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

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.