Extremal behaviour of randomly perturbed dynamical systems

In this talk, we will explain how we get laws of rare events for randomly perturbed dynamical systems using the link between Extreme Value Laws (EVL) and Hitting/Return Time Statistics (HTS/RTS). Mainly, we will consider random perturbations of uniformly expanding systems, such as piecewise expanding maps of the circle, and show that for additive absolutely continuous (w.r.t. Lebesgue) noise, the limiting distribution is standard exponential for any point. Our main ingredient will be decay of correlations against all $L^1$ observables in a suitable Banach space and due to the above link we get our results by means of the first return time from a set to itself.