Rare transitions in noisy heteroclinic networks

TL;DR

This paper analyzes how small white noise influences transitions in planar dynamical systems with heteroclinic networks, revealing decay rates, time scale hierarchies, and the role of saddle points using Malliavin calculus.

Contribution

It provides a detailed probabilistic and analytical framework for understanding rare transitions in noisy heteroclinic networks, including local limit theorems for exit distributions.

Findings

Transition probabilities decay as powers of noise magnitude.

Most likely transitions involve long stays near saddle points.

Hierarchies of time scales and accessibility clusters are characterized.

Abstract

We study small white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude. We show that the most likely scenario for the realization of these rare transition events involves spending atypically long times in the neighborhoods of certain saddle points of the network. We describe the hierarchy of time scales and clusters of accessibility associated with these rare transition events. We discuss applications of our results to homogenization problems and to the invariant distribution asymptotics. At the core of our results are local limit theorems for exit distributions obtained via methods of Malliavin calculus.