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7 matches found
V: 06, 76-76, LNM 191 (1971)
CHUNG, Kai Lai
A simple proof of Doob's convergence theorem (Potential theory)
Doob's theorem is a version of the main convergence theorem of potential theory: the limit of a decreasing sequence of excessive functions differs of its regularized version on a semi-polar set
Comment: It is also shown that a function $f$ satisfying $f\ge P_Kf$ for all compact sets $K$ differs from its regularized function on a semi-polar set
Keywords: Excessive functions, Semi-polar sets
Nature: New exposition of known results
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V: 19, 196-208, LNM 191 (1971)
MEYER, Paul-André
Représentation intégrale des fonctions excessives. Résultats de Mokobodzki (Markov processes, Potential theory)
Main result: the convex cone of excessive functions for a resolvent which satisfies the absolute continuity hypothesis is the union of convex compact metrizable ``hats''\ in a suitable topology, and therefore has the integral representation property. The original proof of Mokobodzki, self-contained and unpublished, is given here
Comment: See Mokobodzki's work on cones of potentials, Séminaire Bourbaki, May 1970
Keywords: Minimal excessive functions, Martin boundary, Integral representations
Nature: Exposition
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V: 21, 211-212, LNM 191 (1971)
MEYER, Paul-André
Deux petits résultats de théorie du potentiel (Potential theory)
Excessive functions are characterized by their domination property over potentials. The strong ordering relation between two functions is carried over to their réduites
Comment: See Dellacherie-Meyer Probability and Potentials, Chapter XII, \S2
Keywords: Excessive functions, Réduite, Strong ordering
Nature: Original
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V: 25, 270-274, LNM 191 (1971)
MEYER, Paul-André
Balayage pour les processus de Markov continus à droite, d'après C.T. Shih (Markov processes, Potential theory)
Hunt's fundamental theorem on balayage gives the probabilistic interpretation of the réduite of an excessive function set on a set. It was proved originally within the special class of Hunt's processes, then extended to ``standard'' processes. Using a method of compactification, Shih (Ann. Inst. Fourier, 20-1, 1970) showed it was quite general
Comment: Shih's paper is the origin of the general definition of ``right processes''
Keywords: Excessive functions, Réduite
Nature: Exposition
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V: 26, 275-277, LNM 191 (1971)
REVUZ, Daniel
Remarque sur les potentiels de mesure (Markov processes, Potential theory)
The standard proof of the equivalence between semi-polar sets being polar and a very precise domination principle (Blumenthal-Getoor, Markov Processes and Potential Theory, 1968) uses the assumption that excessive functions are lower semicontinuous. This assumption is weakened
Comment: To be asked
Keywords: Polar sets, Semi-polar sets, Excessive functions
Nature: Original
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VII: 17, 172-179, LNM 321 (1973)
MEYER, Paul-André
Application de l'exposé précédent aux processus de Markov (Markov processes)
This paper is devoted to results of J.F. Mertens on optimal stopping and strongly supermedian functions for a right Markov process (Zeit. für W-theorie, 26, 1973), which are shown to be closely related to those of the preceding paper 716 on general gambling houses. An interesting result of Mokobodzki is included, showing that the extreme points of the convex set of all balayées of a given measure $\lambda$ are the balayées of $\lambda$ on sets
Comment: See related papers by Mertens in Zeit. für W-theorie, 22, 1972 and Invent. Math., 23, 1974. The original result of Mokobodzki appeared in the Sémin. Théorie du Potentiel, 1969-70
Keywords: Excessive functions, Supermedian functions, Réduite
Nature: Exposition, Original proofs
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VII: 20, 205-209, LNM 321 (1973)
MEYER, Paul-André
Remarques sur les hypothèses droites (Markov processes)
The following technical point is discussed: why does one assume among the ``right hypotheses'' that excessive functions are nearly-Borel?
Keywords: Right processes, Excessive functions
Nature: Original
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