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5 matches found
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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VI: 15, 164-167, LNM 258 (1972)
MEYER, Paul-André
Une note sur le théorème du balayage de Hunt (Markov processes, Potential theory)
The theorem involved is the characterization of the réduite $P_A u$ of an excessive function $u$ on a set $A$ as equal (except on a well-defined semipolar set) to the infimum of the excessive functions that dominate $u$ on $A$. This theorem is slightly improved under the absolute continuity hypothesis. The proof rests on the following property of the fine topology: every (nearly Borel) finely closed set is contained in a fine $G_\delta$ which differs from it by a polar set
Keywords: Réduite, Fine topology, Absolute continuity hypothesis
Nature: Original
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VII: 16, 155-171, LNM 321 (1973)
MEYER, Paul-André; TRAKI, Mohammed
Réduites et jeux de hasard (Potential theory)
This paper arose from an attempt (by the second author) to rewrite the results of Dubins-Savage How to Gamble if you Must in the language of standard (countably additive) measure theory, using the methods of descriptive set theory (analytic sets, section theorems, etc). The attempt is successful, since all general theorems can be proved in this set-up. More recent results in the same line, due to Strauch and Sudderth, are extended too. An appendix includes useful comments by Mokobodzki on the case of a gambling house consisting of a single kernel (discrete potential theory)
Comment: This material is reworked in Dellacherie-Meyer, Probabilités et Potentiel, Vol. C, Chapter X
Keywords: Balayage, Gambling house, Réduite, Optimal strategy
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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