Sample Paths Estimates for Stochastic Fast-Slow Systems driven by Fractional Brownian Motion
We analyze the effect of additive fractional noise with Hurst parameter on fast-slow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet, the setting of fractional Brownian motion does not allow us to use the martingale methods from fast-slow systems with Brownian motion. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, tailored to the fast-slow structure of the system, that contains the process with high probability. We prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood. We also illustrate our results in an example arising in climate modeling, where time-correlated noise processes have become of greater relevance recently.
Fast-slow systems naturally arise in the modeling of several phenomena in natural sciences, when processes have widely differing rates [25, 20, 18]. The standard form of a fast-slow system of ordinary differential equations (ODEs) is given by
where are the fast variables, are the slow variables, is a small parameter, and are sufficiently smooth vector fields; for a more detailed technical introduction regarding the analysis of (1) we refer to Section 2.1. Here we just point out the basic aspects from the modeling perspective. First, note that if , then (1) becomes are parametrized set of ODEs, where the -variables are parameters. Taking this viewpoint, all bifurcation problems [16, 27] involving parameters naturally relate to fast-slow dynamics if the parameters vary slowly, which is often a natural assumption in applications. Second, in practice, we also want to couple many dynamical systems. The resulting large/complex system is often multiscale in time and space. For example, in the context of climate modeling [8, 21] coupled processes can evolve on temporal scales of seconds up to millennial scales. Third, fast-slow systems are the core class of dynamical problems to understand singular perturbations , i.e., roughly speaking singular perturbations problems with small parameters are those, which degenerate in the limit of the small parameter into a different class of equations. Combining all these observations, it is not surprising that fast-slow systems have become an important tool in more theoretical as well as application-oriented parts of nonlinear dynamics .
However, when dealing with real life phenomena certain random influences have to be taken into account and quantified in a suitable way . The most common stochastic process used to describe uncertainty is Brownian motion . One of its key features is the memory-less or Markov property, which means that the behavior of this process after a certain time only depends on the situation at the current time . In certain applications it may be desirable to model long-range dependencies and to take into account the evolution of the process up to time . One of the most famous example is constituted by fractional Brownian motion (fBm) ; see  for its first use. A fBm is a centered stationary Gaussian processes parameterized by the so-called Hurst index/parameter . For one recovers classical Brownian motion. However, for and , fBm exhibits a totally different behavior compared to Brownian motion. Its increments are no longer independent, but positively correlated for and negative correlated for . The Hurst index does not only influence the structure of the covariance but also the regularity of the trajectories. Fractional Brownian motion has been used to model a wide range of phenomena such as network traffic , stock prices and financial markets [29, 40], activity of neurons [36, 10], dynamics of the nerve growth , fluid dynamics , as well as various phenomena in geoscience [30, 23, 35]. However, the mathematical analysis of stochastic systems involving fBm is a very challenging task. Several well-known results for classical Brownian motion are not available. For instance, the distribution of the hitting time of a level is explicitly known for a Brownian motion, whereas for fBm, one has only an asymptotic statement, according to which
as goes to infinity, see . Furthermore, since fBm is not a semi-martingale, Itô-calculus breaks down. Therefore, it is highly non-trivial to define an appropriate integral with respect to the fBm. This issue has been intensively investigated in the literature. There are numerous approaches that exploit the regularity of the trajectories of the fBm in order to develop a completely path-wise integration theory and to analyze differential equations. For more details, see [28, 12, 14, 15, 19] and the references specified therein. Furthermore, another ansatz employed to define stochastic integrals with respect to fBm relies on the stochastic calculus of variations (Malliavin calculus) developed in . In summary, fBm is a natural candidate process to aim to improve our understanding of correlated stochastic dynamics.
Our objective here is to combine the study of fast-slow systems and fBm by starting to study stochastic differential equations of the form
where we start with the case of additive noise for the fast variable(s) and assume there is a single regularly slowly-drifting variable . For , i.e., for Brownian motion, there is a very detailed theory, how to analyze stochastic fast-slow systems . One particular building block - initially developed by Berglund and Gentz - uses a sample paths viewpoint . This approach has recently been extended to broader classes of spatial stochastic fast-slow systems  and it has found many successful applications; see e.g. [1, 24, 38, 41]. Therefore, it is evident that one should also consider the case of correlated noise in the fast-slow setup [46, 17].
Our key goal is to derive sample paths estimates for fast-slow systems driven by fBm with Hurst index . We restrict ourselves to the case of additive noise and establish the theory for the normally hyperbolic stable case. Due to the technical challenges mentioned above, we need to derive sharp estimates for the exit times for processes solving certain equations driven by fBm. Exploring various properties of general Gaussian processes, we propose two variants to obtain optimal sample paths estimates. Then we apply our theory to a climate model describing the North-Atlantic thermoline circulation.
This work is structured as follows. In Section 2 we introduce basic notions from the theory of fast-slow systems and fractional Brownian motion. Furthermore, we state important estimates for the exit times of Gaussian processes which will be required later on. In Section 3, we generalize the theory of  by first deriving an attracting invariant manifold of the variance using the fast-slow structure of the system. Based on this manifold we define a region, where the linearization of the process is contained with high probability. In order to prove such statements, we first derive a suitable nonlocal Lyapunov-type equation for the covariance of the solution of a linear equation driven by fBm, the so-called fractional Ornstein-Uhlenbeck process. Thereafter we analyze two variants which entail sharp estimates for the exit times of this process. Furthermore, we consider more complicated dynamics and provide extensions of our results to the non-linear case, more complicated slow dynamics and finally discuss the case of fully coupled dynamics. We apply our theory to a model for the North-Atlantic thermohaline circulation and provide some simulations. Section 4 generalizes the sample paths estimates to higher dimensions in the autonomous linear case. Our strategy is based on diagonalization techniques, which allow us to go back to the one-dimensional case and apply the results developed in Section 3. For completeness, we provide an appendix which contains a detailed proof regarding the limit superior of a non-autonomous fractional Ornstein-Uhlenbeck processes. We conclude in Section 5 with an outlook of possible continuations of our results.
2.1 Deterministic Fast-Slow Systems
In this section, we will briefly introduce the terminology of fast-slow systems. We restrict ourselves to the most important results tailored to our problem in the upcoming sections. For further details, see . For the definition of the setting, all of the equations are to be understood formally. We will later add regularity assumptions sufficient to deduce important results. These also imply that the formal computation we will have performed before are valid.
A fast-slow system is an (ODE) of the form
where , are the unknown functions of the slow time variable , the vector fields are , and is a small parameter. The variables are called fast variables, while variables are called slow variables. Transforming into another time scale by defining the fast time yields the equivalent system
Depending on the situation both formulations in fast and slow time may be of use. In particular, under certain assumptions, considering them for indicates a lot of information for the underlying dynamics for the case . The process for is called singular limit. The singular limit of (3) for
is called fast subsystem. The resulting system of the slow time formulation of the fast-slow system (4) for
is called slow subsystem. The set
is called critical set. If is a manifold, it is also called critical manifold. From now on, we assume that is a manifold given by a graph of the slow variables, i.e.,
where is an open subset.
Let , , and their derivatives up to order be uniformly bounded. Assume that is uniformly hyperbolic. Then for an there exists a locally invariant -smooth manifold
for all , where with respect to the fast variables. Furthermore, the local stability properties of are the same as the ones for .
2.2 Fractional Brownian Motion
In this section we state important properties of fBm, which will be required later on. For further details see [32, 3] and the references specified therein. We fix a complete probability space and use the abbreviation a.s. for almost surely.
Let . A one-dimensional fractional Brownian motion (fBm) of Hurst index/parameter is a continuous centered Gaussian process with covariance
Note that for the covariance of fBm satisfies
We further observe that:
for one obtains the Brownian motion;
for then a.s. for all . Due to this reason one always considers .
The following result regarding the structure of the covariance of fBm holds true, see [32, Section 2.3].
Let . Then, the covariance of fBm has the integral representation
where the integral kernel is given by
for a positive constant depending exclusively on the Hurst parameter.
We remark that for suitable square integrable kernels, one obtains different stochastic processes, for instance the multi-fractional Brownian motion or the Rosenblatt process, see . We now focus on the most important properties of fBm. For the complete proofs of the following statements, see [32, Chapter 2].
Proposition 2.5 (Correlation of the increments).
Let be a fBm of Hurst index . Then its increments are:
positively correlated for ;
independent for ;
negatively correlated for .
Particularly, for fBm exhibits long-range dependence, i.e.
Let be a fBm of Hurst index . Then:
i.e. fBm is self-similar with Hurst index .
[Stationarity of increments] For all
[Regularity of the increments] fBm has a version which is a.s. Hölder continuous of exponent .
We conclude this section emphasizing the following result, which makes fBm very interesting from the point of view of applications, see [32, Section 2.4 and 2.5].
Let be a fractional Brownian motion with Hurst index . Then is neither a semi-martingale nor a Markov process.
2.2.1 Integration Theory for
Since fBm is not a semi-martingale, the standard Itô calculus is not applicable. Due to this reason, the construction of a stochastic integral of a random function with respect to fBm has been a challenging question, see [6, 3] and the references specified therein. However, for deterministic integrands and for the theory essentially simplifies. We deal exclusively with this case and indicate for the sake of completeness the theory of Wiener integrals of deterministic functions with respect to fBm, see . Let and
be the set of step functions on . For define the linear mapping
Observe that defines a Gaussian random variable with
The representation of the variance can be easily verified by noting the following identity
Note that is crucial here. For we can bound the -norm of as follows
where we have obtained the estimate by applying Hölder’s inequality and Young’s inequality for convolutions [4, Theorem 3.9.4]. The boundedness claim now follows as for . This means that is a bounded linear operator defined on the dense subspace , so it can be uniquely extended to a bounded operator
This discussion justifies the following definition:
For and we set
The integral process is by construction centered Gaussian. Regarding (7), its covariance can be immediately computed as follows.
Proposition 2.9 (Covariance of the integral).
Let and for . Then
2.2.2 Stochastic Differential Equations Driven by Fractional Brownian Motion
After establishing a suitable stochastic integral with respect to the fractional Brownian motion, we consider stochastic differential equations (SDEs) given by:
Its solution satisfies the integral formulation
where the stochastic integral was constructed in Section 2.2.1. Under certain classical regularity assumptions, existence and uniqueness of solutions for (9) can be proven. For more details, see [3, Theorem D.2.4].
Let be globally Lipschitz in both variables, with and globally Lipschitz. Then for every the SDE (9) has a unique continuous solution on a.s..
For our aims, we consider time-dependent linear drift, i.e., is linear with for every and . In this case, the solution of (9) is given by the variation of constants formula/Duhamel’s formula and is called non-autonomous fractional Ornstein-Uhlenbeck process.
Theorem 2.11 (Non-autonomous Fractional Ornstein-Uhlenbeck Process).
Let . Suppose that is globally Lipschitz and uniformly bounded, and with as well as globally Lipschitz. Then there exists an a.s. unique solution to the stochastic differential equation
which satisfies the variation of constants formula
Note that all the results discussed in this subsection extend to higher dimensions, since all previous steps can be done component-wise. Namely, for we mention.
We call an -dimensional fractional Brownian motion if , where is a basis in and , , are independent one-dimensional fractional Brownian motions with the same Hurst index .
Naturally, existence and uniqueness of SDEs in higher dimension carry over from Theorem 2.10 under the same assumptions respectively. In particular, for coefficients with , satisfying the same assumptions as in Theorem 2.11, the solution of (10) is given by
where denotes the fundamental solution of and is an -dimensional fractional Brownian motion.
2.3 Useful Estimates of Gaussian Processes
The fact that fBm is not a semi-martingale restricts the repository of known inequalities (such as Doob or Burkholder-Davies-Gundy) to establish sample paths estimates. A crucial property of fBm we shall exploit is its Gaussianity. In this section we will describe some useful estimates for exit times of certain Gaussian processes, which will be helpful for our analysis in the upcoming sections.
We first state the next auxiliary result regarding the Laplace transform of a Gaussian process. This was established in  by means of Malliavin calculus.
(Proposition 3.5 ) Let be a centered Gaussian process with and covariance function satisfying the following conditions:
exists and is continuous as a function on ,
for all ,
for all ,
Then for any :
In addition, we require the following form of Chebychev’s inequality.
Let be measurable, a random variable and . Then
Under these assumptions we have
Taking expectation in the above inequality yields the result. ∎
Let and be a centered Gaussian process with satisfying the assumptions i)-iv) of Lemma 2.13. Then, for its exit time , the following estimate holds:
The previous lemma established a Bernstein-type inequality solely relying on certain properties of the covariance function of Gaussian processes. Another useful estimate is given by [34, Theorem D.4], which is based on Slepian’s Lemma .
Let and be a centered Gaussian process with a.s. continuous trajectories. Assume that is a.s. mean-square Hölder continuous, i.e. there are constants and such that
Then there exists a constant such that for and
This estimate can be sharpened if we restrict ourselves to the interval of interest.
Let and be a centered Gaussian process with a.s. continuous trajectories. Assume that is a.s. mean-square Hölder continuous, i.e. there are constants and , such that
Then there exists a constant such that for and
with satisfies the assumptions of Theorem 2.16 on . ∎
3 The One-Dimensional Case
In this section, we investigate the dynamics of a planar stochastic fast-slow system driven by fractional Brownian motion with Hurst parameter :
Its equivalent formulation in slow time, i.e. for is
using the self-similarity of fBm (6). We are interested in the normally hyperbolic stable case and therefore make the following assumptions.
Regularity: The functions and , as well as all their existing derivatives up to order two are uniformly bounded on an interval or , , by a constant .
Critical manifold: There is an such that
for all .
Stability: For there is such that
for all .
has an asymptotically slow manifold for small enough due to Fenichel-Tikhonov (Theorem 2.2). We expect that, given small noise , the trajectories of (12) starting sufficiently close to remain in a properly chosen neighborhood of for a long time with high probability. Our goal will be to make this idea rigorous by pursuing the following steps. We first linearize the system around the slow manifold to get an SDE describing the deviations induced by the noise. This helps us obtain a simple description of a suitable neighborhood by using the fast-slow structure inherited by the variance of the system. Then, using this neighborhood, we deduce sample paths estimates for the linear case starting on the slow manifold. To complete the discussion we generalize the result to the non-linear case starting sufficiently close to the slow manifold, that is, such that in the deterministic case solutions are still attracted by the slow manifold. This general strategy inspired by , where a similar system driven by Brownian motion (Hurst parameter ) is analyzed. Yet, the several techniques used in  do not generalize to fBm.
3.1 The Linearized System
The deterministic system
has an asymptotically stable slow manifold due to Fenichel-Tikhonov (Theorem 2.2). As already outlined, our first step is to examine the behavior of the linearized system around . For a solution of we set . Then satisfies the equation
by Taylor’s remainder theorem. Due to the uniform boundedness of the derivatives of one can show that the -term is negligible on finite time scales as Therefore, we restrict ourselves without loss of generality to the analysis of the linearization
Examining the process starting on the slow manifold now corresponds to investigating the unique explicit solution of (14) for initial value , which is given by the fractional Ornstein-Uhlenbeck process (recall Theorem 2.11)
where . In order to define a proper neighborhood, where the fractional Ornstein-Uhlenbeck process is going to stay with high probability, we use the variance as an indicator for the deviations at time . According to Proposition 2.9, the variance is given by
As we would like to see dynamics of , we rescale it by to get rid of the small parameter , which only changes the order of magnitude of the system. It turns out that inherits the fast-slow structure from the SDE, which yields a particularly simple approximation of the variance.
The so-called renormalized variance satisfies the fast-slow ODE
In particular, there is a (globally) asymptotically stable slow manifold of the system of the form
In order to be able to take the singular limit and apply Fenichel-Tikhonov (Theorem 2.2) we need to prove sufficient regularity in ; continuous differentiability will be enough for the approximation of the slow manifold with the critical manifold up to order . To do this, rewrite the integral by substituting
To see that the right hand side of (15) is continuously differentiable in it is sufficient to check it for the integral term
which has an existing limit for because the exponential term goes to faster than the polynomial term diverges. Now taking the singular limit gives the slow subsystem
The critical manifold is hence given by
Using integration by parts we can rewrite , so that the critical manifold can also be written as
As expected, the critical manifold depends on the Hurst parameter . For we have . This means that the only possible structural change of the critical manifold under variation of is induced by the factor . There are two cases. For the critical manifold of the variance increases, as is increasing, while for it decreases. This behavior is different to comparable continuous-time Markovian dynamical systems and shows the strong influence of the time-correlated noise. Furthermore, as for , the slow subsystem for reads
which coincides with the slow subsystem we would obtain in the case of Brownian motion noise, which exactly corresponds to .
The proof of Proposition 3.2 only shows that is in and in the time . Depending on the properties of we expect to even have higher regularity. However, this fact is not required in the following considerations.
Proposition 3.2 already states that the slow manifold is a good indicator for the size of the set we are looking for as (as a solution of (15) with initial datum ) is attracted by the slow manifold. In this particular case we can explicitly state the exponentially fast approach due to the structure of the linear equation
where . Even more is known about the properties of . Due to the uniform boundedness assumption on and we get that the difference between and is actually in uniform This implies that for small enough there are and such that
The goal is now to prove that the stochastic process is concentrated in sets of the form
To get a better understanding of what to expect, note that the probability that leaves at time can be bounded by using the inequality , which holds for any centered Gaussian random variable . This further leads to
Of course, the probability that has exited in the interval at least once