Scaling Limit for the Diffusion Exit Problem

TL;DR

This paper establishes a scaling limit for the exit problem of a small noise dynamical system with hyperbolic behavior, using a novel pathwise approach that overcomes limitations of traditional large deviation estimates.

Contribution

It introduces a pathwise method combining normal forms and geometric arguments to analyze exit problems where large deviation principles are insufficient.

Findings

Proves a new scaling limit for the exit problem in hyperbolic systems.

Develops a pathwise approach that captures dynamics beyond large deviation estimates.

Provides state-of-the-art results in the analysis of small noise exit problems.

Abstract

The objective of this dissertation is to prove a scaling limit for the exit of a domain problem of a small noise system with underlying hyperbolic dynamics. In this case, Large Deviation kind of estimates fail to provide a complete picture of the dynamics of the system under consideration. This is so because the rate function given by the Large Deviation Principle has several minimizing trajectories hence making them indistinguishable at the exponential level. We propose a pathwise approach based on the theory of normal forms combined with geometrical arguments. We prove a scaling limits and provide the state of the art results in related problems.