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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 | 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 (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 | Services: | Forum | Review | PDF | Favorites |
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