Normal forms approach to diffusion near hyperbolic equilibria

TL;DR

This paper analyzes the exit behavior of small noise perturbations near hyperbolic equilibria in planar systems, providing explicit limiting laws for exit distributions and times using normal forms theory.

Contribution

It introduces a detailed analysis of exit distributions and times near hyperbolic points under small noise, completing the theory of noisy heteroclinic networks in two dimensions.

Findings

Explicit limiting laws for exit distribution and time derived

Normal forms theory applied to stochastic exit problems

Completes the theoretical framework for noisy heteroclinic networks

Abstract

We consider the exit problem for small white noise perturbation of a smooth dynamical system on the plane in the neighborhood of a hyperbolic critical point. We show that if the distribution of the initial condition has a scaling limit then the exit distribution and exit time also have a joint scaling limit as the noise intensity goes to zero. The limiting law is computed explicitly. The result completes the theory of noisy heteroclinic networks in two dimensions. The analysis is based on normal forms theory.