The Blumenthal-Getoor-McKean Theorem Revisited

The Blumenthal-Getoor-McKean theorem [BGM] (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 [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 [S].

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