Escape rate and conditional escape rate from a probabilistic point of view

We prove that for a sequence of nested sets $\{U_n\}$ with $\Lambda = \cap_n U_n$ a measure zero set, the localized escape rate converges to the extremal index of $\mathrm{\Lambda }$, provided that the dynamical system is $\varphi$-mixing at polynomial speed. We also establish the general equivalence between the local escape rate for entry times and the local escape rate for returns.