# Scaling Limit of Small Random Perturbation of Dynamical Systems

## TL;DR

This paper proves that small random perturbations of reversible dynamical systems with multiple stable states converge to a Markov chain on neighborhoods of the deepest equilibria, resolving a long-standing problem since the 1970s.

## Contribution

It introduces a novel method to analyze the convergence by reducing the problem to solving a Poisson equation with innovative test functions.

## Key findings

- Convergence of perturbed systems to Markov chains is established.
- A new analytical approach using Poisson equations is developed.
- The method simplifies analysis of complex stochastic dynamical systems.

## Abstract

In this article, we prove that a small random perturbation of dynamical system with multiple stable equilibria converges to a Markov chain whose states are neighborhoods of the deepest stable equilibria, under a suitable time-rescaling, provided that the perturbed dynamics is reversible in time. Such a result has been anticipated from 1970s, when the foundation of mathematical treatment for this problem has been established by Freidlin and Wentzell. We solve this long-standing problem by reducing the entire analysis to an investigation of the solution of an associated Poisson equation, and furthermore provide a method to carry out this analysis by using well-known test functions in a novel manner.