stochastic integral equation By applying Fubini's theorem and making a suitable change of variable (5.1) can be written as where X(t−) = X(lim s↑t ) and M − (t) = (0,t] µ(du). This adjustment is necessary for the functional to define a predictable process (measurable with respect to the filtration generated by adapted, left continuous processes) and hence be integrable with respect to general semimartingales (see Protter [8] for details). Proof. This theorem is a natural specialisation of a result of Métivier and Pellaumail [7,Theorem 5].
In order to apply the aforementioned result we must check that the functional from (5.3) and also the constant functional a(s, ω, X) = 1 obey the following pair of conditions: firstly for any regular processes (adapted with cádlag paths) X and Y , for each d > 0 there exists a constant L d > 0 such that Secondly, for any regular process X there exists C > 0 such that When the functional a is constant the conditions above are trivially satisfied so suppose now that a is given by (5.3) and proceed to verify condition (5.4). Let X and Y be any two regular processes satisfying sup 0≤s<t |X(s)| ≤ d (resp. Y ), fix t ∈ R + and estimate as follows: where we have used both (A1) and (L). Now check (5.5); assume X is a regular process and fix t ∈ R + . The following inequality is a straightforward consequence of (A1) and (GL): Remark 5.2. Note that the condition (GL) will always be satisfied in this section since the hypotheses (A2) and (A4) will be imposed throughout. The assumption (A1) is also present throughout so the only additional hypothesis imposed by Theorem 5.1 is that of local Lipschitz continuity on the nonlinear function f .
We pause now to consider the method by which the results of this section are proven and to illustrate that this presents a framework for generating similar pathwise asymptotic results for a wide range of suitable stochastic forcing terms. We remark that because our method of proof relies principally on building appropriate comparison equations we are not concerned about the pathwise regularity of the solution to (1.2) and hence can treat quite irregular forcing processes.
Given an adapted stochastic process (Z t ) t≥0 which is the forcing term in (1.2) the general approach we take is as follows: (i.) Establish the existence and uniqueness of strong solutions to the equation in question.
(ii.) Prove pathwise asymptotic bounds on the size of the process Z in terms of a well-behaved deterministic function, say γ, on which we can formulate functional hypotheses in terms of L f (·). These bounds should be in the spirit of (F1) or (F2).
(iii.) Construct an upper comparison solution (pathwise) in terms of γ which majorizes the solution to the (1.2); this essentially reduces the stochastic problem to a deterministic one. (iv.) Conclude the argument using suitable hypotheses on L f (γ) and the results of Section 3.

Brownian Noise.
In this section we suppose that Z(t) = t 0 σ(s)dB(s), where B is a standard one-dimensional Brownian motion, and define Analogously to the deterministic case, we classify the behaviour of solutions to (1.2) according to whether the number L f (Σ) is zero, finite or infinite.
The existence and uniqueness of solutions of (1.2) is naturally simpler in the case of Brownian noise. In particular, there is a unique, continuous (strong) solution to (1.2) with Brownian noise if (A1) holds and the nonlinearity is locally Lipschitz continuous with a global linear bound (see Mao [6,Ch. 5] for example).
Our first result regarding (1.2) is in some sense a stochastic analogue of Theorem 3.1. We employ a simple, easily verifiable condition on the perturbation term, in the spirit of (3.1), and this is sufficient to conclude that the solution to (1.2) can exhibit growth no faster than the solution of (1.4) (in the sense that such an event has probability zero).
Throughout this section we make the standing assumption that and X denotes the unique, continuous, adapted process obeying (1.2).
When formulating functional conditions on (1.2) to preserve growth of the type (1.5) it is necessary to distinguish between the cases σ ∈ L 2 (0, ∞) and σ / ∈ L 2 (0, ∞). When σ ∈ L 2 (0, ∞) the martingale term in (1.8), t 0 σ(s)dB(s), will tend to an a.s. finite random variable and in this case we clearly expect to retain the growth rate of solutions of (1.4). However, when σ / ∈ L 2 (0, ∞) the martingale term is recurrent on R and has large fluctuations of order Σ(t) (see Revuz and Yor [9, Ch. V, Ex. 1.15]). The following result shows that when σ / ∈ L 2 (0, ∞) and L f (Σ) = 0 the fluctuations of the martingale term are sufficiently small that the solution to (1.2) cannot grow faster than that of the ordinary differential equation (1.4).
An interesting special case of Theorem 5.4, which is likely to be important in applications, is when the function σ is a nonzero constant. In this case we can additionally show that the size of solution to (1.2) becomes unbounded with probability one. As in the deterministic case when the perturbation is of intermediate or critical magnitude, in the sense that L f (Σ) ∈ (0, ∞), we expect the solution to inherit characteristics of both the perturbation and the ordinary differential equation (1.4). Our next result demonstrates that this is indeed the case by showing that if the solution to (1.2) grows then its growth rate is at most of the same order of size as that of the solution to (1.4), possibly with a different multiplier which we can bound in terms of L f (Σ). .
we show that if the the solution to (1.2) fluctuates, then these fluctuations are at most of order Σ(t) times a multiplier which we can bound in terms of L f (Σ). As in Theorem 3.6 we are unable to extend this argument to L f (Σ) ∈ (0, 1) for technical reasons which become apparent in the relevant construction. We speculate that the bounds achieved here, while useful, are suboptimal in general. Furthermore, with additional hypotheses on the nonlinearity, the authors are confident that this lack of sharpness can be quantified precisely in further work but adding additional restrictive hypotheses would not be in the spirit of this paper.
We remark that the nonnegativity of the measure µ no longer plays an important role in the results above; primarily because we are reduced to proving upper bounds on the growth rate of solutions once solutions are no longer necessarily of one sign. For ease of exposition we have left the hypothesis (A1) in place but it could equally well be replaced by the hypothesis that µ is a Borel measure with finite total variation norm, i.e. |µ| = M ∈ (0, ∞), with the results above unchanged.
Remark 5.8. Under the hypotheses of Theorem 5.7 we can additionally conclude that This leaves open the question of recurrence, or in other words, whether or not the process actually fluctuates, for L f (Σ) ∈ (1, 2).
Finally, when the perturbation term is so large that L f (Σ) = ∞ we expect this exogenous force to dominate the system and this intuition is confirmed by our next result. In particular, we prove that the solution to (1.2) is recurrent on R and that its fluctuations are precisely of order Σ.
and furthermore 5.2. Lévy Noise. We now assume that the semimartingale Z in (1.2) is an α-stable Lévy process; the results which follow further emphasize the fact that our methods do not rely on the path continuity of the process in any essential way. For the readers convenience we recall the relevant definitions from the theory of Lévy processes.
Definition 5.10. If Z = (Z) t≥0 is a Lévy process, then it's characteristic function F Z is given by where Ψ : R → C is of the form Ψ is called the characteristic exponent of the process Z.
The number a in (5.8) corresponds to the linear "drift" coefficient of the Lévy process in question, σ is called the Gaussian coefficient and corresponds to the Brownian or continuous random component; Π is called the Lévy measure and represents the pure jump part of the process. A Lévy process is uniquely specified by the triple (a, σ, Π).
Stable processes are closely related to the class of stable distributions which gain their importance as "attractors" for normalised sums of independent and identically distributed random variables. In particular, a sum of random variables with power law decay in the tails, proportional to |x| −1−α , will tend to a stable distribution if 0 < α < 2 and to a normal distribution if α ≥ 2. Integrability of the Lévy measure forces us to consider α ∈ (0, 2] and in this section we also ignore the case α = 2 since this corresponds to the case of Brownian noise (which was considered in detail in Section 5.1). We tacitly exclude the degenerate case when Z is a pure drift process and assume for the remainder of this section that Z is an α-stable process with α ∈ (0, 2). (L1) Let X denote the unique, strong solution to (1.2) throughout.
The interested reader can consult Bertoin [3, Ch. VIII] for further details of stable processes, including the asympotic properties employed in the proofs of the following results.
Our first result is a stochastic analogue of Theorem 3.2 and provides a sufficient condition to retain growth to infinity no faster than the solution of (1.4) in the presence of α-stable noise. a.s., In Section 6 we provide a simple example of the application of the results above which serves to highlight their practical utility and ease of use.

Deterministic Trends.
With a view to potential applications it is clearly of interest to extend the preceding stochastic results to allow for the addition of underlying deterministic "trends", or in other words, both stochastic and deterministic forcing terms. To this end we briefly consider equations of the form where B is a standard one dimensional Brownian motion and h a deterministic function obeying We deal with the case of Brownian noise for illustrative purposes but this is not an essential component of the following discussion and that similar results can be proven with more general noise terms.
The following results, which are stated without proof, rely on simple size restrictions on H 0 and Σ of the form When λ = ∞ it is clear from scanning the relevant proofs that the results of Section 5.1 are essentially unchanged. Similarly, when λ = 0 the results of Section 5.1 can be amended by simply changing hypotheses on L f (Σ) to the corresponding assumption on L f (H 0 ) with an analogous modification to the conclusion; the sample results below are indicative of this process. The case λ ∈ (0, ∞) can be treated similarly but due to the number of free parameters cases quickly proliferate so for the sake of brevity we omit the details of this extension.
Example 6.2. Let Z be an α-stable process with index α ∈ (0, 2) and, as in the previous example, suppose we have a power-type nonlinearity given by Let µ be a measure obeying (A1). First let the function γ + be given by By construction, γ + is increasing, positive and satisfies β)) .
If the interval (1/α, 1/(1 − β)) is nonempty, then we can take γ in the statement of Theorem 5.12 to be γ + with ǫ ∈ (1/α, 1/(1 − β)). Hence the solution of (1.2) obeys This essentially means that if the nonlinearity is sufficiently strong we cannot experience growth in the solution of (1.2) faster than that seen in (1.4) with positive probability. The restriction β > 1 − α is intuitive in the following sense: the smaller α is the more mass there is in the tail of the Lévy measure associated with Z and hence the partial maxima of Z will tend to grow faster the smaller the value of α; when α is small we require a stronger nonlinearity (larger value of β) to retain the unperturbed growth rate. When α ≥ 1 we always retain the growth rate of the unperturbed equation.
If we take ǫ = 1/(1 − β), then L f (γ + ) = 1/M (1 − β) and we can apply Theorem 5.13 to yield where we require β > (M − 1)/M to ensure that L f (γ + ) > 1. Theorem 5.13 also yields lim sup t→∞ |X(t)| t ǫ = 0 a.s., for each ǫ > max In other words, the solution of (1.2) is o(t ǫ ) with probability one for ǫ sufficiently large (in terms of both the noise and nonlinearity). Next define the function γ − by Note that γ − is positive, increasing and obeys ∞ 0 γ − (t) −α dt = ∞. Since we aim to apply Theorem 5.13 we are only interested in the case L f (γ − ) ∈ (1, ∞]. It is straightforward to show that and lim sup t→∞ |X(t)| t δ = ∞ a.s. for each δ such that We take this opportunity to remark upon a limitation of Theorem 5.13 (and it's deterministic counterpart Theorem 3.5). By comparing (6.2) and (6.3) the reader can see that it is not possible to have both L f (γ + ) ∈ (1, ∞) and L f (γ − ) ∈ (1, ∞) simultaneously in this example; indeed this case is difficult to engineer and only possible in limited circumstances (such as when the nonlinearity is regularly varying with unit index).

Proofs of Miscellaneous Results
Proof of Lemma 2.3.
establishing the first part of (2.2). To prove the second claim estimate as follows Taking the limsup and using the first claim completes the proof.
Proof of Lemma 2.4. By hypothesis, for all ǫ > 0 there exists T (ǫ) > 0 such that for all t ≥ T (ǫ)
Letting y(t) = x(t + T ) for t ≥ 2T 3 , it is straightforward to show that
Proof of Theorem 3.3. The required lower bound, lim inf t→∞ F (x(t))/M t ≥ 1, can be derived exactly as in Theorem 3.2. For the upper bound begin by recalling the estimate (8.4) from the proof of Theorem 3.2: Remark 8.1. The stronger hypothesis (A4) can be used to improve the estimate above . We state this improvement here for convenience. Using the mean value theorem, (A4) and the first part of Lemma 2.3, estimate as follows: where θ t ∈ [0, 1] results from using the mean value theorem. The differential inequality above is now linear in I ǫ (t) and can be solved explicitly; we will return to this estimate frequently.
Then, since x(t) > H(t) for all t ≥ 0, and it follows immediately that Furthermore, because φ preserves asymptotic equivalence (see Lemma 2.4 and note that it requires (A4)), Using the facts collected above compute as follows lim sup Returning to (8.8) and using the limit above yields Finally, let ǫ → 0 + to give the desired conclusion.
Proof of Theorem 3.4 (b.) Follow the argument of Theorem 3.4 (a.) exactly to equation (8.15), which we recall below.
Proof of Theorem 3.6 (a.). Case (a.) follows from Theorem 3.3 and by taking γ = H in Theorem 3.4. Similarly, the first limit in (3.6) is obtained by choosing γ = H in Theorem 3.4. Note that L f (H) ∈ (1, ∞) and our positivity assumptions imply that H is asymptotically increasing.
Proof of Theorem 3.6 (b.). The first limit in (3.6) follows from positivity of H, which implies lim inf t→∞ x(t)/H(t) ≥ 1 directly from (1.1), and setting γ = H in case (b.) of Theorem 3.4. The proof of the second limit in (3.6) is straightforward. By hypothesis and Proposition 1, F (H(t))/t → ∞ as t → ∞. Therefore, for every N > 1 there is T (N ) > 0 such that H(t) > F −1 (N t) for t ≥ T (N ). But H positive implies x(t) > H(t). Thus x(t) > F −1 (N t), or F (x(t))/t > N , for all t ≥ T (N ). Hence lim inf t→∞ F (x(t))/t ≥ N . Letting N → ∞ gives the second part of (3.6).

Proofs of Results with Brownian Noise
Proof of Theorem 5.3. The proof of this result follows directly from the argument used in the proof of Theorem 5.4 and the law of the iterated logarithm for continuous local martingales.
Proof of Theorem 5.6. Let ǫ ∈ (0, 1) be arbitrary and follow the line of argument from the proof of Theorem 5.7 to obtain where A ǫ = M T sup s∈[0,T1] |X(s)|. We define the upper comparison solution X ǫ as in (9.14) by Now by (9.11) there exists T 1 (ǫ) > T such that Hence there exists T 2 (ǫ) > T 1 such that For t ≥ T 2 , using (9.10), calculate as follows .
Let ǫ → 0 + and note that by construction |X(t)| ≤ X ǫ (t) for all t ≥ T . Therefore, as required.
Now apply the argument from the proof of Theorem 3.6 beginning at (8.12). Following this line of argument shows that lim sup Returning to (9.14) this yields , t ≥ T * .

Proofs of Results with Stable Lévy Noise
Proof of Theorem 5.12. First note that Φ(γ + (t)) γ + (t) = 0, for any nonnegative constants A and B. With the estimate above in hand and the proof proceeds as in that of Theorem 5.7 but we arrive at a slightly different initial upper estimate to that derived in equation (9.12) since we employ (10.1) for the asymptotics of Z. By following exactly the steps from the proof of Theorem 5.7 we obtain lim sup t→∞ |X ǫ (t)|/γ + (t) < ∞ with probability one and hence lim sup t→∞ |X(t)| γ + (t) < ∞ a.s. (10.4) With the usual notation that J(t) = t 0 M (t − s)f (X(s)) ds write To finally derive the required bound on lim sup t→∞ |X(t)|/γ + (t) estimate |J(t)| using (10.4) (as was done in the proof of Theorem 5.7, for example); conclude by plugging in this estimate above and using (10.1). The proof is essentially the same when L f (γ + ) = ∞. To show that lim sup t→∞ |X(t)|/γ + (t) = 0 a.s. proceed as before in applying the argument of Theorem 5.7 but note now that this will give lim sup t→∞ X ǫ (t)/γ + (t) ≤ 3ǫ for the comparison solution. The conclusion now follows readily.
Using the fact that f • x is sub-exponential and increasing we have