# The first passage problem for stable linear delay equations perturbed by power law L\'evy noise

## TL;DR

This paper analyzes how small power-law Lévy noise affects the first passage times in linear delay differential equations, revealing memory loss, non-linear acceleration, and specific asymptotic behaviors.

## Contribution

It introduces a probabilistic approach to solve the first passage problem for delay equations with Lévy noise and uncovers novel delay-induced acceleration effects.

## Key findings

- Mean exit time scales with noise amplitude power
- Asymptotic loss of memory in non-Markovian system
- Delay-induced non-linear acceleration observed

## Abstract

This article studies a linear scalar delay differential equation subject to small multiplicative power tail L\'evy 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 L\'evy flights.