The first passage problem for stable linear delay equations perturbed by power law Lévy noise

Abstract

This article studies a linear scalar delay differential equation subject to small multiplicative power tail Lévy noise. We solve the first passage (the Kramers) problem with probabilistic methods and discover an asymptotic loss of memory in this non-Markovian system. Furthermore, the mean exit time increases as the power of the small noise amplitude, whereas the pre-factor accounts for memory effects. In particular, we discover a non-linear delay-induced exit acceleration due to a non-normal growth phenomenon. Our results are illustrated in the well-known linear delay oscillator driven by $\alpha$-stable Lévy flights.

3 figures

References in corpus (4)

• Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond
• On the noise-induced passage through an unstable periodic orbit I: Two-level model
• First exit times for Lévy-driven diffusions with exponentially light jumps
• A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema