White noise perturbation of locally stable dynamical systems

TL;DR

This paper investigates how small persistent white noise perturbations affect the stability of a locally stable equilibrium in a deterministic dynamical system over long time intervals, revealing conditions for stochastic stability.

Contribution

It provides a detailed analysis of stochastic stability under persistent white noise perturbations for systems with locally stable equilibria, extending understanding of long-term behavior.

Findings

Trajectories may leave bounded domains with probability one under persistent noise.

Stability analysis over long time scales $O(\mu^{-N})$ for small perturbation parameter $\mu$.

Conditions under which the equilibrium remains stochastically stable despite noise.

Abstract

The influence of small random perturbations on a deterministic dynamical system with a locally stable equilibrium is considered. The perturbed system is described by the It\^{o} stochastic differential equation. It is assumed that the noise does not vanish at the equilibrium. In this case the trajectories of the stochastic system may leave any bounded domain with probability one. We analyze the stochastic stability of the equilibrium under a persistent perturbation by white noise on an asymptotically long time interval $0\leq t\leq \mathcal O(\mu^{-N})$, where $0<\mu\ll 1$ is a perturbation parameter.