This article studies the behavior of stochastic reaction-diffusion equations driven by additive regularly varying pure jump L'evy noise in the limit of small noise intensity. It is shown that the law of the suitably renormalized first exit times from the domain of attraction of a stable state converges to an exponential law of parameter 1 in a strong sense of Laplace transforms, including exponential moments. As a consequence, the expected exit times increase polynomially in the inverse intensity, in contrast to Gaussian perturbations, where this growth is known to be of exponential rate.
"Asymptotic first exit times of the Chafee-Infante equation with small heavy-tailed Lévy noise." Electron. Commun. Probab. 16 213 - 225, 2011. https://doi.org/10.1214/ECP.v16-1622