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

Abstract

We consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance ${\epsilon }^2$. When $\epsilon =0$ such equations have an unstable non-hyperbolic fixed point and the drift near such a point has a power law behavior. For $\epsilon >0$ small, the fixed point property disappears, but it is replaced by a random escape or transit time which diverges as $\epsilon \searrow 0$. 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.

Published at http://dx.doi.org/10.3150/14-BEJ643 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Application of Tauberian Theorem to the Exponential Decay of the Tail Probability of a Random Variable
• Exit asymptotics for small diffusion about an unstable equilibrium

Cited by in corpus (1)

• Gradual convergence for Langevin dynamics on a degenerate potential