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29 March 2024
 
  » arxiv » 1307.4255

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Weak noise and non hyperbolic unstable fixed points: sharp estimates on transit and exit times
Giambattista Giacomin ; Mathieu Merle ;
Date 16 Jul 2013
AbstractWe consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance $varepsilon^2$ When $varepsilon = 0$ such equations have (at least) 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 $varepsilon o 0$. 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 an 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.
Source arXiv, 1307.4255
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