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Naturality, standardness, and weak duality for Markov processes. (English) Zbl 0553.60070

This paper is comprised of three parts: ”Naturality” (Part I), ”Processes with an excessive initial measure” (Part II), and ”Weak duality” (Part III). Under study is a Borel right process X, with lifetime \(\zeta\). Part I introduces the natural \(\sigma\)-algebra (the trace of the predictable \(\sigma\)-algebra on \(]0,\zeta])\) and discusses natural processes, potentials, projections, etc. One section is devoted to the special case of X standard, and contains a characterization of standardness in terms of hitting times. Part II contains some material preliminary to the study in part III, including a nice discussion of the Revuz measure of a homogeneous random measure.
This discussion continues into part III, where X is assumed to be in weak duality with another Borel right process \(\hat X.\) A key result replaces the classical duality formula \[ u_ A^{\alpha}(x)=\int u^{\alpha}(x,y)\nu_ A(dy) \] for the \(\alpha\)-potential of a natural additive functional A with Revuz measure \(\nu_ A\) [D. Revuz, Trans. Am. Math. Soc. 148, 501-531 (1970; Zbl 0266.60053)]. Also, in part III are a switching identity which generalizes G. A. Hunt’s original identity for hitting operators [Ill. J. Math. 2, 151-213 (1958; Zbl 0100.138)], a characterization of Revuz measures, a discussion of the relationship between capacity and co-capacity, and a weak duality version of M. Nagasawa’s theorem on time reversal [Nagoya Math. J. 24, 177-204 (1964; Zbl 0133.107)]. Many arguments exploit the stationary process associated with the dual pair \(X\), \(\hat X\) [see the reviewer, Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 139-156 (1970; Zbl 0406.60067)].
Reviewer: J.Mitro

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
60J45 Probabilistic potential theory
60J55 Local time and additive functionals
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