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The Blumenthal-Getoor-McKean theorem revisited. (English) Zbl 0744.60090

Stochastic processes, Semin., San Diego/CA (USA) 1989, Prog. Probab. 18, 35-57 (1990).
From the introduction: The Blumenthal-Getoor-McKean theorem [see R. M. Blumenthal, R. K. Getoor and H. P. McKean jun., Ill. J. Math. 6, 402–420 (1962; Zbl 0133.40903) and ibid. 7, 540–542 (1963; Zbl 0211.48602), hereafter referred to as BGM] states that if \(X\) and \(\tilde X\) are two Markov processes with the same hitting distributions, then they may be time changed into each other. This is a deliberately loose statement and one needs to specify the precise hypotheses on \(X\) and \(\tilde X\) and also exactly what the conclusion means before it makes mathematical sense. In §V-5 of [R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory. New York etc.: Academic Press (1968; Zbl 0169.49204)], hereafter referred to as BG, a precise statement and proof are given when \(X\) and \(\tilde X\) are standard processes as defined in [BG]. It is stated in several places in the literature that the proof in [BG] carries over to the case in which \(X\) and \(\tilde X\) are right processes. However, a careful reading of that proof reveals that the quasi-left-continuity (qlc) of \(X\) and \(\tilde X\) is used in a crucial manner at two points: the proofs of (V-5.4) and (V-5.20) in [BG]. The purpose of this paper is to give a careful proof of BGM for arbitrary right processes \(X\) and \(\tilde X\) as defined in [M. J. Sharpe, General theory of Markov processes. Boston, MA: Academic Press (1988; Zbl 0649.60079)].
[For the entire collection see Zbl 0686.00017.]

MSC:

60J45 Probabilistic potential theory
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