Abstract
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].
The research of all three authors was supported, in part, by NSF Grant DMS87-21347.
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References
R. M. Blumenthal and R. K. Getoor. Markov Processes and Potential Theory. Academic Press, New York, 1968.
R. M. Blumenthal, R. K. Getoor, and H. R. McKean, Jr. Markov processes with identical hitting distributions. Ill. J. Math., 6 (1962), 402–420, and supplement Ill. J. Math., 7 (1963), 540–542.
R. V. Chacon and B. Jamison. A fundamental property of Markov processes with an application to equivalence under time changes. Israel J. Math., 33 (1979), 241–269.
R. V. Chacon and B. Jamison. Processes with state dependent hitting probabilities. Adv. in Math., 32 (1979), 1–35.
C. Dellacherie et P. A. Meyer. Probabilités et Potentiel, II. Hermann, Paris, 1980.
C. Dellacherie et P. A. Meyer. Probabilités et Potentiel, IV. Hermann, Paris, 1987.
P. J. Fitzsimmons. Markov processes with identical hitting probabilities. Math. Z., 192 (1986), 547–554.
P. J. Fitzsimmons. On the identification of Markov processes by the distribution of hitting times. Sem. Stoch. Proc, 1986, 15–19. Birkhäuser, Boston, 1987.
P. J. Fitzsimmons. Penetration times and Skorohod stopping. Sém. de Prob. XXII, Lecture Notes in Math., 1321, Springer, Berlin-Heidelberg-New York, 1988.
P. J. Fitzsimmons and B. Maisonneuve. Excessive measures and Markov processes with random birth and death. Probab. Th. Rel. Fields, 72 (1986), 319–336.
R. K. Getoor. Some remarks on continuous additive functionals. Ann. Math. Stat., 38 (1967), 1655–1660.
R. K. Getoor. Markov Processes: Ray Processes and Right Processes. Lecture Notes in Math., 440, Springer, Berlin-Heidelberg-New York, 1975.
R. K. Getoor. Transience and recurrence of Markov processes. Sém. de Prob. XIV, Lecture Notes in Math., 784, Springer, Berlin-Heidelberg-New York, 1980.
R. K. Getoor and J. Glover. Markov processes with identical excessive measures. Math. Z., 184 (1983), 287–300.
R. K. Getoor and M. J. Sharpe. Balayage and multiplicative functionals. Z. Wahrscheinlichkeitstheorie verw. Geb., 28 (1974), 139–164.
R. K. Getoor and M. J. Sharpe. Naturality, standardness, and weak duality for Markov processes. Z. Wahrscheinlichkeitstheorie verw. Geb., 67 (1984), 1–62.
R. K. Getoor and J. Steffens. The energy functional, balayage, and capacity. Ann. Inst. Henri Poincaré, 23 (1987), 321–357.
R. K. Getoor and J. Steffens. More about capacity and excessive measures. Sem. Stoch. Proc, 1987, 135–157. Birkhäuser, Boston, 1988.
J. Glover. Representing last exit potentials as potentials of measures. Z. Wahrscheinlichkeitstheorie verw. Geb., 61 (1982), 17–30.
J. Glover. Markov processes with identical hitting probabilities. Trans. Amer. Math. Soc., 275 (1983), 131–141.
J. Glover. Identifying Markov processes up to time change. Sem. Stoch. Proc., 1982, 171–194. Birkhäuser, Boston, 1983.
P. A. Meyer. Le schéma de remplissage en temps continu, d’après H. Rost. Sém. de Prob. VI, Lecture Notes in Math., 258, Springer, Berlin-Heidelberg-New York, 1972.
P. A. Meyer. Convergence faible et compacité des temps d’arrêt, d’après Baxter et Chacon. Sém. de Prob. XII, Lecture Notes in Math., 649, Springer, Berlin-Heidelberg-New York, 1978.
M. Rao. A note on Revuz measure. Sém. de Prob. XIV, Lecture Notes in Math., 784, Springer, Berlin-Heidelberg-New York, 1980.
H. Rost. The stopping distributions of a Markov process. Invent. Math., 14 (1971), 1–16.
M. J. Sharpe. General Theory of Markov Processes. Academic Press, San Diego, 1988.
J. B. Walsh. On the Chacon-Jamison theorem. Z. Wahrscheinlichkeitstheorie verw. Geb., 68 (1984), 9–28.
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Fitzsimmons, P.J., Getoor, R.K., Sharpe, M.J. (1990). The Blumenthal-Getoor-McKean Theorem Revisited. In: Çinlar, E., Chung, K.L., Getoor, R.K., Fitzsimmons, P.J., Williams, R.J. (eds) Seminar on Stochastic Processes, 1989. Progress in Probability, vol 18. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3458-6_4
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