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 measuresNature: Exposition,
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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 processesNature: Original Retrieve article from Numdam
