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VIII: 13, 172-261, LNM 381 (1974)

**MAISONNEUVE, Bernard**; **MEYER, Paul-André**

Ensembles aléatoires markoviens homogènes (5 talks) (Markov processes)

This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see 412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (``left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (*Ann. Prob.* **1**, 1973; *Indiana Math. J.* **23**, 1973). The basic and earlier paper of Dynkin on the same subject (* Teor. Ver. Prim.* **16**, 1971) was not known to the authors.\par Chapter III is devoted to the original work of Maisonneuve on incursions. Roughly, the incursion at time $t$ is trivial if $t\in M$, and if $t\notin M$ it consists of the post-$t$ part of the excursion straddling $t$. Thus the incursion process is a path valued, non adapted process. It is only adapted to the filtration ${\cal F}_{D_t}$ where $D_t$ is the first hitting time of $M$ after $t$. Contrary to the Ito theory of excursions, no change of time using a local time is performed. The main result is the fact that, if a suitable regeneration property is assumed only on the set $M$ then, in a suitable topology on the space of paths, this process is a right-continuous strong Markov process. Considerable effort is devoted to proving that it is even a right process (the technique is heavy and many errors have crept in, some of them corrected in 932-933).\par Chapter IV makes the connection between II and III: the main results of Chapter II are proved anew (without balayage or Laplace transforms): they amount to computing the Lévy system of the incursion process. Finally, Chapter V consists of applications, among which a short discussion of the boundary theory for Markov chains

Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as*Systèmes Régénératifs,* *Astérisque,* **15**, 1974, a much simpler version of his own results, and discovered important improvements later on (some of which are included in Dellacherie-Maisonneuve-Meyer, *Probabilités et Potentiel,* Chapter XX, 1992). Along the slightly different line of Dynkin, see El~Karoui-Reinhard, *Compactification et balayage de processus droits,* *Astérisque 21,* 1975. A recent book on excursion theory is Blumenthal, *Excursions of Markov Processes,* Birkhäuser 1992

Keywords: Regenerative systems, Regenerative sets, Renewal theory, Local times, Excursions, Markov chains, Incursions

Nature: Original

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IX: 32, 518-521, LNM 465 (1975)

**MAISONNEUVE, Bernard**; **MEYER, Paul-André**

Ensembles aléatoires markoviens homogènes. Mise au point et compléments (Markov processes)

This paper corrects or simplifies many details in the long paper 713 by the same authors

Comment: See also the next paper 933

Keywords: Regenerative systems, Last-exit decompositions, Excursions

Nature: Original

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IX: 33, 522-529, LNM 465 (1975)

**MAISONNEUVE, Bernard**

Le comportement de dernière sortie (Markov processes)

This paper contains improvements to the paper 813 by Maisonneuve-Meyer, whose results are briefly recalled. Incursion processes and Lévy systems are altogether avoided, last-exist decompositions are derived, and the strong Markov property of the analogue of the age process in renewal theory is proved, as well as a non-homogeneous Markov property for some processes starting at last-exit times. The extension of these results to abstractly defined regenerative systems is mentioned

Comment: More detailed versions of these results appear in Maisonneuve,*Ann. Prob.*, **3**, 1975, *Z. für W-theorie,* **80**, 1989, and in Chapter XX of Dellacherie-Maisonneuve-Meyer, *Probabilités et Potentiel,* Hermann 1992

Keywords: Regenerative systems, Last-exit decompositions, Excursions

Nature: Original

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XV: 13, 191-205, LNM 850 (1981)

**MAISONNEUVE, Bernard**

On Lévy's downcrossing theorem and various extensions (Excursion theory)

Lévy's downcrossing theorem describes the local time $L_t$ of Brownian motion at $0$ as the limit of $\epsilon D_t(\epsilon)$, where $D_t$ denotes the number of downcrossings of the interval $(0,\epsilon)$ up to time $t$. To give a simple proof of this result from excursion theory, easy to generalize, the paper uses the weaker definition of regenerative systems described in 1137. A gap in the related author's paper*Zeit. für W-Theorie,* **52**, 1980 is repaired at the end of the paper

Keywords: Excursions, Lévy's downcrossing theorem, Local times, Regenerative systems

Nature: Original

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Ensembles aléatoires markoviens homogènes (5 talks) (Markov processes)

This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see 412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (``left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (

Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as

Keywords: Regenerative systems, Regenerative sets, Renewal theory, Local times, Excursions, Markov chains, Incursions

Nature: Original

Retrieve article from Numdam

IX: 32, 518-521, LNM 465 (1975)

Ensembles aléatoires markoviens homogènes. Mise au point et compléments (Markov processes)

This paper corrects or simplifies many details in the long paper 713 by the same authors

Comment: See also the next paper 933

Keywords: Regenerative systems, Last-exit decompositions, Excursions

Nature: Original

Retrieve article from Numdam

IX: 33, 522-529, LNM 465 (1975)

Le comportement de dernière sortie (Markov processes)

This paper contains improvements to the paper 813 by Maisonneuve-Meyer, whose results are briefly recalled. Incursion processes and Lévy systems are altogether avoided, last-exist decompositions are derived, and the strong Markov property of the analogue of the age process in renewal theory is proved, as well as a non-homogeneous Markov property for some processes starting at last-exit times. The extension of these results to abstractly defined regenerative systems is mentioned

Comment: More detailed versions of these results appear in Maisonneuve,

Keywords: Regenerative systems, Last-exit decompositions, Excursions

Nature: Original

Retrieve article from Numdam

XV: 13, 191-205, LNM 850 (1981)

On Lévy's downcrossing theorem and various extensions (Excursion theory)

Lévy's downcrossing theorem describes the local time $L_t$ of Brownian motion at $0$ as the limit of $\epsilon D_t(\epsilon)$, where $D_t$ denotes the number of downcrossings of the interval $(0,\epsilon)$ up to time $t$. To give a simple proof of this result from excursion theory, easy to generalize, the paper uses the weaker definition of regenerative systems described in 1137. A gap in the related author's paper

Keywords: Excursions, Lévy's downcrossing theorem, Local times, Regenerative systems

Nature: Original

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