Abrupt convergence for stochastic small perturbations of one dimensional dynamical systems

TL;DR

This paper investigates the abrupt convergence (cut-off phenomenon) in stochastic perturbations of one-dimensional dynamical systems, showing how small noise induces rapid mixing and metastability under certain conditions.

Contribution

It establishes the presence of a profile cut-off and local cut-off phenomena in stochastic small perturbations of 1D dynamical systems with multi-well potentials.

Findings

Presence of profile cut-off in total variation distance

Local cut-off near metastable states

Conditions on potential for cut-off phenomena

Abstract

We study the cut-off phenomenon for a family of stochastic small perturbations of a one dimensional dynamical system. We will focus in a semi-flow of a deterministic differential equation which is perturbed by adding to the dynamics a white noise of small variance. Under suitable hypothesis on the potential we will prove that the family of perturbed stochastic differential equations present a profile cut-off phenomenon with respect to the total variation distance. We also prove a local cut-off phenomenon in a neighborhood of the local minima (metastable states) of multi-well potential.