Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process

We study a bistable gradient system 
perturbed by a stable-like additive process with a periodically varying stability index. 
Among a continuum of intrinsic time scales determined by the values of the stability index 
we single out the characteristic time scale on which the system exhibits the 
metastable behaviour, namely it behaves like a time discrete two-state Markov chain.


1.
Introduction. In this work we study the small noise behaviour of a jump diffusion in a double-well potential driven by a stable-like additive process with independent increments and periodic stability index.
The motivation for this work stems from the observation made by P. Ditlevsen that the abrupt climate transitions between glacial and interstadial states during the last glacial period (the Dansgaard-Oeschger events) can be well understood as a realization of a bistable dynamical system perturbed by noise which contains a heavy-tail α-stable Lévy component, see [4,5]. The proxy signal for the global Earth's temperature is modelled there as a solution of a stochastic differential equation (SDE) The function x → V (x) is a double-well potential with two stable minima m − < 0 < m + identified with the cold and warm climate states and separated by the unstable local maximum m 0 = 0, e.g. a standard quartic potential V (x) = x 4 4 − x 2 2 . The process L is a symmetric α-stable Lévy process with the jump measure ν(dx) = |x| −1−α dx, α ∈ (0, 2). It was noted that big (heavy) jumps of the driving process L induce fast transitions of X ε between the potential wells.
The use of an α-stable noise in (1.1) is rather ad hoc and the values of α can be derived not from the physical laws but rather from the data. Using a histogram analysis, Ditlevsen estimated the stability index as α ≈ 1.75. Hein et al. in [13] studied the empirical p-variation of the time series and obtained α ≈ 0.75. On the other hand, Gairing et al. [10] studied the heavy tailed noises without an assumption 3176 ISABELLE KUHWALD AND ILYA PAVLYUKEVICH of α-stability and discovered that for large |x| the tails of the Lévy measure should decay as ν(dx) ≈ |x| −1−α dx, with α ≈ 3.5.
In the present paper we will continue to work with the technically simpler αstable distributions. One can expect however that our results will hold for noises with polynomial regularly varying tails, see [21].
The metastable behaviour of X ε in the small noise regime, namely the laws of transitions of X ε between the wells, was studied in [20,21,18]. In particular it was shown (see, e.g. Theorem 1.1 in [21]) that the finite dimensional laws of the the jump diffusion (X ε αε −α t ) t≥0 on a new fast time scale converge to those of a continuous time Markov chain on the state space {m − , m + } with the generator (1. 2) This means that in the small noise limit, the transition times normalized by the factor α −1 ε α are exponentially distributed with means |m ± | α . A more thorough analysis (see, e.g. Schulz [31]) of the palaeoclimatic data from the Greenland ice cores which motivated Ditlevsen's conjectures detected a periodic signal with the estimated period of 1470 years. The existence of such a periodic component may be interpreted as a response of the climate system to an external periodic forcing and can be explained within the paradigm of the stochastic resonance, see [1,11,6] for more information and a discussion. In the SDE setting, such dynamics can be explained as follows. Consider a non-autonomous SDE with a time-periodic double-well potential U , U (·, t) = U (·, t + 1), and U (x, t) = d dx U (x, t). The dependence of U on time can be interpreted as a perturbation of a fixed double-well potential by a weak periodic signal, e.g. U (x, t) = V (x) + ax cos(2πt), a > 0 being small. Then, for noise amplitudes ε within a certain range (the optimal tuning amplitudes), the exit times of X ε,T from the potential wells get synchronized with the weak periodic signal. In climate modelling, the weak periodic forcing is often related to Earth's orbital changes or deep ocean processes. Stochastic resonance in diffusions driven by small Gaussian noise (L = B being a standard Brownian motion) was firstly studied in [9] in the framework of large deviations theory, and also in [19,2,14,15,17,16]. The crucial role in the description of the stochastic resonance is played by the exponentially large exit times from the wells which obey the so-called Kramers law. In a time homogeneous case, U (x, t) = V (x) they are of the order exp(2∆V ± /ε 2 ), where ∆V ± are the heights of the potential barriers separating the wells. In the presence of a weak periodic perturbation, the barriers ∆U ± ( t 2T ) are time dependent. However it is intuitively clear that in the so-called adiabatic limit when T is very large, the stochastic resonance can only be observed if the noise amplitude ε and the period 2T are related in such a way that a Brownian particle has enough time to exit the shallow well during the half period (see, e.g. Theorem 1 in [9] and Chapter 3.1 in [16]).
Stochastic resonance in a jump-diffusion (1.3) driven by Lévy processes with heavy tails was studied recently in [25,26,27]. The characteristic feature of the jump-diffusion (1.3) driven by an α-stable Lévy process L is the unique (up to a constant factor) time scale of the order ε −α on which the transitions between the wells occur. It was shown that a non-trivial behaviour of X ε,T can be observed only for the periods of the order 2T (ε) = Const · ε −α . In [25,26,27], the authors determined the "optimal tuning" pre-factor which maximizes the probability to exit the well at time instants when the distance |m ± (t) − m 0 (t)| between the current well's minimum and the saddle point is minimal.
The obvious drawback of the models (1.1) and (1.3) is the unique time scale O(ε −α ) on which the non-trivial metastable behaviour can be observed. In the present paper we are going to modify the equation (1.1) and consider a bistable jump-diffusion driven by a heavy-tail noise which has a continuum of intrinsic time scales of different orders w.r.t. ε. This can be achieved by making the stability index α of the driving process depending on time and/or the current location of the jump-diffusion. Such perturbations are called stable-like.
Elements of the theory of stable-like jump diffusions with a spatially dependent stability index can be found in [23,Chapter 5] and [24,Chapter 7]. A scaling limit behaviour of stable-like processes with a spatially periodic stability index was performed in [7,8]. Large deviations type theorems for general Markov processes with heavy tails were obtained in [12].
In this work we assume that the stability index α depends only on time, namely the function t → α(t) is 1-periodic. The corresponding stochastic process L = A is an additive process, i.e. a process with independent but in general not stationary increments. Such processes resemble very much Lévy processes and we can use many of the methods developed for the Lévy setting. However the dependence of α on time implies the existence of a continuum of intrinsic time scales of the order ε −µ , µ ∈ [min α(t), max α(t)] and the behaviour of the time non-autonomous jump-diffusion differs strongly from its Lévy driven counterpart. In view of our previous results on metastability of Lévy driven diffusions it is not surprising that the transition behaviour is determined by the lowest value α * , i.e. the by the heaviest component of the driving process. However it is rather unexpected that the characteristic time scale turns out to be slightly longer than the time scale ε −α * discovered in [21], namely it is of the order ε −α * | ln ε|.
The paper is organized as follows. In the next Section 2 we formulate the setting and the main result of the paper. Section 3 is devoted to a rather elementary analysis of a simple two-state Markov chain with periodic transition probabilities which mimics the behaviour of the jump diffusion. We present the proof of the main result in Sections 4 and 5. In this paper, the complement of a set A is denoted by A c , and B r (x) stands for an open ball of radius r centred at x.

2.
The setting and the main result. Let V ∈ C 2 (R d , R + ) be a double-well potential. We assume that ∇V (y) = 0 only for y = m − and m + and a saddle m 0 . The Hesse matrix of V at m − and m + is positive definite while the Hessian at m 0 is indefinite and has non-zero eigenvalues. We also assume that lim y →∞ V (y) = +∞ and the derivatives up to order two of y → ln(1 + V (y)) are bounded.
Let (y t (y 0 )) t≥0 be the solution of the ordinary differential equationẏ t = −∇V (y t ) with the initial value y 0 ∈ R d . Define the domains of attraction of the stable points m ± by (2.1) Let Γ := R d \(Ω + ∪ Ω − ) be the separatrix between Ω − and Ω + . On a probability space (Ω, F, P), let N T (dx, ds), T > 0, be a family of Poisson random measures with intensity measures ν t/2T (dx)dt where has no intervals of constancy, and the minimum of t → α(t) on t ∈ [0, 1] is attained at a unique point a ∈ (0, 1), α(a) = α * . Moreover we assume that α (a) > 0, and min t∈[0,1] c(t) > 0. Let γ = γ(t) be a bounded d-dimensional function which has bounded variation on finite time intervals, and σ be a bounded continuous d×d-matrix-valued function. Let B be a d-dimensional Brownian motion independent of the Poisson random measures N T .
which exists under the conditions on V and A T formulated above (see [25] for details). In out notation, we emphasize the dependence of Y ε,T on the noise intensity ε and the period 2T of the stability index α(·).
Denote the exit times of Y ε,T from the domains Ω ± by Our first theorem describes the behaviour of the jump diffusion Y ε,T on the algebraic time scales of the order ε −µ , µ > 0.
The result of this Theorem can be interpreted as follows. If 2T (ε) = ε −µ with µ ≤ α * , then the time horizon of the order ε −µ is too short to observe any jump, so that the jump diffusion stays in a small neighbourhood of one of the wells (see Fig.  1 (left)). If the time scale is of the order ε −µ with µ ≥ α * , the jump-diffusion jumps chaotically (see Fig. 1 (right)), i.e. the waiting times between transitions converge to zero and the current position of the jump-diffusion cannot be predicted. Finally in the intermediate regime µ ∈ (α * , α * ) the intervals of chaotic and deterministic behaviour repeat themselves periodically, namely on the time intervals where α(t) < µ, the jump-diffusion is chaotic, and if α(t) > µ then it is almost constant (see Fig.  1 (middle)). Similar intervals of chaotic behaviour were discovered by Herrmann and Imkeller in [14,15] in the context of stochastic resonance in diffusions.
The main result of this paper formulated in the following theorem recovers another sub-algebraic time scale at which the chaotic behaviour degenerates so that the limiting transition behaviour of the jump-diffusion can be described probabilistically.
Theorem 2.2. Let Y ε,T be a solution of the SDE (2.4) and let the half-period T = T (ε) and the rate λ = λ(ε) be defined as Then for any y ∈ Ω ± and t ≥ 0 the following limit holds true: where
In the next section we perform an elementary analysis of a two-state continuous time Markov chain which dynamics reminds of those of the jump discussion Y ε,T in a symmetric potential V .

BISTABILITY OF A PERIODIC STABLE-LIKE JUMP-DIFFUSION 3181
The following asymptotics is obtained as a result of a Laplace type expansion of an asymptotic integral, see Chapter 3.7 in [28].
Lemma 3.1. Let α ∈ C 2 (R, R + ) be 1-periodic with min t∈[0,1] α(t) = α(a) = α * > 0 for the unique a ∈ (0, 1). Let f ∈ C(R, R) be also 1-periodic with f (a) > 0. Then for any t ≥ 0 where the function g is defined in (2.8). Proof. The statement of the lemma is obtained by a straightforward analysis of the conditional density of τ ε,T n given τ ε,T n−1 = s which equals to In particular for any t ≥ s  (ii) and (iii) Due to the continuity of t → α(t), there is δ * > 0 such that max t∈(us,µ,us,µ+δ * ] (α(t) − µ) < 0. Consequently, for any δ ∈ (0, δ * ] we get Analogously for all δ small enough As we see, for different scalings 2T (ε) = ε −µ , the behaviour of the Markov chain Z ε,T can be almost deterministic, chaotic or repeatedly deterministic or chaotic. In the next proposition we determine such an sub-algebraic optimal period 2T (ε) such that the intervals of chaotic behaviour degenerate to points t k = a + k, k ≥ 0.
Motivated by a question of the best synchronization of the transitions of Z ε,T with the periodic signal α(·/2T ) let us determine the optimal half-period T = T (ε) at which the probability for Z ε,T to change a state is maximal. The proof of the next proposition is straightforward. Proposition 3.3. For any δ > 0, ε > 0, and k ≥ 0 the map T → P −1 (τ ε,T 1 ∈ [k + a − δ, k + a + δ]) attains its largest value at the optimal half-period length As δ → 0, the optimal half-period T k,δ (ε) tends to the value Moreover, the following asymptotics holds as ε → 0 (Laplace's method): In other words, the most synchronized behaviour of Z ε,T is attained on a time scale of the order T = T (ε) = ε −α * | ln ε|.
The next two sections will be devoted to the proof of the Theorems 2.1 and 2.2.

4.
The basic properties of the jump-diffusion Y ε,T . For the additive process A T defined in (2.3), let us consider another Lévy-Itô decomposition into a big and small jump components. For ρ ∈ (0, 1) and ε ∈ (0, 1], denote (4.1) Note that since the jump measure ν t (dx) is symmetric, the drift components of the processes A ε,T and A T coincide. Denote the jump times of the rescaled non-autonomous compound Poisson process εQ ε,T by (τ ε,T j ) j≥0 with τ ε,T 0 ≡ 0 and the corresponding jump sizes by denote the instant jump intensity. Note that β ε (t) can be estimated as a function of ε uniformly over t as Since εQ ε,T is an additive process its inter-jump times τ ε,T j+1 − τ ε,T j , j ≥ 1, are independent and have the following conditional probability density: The conditional probability law of the jump sizes W ε,T of the process εQ ε,T is given by (4.5) and the set ε −1 B : We impose no condition of the increase rate of the potential V at infinity and thus localize the dynamics of Y ε,T in a large domain which contains the local minima of the potential V . We are not able to control the behaviour of Y ε,T near the separatrix Γ and thus also exclude a small neighbourhood of Γ from O ±,L . Namely, for any δ > 0 we define the reduced domains of attraction Let Y ε,T be the unique strong solution of the SDE (2.4), and let F = F ε,T be its augmented own filtration. Between the two successive jumps of the compound Poisson process Q ε,T the dynamics of Y ε,T is described by the SDE with τ ε,T j = s, t ∈ [0, τ ε,T j+1 −τ ε,T j ) and y = Y ε,T τ ε,T j . The driving process ε( A ε,T s+· − A ε,T s ) consists of a continuous component of the order ε and a pure jump martingale with jumps which do not exceed the threshold ε 1−ρ . Thus the jump diffusion Y ε,T s,t (y) can be seen as a small noise perturbation of the deterministic dynamical systeṁ y = −∇V (y). The next Lemma shows that Y ε,T s,t (y) indeed follows the deterministic solution with high probability. Lemma 4.1. For s ≥ 0, let τ ε,T s be a random variable independent of ( A ε,T s+t ) t≥0 with the probability density (4.4). Then for any L > 0 big enough and δ > 0 small enough there are ε 0 , p 0 , γ 0 > 0 such that the following inequality holds true for all ε < ε 0 , p < p 0 , γ < γ 0 : An analogous statement was proven for SDE driven by a heavy tail Lévy process in [29]. The extension of the proof to the case of the additive noise can be found in [25]. The proof is based on the combination of an exponential inequality for martingales (see, e.g. Theorem 23.17 in [22]), Gronwall's lemma, and estimates with the help of Lyapunov functions in the neighbourhoods of the local minima of V . We emphasize that the estimate is uniform in T and s due to the following uniform estimate for the tail of the random variable τ ε,T and a uniform estimate for the martingale component of A ε,T s+· over t ∈ [0, ε −κ ].
For the argument in the next section, for L > 0, δ > 0, ε, γ > 0 from Lemma 4.1, we introduce the relaxation time R γ (ε) = R γ (ε; L, δ) such that the trajectory y t (y) satisfies sup The non-degeneracy of the potential V at the local minima m ± imply that R γ (ε) ≤ C| ln ε| for small ε > 0.
Proof. From now on, we will consider only the "left" well Ω − and prove for the exit time σ The estimate from above can be obtained analogously, but is a bit more involved technically. The proof can be found in [25]. Choose ρ ∈ ( 2 3 , 1) and define k ε = ε −κ for some κ ∈ (α * (1 − ρ), 1 2 α * ρ). For y ∈ Ω −,L (δ 1 , δ 2 ) and W ∈ R d define the events We exploit the fact that on the event E τ ε,T j ,τ ε,T j+1 (y) the small jump process Y ε,T (y) follows the deterministic trajectory y · (y). If the inter-jump time τ ε,T j+1 − τ ε,T j exceeds the relaxation time R γ (ε), we obtain that Y ε,T τ ε,T j+1 − (y) belongs to a small neighbourhood of m − , and thus we can determine the location of Y ε,T τ ε,T j+1 (y) ≈ m − + W ε,T j+1 . To make the last approximate relation precise we introduce another smaller reduced domain Ω −,L (δ 1 , δ 2 , δ 2 ) defined analogously to (5.2) as Then we obtain The last error term is estimated with the help of Lemma 4.1 as ke −1/ε p . Using the density (4.4) of the inter-jump times τ ε,T j+1 − τ ε,T j , j = 0, . . . , k − 1, and recalling the explicit formula (4.5) for the law of W ε,T j we disintegrate (5.8) We estimate the term which involves the logarithmic return times and extend the finite sum to an infinite series to obtain First we estimate the error terms in (5.9). The second sum is an error term of order ε α * ρ−2κ | ln ε|. Indeed, taking into account that for any s ≥ 0, ν t (B c 1 (0)) ε ρα * | ln ε|. (5.10) Summing up these errors over k = 1, . . . , k 2 ε = O(ε −2κ ), we get the total error estimate.
The third term in (5.9) is the probability that the compound Poisson process Q ε,T makes more than k ε jumps on the interval [0, 2T t]. With the help of the elementary inequality P(X ≥ k) ≤ c k /k!, k ≥ 0, which holds for a Poissonian random variable X with intensity c > 0, we estimate the last term in (5. Recalling that β ε (t) ≤ Const · ε α * ρ and the definition of T = T (ε) from (2.6), it is easy to check with the help of Stirling's formula that for κ > α * (1 − ρ) the r.h.s. of (5.11) converges to zero. Finally, the fourth term is easily estimated by k 2 ε e −1/ε p and converges to 0 as ε → 0.
The first series in (5.9) contributes to the main part of the probability we look for. To treat this term, we recall the identity R . . . which holds for any c > 0, n ≥ 1, and any non-negative integrable function f . Since the jump measure is self-similar, i.e. ν t ( 1 ε B) = ε α(t) ν t (B) for any B ∈ B(R d ), we get that the first term in (5.9) equals to Choose L large and δ 1 , δ sufficiently small so that ν r (Ω c −,L (δ 1 , δ 2 , δ 2 ) − m − ) ≤ ν r (Ω c − −m − )+η for r ∈ [0, 1]. Eventually, we recall again that 2T (ε) = ε −α * | ln ε| to get for small ε that  We evaluate the integral in the exponent in the limit ε → 0 with the help of the Laplace method (Lemma 3.1) and find that for some C > 0 and ε small enough P y (σ ε,T − λ(ε) ≤ t) ≥ 1 − e −c−g(t) − Cη, (5.14) what finishes the proof.
Proof. (Theorem 2.1) For definiteness we consider the "left" well Ω − . For any t > 0 and η > 0, we perform the localization and consider the reduced domains of attraction and the corresponding exit times σ ε,T ± and τ ε,T