Weak noise and non-hyperbolic unstable fixed points: Sharp estimates on transit and exit times

TL;DR

This paper analyzes the asymptotic behavior of escape times near non-hyperbolic unstable fixed points in stochastic differential equations with small noise, revealing universal laws and sharp tail estimates.

Contribution

It provides the first rigorous analysis of transit times near non-hyperbolic fixed points with small noise, establishing universal limit laws and precise tail estimates.

Findings

Transit times converge to a universal limit distribution as noise vanishes.

Tail properties of the limit laws are quantitatively characterized.

Results apply broadly to systems with power-law drift near unstable fixed points.

Abstract

We consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance $\varepsilon ^2$. When $\varepsilon =0$ such equations have an unstable non-hyperbolic fixed point and the drift near such a point has a power law behavior. For $\varepsilon >0$ small, the fixed point property disappears, but it is replaced by a random escape or transit time which diverges as $\varepsilon \searrow0$. We show that this random time, under suitable (easily guessed) rescaling, converges to a limit random variable that essentially depends only on the power exponent associated to the fixed point. Such random variables, or laws, have therefore a universal character and they arise of course in a variety of contexts. We then obtain quantitative sharp estimates, notably tail properties, on these universal laws.