# Exponentially-ergodic Markovian noise perturbations of delay differential equations at Hopf bifurcation

## TL;DR

This paper studies how exponentially ergodic noise affects delay differential equations near Hopf bifurcation, showing that under certain conditions, the critical modes behave like a diffusion process as noise diminishes.

## Contribution

It demonstrates the convergence of critical eigenmodes of scalar delay differential equations under exponentially ergodic noise to a diffusion process as noise strength decreases.

## Key findings

- Critical eigenmodes converge to a diffusion process
- Results are proven for scalar DDEs only
- Noise perturbations influence bifurcation behavior

## Abstract

We consider noise perturbations of delay differential equations (DDE) experiencing Hopf bifurcation. The noise is assumed to be exponentially ergodic, i.e. transition density converges to stationary density exponentially fast uniformly in the initial condition. We show that, under an appropriate change of time scale, as the strength of the perturbations decreases to zero, the law of the critical eigenmodes converges to the law of a diffusion process (without delay). We prove the result only for scalar DDE. For vector-valued DDE without proofs see Phys.Rev.E v93, 062104.