Asymptotic Exit Times

Abstract

We shall now use the small deviations estimates of Chap. 4 in order to give a precise account of the exit times of the system described by our Chafee–Infante equation with small Lévy noise in H of the reduced domains of attraction of the stable states ϕ ^± defined in Chap. 2. Our main line of reasoning will be based on the splitting of small and large jumps proposed there. In fact, the Chafee–Infante equation perturbed by small jumps being subject to only small deviations from the solution of the deterministic system before the first big jump, as shown in Chap. 4, and the time needed for relaxation in a small neighborhood of ϕ ^± being only of logarithmic order in $\varepsilon$, exits will happen at times of big jumps that are big enough to leave the reduced domains of attraction. To characterize the asymptotic law of the exit time, we shall compute the asymptotics of its Laplace transform. Making these heuristic arguments mathematically rigorous will be the main task of this chapter.