On the approaching time towards the attractor of differential equations perturbed by small noise

TL;DR

This paper analyzes the asymptotic times for points and sets to approach attractors in small-noise stochastic differential equations, revealing exponential versus linear growth rates.

Contribution

It introduces a novel comparison of approach times for points and sets to attractors under small noise, using large deviation techniques and process comparison.

Findings

Set approach time increases exponentially as noise diminishes.

Point approach time increases linearly with decreasing noise.

Different rates of approach are established for points and sets.

Abstract

We estimate the time a point or set, respectively, requires to approach the attractor of a radially symmetric gradient type stochastic differential equation driven by small noise. Here, both of these times tend to infinity as the noise gets small. However, the rates at which they go to infinity differ significantly. In the case of a set approaching the attractor, we use large deviation techniques to show that this time increases exponentially. In the case of a point approaching the attractor, we apply a time change and compare the accelerated process to another process and obtain that this time increases merely linearly.