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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)
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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