I am reading a paper (Predictability of escape for a stochastic saddle-node bifurcation: When rare events are typical, Herbert, Bouchet, 2017 Physical Review E), which is related to Kramers escape rate of a Langevin system.
I couldn't understand how to determine an asymptotic behavior of the moments of the first-passage time when the noise, $\epsilon$, goes to infinity, where the PDF of the first-passage time is
$P(t) = \sqrt{-t}/\pi * \exp [ -4(-t)^{3/2}/(3\epsilon) ] * \exp[-\epsilon/(2\pi)*\exp[-4(-t)^{3/2}/(3\epsilon)]]$
and it is derived to have n-th moments of
$E[t^n] = k (-1)^n/(2\pi)*(3\epsilon \ln(\epsilon)/4)^{2n/3}$
Any help will be sincerely appreciated.