Potential processes

Author:
R. V. Chacon

Journal:
Trans. Amer. Math. Soc. **226** (1977), 39-58

MSC:
Primary 60J45; Secondary 31D05

MathSciNet review:
0501374

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Abstract: The prototype of a potential process is a stochastic process which visits the same points in the same order as a Markov process, but at a rate obtained from a nonanticipating time change. The definition of a potential process may be given intrinsically and most generally without mention of a Markov process, in terms of potential theory. The definition may be given more directly and less generally in terms of potentials which arise from Markov processes, or more directly than this, as suitably time-changed Markov processes. The principal purpose of studying the class of potential processes, which may be shown to include martingales as well as Markov processes themselves, is to give a unified treatment to a wide class of processes which has potential theory at its core. That it is possible to do so suggests that potential rather than martingale results are central to the study of Markov processes. Furthermore, this also suggests that it is not the Markov property itself which makes Markov processes tractable, but rather the potential structure which can be constructed with the assistance of the Markov property. The general theory of potential processes is developed in a forthcoming paper. It will be shown there that a Markov process subject to an ordinary continuous nonanticipating time change is a *local* potential process. It may be seen, by examining examples, that it is necessary to consider randomized stopping times and randomized nonanticipating time changes in the general case. In the forthcoming paper a more general notion than randomized nonanticipating time changes is used to obtain a characterization of potential processes. It is an open problem whether randomization itself is sufficient in the general case, and whether ordinary nonanticipating time changes are sufficient for continuous parameter martingales and Brownian motion on the line. The emphasis in the present paper will be on developing the theory of discrete parameter martingales as a special case of the general theory.

**[1]**D. G. Austin, G. A. Edgar, and A. Ionescu Tulcea,*Pointwise convergence in terms of expectations*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**30**(1974), 17–26. MR**0358945****[2]**J. R. Baxter and R. V. Chacon,*Potentials of stopped distributions*, Illinois J. Math.**18**(1974), 649–656. MR**0358960****[3]**Leo Breiman,*On the tail behavior of sums of independent random variables*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**9**(1967), 20–25. MR**0226707****[4]**K. È. Dambis,*On decomposition of continuous submartingales*, Teor. Verojatnost. i Primenen.**10**(1965), 438–448 (Russian, with English summary). MR**0202179****[5]**J. L. Doob,*Stochastic processes*, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR**0058896****[6]**Lester E. Dubins and Gideon Schwarz,*On continuous martingales*, Proc. Nat. Acad. Sci. U.S.A.**53**(1965), 913–916. MR**0178499****[7]**Lester E. Dubins,*On a theorem of Skorohod*, Ann. Math. Statist.**39**(1968), 2094–2097. MR**0234520****[8]**W. Hall,*On the Skorohod embedding theorem*, J. Appl. Probability**7**(1970).**[9]**J. Kiefer,*Skorohod embedding of multivariate RV’s, and the sample DF*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**24**(1972), no. 1, 1–35. MR**1554013**, 10.1007/BF00532460**[10]**Frank B. Knight,*A reduction of continuous square-integrable martingales to Brownian motion*, Martingales (Rep. Meeting, Oberwolfach, 1970) Springer, Berlin, 1971, pp. 19–31. Lecture Notes in Math., Vol. 190. MR**0370741****[11]**Itrel Monroe,*On embedding right continuous martingales in Brownian motion*, Ann. Math. Statist.**43**(1972), 1293–1311. MR**0343354****[12]**V. A. Rohlin,*On the fundamental ideas of measure theory*, Mat. Sbornik N.S.**25(67)**(1949), 107–150 (Russian). MR**0030584****[13]**D. H. Root,*The existence of certain stopping times on Brownian motion*, Ann. Math. Statist.**40**(1969), 715–718. MR**0238394****[14]**Stanley Sawyer,*A remark on the Skorohod representation*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**23**(1972), 67–74. MR**0310939****[15]**Gordon Simons,*A martingale decomposition theorem*, Ann. Math. Statist.**41**(1970), 1102–1104. MR**0261678****[16]**Hiroshi Kunita and Shinzo Watanabe,*On square integrable martingales*, Nagoya Math. J.**30**(1967), 209–245. MR**0217856**

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DOI:
https://doi.org/10.1090/S0002-9947-1977-0501374-5

Keywords:
Potential processes

Article copyright:
© Copyright 1977
American Mathematical Society