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We consider certain one dimensional ordinary stochastic differentialequations driven by additive Brownian motion of variance $\varepsilon ^2$. When$\varepsilon =0$ such equations have an unstable non-hyperbolic fixed point andthe 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 randomescape or transit time which diverges as $\varepsilon \searrow0$. We show thatthis random time, under suitable (easily guessed) rescaling, converges to alimit random variable that essentially depends only on the power exponentassociated to the fixed point. Such random variables, or laws, have therefore auniversal character and they arise of course in a variety of contexts. We thenobtain quantitative sharp estimates, notably tail properties, on theseuniversal laws.

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