Estimates of Exit Events by Large Jump and Perturbation Events

First exit times of the Chafee-Infante equation with Lévy noise 219 4 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.

4.1 Estimates of Exit Events by Large Jump and Perturbation Events

To this end, in this subsection we first estimate exit events of X ǫ 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 ∈ ◆, t ∈ [0, t k ], x ∈ H we have X ǫ t + T k−1 ; x = Y ǫ t; X ǫ 0; x ◦ θ T k−1 + ǫW k 1{t = t k }. 4.1 In the following two lemmas we estimate certain events connecting the behavior of X ǫ in the domains of the type D ± ǫ γ with the large jumps η ǫ in the reshifted domains of the type D ± ǫ γ . We introduce for ǫ 0 and x ∈ ˜ D ± ǫ γ the events A x :={Y ǫ s; x ∈ D ± ǫ γ for s ∈ [0, T 1 ] and Y ǫ T 1 ; x + ǫW 1 ∈ D ± ǫ γ }, B x :={Y ǫ s; x ∈ D ± ǫ γ for s ∈ [0, T 1 ] and Y ǫ T 1 ; x + ǫW 1 ∈ D ± ǫ γ }, C x :={Y ǫ s; x ∈ D ± ǫ γ f. s ∈ [0, T 1 ] a. Y ǫ T 1 ; x + ǫW 1 ∈ D ± ǫ γ \ ˜ D ± ǫ γ }, A − x :={Y ǫ s; x ∈ D ± ǫ γ for s ∈ [0, T 1 ] and Y ǫ T 1 ; x + ǫW 1 ∈ ˜ D ± ǫ γ }. 4.2 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. Lemma 4.1 Partial estimates of the major events. Let T r ec , κ 0 be given by Proposition 2.4 and assume that Hypotheses H.1 and H.2 are satisfied. For ρ ∈ € 1 2 , 1 Š , γ ∈ 0, 1 − ρ there exists ǫ 0 so that the following inequalities hold true for all 0 ǫ 6 ǫ and x ∈ D ± ǫ γ i 1A x 1E x 1{T 1 T r ec + κγ| ln ǫ|} 6 1{ǫW 1 ∈ D ± }, 4.3 ii 1B x 1E x 1{T 1 T r ec + κγ| ln ǫ|} 6 1{ǫW 1 ∈ D ± ǫ γ , ǫ 2 γ }, 4.4 iii 1C x 1E x 1{T 1 T r ec + κγ| ln ǫ|} 6 1{ǫW 1 ∈ D ∗ ǫ γ }. 4.5 Additionally, for x ∈ D ± ǫ γ we have i v 1B x 1E x 1{kǫW 1 k 6 12ǫ 2 γ }1{T 1 T r ec + κγ| ln ǫ|} = 0, 4.6 v 1C x 1E x 1{kǫW 1 k 6 12ǫ 2 γ }1{T 1 T r ec + κγ| ln ǫ|} = 0. 4.7 In the opposite sense for x ∈ ˜ D ± ǫ γ vi 1E x 1{T 1 T r ec + κγ| ln ǫ|}1{ǫW 1 ∈ D ± } 6 1B x , 4.8 vii 1E x 1{T 1 T r ec + κγ| ln ǫ|}1{ǫW 1 ∈ D ± ǫ γ , ǫ 2 γ , ǫ 2 γ } 6 1A − x . 4.9 With the help of Lemma 4.1 we can show the following crucial estimates. Lemma 4.2 Full estimates of the major events. Let T r ec , κ 0 be given by Proposition 2.4 and Hypotheses H.1 and H.2 be satisfied. For ρ ∈ € 1 2 , 1 Š , γ ∈ 0, 1 − ρ there exists ǫ 0 such that the following inequalities hold true for all 0 ǫ 6 ǫ , κ 0 and x ∈ D ± ǫ γ i x 1A x 6 1{ ǫW 1 ∈ D ± } + 1{kǫW 1 k 1 2 ǫ 2 γ }1{T 1 T r ec + κγ| ln ǫ|} + 1E c x , x 1B x 6 1{ ǫW 1 ∈ D ± ǫ γ , ǫ 2 γ } + 1{T 1 T r ec + κγ| ln ǫ|} + 1E c x , 220 Electronic Communications in Probability x i sup y∈ ˜ D ± ǫ γ 1{Y ǫ s; y ∈ D ± ǫ γ for some s ∈ 0, T 1 } 6 sup y∈ ˜ D ± ǫ γ 1E c y , x ii 1A x 1{Y ǫ s; X ǫ 0, x ◦ θ T 1 ∈ D ± ǫ γ for some s ∈ 0, T 1 } 6 1 ¦ ǫW 1 ∈ D ∗ ǫ γ © + 1{T 1 T r ec + κγ| ln ǫ|} + sup y∈ ˜ D ± ǫ γ 1E c y ◦ θ T 1 + 1E c x . In the opposite sense for x ∈ ˜ D ± ǫ γ x iii 1A − x 1{ǫW 1 ∈ D ± ǫ γ , ǫ 2 γ , ǫ 2 γ } − 1{T 1 T r ec + κγ| ln ǫ|} − 2 1E c x , x i v 1B x 1{ǫW 1 ∈ D ± }1 − 1{T 1 T r ec + κγ| ln ǫ|} − 1E c x . The next lemma ensures that after having relaxed to B ǫ 2 γ φ ± the solution X ǫ jumps close to the separatrix only with negligible probability for ǫ → 0+. Lemma 4.3 Asymptotic behavior of large jump events. Let Hypotheses H.1 and H.2 be satis- fied and 1 2 ρ 1 − 2γ. Then for any C 0 there is ǫ = ǫ C 0 such that for all 0 ǫ 6 ǫ I µ € D ± c Š µB c 1 − C ǫ α1−ρ 6 λ ± ǫ β ǫ 6 ‚ µD ± c µB c 1 + C Œ ǫ α1−ρ , I I P € kǫW 1 k 12ǫ 2 㠊 6 4 ǫ α1−ρ−2γ , I I I P € ǫW 1 ∈ ˜ D ± ǫ γ c Š 6 1 + C λ ± ǫ β ǫ , I V P € ǫW 1 ∈ D ∗ ǫ 㠊 6 C λ ± ǫ β ǫ , V PǫW 1 ∈ D c ǫ γ , ǫ 2 γ , ǫ 2 γ 6 1 + C λ ± ǫ β ǫ . A detailed proof is given in [ 4 ].

4.2 Asymptotic Exit Times from Reduced Domains of Attraction