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**Duality under a new setting.**
*(English)*
Zbl 0558.60056

Stochastic processes, Semin. Gainesville/Fla. 1983, Prog. Probab. Stat. 7, 23-38 (1984).

[For the entire collection see Zbl 0546.00025.]

Given a Hunt process (or more generally a standard process) whose resolvent possesses a dual resolvent, it is well-known that [see R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory (1968; Zbl 0169.492)] the existence in this case of a Green function (w.r.t. to the given reference measure) gives rise to a rich probabilistic potential theory. However the problem of existence and uniqueness of integral representations of potentials (and co-potentials) requires extra hypotheses which guarantee that the Green potentials \(x\to u(x,y)\) (resp. \(y\to u(x,y))\) are extreme for \(\forall y\in E\) (resp. \(\forall x\in E)\), where u(x,y) is the Green function and where E is the state space of the process.

In a previous work K. L. Chung and K. M. Rao [Ann. Inst. Fourier 30, 167-198 (1980; Zbl 0424.31004)] have studied some standard problems (such as the existence of dual semi-group and dual process, Riesz decomposition theorem...) under a set of hypotheses which are slightly different of that of Blumenthal-Getoor and they show there the existence of an exceptional set z (which turns out to be polar if Hunt-hypothesis (B) is satisfied): roughly speaking z is the set of points in E such that \(x\to u(x,y)\) is not extreme.

The main results of the paper under review are concerned with the sufficient conditions which guarantee that \(z=\Phi\). Under this new setting the authors studied the standard properties of the dual process and prove tht the dual process is actually a Hunt process. It is interesting to note that the proportionality of potentials (and w- potentials) with the same one-point support under the hypothesis of duality of two Hunt resolvents has been studied in Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 147-155 (1975; Zbl 0301.60054) by K. Janssen and the reviewer. It seems that one of the essential hypotheses (hypothesis (iii) of the paper under review) is closely related to the continuity principle which turns out to be equivalent to \(u(x,y)=f_ Gu(x,y)\), \(x\in E\), \(y\in G\subset E\) (finely open). (See also the last quoted paper.)

Given a Hunt process (or more generally a standard process) whose resolvent possesses a dual resolvent, it is well-known that [see R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory (1968; Zbl 0169.492)] the existence in this case of a Green function (w.r.t. to the given reference measure) gives rise to a rich probabilistic potential theory. However the problem of existence and uniqueness of integral representations of potentials (and co-potentials) requires extra hypotheses which guarantee that the Green potentials \(x\to u(x,y)\) (resp. \(y\to u(x,y))\) are extreme for \(\forall y\in E\) (resp. \(\forall x\in E)\), where u(x,y) is the Green function and where E is the state space of the process.

In a previous work K. L. Chung and K. M. Rao [Ann. Inst. Fourier 30, 167-198 (1980; Zbl 0424.31004)] have studied some standard problems (such as the existence of dual semi-group and dual process, Riesz decomposition theorem...) under a set of hypotheses which are slightly different of that of Blumenthal-Getoor and they show there the existence of an exceptional set z (which turns out to be polar if Hunt-hypothesis (B) is satisfied): roughly speaking z is the set of points in E such that \(x\to u(x,y)\) is not extreme.

The main results of the paper under review are concerned with the sufficient conditions which guarantee that \(z=\Phi\). Under this new setting the authors studied the standard properties of the dual process and prove tht the dual process is actually a Hunt process. It is interesting to note that the proportionality of potentials (and w- potentials) with the same one-point support under the hypothesis of duality of two Hunt resolvents has been studied in Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 147-155 (1975; Zbl 0301.60054) by K. Janssen and the reviewer. It seems that one of the essential hypotheses (hypothesis (iii) of the paper under review) is closely related to the continuity principle which turns out to be equivalent to \(u(x,y)=f_ Gu(x,y)\), \(x\in E\), \(y\in G\subset E\) (finely open). (See also the last quoted paper.)

Reviewer: X.L.Nguyen