First entrance and exit times and the intrinsic topology in the state space

Abstract

Let X = (x t , ζ ,ℳ t P x ) be a Markov process on a state space (E,ℬ). Forevery ω ∈ Ω t we denote by F 8 t = F8t (ω) the range of the sample function during the period [s,t] (i.e. the set of points x u (ω) with u ∈ [s, t]). We call the function $$\tau \left( \omega \right) = \sup \left\{ {t{:^8}_t \cap \Gamma = \emptyset } \right\} = \sup \left\{ {t:{F^8}_t \subseteq E\backslash \Gamma } \right\}$$ (4.1) the first entrance time after time s into the set Г or the first exit time after time s from E\Г *. Let ℱ be any class of subsets of the space E. We call the upper bound of the first exit times after time s from all sets Г∈ ℱ the first exit time after time s from ℱ.