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IX: 01, 2-96, LNM 465 (1975)

**MEYER, Paul-André**; **SAM LAZARO, José de**

Questions de théorie des flots (7 chapters) (Ergodic theory)

This is part of a seminar given in the year 1972/73. A flow is meant to be a one-parameter group $(\theta_t)$ of 1--1 measure preserving transformations of a probability space. The main topic of this seminar is the theory of filtered flows, i.e., a filtration $({\cal F}_t)$ ($t\!\in\!**R**$) is given such that $\theta_s ^{-1}{\cal F}_t={\cal F}_{s+t}$, and particularly the study of *helixes,* which are real valued processes $(Z_t)$ ($t\!\in\!**R**$) such that $Z_0=0$, which for $t\ge0$ are adapted, and on the whole line have homogeneous increments ($Z_{s+t}-Z_t=Z_t\circ \theta_s$). Two main classes of helixes are considered, the increasing helixes, and the martingale helixes. Finally, a filtered flow such that ${\cal F}_{-\infty}$ is degenerate is called a K-flow (K for Kolmogorov). Chapter~1 gives these definitions and their simplest consequences, as well as the definition of (continuous time) point processes, and the Ambrose construction of (unfiltered) flows from discrete flows as *flows under a function.* Chapter II shows that homogeneous discrete point processes and flows under a function are two names for the same object (Hanen, *Ann. Inst. H. Poincaré,* **7**, 1971), leading to the definition of the Palm measure of a discrete point process, and proves the classical (Ambrose-Kakutani) result that every flow with reasonable ergodicity properties can be interpreted as a flow under a function. A discussion of the case of filtered flows follows, with incomplete results. Chapter III is devoted to examples of flows and K-flows (Totoki's theorem). Chapter IV contains the study of increasing helixes, their Palm measures, and changes of times on flows. Chapter V is the original part of the seminar, devoted to the (square integrable) martingale helixes, their brackets, and the fact that in every K-flow these martingale helixes generate all martingales by stochastic integration. The main tool to prove this is a remark that every filtered K-flow can be interpreted (in a somewhat loose sense) as the flow of a stationary Markov process, helixes then becoming additive functionals, and standard Markovian methods becoming applicable. Chapter VI is devoted to spectral multiplicity, the main result being that a filtered flow, whenever it possesses one martingale helix, possesses infinitely many orthogonal helixes (orthogonal in a weak sense, not as martingales). Chapter VII is devoted to an independent topic: approximation in law of any ergodic stationary process by functionals of the Brownian flow (Nisio's theorem)

Comment: This set of lectures should be completed by the paper of Benveniste 902 which follows it, by an (earlier) paper by Sam Lazaro-Meyer (*Zeit. für W-theorie,* **18**, 1971) and a (later) paper by Sam Lazaro (*Zeit. für W-theorie,* **30**, 1974). Some of the results presented were less original than the authors believed at the time of the seminar, and due acknowledgments of priority are given; for an additional one see 1031. Related papers are due to Geman-Horowitz (*Ann. Inst. H. Poincaré,* **9**, 1973). The theory of filtered flows and Palm measures had a striking illustration within the theory of Markov processes as Kuznetsov measures (Kuznetsov, *Th. Prob. Appl.*, **18**, 1974) and the interpretation of ``Hunt quasi-processes'' as their Palm measures (Fitzsimmons, *Sem. Stoch. Processes 1987*, 1988)

Keywords: Filtered flows, Kolmogorov flow, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures

Nature: Exposition, Original additions

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IX: 02, 97-153, LNM 465 (1975)

**BENVENISTE, Albert**

Processus stationnaires et mesures de Palm du flot spécial sous une fonction (Ergodic theory, General theory of processes)

This paper takes over several topics of 901, with important new results and often with simpler proofs. It contains results on the existence of ``perfect'' versions of helixes and stationary processes, a better (uncompleted) version of the filtration itself, a more complete and elegant exposition of the Ambrose-Kakutani theorem, taking the filtration into account (the fundamental counter is adapted). The general theory of processes (projection and section theorems) is developed for a filtered flow, taking into account the fact that the filtrations are uncompleted. It is shown that any bounded measure that does not charge ``polar sets'' is the Palm measure of some increasing helix (see also Geman-Horowitz (*Ann. Inst. H. Poincaré,* **9**, 1973). Then a deeper study of flows under a function is performed, leading to section theorems of optional or previsible homogeneous sets by optional or previsible counters. The last section (written in collaboration with J.~Jacod) concerns a stationary counter (discrete point process) in its natural filtration, and its stochastic intensity: here it is shown (contrary to the case of processes indexed by a half-line) that the stochastic intensity does not determine the law of the counter

Keywords: Filtered flows, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures, Perfection, Point processes

Nature: Original

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IX: 03, 154-205, LNM 465 (1975)

**NANOPOULOS, Photius**

Mesures d'information et représentation de semi-groupes associés (Miscellanea)

Several attempts have been made to define the information content of an event on a measurable space independently from its probability. This paper develops a point of view of Kampé de Fériet-Forte,*C.R.A.S. Paris,* **265A**, 1969

Keywords: Information theory

Nature: Original

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IX: 04, 206-212, LNM 465 (1975)

**CHOU, Ching Sung**

Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)

A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities

Keywords: Inequalities, Burkholder inequalities

Nature: Exposition

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IX: 05, 213-225, LNM 465 (1975)

**CHOU, Ching Sung**

Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (Martingale theory)

The methods developed in discrete time by Garsia*Martingale Inequalities: Seminar Notes on Recent Progress,* Benjamin, 1973, are extended to continuous time

Comment: See Lenglart-Lépingle-Pratelli 1404. These methods have now become standard, and can be found in a number of books

Keywords: Inequalities, Burkholder inequalities

Nature: Original

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IX: 06, 226-236, LNM 465 (1975)

**CHOU, Ching Sung**; **MEYER, Paul-André**

Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels (General theory of processes)

Dellacherie has studied in 405 the filtration generated by a point process with one single jump. His study is extended here to the filtration generated by a discrete point process. It is shown in particular how to construct a martingale which has the previsible representation property

Comment: In spite or because of its simplicity, this paper has become a standard reference in the field. For a general account of the subject, see He-Wang-Yan,*Semimartingale Theory and Stochastic Calculus,* CRC~Press 1992

Keywords: Point processes, Previsible representation

Nature: Original

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IX: 07, 237-238, LNM 465 (1975)

**MEYER, Paul-André**

Complément sur la dualité entre $H^1$ et $BMO$ (Martingale theory)

Fills a gap in the proof of the duality theorem in 714

Keywords: $BMO$

Nature: Correction

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IX: 08, 239-245, LNM 465 (1975)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

Un nouveau théorème de projection et de section (General theory of processes)

Optional section and projection theorems are proved without assuming the ``usual conditions'' on the filtration

Comment: This paper is obsolete. As stated at the end by the authors, the result could have been deduced from the general theorem in Dellacherie 705. The result takes its definitive form in Dellacherie-Meyer,*Probabilités et Potentiel,* theorems IV.84 of vol. A and App.1, \no~6

Keywords: Section theorems, Optional processes, Projection theorems

Nature: Original

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IX: 09, 246-267, LNM 465 (1975)

**DACUNHA-CASTELLE, Didier**

Processus et espaces de Banach invariants par réarrangement (Miscellanea)

The first topic of this paper is the class of processes on an interval with exchangeable increments, which includes all processes with independent and stationary increments, but also the standard Brownian bridge (for several results quoted here, see Kallenberg,*Z. für W-theorie,* **27**, 1973). The results are applied to problems in the classification of Banach spaces, in the spirit of Dacunha-Castelle-Schreiber, *Ann. Inst. H. Poincaré,* **10**, 1974

Keywords: Banach spaces, Exchangeable random variables

Nature: Original

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IX: 10, 268-284, LNM 465 (1975)

**DACUNHA-CASTELLE, Didier**

Sous-espaces symétriques des espaces d'Orlicz (Miscellanea)

Will be asked from the author

Comment: To be asked from the author. See Dacunha-Castelle-Schreiber,*Ann. Inst. H. Poincaré,* **10**, 1974

Keywords: Banach spaces

Nature: Original

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IX: 11, 285-293, LNM 465 (1975)

**ARTZNER, Philippe**

Quelques résultats de décomposabilité en algèbre linéaire et en algèbre quadratique aléatoires (Miscellanea)

It is shown that if a random quadratic form on $**R**^4$ is a.s. of rank 3 and has an absolutely continuous law, then its law is indecomposable. To prove this, it is shown that the sum of two independent planes in $**R**^4$ cannot at the same time be a.s. a hyperplane and have an absolutely continuous law (with respect to the natural measure for hyperplanes)

Keywords: Independent random subspaces, Independent quadratic forms

Nature: Original

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IX: 12, 294-304, LNM 465 (1975)

**SIGMUND, Karl**

Propriétés générales et exceptionnelles des états statistiques des systèmes dynamiques stables (Ergodic theory)

This is an introduction to important problems concerning discrete dynamical systems: any homeomorphism of a compact metric space has invariant probability measures, which form a non-empty compact convex set. Which properties of these measures are ``generic'' or ``exceptional'' in the sense of Baire category? No proofs are given

Keywords: Dynamical systems, Invariant measures

Nature: Exposition

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IX: 13, 305-317, LNM 465 (1975)

**FÖLLMER, Hans**

Phase transition and Martin boundary (Miscellanea)

To be completed

Comment: To be completed

Keywords: Random fields, Martin boundary

Nature: Original

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IX: 14, 318-335, LNM 465 (1975)

**FERNIQUE, Xavier**

Des résultats nouveaux sur les processus gaussiens (Gaussian processes)

Given a centered Gaussian process indexed by an arbitrary set~$T$, a major problem has been to find conditions implying that the sample functions are a.s. bounded, or a.s. continuous in the natural metric associated with the covariance. Here new necessary conditions for boundedness are given, which turn out to be sufficient in the case of stationary processes on $**R**^n$. The conditions given here involve the existence of a majorizing measure, an idea which became crucial in the theory

Comment: For a systematic account of the theory around the time this paper was written, see Fernique's lectures in*École d'Été de Saint-Four~IV*, LNM **480**, 1974. For the definitive solution, see chapter 11 of Ledoux-Talagrand *Probability in Banach spaces,* Springer 1991

Keywords: Gaussian processes, Sample path regularity

Nature: Original

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IX: 15, 336-372, LNM 465 (1975)

**DELLACHERIE, Claude**

Ensembles analytiques, théorèmes de séparation et applications (Descriptive set theory)

According to the standard (``first'') separation theorem, in a compact metric space or any space which is Borel isomorphic to it, two disjoint analytic sets can be separated by Borel sets, and in particular any bianalytic set (analytic and coanalytic i.e., complement of analytic) is Borel. Not so in general metric spaces. That the same statement holds in full generality with ``bianalytic'' instead of ``Borel'' is the second separation theorem, which according to the general opinion was considered much more difficult than the first. This result and many more (on projections of Borel sets with compact sections or countable sections, for instance) are fully proved in this exposition

Comment: See also the next paper 916, the set of lectures by Dellacherie in C.A. Rogers,*Analytic Sets,* Academic Press 1981, and chapter XXIV of Dellacherie-Meyer, *Probabilités et potentiel *

Keywords: Second separation theorem

Nature: Exposition

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IX: 16, 373-389, LNM 465 (1975)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

Ensembles analytiques et temps d'arrêt (Descriptive set theory)

This is a sequel to the preceding paper 915. Instead of using the language of trees to prove the second separation theorem, a language more familiar to probabilists is used, in which the space of stopping times on $**N**^**N**$ is given a compact metric topology and the space of non-finite stopping times appears as the universal analytic, non-Borel set, from which all analytic sets can be constructed. Many proofs become very natural in this language

Comment: See also the next paper 917, the set of lectures by Dellacherie in C.A. Rogers,*Analytic Sets,* Academic Press 1981, and chapter XXIV of Dellacherie-Meyer, *Probabilités et potentiel *

Keywords: Second separation theorem, Stopping times

Nature: Original

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IX: 17, 390-405, LNM 465 (1975)

**DELLACHERIE, Claude**

Jeux infinis avec information complète et temps d'arrêt (Descriptive set theory)

This is a sequel to the preceding paper 916. It shows how well the language of stopping times applies, not only to the second separation theorem, but to the Gale-Stewart theorem on the determinacy of open games

Comment: The original remark on the relation between game determinacy and separation theorems, due to Blackwell (1967), led to a huge literature. More details can be found in chapter XXIV of Dellacherie-Meyer,*Probabilités et potentiel *

Keywords: Determinacy of games, Gale and Stewart theorem

Nature: Original

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IX: 18, 406-407, LNM 465 (1975)

**DELLACHERIE, Claude**

Une remarque sur les espaces sousliniens de Bourbaki (Descriptive set theory)

The paper claims to prove that given any space with a Souslin topology i.e., which is Hausdorff and a continuous image of a Polish space, it is possible to strengthen the topology so that it becomes metrizable without losing the Souslin property. The author discovered later a serious mistake in his proof

Comment: The problem is still open, and interesting

Keywords: Analytic sets

Nature: False

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IX: 19, 408-419, LNM 465 (1975)

**STRICKER, Christophe**

Mesure de Föllmer en théorie des quasimartingales (Martingale theory)

The Föllmer measure associated with a positive supermartingale, or more generally a quasimartingale (Föllmer,*Z. für W-theorie,* **21**, 1972; *Ann. Prob.* **1**, 1973) is constructed using a weak limit procedure instead of a projective limit

Comment: On Föllmer measures see 611. This paper corresponds to an early stage in the theory of quasimartingales, for which the main reference was Orey,*Proc. Fifth Berkeley Symp.*, **2**

Keywords: Quasimartingales, Föllmer measures

Nature: Original

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IX: 20, 420-424, LNM 465 (1975)

**STRICKER, Christophe**

Une caractérisation des quasimartingales (Martingale theory)

An integral criterion is shown to be equivalent to the usual definition of a quasimartingale using the stochastic variation

Keywords: Quasimartingales

Nature: Original

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IX: 21, 425-436, LNM 465 (1975)

**ÉMERY, Michel**

Primitive d'une mesure sur les compacts d'un espace métrique (Measure theory)

It is well known that the set ${\cal K}$ of all compact subsets of a compact metric space has a natural compact metric topology. The ``distribution function'' of a positive measure on ${\cal K}$ associates with every $A\in{\cal K}$ the measure of the subset $\{K\subset A\}$ of ${\cal K}$. It is shown here (following A.~Revuz,*Ann. Inst. Fourier,* **6**, 1955-56) that the distribution functions of measures are characterized by simple algebraic properties and right continuity

Comment: This elegant theorem apparently never had applications

Keywords: Distribution functions on ordered spaces

Nature: Exposition

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IX: 22, 437-442, LNM 465 (1975)

**MOKOBODZKI, Gabriel**

Relèvement borélien compatible avec une classe d'ensembles négligeables. Application à la désintégration des mesures (Measure theory)

This is a beautiful application of the continuum ``hypothesis'' (axiom). It is shown that if $(E, {\cal E})$ is a separable measurable space (or more generally ${\cal E}$ has the power of the continuum) and ${\cal N}$ is a family of negligible sets within ${\cal E}$, then the space of classes of bounded measurable functions mod.~${\cal N}$ has a linear, isometric, and multiplicative lifting. The proof is rather simple. Essentially the same theorem was discovered independently by Chatterji, see*Vector and Operator Valued Measures,* Academic Press 1973

Comment: The same proof leads to a slightly stronger and useful result (Meyer 2711): if $E$ is a compact metric space and if any two continuous functions equal a.e. are equal everywhere, the lifting can be taken to be the identity on continuous functions, and to be local, i.e., the liftings of two Borel functions equal a.e. in an open set are equal everywhere in this set

Keywords: Continuum axiom, Lifting theorems, Negligible sets

Nature: Original

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IX: 23, 443-463, LNM 465 (1975)

**GETOOR, Ronald K.**

On the construction of kernels (Measure theory)

Given two measurable spaces $(E, {\cal E})$ $(F, {\cal F})$ and a family ${\cal N}\subset{\cal E}$ of negligible sets, a pseudo-kernel $T$ is a mapping from bounded measurable functions on $F$ to classes mod.${\cal N}$ of bounded measurable functions on $E$, which has all a.e. the properties (positivity, countable additivity) of a kernel. Regularizing $T$ consists in finding a true kernel $\hat T$ such that $\hat Tf$ belongs to the class $Tf$ for every measurable bounded $f$ on $F$. The regularization is easy whenever $F$ is compact metric. Then the result is extended to the case of a Lusin space, and to the case of a U-space (Radon space) assuming ${\cal N}$ consists of the negligible sets for a family of measures on $E$. An application is given to densities of continuous additive functionals of a Markov process

Comment: The author states that his paper is purely expository. This is not true, though the proof is a standard one in the theory of conditional distributions. For a deeper result, see Dellacherie 1030. For a presentation in book form, see Dellacherie-Meyer,*Probabilités et Potentiel C,* chapter XI **41**

Keywords: Pseudo-kernels, Regularization

Nature: Original

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IX: 24, 464-465, LNM 465 (1975)

**MEYER, Paul-André**

Une remarque sur la construction de noyaux (Measure theory)

With the notation of the preceding report 923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity

Comment: See Dellacherie 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

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IX: 25, 466-470, LNM 465 (1975)

**MEYER, Paul-André**; **YAN, Jia-An**

Génération d'une famille de tribus par un processus croissant (General theory of processes)

The previsible and optional $\sigma$-fields of a filtration $({\cal F}_t)$ on $\Omega$ are studied without the usual hypotheses: no measure is involved, and the filtration is not right continuous. It is proved that if the $\sigma$-fields ${\cal F}_{t-}$ are separable, then so is the previsible $\sigma$-field, and the filtration is the natural one for a continuous strictly increasing process. A similar result is proved for the optional $\sigma$-field assuming $\Omega$ is a Blackwell space, and then every measurable adapted process is optional

Comment: Making the filtration right continuous generally destroys the separability of the optional $\sigma$-field

Keywords: Previsible processes, Optional processes

Nature: Original

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IX: 26, 471-485, LNM 465 (1975)

**NAGASAWA, Masao**

Multiplicative excessive measures and duality between equations of Boltzmann and of branching processes (Markov processes, Statistical mechanics)

The author investigates the connection between the branching Markov processes constructed over some given Markov processes and a non-linear equation close to Boltzmann's equation. A special class of excessive measures for the branching Markov process is described and studied, as well as the corresponding dual processes

Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 1011

Keywords: Boltzmann equation, Branching processes

Nature: Original

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IX: 27, 486-493, LNM 465 (1975)

**WEIL, Michel**

Surlois d'entrée (Markov processes)

The results presented here are due to T. Leviatan (*Ann. Prob.*, **1**, 1973) and concern the construction of a Markov process with creation of mass, corresponding to a given transition semigroup $(P_t)$ and a given ``super-entrance law'' $(\mu_t)$, consisting of bounded measures such that $\mu_{s+t}\ge\mu_s P_t$. The proof is a clever argument of projective limits. The paper mentions briefly the relation with earlier results of Helms (*Z. für W-theorie,* **7**, 1967)

Comment: This beautiful paper was superseded by the (slightly later) fundamental paper of Kuznetsov (1974)

Keywords: Entrance laws, Creation of mass, Kuznetsov measures

Nature: Exposition

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IX: 28, 494-494, LNM 465 (1975)

**DELLACHERIE, Claude**

Correction à ``Intégrales stochastiques par rapport...'' (General theory of processes)

This paper completes a gap in the simple proof of the previsible representation property of the Wiener process, given by Dellacherie 805

Comment: Another way of filling this gap is given by Ruiz de Chavez 1821. The same gap for the Poisson process is corrected in 2002

Keywords: Previsible representation

Nature: Original

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IX: 29, 495-495, LNM 465 (1975)

**DELLACHERIE, Claude**

Une propriété des ensembles semi-polaires (Markov processes)

It is shown that semi-polar sets are exactly those which have potential 0 for all continuous additive functionals (or for all time-changed processes)

Keywords: Semi-polar sets

Nature: Original

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IX: 30, 496-514, LNM 465 (1975)

**SHARPE, Michael J.**

Homogeneous extensions of random measures (Markov processes)

Homogeneous random measures are the appropriate definition of additive functionals which may explode. The problem discussed here is the extension of such a measure given up to a terminal time into a measure defined up to the lifetime

Comment: The subject is taken over in a systematic way in Sharpe,*General Theory of Markov processes,* Academic Press 1988

Keywords: Homogeneous random measures, Terminal times, Subprocesses

Nature: Original

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IX: 31, 515-517, LNM 465 (1975)

**HEATH, David C.**

Skorohod stopping in discrete time (Markov processes, Potential theory)

Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times

Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath 811. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

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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IX: 34, 530-533, LNM 465 (1975)

**MEYER, Paul-André**

Sur la démonstration de prévisibilité de Chung et Walsh (Markov processes)

A new proof of the result that for a Hunt process, the previsible stopping times are exactly those at which the process does not jump was given by Chung-Walsh (*Z. für W-theorie,* **29**, 1974). Their idea is used here in a modified way, using a formula of Dawson which ``explicitly'' computes conditional expectations and projections. Then it is extended to Ray processes

Comment: The contents of this paper became Chapter XIV**44**--47 in Dellacherie-Meyer, *Probabilités et Potentiel*

Keywords: Hunt processes, Previsible times

Nature: Exposition

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IX: 35, 534-554, LNM 465 (1975)

**EL KAROUI, Nicole**

Processus de réflexion dans ${\bf R}^n$ (Diffusion theory)

In the line of the seminar on diffusions 419 this talk presents the theory of diffusions in a half space with continuous coefficients and a boundary condition on the boundary hyperplane involving a reflexion part, but more general than the pure reflexion case considered by Stroock-Varadhan (*Comm. Pure Appl. Math.*, **24**, 1971). The point of view is that of martingale problems

Comment: This talk is a late publication of work done by the author in 1971

Keywords: Boundary reflection, Local times

Nature: Original

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IX: 36, 555-555, LNM 465 (1975)

**MEYER, Paul-André**

Une remarque sur les processus de Markov (Markov processes)

It is shown that, under a fixed measure $**P**^{\mu}$, the optional processes and times relative to the uncompleted filtrations $({\cal F}_{t+}^{\circ})$ and $({\cal F}_{t}^{\circ})$ are undistinguishable from each other

Comment: No applications are known

Keywords: Stopping times

Nature: Original

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IX: 37, 556-564, LNM 465 (1975)

**MEYER, Paul-André**

Retour aux retournements (Markov processes, General theory of processes)

The first part of the talk is devoted to an important correction to the theorem on p.285 of 814 (Azéma's theory of cooptional and coprevisible sets and its application to Markov processes): the definition of left additive functionals should allow an a.s. explosion at time 0 for initial points belonging to a polar set. The second part belongs to the general theory of processes: if the index set is not the right half-line as usual but the left half-line, the section and projection theorems must be modified in a not quite trivial way

Comment: See Chapter XXVIII of Dellacherie-Maisonneuve-Meyer*Probabilités et potentiel *

Keywords: Time reversal, Cooptional processes, Coprevisible processes, Homogeneous processes

Nature: Exposition, Original additions

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IX: 38, 565-588, LNM 465 (1975)

**WALDENFELS, Wilhelm von**

Interval partitions and pair interactions (Miscellanea)

To be completed

Comment: To be completed

Nature: Original

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IX: 39, 589-589, LNM 465 (1975)

**KHALILI-FRANÇON, Elisabeth**

Correction à ``Processus de Galton-Watson'' (Markov processes)

Proposition 1(ii) p.126 of 713 is true only in the case $q=0$

Keywords: Branching processes

Nature: Correction

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Questions de théorie des flots (7 chapters) (Ergodic theory)

This is part of a seminar given in the year 1972/73. A flow is meant to be a one-parameter group $(\theta_t)$ of 1--1 measure preserving transformations of a probability space. The main topic of this seminar is the theory of filtered flows, i.e., a filtration $({\cal F}_t)$ ($t\!\in\!

Comment: This set of lectures should be completed by the paper of Benveniste 902 which follows it, by an (earlier) paper by Sam Lazaro-Meyer (

Keywords: Filtered flows, Kolmogorov flow, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures

Nature: Exposition, Original additions

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IX: 02, 97-153, LNM 465 (1975)

Processus stationnaires et mesures de Palm du flot spécial sous une fonction (Ergodic theory, General theory of processes)

This paper takes over several topics of 901, with important new results and often with simpler proofs. It contains results on the existence of ``perfect'' versions of helixes and stationary processes, a better (uncompleted) version of the filtration itself, a more complete and elegant exposition of the Ambrose-Kakutani theorem, taking the filtration into account (the fundamental counter is adapted). The general theory of processes (projection and section theorems) is developed for a filtered flow, taking into account the fact that the filtrations are uncompleted. It is shown that any bounded measure that does not charge ``polar sets'' is the Palm measure of some increasing helix (see also Geman-Horowitz (

Keywords: Filtered flows, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures, Perfection, Point processes

Nature: Original

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IX: 03, 154-205, LNM 465 (1975)

Mesures d'information et représentation de semi-groupes associés (Miscellanea)

Several attempts have been made to define the information content of an event on a measurable space independently from its probability. This paper develops a point of view of Kampé de Fériet-Forte,

Keywords: Information theory

Nature: Original

Retrieve article from Numdam

IX: 04, 206-212, LNM 465 (1975)

Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)

A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities

Keywords: Inequalities, Burkholder inequalities

Nature: Exposition

Retrieve article from Numdam

IX: 05, 213-225, LNM 465 (1975)

Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (Martingale theory)

The methods developed in discrete time by Garsia

Comment: See Lenglart-Lépingle-Pratelli 1404. These methods have now become standard, and can be found in a number of books

Keywords: Inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

IX: 06, 226-236, LNM 465 (1975)

Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels (General theory of processes)

Dellacherie has studied in 405 the filtration generated by a point process with one single jump. His study is extended here to the filtration generated by a discrete point process. It is shown in particular how to construct a martingale which has the previsible representation property

Comment: In spite or because of its simplicity, this paper has become a standard reference in the field. For a general account of the subject, see He-Wang-Yan,

Keywords: Point processes, Previsible representation

Nature: Original

Retrieve article from Numdam

IX: 07, 237-238, LNM 465 (1975)

Complément sur la dualité entre $H^1$ et $BMO$ (Martingale theory)

Fills a gap in the proof of the duality theorem in 714

Keywords: $BMO$

Nature: Correction

Retrieve article from Numdam

IX: 08, 239-245, LNM 465 (1975)

Un nouveau théorème de projection et de section (General theory of processes)

Optional section and projection theorems are proved without assuming the ``usual conditions'' on the filtration

Comment: This paper is obsolete. As stated at the end by the authors, the result could have been deduced from the general theorem in Dellacherie 705. The result takes its definitive form in Dellacherie-Meyer,

Keywords: Section theorems, Optional processes, Projection theorems

Nature: Original

Retrieve article from Numdam

IX: 09, 246-267, LNM 465 (1975)

Processus et espaces de Banach invariants par réarrangement (Miscellanea)

The first topic of this paper is the class of processes on an interval with exchangeable increments, which includes all processes with independent and stationary increments, but also the standard Brownian bridge (for several results quoted here, see Kallenberg,

Keywords: Banach spaces, Exchangeable random variables

Nature: Original

Retrieve article from Numdam

IX: 10, 268-284, LNM 465 (1975)

Sous-espaces symétriques des espaces d'Orlicz (Miscellanea)

Will be asked from the author

Comment: To be asked from the author. See Dacunha-Castelle-Schreiber,

Keywords: Banach spaces

Nature: Original

Retrieve article from Numdam

IX: 11, 285-293, LNM 465 (1975)

Quelques résultats de décomposabilité en algèbre linéaire et en algèbre quadratique aléatoires (Miscellanea)

It is shown that if a random quadratic form on $

Keywords: Independent random subspaces, Independent quadratic forms

Nature: Original

Retrieve article from Numdam

IX: 12, 294-304, LNM 465 (1975)

Propriétés générales et exceptionnelles des états statistiques des systèmes dynamiques stables (Ergodic theory)

This is an introduction to important problems concerning discrete dynamical systems: any homeomorphism of a compact metric space has invariant probability measures, which form a non-empty compact convex set. Which properties of these measures are ``generic'' or ``exceptional'' in the sense of Baire category? No proofs are given

Keywords: Dynamical systems, Invariant measures

Nature: Exposition

Retrieve article from Numdam

IX: 13, 305-317, LNM 465 (1975)

Phase transition and Martin boundary (Miscellanea)

To be completed

Comment: To be completed

Keywords: Random fields, Martin boundary

Nature: Original

Retrieve article from Numdam

IX: 14, 318-335, LNM 465 (1975)

Des résultats nouveaux sur les processus gaussiens (Gaussian processes)

Given a centered Gaussian process indexed by an arbitrary set~$T$, a major problem has been to find conditions implying that the sample functions are a.s. bounded, or a.s. continuous in the natural metric associated with the covariance. Here new necessary conditions for boundedness are given, which turn out to be sufficient in the case of stationary processes on $

Comment: For a systematic account of the theory around the time this paper was written, see Fernique's lectures in

Keywords: Gaussian processes, Sample path regularity

Nature: Original

Retrieve article from Numdam

IX: 15, 336-372, LNM 465 (1975)

Ensembles analytiques, théorèmes de séparation et applications (Descriptive set theory)

According to the standard (``first'') separation theorem, in a compact metric space or any space which is Borel isomorphic to it, two disjoint analytic sets can be separated by Borel sets, and in particular any bianalytic set (analytic and coanalytic i.e., complement of analytic) is Borel. Not so in general metric spaces. That the same statement holds in full generality with ``bianalytic'' instead of ``Borel'' is the second separation theorem, which according to the general opinion was considered much more difficult than the first. This result and many more (on projections of Borel sets with compact sections or countable sections, for instance) are fully proved in this exposition

Comment: See also the next paper 916, the set of lectures by Dellacherie in C.A. Rogers,

Keywords: Second separation theorem

Nature: Exposition

Retrieve article from Numdam

IX: 16, 373-389, LNM 465 (1975)

Ensembles analytiques et temps d'arrêt (Descriptive set theory)

This is a sequel to the preceding paper 915. Instead of using the language of trees to prove the second separation theorem, a language more familiar to probabilists is used, in which the space of stopping times on $

Comment: See also the next paper 917, the set of lectures by Dellacherie in C.A. Rogers,

Keywords: Second separation theorem, Stopping times

Nature: Original

Retrieve article from Numdam

IX: 17, 390-405, LNM 465 (1975)

Jeux infinis avec information complète et temps d'arrêt (Descriptive set theory)

This is a sequel to the preceding paper 916. It shows how well the language of stopping times applies, not only to the second separation theorem, but to the Gale-Stewart theorem on the determinacy of open games

Comment: The original remark on the relation between game determinacy and separation theorems, due to Blackwell (1967), led to a huge literature. More details can be found in chapter XXIV of Dellacherie-Meyer,

Keywords: Determinacy of games, Gale and Stewart theorem

Nature: Original

Retrieve article from Numdam

IX: 18, 406-407, LNM 465 (1975)

Une remarque sur les espaces sousliniens de Bourbaki (Descriptive set theory)

The paper claims to prove that given any space with a Souslin topology i.e., which is Hausdorff and a continuous image of a Polish space, it is possible to strengthen the topology so that it becomes metrizable without losing the Souslin property. The author discovered later a serious mistake in his proof

Comment: The problem is still open, and interesting

Keywords: Analytic sets

Nature: False

Retrieve article from Numdam

IX: 19, 408-419, LNM 465 (1975)

Mesure de Föllmer en théorie des quasimartingales (Martingale theory)

The Föllmer measure associated with a positive supermartingale, or more generally a quasimartingale (Föllmer,

Comment: On Föllmer measures see 611. This paper corresponds to an early stage in the theory of quasimartingales, for which the main reference was Orey,

Keywords: Quasimartingales, Föllmer measures

Nature: Original

Retrieve article from Numdam

IX: 20, 420-424, LNM 465 (1975)

Une caractérisation des quasimartingales (Martingale theory)

An integral criterion is shown to be equivalent to the usual definition of a quasimartingale using the stochastic variation

Keywords: Quasimartingales

Nature: Original

Retrieve article from Numdam

IX: 21, 425-436, LNM 465 (1975)

Primitive d'une mesure sur les compacts d'un espace métrique (Measure theory)

It is well known that the set ${\cal K}$ of all compact subsets of a compact metric space has a natural compact metric topology. The ``distribution function'' of a positive measure on ${\cal K}$ associates with every $A\in{\cal K}$ the measure of the subset $\{K\subset A\}$ of ${\cal K}$. It is shown here (following A.~Revuz,

Comment: This elegant theorem apparently never had applications

Keywords: Distribution functions on ordered spaces

Nature: Exposition

Retrieve article from Numdam

IX: 22, 437-442, LNM 465 (1975)

Relèvement borélien compatible avec une classe d'ensembles négligeables. Application à la désintégration des mesures (Measure theory)

This is a beautiful application of the continuum ``hypothesis'' (axiom). It is shown that if $(E, {\cal E})$ is a separable measurable space (or more generally ${\cal E}$ has the power of the continuum) and ${\cal N}$ is a family of negligible sets within ${\cal E}$, then the space of classes of bounded measurable functions mod.~${\cal N}$ has a linear, isometric, and multiplicative lifting. The proof is rather simple. Essentially the same theorem was discovered independently by Chatterji, see

Comment: The same proof leads to a slightly stronger and useful result (Meyer 2711): if $E$ is a compact metric space and if any two continuous functions equal a.e. are equal everywhere, the lifting can be taken to be the identity on continuous functions, and to be local, i.e., the liftings of two Borel functions equal a.e. in an open set are equal everywhere in this set

Keywords: Continuum axiom, Lifting theorems, Negligible sets

Nature: Original

Retrieve article from Numdam

IX: 23, 443-463, LNM 465 (1975)

On the construction of kernels (Measure theory)

Given two measurable spaces $(E, {\cal E})$ $(F, {\cal F})$ and a family ${\cal N}\subset{\cal E}$ of negligible sets, a pseudo-kernel $T$ is a mapping from bounded measurable functions on $F$ to classes mod.${\cal N}$ of bounded measurable functions on $E$, which has all a.e. the properties (positivity, countable additivity) of a kernel. Regularizing $T$ consists in finding a true kernel $\hat T$ such that $\hat Tf$ belongs to the class $Tf$ for every measurable bounded $f$ on $F$. The regularization is easy whenever $F$ is compact metric. Then the result is extended to the case of a Lusin space, and to the case of a U-space (Radon space) assuming ${\cal N}$ consists of the negligible sets for a family of measures on $E$. An application is given to densities of continuous additive functionals of a Markov process

Comment: The author states that his paper is purely expository. This is not true, though the proof is a standard one in the theory of conditional distributions. For a deeper result, see Dellacherie 1030. For a presentation in book form, see Dellacherie-Meyer,

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

IX: 24, 464-465, LNM 465 (1975)

Une remarque sur la construction de noyaux (Measure theory)

With the notation of the preceding report 923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity

Comment: See Dellacherie 1030

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

IX: 25, 466-470, LNM 465 (1975)

Génération d'une famille de tribus par un processus croissant (General theory of processes)

The previsible and optional $\sigma$-fields of a filtration $({\cal F}_t)$ on $\Omega$ are studied without the usual hypotheses: no measure is involved, and the filtration is not right continuous. It is proved that if the $\sigma$-fields ${\cal F}_{t-}$ are separable, then so is the previsible $\sigma$-field, and the filtration is the natural one for a continuous strictly increasing process. A similar result is proved for the optional $\sigma$-field assuming $\Omega$ is a Blackwell space, and then every measurable adapted process is optional

Comment: Making the filtration right continuous generally destroys the separability of the optional $\sigma$-field

Keywords: Previsible processes, Optional processes

Nature: Original

Retrieve article from Numdam

IX: 26, 471-485, LNM 465 (1975)

Multiplicative excessive measures and duality between equations of Boltzmann and of branching processes (Markov processes, Statistical mechanics)

The author investigates the connection between the branching Markov processes constructed over some given Markov processes and a non-linear equation close to Boltzmann's equation. A special class of excessive measures for the branching Markov process is described and studied, as well as the corresponding dual processes

Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 1011

Keywords: Boltzmann equation, Branching processes

Nature: Original

Retrieve article from Numdam

IX: 27, 486-493, LNM 465 (1975)

Surlois d'entrée (Markov processes)

The results presented here are due to T. Leviatan (

Comment: This beautiful paper was superseded by the (slightly later) fundamental paper of Kuznetsov (1974)

Keywords: Entrance laws, Creation of mass, Kuznetsov measures

Nature: Exposition

Retrieve article from Numdam

IX: 28, 494-494, LNM 465 (1975)

Correction à ``Intégrales stochastiques par rapport...'' (General theory of processes)

This paper completes a gap in the simple proof of the previsible representation property of the Wiener process, given by Dellacherie 805

Comment: Another way of filling this gap is given by Ruiz de Chavez 1821. The same gap for the Poisson process is corrected in 2002

Keywords: Previsible representation

Nature: Original

Retrieve article from Numdam

IX: 29, 495-495, LNM 465 (1975)

Une propriété des ensembles semi-polaires (Markov processes)

It is shown that semi-polar sets are exactly those which have potential 0 for all continuous additive functionals (or for all time-changed processes)

Keywords: Semi-polar sets

Nature: Original

Retrieve article from Numdam

IX: 30, 496-514, LNM 465 (1975)

Homogeneous extensions of random measures (Markov processes)

Homogeneous random measures are the appropriate definition of additive functionals which may explode. The problem discussed here is the extension of such a measure given up to a terminal time into a measure defined up to the lifetime

Comment: The subject is taken over in a systematic way in Sharpe,

Keywords: Homogeneous random measures, Terminal times, Subprocesses

Nature: Original

Retrieve article from Numdam

IX: 31, 515-517, LNM 465 (1975)

Skorohod stopping in discrete time (Markov processes, Potential theory)

Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times

Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath 811. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

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

IX: 34, 530-533, LNM 465 (1975)

Sur la démonstration de prévisibilité de Chung et Walsh (Markov processes)

A new proof of the result that for a Hunt process, the previsible stopping times are exactly those at which the process does not jump was given by Chung-Walsh (

Comment: The contents of this paper became Chapter XIV

Keywords: Hunt processes, Previsible times

Nature: Exposition

Retrieve article from Numdam

IX: 35, 534-554, LNM 465 (1975)

Processus de réflexion dans ${\bf R}^n$ (Diffusion theory)

In the line of the seminar on diffusions 419 this talk presents the theory of diffusions in a half space with continuous coefficients and a boundary condition on the boundary hyperplane involving a reflexion part, but more general than the pure reflexion case considered by Stroock-Varadhan (

Comment: This talk is a late publication of work done by the author in 1971

Keywords: Boundary reflection, Local times

Nature: Original

Retrieve article from Numdam

IX: 36, 555-555, LNM 465 (1975)

Une remarque sur les processus de Markov (Markov processes)

It is shown that, under a fixed measure $

Comment: No applications are known

Keywords: Stopping times

Nature: Original

Retrieve article from Numdam

IX: 37, 556-564, LNM 465 (1975)

Retour aux retournements (Markov processes, General theory of processes)

The first part of the talk is devoted to an important correction to the theorem on p.285 of 814 (Azéma's theory of cooptional and coprevisible sets and its application to Markov processes): the definition of left additive functionals should allow an a.s. explosion at time 0 for initial points belonging to a polar set. The second part belongs to the general theory of processes: if the index set is not the right half-line as usual but the left half-line, the section and projection theorems must be modified in a not quite trivial way

Comment: See Chapter XXVIII of Dellacherie-Maisonneuve-Meyer

Keywords: Time reversal, Cooptional processes, Coprevisible processes, Homogeneous processes

Nature: Exposition, Original additions

Retrieve article from Numdam

IX: 38, 565-588, LNM 465 (1975)

Interval partitions and pair interactions (Miscellanea)

To be completed

Comment: To be completed

Nature: Original

Retrieve article from Numdam

IX: 39, 589-589, LNM 465 (1975)

Correction à ``Processus de Galton-Watson'' (Markov processes)

Proposition 1(ii) p.126 of 713 is true only in the case $q=0$

Keywords: Branching processes

Nature: Correction

Retrieve article from Numdam