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IV: 11, 132-132, LNM 124 (1970)

**SAM LAZARO, José de**

Théorème de Stone et espérances conditionnelles (Ergodic theory)

It is shown that the spectral projections of the unitary group arising from a group of measure preserving transformations must be complex operators, and in particular cannot be conditional expectations

Comment: This remark arose from the work on flows in Sam Lazaro-Meyer,*Z. für W-theorie,* **18**, 1971

Keywords: Flows, Spectral representation

Nature: Original

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V: 23, 237-250, LNM 191 (1971)

**MEYER, Paul-André**

Travaux de H. Rost en théorie du balayage (Potential theory, Ergodic theory)

The ``filling scheme'' is a technique used in ergodic theory to prove Hopf's maximal Lemma and the Chacon-Ornstein theorem, studied in detail by H.~Rost (*Zeit. für W-theorie,* **15**, 1970; *Ann. Inst. Fourier,* **21**, 1971): it provides a solution to Skorohod's imbedding problem for measures on discrete time Markov processes. Here it is also used to prove Brunel's Lemma in pointwise ergodic theory

Comment: Extension to continuous time in Meyer 612. See also 806, 1012. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Filling scheme, Brunel's lemma, Skorohod imbedding

Nature: Exposition, Original additions

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V: 27, 278-282, LNM 191 (1971)

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

Une remarque sur le flot du mouvement brownien (Brownian motion, Ergodic theory)

It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense

Comment: See Sam Lazaro-Meyer,*Z. für W-theorie,* **18**, 1971

Keywords: Brownian motion, Wiener chaos, Screw-lines

Nature: Original

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VI: 12, 130-150, LNM 258 (1972)

**MEYER, Paul-André**

Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)

The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (*Invent. Math.,* **14**, 1971) the construction is extended to continuous time Markov processes. In the transient case, the results are translated in potential-theoretic language, and proved using techniques due to Mokobodzki. Then the general case follows from this result applied to a space-time extension of the semi-group

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Filling scheme, Balayage of measures, Skorohod imbedding

Nature: Exposition, Original additions

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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: 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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X: 31, 578-578, LNM 511 (1976)

**MEYER, Paul-André**

Un point de priorité (Ergodic theory)

An important remark in Sam Lazaro-Meyer 901, on the relation between Palm measures and the Ambrose-Kakutani theorem that any flow can be interpreted as a flow under a function, was made earlier by F.~Papangelou (1970)

Keywords: Palm measures, Flow under a function, Ambrose-Kakutani theorem

Nature: Acknowledgment

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XI: 02, 21-26, LNM 581 (1977)

**BENVENISTE, Albert**

Application d'un théorème de G. Mokobodzki à la théorie des flots (Ergodic theory, General theory of processes)

The purpose of this paper is to extend to the theory of filtered flows (for which see 901 and 902) the dual version of the general theory of processes due to Azéma (for which see 814 and 937), in particular the association with any measurable process of suitable projections which are homogeneous processes. An important difference here is the fact that the time set is the whole line. Here the class of measurable processes which can be projected is reduced to a (not very explicit) class, and a commutation theorem similar to Azéma's is proved. The proof uses the technique of*medial limits * due to Mokobodzki (see 719), which in fact was developed precisely at the author's request to solve this problem

Keywords: Filtered flows, Stationary processes, Projection theorems, Medial limits

Nature: Original

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XI: 11, 120-131, LNM 581 (1977)

**MEYER, Paul-André**

Résultats récents de A. Benveniste en théorie des flots (Ergodic theory)

A filtered flow is said to be diffuse if there exists a r.v. ${\cal F}_0$-measurable $J$ such that given any ${\cal F}_0$-measurable r.v.'s $T$ and $H$, $P\{J\circ\theta_T=H, 0<T<\infty\}=0$. The main result of the paper is the fact that a diffuse flow contains all Lévy flows (flows of increments of Lévy processes, no invariant measure is involved). In particular, the Brownian flow contains a Poisson counter

Comment: This result on the whole line is similar to 1106, which concerns a half-line. The original paper of Benveniste appeared in*Z. für W-theorie,* **41**, 1977/78

Keywords: Filtered flows, Poisson flow

Nature: Exposition

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XIV: 46, 475-488, LNM 784 (1980)

**WEBER, Michel**

Sur un théorème de Maruyama (Gaussian processes, Ergodic theory)

Given a stationary centered Gaussian process $X$ with spectral measure $\mu$, a new proof is given of the fact that if $\mu$ is continuous, the flow of $X$ is weakly mixing

Keywords: Stationary processes

Nature: Original

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Théorème de Stone et espérances conditionnelles (Ergodic theory)

It is shown that the spectral projections of the unitary group arising from a group of measure preserving transformations must be complex operators, and in particular cannot be conditional expectations

Comment: This remark arose from the work on flows in Sam Lazaro-Meyer,

Keywords: Flows, Spectral representation

Nature: Original

Retrieve article from Numdam

V: 23, 237-250, LNM 191 (1971)

Travaux de H. Rost en théorie du balayage (Potential theory, Ergodic theory)

The ``filling scheme'' is a technique used in ergodic theory to prove Hopf's maximal Lemma and the Chacon-Ornstein theorem, studied in detail by H.~Rost (

Comment: Extension to continuous time in Meyer 612. See also 806, 1012. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Filling scheme, Brunel's lemma, Skorohod imbedding

Nature: Exposition, Original additions

Retrieve article from Numdam

V: 27, 278-282, LNM 191 (1971)

Une remarque sur le flot du mouvement brownien (Brownian motion, Ergodic theory)

It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense

Comment: See Sam Lazaro-Meyer,

Keywords: Brownian motion, Wiener chaos, Screw-lines

Nature: Original

Retrieve article from Numdam

VI: 12, 130-150, LNM 258 (1972)

Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)

The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Filling scheme, Balayage of measures, Skorohod imbedding

Nature: Exposition, Original additions

Retrieve article from Numdam

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

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

Retrieve article from Numdam

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

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

X: 31, 578-578, LNM 511 (1976)

Un point de priorité (Ergodic theory)

An important remark in Sam Lazaro-Meyer 901, on the relation between Palm measures and the Ambrose-Kakutani theorem that any flow can be interpreted as a flow under a function, was made earlier by F.~Papangelou (1970)

Keywords: Palm measures, Flow under a function, Ambrose-Kakutani theorem

Nature: Acknowledgment

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XI: 02, 21-26, LNM 581 (1977)

Application d'un théorème de G. Mokobodzki à la théorie des flots (Ergodic theory, General theory of processes)

The purpose of this paper is to extend to the theory of filtered flows (for which see 901 and 902) the dual version of the general theory of processes due to Azéma (for which see 814 and 937), in particular the association with any measurable process of suitable projections which are homogeneous processes. An important difference here is the fact that the time set is the whole line. Here the class of measurable processes which can be projected is reduced to a (not very explicit) class, and a commutation theorem similar to Azéma's is proved. The proof uses the technique of

Keywords: Filtered flows, Stationary processes, Projection theorems, Medial limits

Nature: Original

Retrieve article from Numdam

XI: 11, 120-131, LNM 581 (1977)

Résultats récents de A. Benveniste en théorie des flots (Ergodic theory)

A filtered flow is said to be diffuse if there exists a r.v. ${\cal F}_0$-measurable $J$ such that given any ${\cal F}_0$-measurable r.v.'s $T$ and $H$, $P\{J\circ\theta_T=H, 0<T<\infty\}=0$. The main result of the paper is the fact that a diffuse flow contains all Lévy flows (flows of increments of Lévy processes, no invariant measure is involved). In particular, the Brownian flow contains a Poisson counter

Comment: This result on the whole line is similar to 1106, which concerns a half-line. The original paper of Benveniste appeared in

Keywords: Filtered flows, Poisson flow

Nature: Exposition

Retrieve article from Numdam

XIV: 46, 475-488, LNM 784 (1980)

Sur un théorème de Maruyama (Gaussian processes, Ergodic theory)

Given a stationary centered Gaussian process $X$ with spectral measure $\mu$, a new proof is given of the fact that if $\mu$ is continuous, the flow of $X$ is weakly mixing

Keywords: Stationary processes

Nature: Original

Retrieve article from Numdam