Markov Processes: Ray Processes and Right Processes |
Contents
PRELIMINARIES | 1 |
RAY RESOLVENTS AND SEMIGROUPS | 8 |
INCREASING SEQUENCES OF SUPERMARTINGALES | 17 |
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a-excessive function algebra bẸ Borel subsets branch points compact metric space completing the proof Consequently continuous functions convex cone d₂ defined definition denote E₁ E₂ evanescent exists finite measure fixed follows g(YT hence homeomorphism Hunt process hypothesis implies increasing sequence indistinguishable isometry Laplace transforms left limits LEMMA Let f Let g Markov processes Markov property measurable space measure on E,E monotone class theorem nearly Borel notation O-algebra original topology Pf(x Po(x previsible stopping probability on E,E PROPOSITION proving quasi-left-continuous R₁ R₂ Ray process Ray resolvent Ray topology resp right process Section semigroup separates the points standard process statement strong Markov sub-Markov subspace subspace topology suffices to show supermartingale suppose surely pH theorem totally bounded totally bounded metric U-space universally measurable α α