# Large deviations in presence of small noise for delay differential equations at an instability

## TL;DR

This paper investigates the large deviations behavior of delay differential equations near instability under small noise, leveraging spectral theory and classical large deviations principles.

## Contribution

It extends large deviations analysis to DDEs at instability by connecting spectral theory with Freidlin-Wentzell results.

## Key findings

- Projection onto zero root space simplifies analysis
- Large deviations follow classical Freidlin-Wentzell results for the projected process
- Spectral theory facilitates understanding of noise effects in DDEs at instability

## Abstract

We consider delay differential equations (DDE) that are on the verge of an instability, i.e. the characteristic equation for the linearized equation has one root as zero and all other roots have negative real parts. In presence of small mean-zero noise, we study the large deviations from the corresponding deterministic system. Using spectral theory for DDE it is easy to see that, the projection on to the one dimensional space corresponding to the zero root is exponentially equivalent with the original process. For the one-dimensional process we make the observation that the results of Freidlin-Wentzell apply.