Scaling Limit for the Diffusion Exit Problem, a Survey

TL;DR

This survey reviews the Freidlin-Wentzell theory, which analyzes the probabilities and most likely paths of rare events in perturbed dynamical systems, highlighting recent advances and applications in algorithms.

Contribution

It provides a comprehensive overview of recent extensions of Freidlin-Wentzell theory and its applications in Monte Carlo methods and simulated annealing.

Findings

Distinguishes between exponentially equivalent paths using new approaches

Highlights applications in Monte Carlo algorithms

Provides insights into the influence of the theory on optimization schemes

Abstract

In this review, an outline of the so called Freidlin-Wentzell theory and its recent extensions is given. Broadly, this theory studies the exponential rate at which the probabilities of rare events related to random perturbation of ODE decays. The typical situation is when an ODE has several stable equilibria, in which case, the theory predicts the most likely paths in which the randomly perturbed system goes from one equilibria to another. In recent developments I will outline how recent approaches allows to distinguish between paths that are otherwise exponentially equivalent and provide an overview of applications of this theory. In particular, we outline the influence of this theory in Monte Carlo Algorithms and Simulated Annealing schemes.