Notation and Preliminaries

Abstract

We shall assume once and for all that

$$X = (\Omega ,\mathcal{F},{{\mathcal{F}}_{t}},{{X}_{t}},{{\theta }_{t}},{{P}^{x}})$$

is a right Markov process as defined in §8 of [S] with state space (E,ε),semigroup (P _t ), and resolvent (U^q). To be explicit E is a separable Radon space and ε is the Borel σ -lgebra of E. A cemetery point △ is adjoined to E as an isolated point and E_△: = E ∪ {△}, ε_△: = σ(ε ∪ {△}). (The symbol “: =” should be read as “is defined to be”.) We suppose that [S, (20.5)] holds; that is, X _t (ω) = △ implies that X _s (ω) = △ for all s ≥ t and that there is a point [△] in Ω (the dead path) with X _t ([△]) = △ for all t ≥ 0. Of course, ζ= inf {t: X _t = △} is the lifetime of X. The filtration (F,F _t ) is the augmented natural filtration of X, [S, (3.3)]. We shall always use ε to denote the Borel σ -algebra of E in the original topology of E. Beginning in §20, Sharpe uses ε to denote the Borel σ -algebra of E in the Ray topology. We shall not use this convention. We shall write ε^r for the σ -algebra of Ray Borel sets. These assumptions on X are weaker than those in [G] or [DM, XVI4]. Beginning in §6 we shall make an additional assumption on X. (See (6.2)). To avoid trivialities we assume throughout this monograph that X_∞(ω) = △, θ₀ ω = ω, and θ _∞ ω = [△] for all ω ∈ Ω.