V: 17, 177-190, LNM 191 (1971)
MEYER, Paul-André
Processus de Poisson ponctuels d'après K. Ito (
Markov processes,
Point processes)
Presents (a preliminary form of) the celebrated paper of Ito (
Proc. Sixth Berkeley Symposium, 3, 1972) on excursion theory, with an extension (the use of possibly unbounded entrance laws instead of initial measures) which has become part of the now classical theory
Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form
Keywords: Poisson point processes,
Excursions,
Local timesNature: Exposition,
Original additions
Retrieve article from Numdam
VII: 15, 146-154, LNM 321 (1973)
MEYER, Paul-André
Chirurgie sur un processus de Markov, d'après Knight et Pittenger (
Markov processes)
This paper presents a remarkable result of Knight and Pittenger, according to which excising from the sample path of a Markov process all the excursions from a given set $A$ which meet another set $B$ preserves the Markov property
Comment: The original paper appeared in
Zeit. für W-theorie, 23, 1972
Keywords: Transformations of Markov processes,
ExcursionsNature: Exposition Retrieve article from Numdam
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,
IncursionsNature: Original Retrieve article from Numdam
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
933Keywords: Regenerative systems,
Last-exit decompositions,
ExcursionsNature: Original Retrieve article from Numdam
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,
ExcursionsNature: Original Retrieve article from Numdam
X: 14, 216-234, LNM 511 (1976)
WILLIAMS, David
The Q-matrix problem (
Markov processes)
This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched
Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams
Diffusions, Markov Processes and Martingales, vol. 1 (second edition), Wiley 1994. See also
1024Keywords: Markov chains,
Ray compactification,
Local times,
ExcursionsNature: Original Retrieve article from Numdam
XIII: 43, 490-494, LNM 721 (1979)
WILLIAMS, David
Conditional excursion theory (
Brownian motion,
Markov processes)
To be completed
Keywords: ExcursionsNature: Original Retrieve article from Numdam
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 systemsNature: Original Retrieve article from Numdam
XV: 16, 227-250, LNM 850 (1981)
ROGERS, L.C.G.
Williams' characterization of the Brownian excursion law: proof and applications (
Brownian motion)
In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams
Diffusions, Markov Processes and Martingales, Wiley 1979, II.67), and it collects a number of applications, including the Azéma-Yor approach to Skorohod's imbedding theorem (
1306)
Keywords: Excursions,
Explicit laws,
Bessel processes,
Skorohod imbeddingNature: Original Retrieve article from Numdam
XLII: 12, 331-363, LNM 1978 (2009)
DEBS, Pierre
Penalisation of the standard random walk by a function of the one-sided maximum of the local time or of the duration of the excursions (
Theory of stochastic processes)
Keywords: Penalisation,
Excursions,
Local timesNature: Original
