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Notation and Preliminaries

  • R. K. Getoor
Part of the Probability and Its Applications book series (PA)

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) = △, θ0ω = ω, and θ ω = [△] for all ω ∈ Ω.

Keywords

Markov Process Measurable Space Special Mention Finite Measure Semi Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • R. K. Getoor
    • 1
  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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