Séminaire de probabilités de Strasbourg - Tome IX

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