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 with the power of the small noise amplitude, whereas the prefactor accounts for memory effects. In particular, we discover a nonlinear delay-induced exit acceleration due to a non-normal growth phenomenon. Our results are illustrated by the example of a linear delay oscillator driven by α-stable Lévy flights.