# Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium

## TL;DR

This paper studies the exit times of one-dimensional dynamical systems with rapid random switching near an unstable equilibrium, revealing a limit theorem with a deterministic and a stochastic component.

## Contribution

It introduces a limit theorem for exit times in systems with fast switching near unstable points, extending understanding of stochastic stability and universality classes.

## Key findings

- Exit time follows a limit theorem with a logarithmic deterministic term.
- Random correction to exit time converges in distribution.
- Results connect fast switching systems to small white-noise perturbation behavior.

## Abstract

We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit time satisfies a limit theorem with a logarithmic deterministic term and a random correction converging in distribution. Thus this setting is in the universality class of the unstable equilibrium exit under small white-noise perturbations.