We consider a random discrete time system in which the evolution of a stochastic differential equation is sampled at a sequence of discrete times. We set up a functional analytic framework for which we can prove the existence of a spectral gap and estimate the behavior of the leading eigenvalue of the related transfer operator as the system is perturbed by putting a ”hole” in it that corresponds to a rare event. By doing so, we derive the distribution of the hitting times corresponding to the rare event and the extreme value theory associated with it. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)