Malliavin calculus approach to long exit times from an unstable equilibrium

TL;DR

This paper applies Malliavin calculus to analyze the asymptotic behavior of exit times from an unstable equilibrium in a stochastic differential equation, providing detailed tail estimates and exit distributions.

Contribution

It introduces a Malliavin calculus-based method to derive precise asymptotics for exit times and distributions near an unstable equilibrium under small noise perturbations.

Findings

Derived asymptotic tail estimates for exit times

Characterized exit distributions conditioned on long exits

Provided a new analytical framework for unstable equilibria

Abstract

For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. Using Malliavin calculus tools, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.