Asymptotic first exit times of the Chafee-Infante equation with small heavy-tailed L\'evy noise

We study the first exit times form a reduced domain of attraction of a stable fixed of the Chafee-Infante equation when perturbed by a heavy tailed L\'evy noise with small intensity.

this setting, our result suggests a probabilistic interpretation of fast transitions between different climate states corresponding to the stable equilibria observed in ice core time series of temperature proxies of [3].
In the following sections we outline the partially tedious and complex arguments needed to describe the asymptotic properties of the exit times. Detailed proofs in particular of the more technical parts are given in the forthcoming [4]. and which is regularly varying with index α = −β ∈ (0, 2) and limiting measure µ ∈ M 0 (H). For a more comprehensive account we refer to [2] and [12].
(2.1) We summarize some results for the solution u(t; x) = X 0 (t; x) of the deterministic Chafee-Infante equation (ChI). It is well-known that the solution flow (t, x) → u(t; x) is continuous in t and x and defines a dynamical system in H. Furthermore the solutions are extremely regular for any positive time, i.e. u(t) ∈ C ∞ (0, 1) for t > 0. The attractor of (ChI) is explicitly known to be contained in the unit ball with respect to the norm | · | ∞ (see for instance [7], Chapter 5.6).
Proposition 2.1. For λ > 0 denote by E λ the set of fixed points of (ChI). Then for any λ > 0 and initial value x ∈ H there exists a stationary state ψ ∈ E λ of the system (ChI) such that lim t→∞ u(t; x) = ψ.
Furthermore if π 2 < λ = (kπ) 2 , k ∈ N, there are two stable fixed points and all elements of E λ are hyperbolic. In addition, the stable and the unstable manifolds of any unstable fixed point of E λ intersect transversally.
This relies on the fact that there is an energy functional, which may serve as a Lyapunov function for the system. A proof of the first part can be found in [8], [11], and of the second part in [10].
Definition 2.2. For λ > π 2 the solution of system (ChI) has two stable stationary states denoted by φ + and φ − . The full domains of attraction are given by and the separatrix by Due to the Morse-Smale property the separatrix is a closed C 1 -manifold without boundary in H of codimension 1 separating D + from D − , and containing all unstable fixed points. For more refined results we refer to [17] and references therein.
Proposition 2.4. Given the Chafee-Infante parameter π 2 < λ = (kπ) 2 for all k ∈ N there exist a finite time T rec = T rec (λ) > 0 and a constant κ = κ(λ) > 0, which satisfy the following. For each γ > 0 there is ε 0 = ε 0 (γ) > 0, such that for all 0 < ε ε 0 , T rec + κγ| ln ε| t and x ∈ D ± (ε γ ) This results relies on the hyperbolicity of the fixed points and the fine dynamics of the deterministic solution flow. In [4] it is proved in the stronger Hilbert space topology of H. The preceding theorem follows then as a corollary. We denote the jump increment of L at time t 0 by ∆ t L := L(t) − L(t−), and decompose the process L for ρ ∈ (0, 1) and ε > 0 in the following way. We call η ε the "large jump" compound Poisson process with intensity β ε := ν (ε −ρ B c 1 (0)) and jump probability measure ν(· ∩ ε −ρ B c 1 (0))/β ε , and the complementary "small jump" process ξ ε := L − η ε . The process ξ ε is a mean zero martingale in H thanks to the symmetry of ν with finite exponential moments. We define the jump times of η ε as and the times between successive large jumps of η ε t recursively as t 0 = 0 and . We shall denote the k-th large jump by W 0 = 0 and W k = ∆ T k L for k 1.
A proof can be found in [16], Chapter 10. By localization this notion of solution is extended to the heavy-tailed process L. In [4] this will be carried out in detail.
Definition 2.7. For γ ∈ (0, 1), ε > 0, and the càdlàg mild solution X ε (·; x) of (2.1) with initial position x ∈D ± (ε γ ) we define the first exit time from the reduced domain of attraction We now introduce the following two hypotheses, which will be required in our main theorem. They are natural conditions on the regularly varying Lévy measure ν with respect to the underlying deterministic dynamics in terms of its limit measure µ. See [12] for the relationship between ν and µ, and (2.9) below for the particular scaling function 1 needed here. (H.2) Non-degenerate limiting measure: For α ∈ (0, 2) and Γ > 0 according to Proposition 3.4 let While (H.1) ensures that there actually are transitions also by "large" jumps with positive probability, (H.2) implies that the slow deterministic dynamics close to the separatrix does not distort the generic exit scenario of X ε . For comparable finite dimensional situations with absolutely continuous Lévy measure ν dx these hypotheses are always satisfied. For ε > 0 we define the characteristic rate of the system (2.1) by According to [2] and [12] for ν chosen above there is a slowly varying function We may now state the main theorem.
Theorem 2.8. Given the Chafee-Infante parameter π 2 < λ = (kπ) 2 for all k ∈ N, we suppose that Hypotheses (H.1) and (H.2) are satisfied. Then for The supremum in the formula can be replaced by the infimum.
The theorem states that in the limit of small ε, suitably renormalized exit times from reduced domains of attraction have unit exponential laws.

The Small Deviation of the Small Noise Solution
This section is devoted to a small deviations' estimate. It quantifies the fact, that in the time interval between two adjacent large jumps the solution of the Chafee-Infante equation perturbed by only the small noise component deviates from the solution of the deterministic equation by only a small ε-dependent quantity, with probability converging to 1 in the small noise limit ε → 0. Define the stochastic convolution ξ * with respect to the small jump part ξ ε by ξ * (t) = t 0 S(t−s)dξ ε (s) for t 0 (see [16]). In order to control the deviation for Y ε − u for small ε > 0, we decompose Y ε = u + εξ * + R ε . By standard methods we obtain in [4] the following lemmas.  For ρ ∈ (0, 1), γ > 0, p > 0 and 0 < Θ < 1 there are constants C > 0 and ε 0 > 0 such that for 0 < ε ε 0 and T 0 P sup Define for T > 0, Γ > 0 and γ > 0 the small convolution event By perturbation arguments, the stability of φ ± , Proposition 2.4 and Lemma 3.1 we may estimate the remainder term R ε for small ε.
We next combine Proposition 2.4, Lemma 3.1 and Lemma 3.2, to obtain the following proposition on small deviations on deterministic time intervals.
There is a constant Γ > 0 such that for 0 < α < 2 given the conditions there exist ε 0 > 0 and C > 0 such that for any T > 0, 0 < ε ε 0 and This can be generalized to the first jump time T 1 replacing T .

Corollary 3.1. Given the assumptions of Proposition 3.4 there is a constant
Corollary 3.2. Let C > 0, and let the assumptions of Proposition 3.4 be satisfied. Then there is a constant ε 0 > 0 such that for all 0 < ε ε 0 , θ > −1

Asymptotic first exit times
In this section we derive estimates on exit events which then enable us to obtain upper and lower bounds for the Laplace transform of the exit times in the small noise limit.

Estimates of Exit Events by Large Jump and Perturbation Events
To this end, in this subsection we first estimate exit events of X ε by large jump exits on the one hand, and small deviations on the other hand. Denote the shift by time t on the space of trajectories by θ t , t 0. For any k ∈ N, t ∈ [0, t k ], x ∈ H we have In the following two lemmas we estimate certain events connecting the behavior of X ε in the domains of the type D ± (ε γ ) with the large jumps η ε in the reshifted domains of the type D ± 0 (ε γ ). We introduce for ε > 0 and x ∈D ± (ε γ ) the events We exploit the definitions of the reduced domains of attraction in order to obtain estimates of solution path events by events only depending on the driving noise. For ρ ∈ 1 2 , 1 , γ ∈ (0, 1 − ρ) there exists ε 0 > 0 so that the following inequalities hold true for all 0 < ε ε 0 and x ∈ D ± (ε γ ) In the opposite sense for x ∈D ± (ε γ ) With the help of Lemma 4.1 we can show the following crucial estimates.
A detailed proof is given in [4].

Asymptotic Exit Times from Reduced Domains of Attraction
We next exploit the estimates obtained in the previous subsection and combine them with the small deviations result of section 3, to identify the exit times from the reduced domains of attraction with large jumps from small neighborhoods of the stable equilibria that are large enough to cross the separatrix.
Proof. By (H.2) Γ > 0 can be chosen large enough to fulfill the hypotheses of Proposition 3.4. Let C > 0 be given. We drop the superscript ±. Since the jumps of the noise process L exceed any fixed barrier P-a.s., τ x (ε) is P-a.s. finite.
Therefore we may rewrite the Laplace transform of τ x (ε) for ε > 0, giving Using the strong Markov property, the independence and stationarity of the increments of the large jumps W i we obtain for k 1 In the subsequent Claims 1-4 we estimate the preceding factors with the help of Lemma 4.2.

Claim 1:
There exists ε 0 > 0 such that for all 0 < ε ε 0 In fact: in the inequality of Lemma 4.2 ix) we can pass to the supremum in y ∈ D(ε γ ), and integrate to obtain, using the independence of (W i ) i∈N and The terms K 1 , K 3 and K 4 can be calculated explicitly, for K 2 we apply Lemma 4.3 II). For K 5 we use Corollary 3.2 and Lemma 4.3 I) ensuring that there is ε 0 so that we have for 0 < ε ε 0 (4.11)

Claim 2:
There is ε 0 > 0 such that for all 0 < ε ε 0 Indeed, in a similar manner and with the help of Lemma 4.2 x) and Lemma 4.3 III) we obtain that there is ε 0 > 0 such that for all 0 < ε ε 0 In order to treat the summands of the second sum of (4.10) we have to distinguish the cases θ 0 and θ ∈ (−1, 0), as well as k = 1 and k 2. Let us first discuss the case θ 0.

Claim 3:
There is ε 0 > 0 such that for all 0 < ε ε 0 This statement is proved by means of Lemma 4.2 xi) and Corollary 3.1.

Claim 4:
There exists ε 0 > 0 such that for any k 2 To show this, we use the strong Markov property and Lemma 4.2 xii), as in the estimate for the first summand to get for k 2 and θ 0 Lemma 4.2 xii) and Lemma 4.3 IV ) provide the existence of ε 0 > 0 such that for 0 < ε ε 0 It remains to discuss the case θ ∈ (−1, 0) in a similar way. This is detailed in [4].
The series converges if and only if C < θ + 1. exp −θλ ± (ε)τ ± x (ε) Proof. Again we omit the superscript ± and fix Γ > 0 large enough due to (H.2). Omitting the term I 2 in equation (4.10), we obtain the estimate We treat the terms appearing in (4.13) in a similar way as for the upper estimate.