Noisy heteroclinic networks

TL;DR

This paper analyzes how white noise perturbations affect dynamics near heteroclinic networks, showing convergence to a jump process and providing asymptotics for exit measures.

Contribution

It introduces a novel analysis of stochastic perturbations in heteroclinic networks, revealing convergence to a piecewise constant process and detailed exit measure asymptotics.

Findings

Diffusion converges to a jump process under logarithmic time rescaling

Explicit asymptotics for exit measures from domains containing starting points

Characterization of stochastic dynamics near heteroclinic networks

Abstract

We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic network. We show that under the logarithmic time rescaling the diffusion converges in distributon in a special topology to a piecewise constant process that jumps between saddle points along the heteroclinic orbits of the network. We also obtain precise asymptotics for the exit measure for a domain containing the starting point of the diffusion.