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I: 06, 72-162, LNM 39 (1967)

**MEYER, Paul-André**

Intégrales stochastiques I--IV (4 talks) (Martingale theory, Stochastic calculus)

This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (*Nagoya Math. J.* **30**, 1967) on square integrable martingales. The filtration is assumed to be free from fixed times of discontinuity, a restriction lifted in the modern theory. A new feature is the definition of the second increasing process associated with a square integrable martingale (a ``square bracket'' in the modern terminology). In the second talk, stochastic integrals are defined with respect to local martingales (introduced from Ito-Watanabe, *Ann. Inst. Fourier,* **15**, 1965), and the general integration by parts formula is proved. Also a restricted class of semimartingales is defined and an ``Ito formula'' for change of variables is given, different from that of Kunita-Watanabe. The third talk contains the famous Kunita-Watanabe theorem giving the structure of martingale additive functionals of a Hunt process, and a new proof of Lévy's description of the structure of processes with independent increments (in the time homogeneous case). The fourth talk deals mostly with Lévy systems (Motoo-Watanabe, *J. Math. Kyoto Univ.*, **4**, 1965; Watanabe, *Japanese J. Math.*, **36**, 1964)

Comment: This paper was a step in the development of stochastic integration. Practically every detail of it has been reworked since, starting with Doléans-Dade-Meyer 409. Note a few corrections in Meyer 312

Keywords: Square integrable martingales, Angle bracket, Stochastic integrals

Nature: Exposition, Original additions

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III: 15, 190-229, LNM 88 (1969)

**MORANDO, Philippe**

Mesures aléatoires (Independent increments)

This paper consists of two talks, on the construction and structure of measures with independent values on an abstract measurable space, inspired by papers of Prekopa (*Acta Math. Acad. Sci. Hung.,* **7**, 1956 and **8**, 1957) and Kingman (*Pacific J. Math.,* **21**, 1967)

Comment: If the measurable space is not ``too'' abstract, it can be imbedded into the line, and the standard theory of Lévy processes (non-homogeneous) can be used. This simple remark reduces the interest of the general treatment: see Dellacherie-Meyer,*Probabilités et potentiel,* Chapter XIII, end of \S4

Keywords: Random measures, Independent increments

Nature: Exposition, Original additions

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IV: 03, 37-46, LNM 124 (1970)

**CHERSI, Franco**

Martingales et intégrabilité de $X\log^+X$ d'après Gundy (Martingale theory)

Gundy's result (*Studia Math.*, **33**, 1968) is a converse to Doob's inequality: for a positive martingale such that $X_n\leq cX_{n-1}$, the integrability of $\sup_n X_n$ implies boundedness in $L\log^+L$. All martingales satisfy this condition on regular filtrations

Comment: The integrability of $\sup_n |\,X_n\,|$ has become now the $H^1$ theory of martingales

Keywords: Inequalities, Regular martingales

Nature: Exposition, Original additions

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V: 03, 21-36, LNM 191 (1971)

**BRETAGNOLLE, Jean**

Résultats de Kesten sur les processus à accroissements indépendants (Markov processes, Independent increments)

The question is to find all Lévy processes for which single points are polar. Kesten's answer (*Mem. Amer. Math. Soc.*, **93**, 1969) is almost complete and in particular proves Chung's conjecture. The proofs in this paper have been considerably reworked

Comment: See also 502 in the same volume

Keywords: Subordinators, Polar sets

Nature: Exposition, Original additions

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V: 17, 177-190, LNM 191 (1971)

**MEYER, Paul-André**

Processus de Poisson ponctuels d'après K. Ito (Markov processes, Point processes)

Presents (a preliminary form of) the celebrated paper of Ito (*Proc. Sixth Berkeley Symposium,* **3**, 1972) on excursion theory, with an extension (the use of possibly unbounded entrance laws instead of initial measures) which has become part of the now classical theory

Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form

Keywords: Poisson point processes, Excursions, Local times

Nature: Exposition, Original additions

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V: 18, 191-195, LNM 191 (1971)

**MEYER, Paul-André**

Démonstration simplifiée d'un théorème de Knight (Martingale theory)

A well known theorem (Dambis, Dubins) asserts that a continuous martingale reduces to Brownian motion when time-changed by its own increasing process. Knight's theorem (LN in M**190**) asserts that this operation performed on $n$ orthogonal martingales yields $n$ independent Brownian motions. The result is extended to Poisson processes

Comment: Still simpler proofs can be given, see 1448 (included in Revuz-Yor*Continuous Martingales and Brownian Motion,* Chapter V)

Keywords: Continuous martingales, Changes of time

Nature: Exposition, Original additions

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V: 22, 213-236, LNM 191 (1971)

**MEYER, Paul-André**

Le retournement du temps, d'après Chung et Walsh (Markov processes)

The paper of Chung and Walsh (*Acta Math.*, **134**, 1970) proved that any right continuous strong Markov process had a reversed left continuous moderate Markov process at any $L$-time, with a suitably constructed dual semigroup. Appendix 1 gives a useful characterization of càdlàg processes using stopping times (connected with amarts). Appendix 2 proves (following Mokobodzki) that any excessive function strongly dominated by a potential of function is such a potential

Comment: The theorem of Chung-Walsh remains the deepest on time reversal (to be supplemented by the consideration of Kuznetsov's measures)

Keywords: Time reversal, Dual semigroups

Nature: Exposition, Original additions

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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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VI: 11, 118-129, LNM 258 (1972)

**MEYER, Paul-André**

La mesure de H. Föllmer en théorie des surmartingales (Martingale theory)

The Föllmer measure of a supermartingale is an extension to very general situation of the construction of $h$-path processes in the Markovian case. Let $\Omega$ be a probability space with a filtration, let $\Omega'$ be the product space $[0,\infty]\times\Omega$, the added coordinate playing the role of a lifetime $\zeta$. Then the Föllmer measure associated with a supermartingale $(X_t)$ is a measure $\mu$ on this enlarged space which satisfies the property $\mu(]T,\infty])=E(X_T)$ for any stopping time $T$, and simple additional properties to ensure uniqueness. When $X_t$ is a class (D) potential, it turns out to be the usual Doléans measure, but except in this case its existence requires some measure theoretic conditions on $\Omega$; which are slightly different here from those used by Föllmer,*Zeit für X-theorie,* **21**, 1970

Keywords: Supermartingales, Föllmer measures

Nature: Exposition, Original additions

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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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VII: 08, 58-60, LNM 321 (1973)

**DELLACHERIE, Claude**

Potentiels de fonctionnelles additives. Un contre-exemple de Knight (Markov processes)

An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point

Keywords: Additive functionals

Nature: Exposition, Original additions

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VII: 13, 122-135, LNM 321 (1973)

**KHALILI-FRANÇON, Elisabeth**

Processus de Galton-Watson (Markov processes)

This paper is mostly a survey of previous results with comments and some alternative proofs

Comment: An erroneous statement is corrected in 939

Keywords: Branching processes, Galton-Watson processes

Nature: Exposition, Original additions

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VII: 22, 217-222, LNM 321 (1973)

**MEYER, Paul-André**

Sur les désintégrations régulières de L. Schwartz (General theory of processes)

This paper presents a small part of an important article of L.~Schwartz (*J. Anal. Math.*, **26**, 1973). The main result is an extension of the classical theorem on the existence of regular conditional probability distributions: on a good filtered probability space, the previsible and optional projections of a process can be computed by means of true kernels

Comment: The contents of this paper have been considerably developed by F.~Knight in his theory of prediction. See 1007

Keywords: Previsible projections, Optional projections, Prediction theory

Nature: Exposition, Original additions

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VIII: 09, 80-133, LNM 381 (1974)

**GEBUHRER, Marc Olivier**

Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)

The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (*Arkiv för Math.*, **6**, 1965-67), which is studied as a Lorentz invariant diffusion process (in the usual sense) on the standard hyperboloid of velocities in special relativity, on which the Lorentz group acts. The Brownian paths themselves are constructed by integration and possess a speed smaller than the velocity of light but no higher derivatives. The second part studies more generally invariant Markov processes on a Riemannian symmetric space of non-compact type, their generators and the corresponding semigroups

Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces

Nature: Exposition, Original additions

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VIII: 14, 262-288, LNM 381 (1974)

**MEYER, Paul-André**

Les travaux d'Azéma sur le retournement du temps (General theory of processes, Markov processes)

This paper is an exposition of a paper by Azéma (*Ann. Sci. ENS,* **6**, 1973) in which the theory ``dual'' to the general theory of processes was developed. It is shown first how the general theory itself can be developed from a family of killing operators, and then how the dual theory follows from a family of shift operators $\theta_t$. A transience hypothesis involving the existence of many ``return times'' permits the construction of a theory completely similar to the usual one. Then some of Azéma's applications to the theory of Markov processes are given, particularly the representation of a measure not charging $\mu$-polar sets as expectation under the initial measure $\mu$ of a left additive functional

Comment: This paper follows (with considerable progress) the line of 602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer,*Probabilités et Potentiel,* Chapter XVIII, 1992

Keywords: Time reversal, Shift operators, Killing operators, Cooptional processes, Coprevisible processes, Additive functionals, Left additive functionals

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: 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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X: 07, 86-103, LNM 511 (1976)

**MEYER, Paul-André**

La théorie de la prédiction de F. Knight (General theory of processes)

This paper is devoted to the work of Knight,*Ann. Prob.* **3**, 1975, the main idea of which is to associate with every reasonable process $(X_t)$ another process, taking values in a space of probability measures, and whose value at time $t$ is a conditional distribution of the future of $X$ after $t$ given its past before $t$. It is shown that the prediction process contains essentially the same information as the original process (which can be recovered from it), and that it is a time-homogeneous Markov process

Comment: The results are related to those of Schwartz (presented in 722), the main difference being that the future is predicted instead of the whole path. Knight has devoted to this subject the*Essays on the Prediction Process,* Hayward Inst. of Math. Stat., 1981, and a book, *Foundations of the Prediction Process,* Oxford Science Publ. 1992

Keywords: Prediction theory

Nature: Exposition, Original additions

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X: 17, 245-400, LNM 511 (1976)

**MEYER, Paul-André**

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

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X: 25, 521-531, LNM 511 (1976)

**BENVENISTE, Albert**

Séparabilité optionnelle, d'après Doob (General theory of processes)

A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given

Comment: Doob's original paper appeared in*Ann. Inst. Fourier,* **25**, 1975. See also 1105

Keywords: Optional processes, Separability, Section theorems

Nature: Exposition, Original additions

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XI: 04, 34-46, LNM 581 (1977)

**DELLACHERIE, Claude**

Les dérivations en théorie descriptive des ensembles et le théorème de la borne (Descriptive set theory)

At the root of set theory lies Cantor's definition of the ``derived set'' $\delta A$ of a closed set $A$, i.e., the set of its non-isolated points, with the help of which Cantor proved that a closed set can be decomposed into a perfect set and a countable set. One may define the index $j(A)$ to be the smallest ordinal $\alpha$ such that $\delta^\alpha A=\emptyset$, or $\omega_1$ if there is no such ordinal. Considering the set $F$ of all closed sets as a (Polish) topological space, ordered by inclusion, $\delta$ as an increasing mapping from $F$ such that $\delta A\subset A$, let $D$ be the set of all $A$ such that $j(A)<\omega_1$ (thus, the set of all countable closed sets). Then $D$ is coanalytic and non-Borel, while the index is bounded by a countable ordinal on every analytic subset of $D$. These powerful results are stated abstractly and proved under very general conditions. Several examples are given

Comment: See a correction in 1241, and several examples in Hillard 1242. The whole subject has been exposed anew in Chapter~XXIV of Dellacherie-Meyer,*Probabilités et Potentiel*

Keywords: Derivations (set-theoretic), Kunen-Martin theorem

Nature: Exposition, Original additions

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XI: 12, 132-195, LNM 581 (1977)

**MEYER, Paul-André**

Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)

This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(**R**^n)$ is $BMO$, using methods adapted from the probabilistic Littlewood-Paley theory (of which this is a kind of limiting case). Some details of the proof are interesting in their own right

Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected

Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms

Nature: Exposition, Original additions

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XI: 13, 196-256, LNM 581 (1977)

**WEBER, Michel**

Classes uniformes de processus gaussiens stationnaires (Gaussian processes)

To be completed

Comment: See the long and interesting review by Berman in*Math. Reviews,* **56**, **13343**

Nature: Exposition, Original additions

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XI: 24, 376-382, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Équations différentielles stochastiques (Stochastic calculus)

This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in*Zeit. für W-theorie,* **36**, 1976 and by Protter in *Ann. Prob.* **5**, 1977. The theory has become now so classical that the paper has only historical interest

Keywords: Stochastic differential equations, Semimartingales

Nature: Exposition, Original additions

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XII: 28, 411-423, LNM 649 (1978)

**MEYER, Paul-André**

Convergence faible et compacité des temps d'arrêt, d'après Baxter et Chacón (General theory of processes)

Baxter and Chacón (*Zeit. für W-theorie,* 40, 1977) introduced a topology on the sets of ``fuzzy'' times and of fuzzy stopping times which turn these sets into compact metrizable spaces---a fuzzy r.v. $T$ is a right continuous decreasing process $M_t$ with $M_{0-}=1$, $M_t(\omega)$ being interpreted for each $\omega$ as the distribution function $P_{\omega}\{T>t\}$. When this process is adapted the fuzzy r.v. is a fuzzy stopping time. A number of properties of this topology are investigated

Comment: See 1536 for an extension to Polish spaces

Keywords: Stopping times, Fuzzy stopping times

Nature: Exposition, Original additions

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XII: 58, 770-774, LNM 649 (1978)

**MEYER, Paul-André**

Sur le lemme de La Vallée Poussin et un théorème de Bismut (Measure theory, General theory of processes)

Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces

Keywords: Uniform integrability, Class (D) processes, Moderate convex functions

Nature: Exposition, Original additions

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XIII: 15, 199-203, LNM 721 (1979)

**MEYER, Paul-André**

Une remarque sur le calcul stochastique dépendant d'un paramètre (General theory of processes)

Call a ``process'' a measurable function $X(u,t,\omega)$ where $t$ and $\omega$ are as usual and $u$ is a parameter ranging over some nice measurable space ${\cal U}$. Say that $X$ is evanescent if $X(.\,,\,.\,,\omega)\equiv0$ for a.a. $\omega$. The problem is to define previsible processes, and previsible projections defined up to evanescent sets. This is achieved following Jacod,*Zeit. für W-Theorie,* **31**, 1975. The main feature is the corresponding use of random measures, previsible random measures, and previsible dual projections

Keywords: Processes depending on a parameter, Previsible processes, Previsible projections, Random measures

Nature: Exposition, Original additions

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XIII: 21, 240-249, LNM 721 (1979)

**MEYER, Paul-André**

Représentations multiplicatives de sousmartingales, d'après Azéma (Martingale theory)

The problem of the multiplicative decomposition of a positive supermartingale $Y$ is relatively easy, but the similar problem for a positive submartingale (find a previsible decreasing process $(C_t)$ such that $(C_tY_t)$ is a martingale) is plagued by the zeros of $Y$. An important idea of Azéma (*Zeit. für W-Theorie,* **45**, 1978) is the introduction of a *multiplicative system * as a two-parameter process $C_{st}$ taking values in $[0,1]$, defined for $s\le t$, such that $C_{tt}=1$, $C_{st}C_{tu}=C_{su}$ for $s\le t\le u$, decreasing and previsible in $t$ for fixed $s$, such that $E[C_{st}Y_t\,|\,{\cal F}_s]=Y_s$ for $s<t$. Then for fixed $s$ the process $(C_{st}Y_t)$ turns out to be a right-continuous martingale on $[s,\infty[$, and what we have done amounts to pasting together all the multiplicative decompositions on zero-free intervals. Existence (and uniqueness of multiplicative systems are proved, though the uniqueness result is slightly different from Azéma's

Keywords: Multiplicative decomposition

Nature: Exposition, Original additions

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XIII: 28, 313-331, LNM 721 (1979)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Inégalités de normes avec poids (Martingale theory)

See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory

Comment: An exponent $1/\lambda$ is missing in formula (4), p.315

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

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XIII: 58, 642-645, LNM 721 (1979)

**MAISONNEUVE, Bernard**

Martingales de valeur absolue donnée, d'après Protter-Sharpe (Martingale theory)

The main difficulty of Gilat's theorem (every positive submartingale $X$ can be interpreted as the absolute value of a martingale, in a suitably enlarged filtration) is due to the zeros of $X$. In the strictly positive case a simple proof was given by Protter and Sharpe (*Ann. Prob.*, **7**, 1979). This proof is further simplified and slightly generalized

Comment: See also 1407

Keywords: Gilat's theorem

Nature: Exposition, Original additions

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XIV: 09, 102-103, LNM 784 (1980)

**MEYER, Paul-André**

Sur un résultat de L. Schwartz (Martingale theory)

the following definition of a semimartingale $X$ in a random open set $A$ is due to L. Schwartz (*Semimartingales dans les variétés...*, Lecture Notes in M. **780**): $A$ can be represented as a countable union of random open sets $A_n$, and for each $n$ there exists an ordinary semimartingale $Y_n$ such $X=Y_n$ on $A_n$. It is shown that if $K\subset A$ is a compact optional set, then there exists an ordinary semimartingale $Y$ such that $X=Y$ on $K$

Comment: The results are extended in Meyer-Stricker*Stochastic Analysis and Applications, part B,* *Advances in M. Supplementary Studies,* 1981

Keywords: Semimartingales in a random open set

Nature: Exposition, Original additions

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XIV: 15, 128-139, LNM 784 (1980)

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

Sur l'intégrale stochastique de processus prévisibles non bornés (Stochastic calculus)

The standard theory of stochastic integration deals with locally bounded previsible processes. The natural definition of the stochastic integral $H.X$ of a previsible process $H$ w.r.t. a semimartingale $X$ consists in assuming the existence of some decomposition $X=M+A$ such that $H.M$ exists in the martingale sense, and $H.A$ in the Stieltjes sense, and then defining $H.X$ as their sum. This turns out to be a very awkward definition. It is shown here to be equivalent to the following one: truncating $H$ at $n$, the standard stochastic integrals $H_n.X$ converge in the topology of semimartingales. This is clearly invariant under changes of law. A counterexample shows that integrability may be lost if the filtration is enlarged

Comment: See also 1417. This is a synthesis of earlier work, much of which is due to Jacod,*Calcul Stochastique et Problèmes de Martingales,* Lect. Notes in M. 714. The contents of this paper appeared in book form in Dellacherie-Meyer, *Probabilités et Potentiel B,* Chap. VIII, \S3. An equivalent definition is given by L. Schwartz in 1530, using the idea of ``formal semimartingales''. For further steps in the same direction, see Stricker 1533

Keywords: Stochastic integrals

Nature: Exposition, Original additions

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XIV: 42, 397-409, LNM 784 (1980)

**GETOOR, Ronald K.**

Transience and recurrence of Markov processes (Markov processes)

From the introduction: The purpose of this paper is to present an elementary exposition of some various conditions that have been used to define transience or recurrence of a Markov process... an elementary and unified discussion of these ideas may be worthwhile

Keywords: Recurrent Markov processes

Nature: Exposition, Original additions

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XV: 06, 103-117, LNM 850 (1981)

**MEYER, Paul-André**

Flot d'une équation différentielle stochastique (Stochastic calculus)

Malliavin showed very neatly how an (Ito) stochastic differential equation on $**R**^n$ with $C^{\infty}$ coefficients, driven by Brownian motion, generates a flow of diffeomorphisms. This consists of three results: smoothness of the solution as a function of its initial point, showing that the mapping is 1--1, and showing that it is onto. The last point is the most delicate. Here the results are extended to stochastic differential equations on $**R**^n$ driven by continuous semimartingales, and only partially to the case of semimartingales with jumps. The essential argument is borrowed from Kunita and Varadhan (see Kunita's talk in the Proceedings of the Durham Symposium on SDE's, LN 851)

Comment: The results on semimartingales with jumps have been proved independently by Uppman. Some dust has been swept under the rugs about the non-explosion of the solution, and the results should be considered valid only in the globally Lipschitz case. See also Uppman 1624 and Léandre 1922

Keywords: Stochastic differential equations, Flow of a s.d.e.

Nature: Exposition, Original additions

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XV: 19, 278-284, LNM 850 (1981)

**ÉMERY, Michel**

Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)

Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential

Keywords: $BMO$

Nature: Exposition, Original additions

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XVI: 06, 95-132, LNM 920 (1982)

**MEYER, Paul-André**

Note sur les processus d'Ornstein-Uhlenbeck (Malliavin's calculus)

With every Gaussian measure $\mu$ one can associate an Ornstein-Uhlenbeck semigroup, for which $\mu$ is a reversible invariant measure. When $\mu$ is Wiener's measure on ${\cal C}(**R**)$, this semigroup is a fundamental tool in Malliavin's own approach to the ``Malliavin calculus''. See for instance Stroock's exposition of it in *Math. Systems Theory,* **13**, 1981. With this semigroup one can associate its generator $L$ which plays the role of the classical Laplacian, and the positive bilinear functional $\Gamma(f,g)= L(fg)-fLg-gLf$---leaving aside domain problems for simplicity---sometimes called ``carré du champ'', which plays the role of the squared classical gradient. As in classical analysis, one can define it as $\sum_i \nabla_i f\nabla i g$, the derivatives being relative to an orthonormal basis of the Cameron-Martin space. We may define Sobolev-like spaces of order one in two ways: either by the fact that $Cf$ belongs to $L^p$, where $C=-\sqrt{-L}$ is the ``Cauchy generator'', or by the fact that $\sqrt{\Gamma(f,f)}$ belongs to $L^p$. A result which greatly simplifies the analytical part of the ``Malliavin calculus'' is the fact that both definitions are equivalent. This is the main topic of the paper, and its proof uses the Littlewood-Paley-Stein theory for semigroups as presented in 1010, 1510

Comment: An important problem is the extension to higher order Sobolev-like spaces. For instance, we could define the Sobolev space of order 2 either by the fact that $C^2f=-Lf$ belongs to $L^p$, and on the other hand define $\Gamma_2(f,g)=\sum_{ij} \nabla_i\nabla_j f \nabla_i\nabla_j g$ (derivatives of order 2) and ask that $\sqrt{\Gamma_2(f,f)}\in L^p$. For the equivalence of these two definitions and general higher order ones, see 1816, which anyhow contains many improvements over 1606. Also, proofs of these results have been given which do not involve Littlewood-Paley methods. For instance, Pisier has a proof which only uses the boundedness in $L^p$ of classical Riesz transforms.\par Another trend of research has been the correct definition of ``higher gradients'' within semigroup theory (the preceding definition of $\Gamma_2(f,g)$ makes use of the Gaussian structure). Bakry investigated the fundamental role of ``true'' $\Gamma_2$, the bilinear form $\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(Lf,g)$, which is positive in the case of the Ornstein-Uhlenbeck semigroup but is not always so. See 1909, 1910, 1912

Keywords: Ornstein-Uhlenbeck process, Gaussian measures, Littlewood-Paley theory, Hypercontractivity, Hermite polynomials, Riesz transforms, Test functions

Nature: Exposition, Original additions

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XVI: 10, 151-152, LNM 920 (1982)

**MEYER, Paul-André**

Sur une inégalité de Stein (Applications of martingale theory)

In his book*Topics in harmonic analysis related to the Littlewood-Paley theory * (1970) Stein uses interpolation between two results, one of which is a discrete martingale inequality deduced from the Burkholder inequalities, whose precise statement we omit. This note states and proves directly the continuous time analogue of this inequality---a mere exercise in translation

Keywords: Littlewood-Paley theory, Martingale inequalities

Nature: Exposition, Original additions

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XVI: 24, 268-284, LNM 920 (1982)

**UPPMAN, Are**

Sur le flot d'une équation différentielle stochastique (Stochastic calculus)

This paper is a companion to 1506, devoted to the main results on the flow of a (Lipschitz) stochastic differential equation driven by continous semimartingales: non-confluence of solutions from different initial points, surjectivity of the mapping, smooth dependence on the initial conditions. The proofs have been greatly simplified

Keywords: Stochastic differential equations, Flow of a s.d.e., Injectivity

Nature: Exposition, Original additions

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XVII: 19, 185-186, LNM 986 (1983)

**ÉMERY, Michel**

Note sur l'exposé précédent (Stochastic calculus)

A small remark on 1718: The event where a semimartingale converges perfectly is also the smallest (modulo negligibility) event where it is a semimartingale up to infinity

Keywords: Semimartingales

Nature: Original additions

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XXIV: 30, 448-452, LNM 1426 (1990)

**ÉMERY, Michel**; **LÉANDRE, Rémi**

Sur une formule de Bismut (Markov processes, Stochastic differential geometry)

This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group

Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes

Nature: Exposition, Original additions

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XXXII: 19, 264-305, LNM 1686 (1998)

**BARLOW, Martin T.**; **ÉMERY, Michel**; **KNIGHT, Frank B.**; **SONG, Shiqi**; **YOR, Marc**

Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)

Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA**7**, 1997). Using his methods, the result is extended to spider martingales. A conjecture of M. Barlow is also proved: if $L$ is an honest time in a (possibly multidimensional) Brownian filtration, then ${\cal F}_{L+}$ is generated by ${\cal F}_{L}$ and at most one event. Last, it is shown that a Walsh's Brownian motion can live in the filtration generated by another Walsh's Brownian motion only if the former is obtained from the latter by aggregating rays

Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,*Astérisque* **282** (2002). A simplified proof of Barlow's conjecture is given in 3304. For more on Théorème 1 (Slutsky's lemma), see 3221 and 3325

Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma

Nature: New exposition of known results, Original additions

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Intégrales stochastiques I--IV (4 talks) (Martingale theory, Stochastic calculus)

This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (

Comment: This paper was a step in the development of stochastic integration. Practically every detail of it has been reworked since, starting with Doléans-Dade-Meyer 409. Note a few corrections in Meyer 312

Keywords: Square integrable martingales, Angle bracket, Stochastic integrals

Nature: Exposition, Original additions

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III: 15, 190-229, LNM 88 (1969)

Mesures aléatoires (Independent increments)

This paper consists of two talks, on the construction and structure of measures with independent values on an abstract measurable space, inspired by papers of Prekopa (

Comment: If the measurable space is not ``too'' abstract, it can be imbedded into the line, and the standard theory of Lévy processes (non-homogeneous) can be used. This simple remark reduces the interest of the general treatment: see Dellacherie-Meyer,

Keywords: Random measures, Independent increments

Nature: Exposition, Original additions

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IV: 03, 37-46, LNM 124 (1970)

Martingales et intégrabilité de $X\log^+X$ d'après Gundy (Martingale theory)

Gundy's result (

Comment: The integrability of $\sup_n |\,X_n\,|$ has become now the $H^1$ theory of martingales

Keywords: Inequalities, Regular martingales

Nature: Exposition, Original additions

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V: 03, 21-36, LNM 191 (1971)

Résultats de Kesten sur les processus à accroissements indépendants (Markov processes, Independent increments)

The question is to find all Lévy processes for which single points are polar. Kesten's answer (

Comment: See also 502 in the same volume

Keywords: Subordinators, Polar sets

Nature: Exposition, Original additions

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V: 17, 177-190, LNM 191 (1971)

Processus de Poisson ponctuels d'après K. Ito (Markov processes, Point processes)

Presents (a preliminary form of) the celebrated paper of Ito (

Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form

Keywords: Poisson point processes, Excursions, Local times

Nature: Exposition, Original additions

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V: 18, 191-195, LNM 191 (1971)

Démonstration simplifiée d'un théorème de Knight (Martingale theory)

A well known theorem (Dambis, Dubins) asserts that a continuous martingale reduces to Brownian motion when time-changed by its own increasing process. Knight's theorem (LN in M

Comment: Still simpler proofs can be given, see 1448 (included in Revuz-Yor

Keywords: Continuous martingales, Changes of time

Nature: Exposition, Original additions

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V: 22, 213-236, LNM 191 (1971)

Le retournement du temps, d'après Chung et Walsh (Markov processes)

The paper of Chung and Walsh (

Comment: The theorem of Chung-Walsh remains the deepest on time reversal (to be supplemented by the consideration of Kuznetsov's measures)

Keywords: Time reversal, Dual semigroups

Nature: Exposition, Original additions

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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

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VI: 11, 118-129, LNM 258 (1972)

La mesure de H. Föllmer en théorie des surmartingales (Martingale theory)

The Föllmer measure of a supermartingale is an extension to very general situation of the construction of $h$-path processes in the Markovian case. Let $\Omega$ be a probability space with a filtration, let $\Omega'$ be the product space $[0,\infty]\times\Omega$, the added coordinate playing the role of a lifetime $\zeta$. Then the Föllmer measure associated with a supermartingale $(X_t)$ is a measure $\mu$ on this enlarged space which satisfies the property $\mu(]T,\infty])=E(X_T)$ for any stopping time $T$, and simple additional properties to ensure uniqueness. When $X_t$ is a class (D) potential, it turns out to be the usual Doléans measure, but except in this case its existence requires some measure theoretic conditions on $\Omega$; which are slightly different here from those used by Föllmer,

Keywords: Supermartingales, Föllmer measures

Nature: Exposition, Original additions

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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

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VII: 08, 58-60, LNM 321 (1973)

Potentiels de fonctionnelles additives. Un contre-exemple de Knight (Markov processes)

An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point

Keywords: Additive functionals

Nature: Exposition, Original additions

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VII: 13, 122-135, LNM 321 (1973)

Processus de Galton-Watson (Markov processes)

This paper is mostly a survey of previous results with comments and some alternative proofs

Comment: An erroneous statement is corrected in 939

Keywords: Branching processes, Galton-Watson processes

Nature: Exposition, Original additions

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VII: 22, 217-222, LNM 321 (1973)

Sur les désintégrations régulières de L. Schwartz (General theory of processes)

This paper presents a small part of an important article of L.~Schwartz (

Comment: The contents of this paper have been considerably developed by F.~Knight in his theory of prediction. See 1007

Keywords: Previsible projections, Optional projections, Prediction theory

Nature: Exposition, Original additions

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VIII: 09, 80-133, LNM 381 (1974)

Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)

The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (

Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces

Nature: Exposition, Original additions

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VIII: 14, 262-288, LNM 381 (1974)

Les travaux d'Azéma sur le retournement du temps (General theory of processes, Markov processes)

This paper is an exposition of a paper by Azéma (

Comment: This paper follows (with considerable progress) the line of 602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer,

Keywords: Time reversal, Shift operators, Killing operators, Cooptional processes, Coprevisible processes, Additive functionals, Left additive functionals

Nature: Exposition, Original additions

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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

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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

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X: 07, 86-103, LNM 511 (1976)

La théorie de la prédiction de F. Knight (General theory of processes)

This paper is devoted to the work of Knight,

Comment: The results are related to those of Schwartz (presented in 722), the main difference being that the future is predicted instead of the whole path. Knight has devoted to this subject the

Keywords: Prediction theory

Nature: Exposition, Original additions

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X: 17, 245-400, LNM 511 (1976)

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

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X: 25, 521-531, LNM 511 (1976)

Séparabilité optionnelle, d'après Doob (General theory of processes)

A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given

Comment: Doob's original paper appeared in

Keywords: Optional processes, Separability, Section theorems

Nature: Exposition, Original additions

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XI: 04, 34-46, LNM 581 (1977)

Les dérivations en théorie descriptive des ensembles et le théorème de la borne (Descriptive set theory)

At the root of set theory lies Cantor's definition of the ``derived set'' $\delta A$ of a closed set $A$, i.e., the set of its non-isolated points, with the help of which Cantor proved that a closed set can be decomposed into a perfect set and a countable set. One may define the index $j(A)$ to be the smallest ordinal $\alpha$ such that $\delta^\alpha A=\emptyset$, or $\omega_1$ if there is no such ordinal. Considering the set $F$ of all closed sets as a (Polish) topological space, ordered by inclusion, $\delta$ as an increasing mapping from $F$ such that $\delta A\subset A$, let $D$ be the set of all $A$ such that $j(A)<\omega_1$ (thus, the set of all countable closed sets). Then $D$ is coanalytic and non-Borel, while the index is bounded by a countable ordinal on every analytic subset of $D$. These powerful results are stated abstractly and proved under very general conditions. Several examples are given

Comment: See a correction in 1241, and several examples in Hillard 1242. The whole subject has been exposed anew in Chapter~XXIV of Dellacherie-Meyer,

Keywords: Derivations (set-theoretic), Kunen-Martin theorem

Nature: Exposition, Original additions

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XI: 12, 132-195, LNM 581 (1977)

Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)

This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(

Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected

Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms

Nature: Exposition, Original additions

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XI: 13, 196-256, LNM 581 (1977)

Classes uniformes de processus gaussiens stationnaires (Gaussian processes)

To be completed

Comment: See the long and interesting review by Berman in

Nature: Exposition, Original additions

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XI: 24, 376-382, LNM 581 (1977)

Équations différentielles stochastiques (Stochastic calculus)

This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in

Keywords: Stochastic differential equations, Semimartingales

Nature: Exposition, Original additions

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XII: 28, 411-423, LNM 649 (1978)

Convergence faible et compacité des temps d'arrêt, d'après Baxter et Chacón (General theory of processes)

Baxter and Chacón (

Comment: See 1536 for an extension to Polish spaces

Keywords: Stopping times, Fuzzy stopping times

Nature: Exposition, Original additions

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XII: 58, 770-774, LNM 649 (1978)

Sur le lemme de La Vallée Poussin et un théorème de Bismut (Measure theory, General theory of processes)

Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces

Keywords: Uniform integrability, Class (D) processes, Moderate convex functions

Nature: Exposition, Original additions

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XIII: 15, 199-203, LNM 721 (1979)

Une remarque sur le calcul stochastique dépendant d'un paramètre (General theory of processes)

Call a ``process'' a measurable function $X(u,t,\omega)$ where $t$ and $\omega$ are as usual and $u$ is a parameter ranging over some nice measurable space ${\cal U}$. Say that $X$ is evanescent if $X(.\,,\,.\,,\omega)\equiv0$ for a.a. $\omega$. The problem is to define previsible processes, and previsible projections defined up to evanescent sets. This is achieved following Jacod,

Keywords: Processes depending on a parameter, Previsible processes, Previsible projections, Random measures

Nature: Exposition, Original additions

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XIII: 21, 240-249, LNM 721 (1979)

Représentations multiplicatives de sousmartingales, d'après Azéma (Martingale theory)

The problem of the multiplicative decomposition of a positive supermartingale $Y$ is relatively easy, but the similar problem for a positive submartingale (find a previsible decreasing process $(C_t)$ such that $(C_tY_t)$ is a martingale) is plagued by the zeros of $Y$. An important idea of Azéma (

Keywords: Multiplicative decomposition

Nature: Exposition, Original additions

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XIII: 28, 313-331, LNM 721 (1979)

Inégalités de normes avec poids (Martingale theory)

See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory

Comment: An exponent $1/\lambda$ is missing in formula (4), p.315

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

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XIII: 58, 642-645, LNM 721 (1979)

Martingales de valeur absolue donnée, d'après Protter-Sharpe (Martingale theory)

The main difficulty of Gilat's theorem (every positive submartingale $X$ can be interpreted as the absolute value of a martingale, in a suitably enlarged filtration) is due to the zeros of $X$. In the strictly positive case a simple proof was given by Protter and Sharpe (

Comment: See also 1407

Keywords: Gilat's theorem

Nature: Exposition, Original additions

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XIV: 09, 102-103, LNM 784 (1980)

Sur un résultat de L. Schwartz (Martingale theory)

the following definition of a semimartingale $X$ in a random open set $A$ is due to L. Schwartz (

Comment: The results are extended in Meyer-Stricker

Keywords: Semimartingales in a random open set

Nature: Exposition, Original additions

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XIV: 15, 128-139, LNM 784 (1980)

Sur l'intégrale stochastique de processus prévisibles non bornés (Stochastic calculus)

The standard theory of stochastic integration deals with locally bounded previsible processes. The natural definition of the stochastic integral $H.X$ of a previsible process $H$ w.r.t. a semimartingale $X$ consists in assuming the existence of some decomposition $X=M+A$ such that $H.M$ exists in the martingale sense, and $H.A$ in the Stieltjes sense, and then defining $H.X$ as their sum. This turns out to be a very awkward definition. It is shown here to be equivalent to the following one: truncating $H$ at $n$, the standard stochastic integrals $H_n.X$ converge in the topology of semimartingales. This is clearly invariant under changes of law. A counterexample shows that integrability may be lost if the filtration is enlarged

Comment: See also 1417. This is a synthesis of earlier work, much of which is due to Jacod,

Keywords: Stochastic integrals

Nature: Exposition, Original additions

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XIV: 42, 397-409, LNM 784 (1980)

Transience and recurrence of Markov processes (Markov processes)

From the introduction: The purpose of this paper is to present an elementary exposition of some various conditions that have been used to define transience or recurrence of a Markov process... an elementary and unified discussion of these ideas may be worthwhile

Keywords: Recurrent Markov processes

Nature: Exposition, Original additions

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XV: 06, 103-117, LNM 850 (1981)

Flot d'une équation différentielle stochastique (Stochastic calculus)

Malliavin showed very neatly how an (Ito) stochastic differential equation on $

Comment: The results on semimartingales with jumps have been proved independently by Uppman. Some dust has been swept under the rugs about the non-explosion of the solution, and the results should be considered valid only in the globally Lipschitz case. See also Uppman 1624 and Léandre 1922

Keywords: Stochastic differential equations, Flow of a s.d.e.

Nature: Exposition, Original additions

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XV: 19, 278-284, LNM 850 (1981)

Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)

Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential

Keywords: $BMO$

Nature: Exposition, Original additions

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XVI: 06, 95-132, LNM 920 (1982)

Note sur les processus d'Ornstein-Uhlenbeck (Malliavin's calculus)

With every Gaussian measure $\mu$ one can associate an Ornstein-Uhlenbeck semigroup, for which $\mu$ is a reversible invariant measure. When $\mu$ is Wiener's measure on ${\cal C}(

Comment: An important problem is the extension to higher order Sobolev-like spaces. For instance, we could define the Sobolev space of order 2 either by the fact that $C^2f=-Lf$ belongs to $L^p$, and on the other hand define $\Gamma_2(f,g)=\sum_{ij} \nabla_i\nabla_j f \nabla_i\nabla_j g$ (derivatives of order 2) and ask that $\sqrt{\Gamma_2(f,f)}\in L^p$. For the equivalence of these two definitions and general higher order ones, see 1816, which anyhow contains many improvements over 1606. Also, proofs of these results have been given which do not involve Littlewood-Paley methods. For instance, Pisier has a proof which only uses the boundedness in $L^p$ of classical Riesz transforms.\par Another trend of research has been the correct definition of ``higher gradients'' within semigroup theory (the preceding definition of $\Gamma_2(f,g)$ makes use of the Gaussian structure). Bakry investigated the fundamental role of ``true'' $\Gamma_2$, the bilinear form $\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(Lf,g)$, which is positive in the case of the Ornstein-Uhlenbeck semigroup but is not always so. See 1909, 1910, 1912

Keywords: Ornstein-Uhlenbeck process, Gaussian measures, Littlewood-Paley theory, Hypercontractivity, Hermite polynomials, Riesz transforms, Test functions

Nature: Exposition, Original additions

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XVI: 10, 151-152, LNM 920 (1982)

Sur une inégalité de Stein (Applications of martingale theory)

In his book

Keywords: Littlewood-Paley theory, Martingale inequalities

Nature: Exposition, Original additions

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XVI: 24, 268-284, LNM 920 (1982)

Sur le flot d'une équation différentielle stochastique (Stochastic calculus)

This paper is a companion to 1506, devoted to the main results on the flow of a (Lipschitz) stochastic differential equation driven by continous semimartingales: non-confluence of solutions from different initial points, surjectivity of the mapping, smooth dependence on the initial conditions. The proofs have been greatly simplified

Keywords: Stochastic differential equations, Flow of a s.d.e., Injectivity

Nature: Exposition, Original additions

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XVII: 19, 185-186, LNM 986 (1983)

Note sur l'exposé précédent (Stochastic calculus)

A small remark on 1718: The event where a semimartingale converges perfectly is also the smallest (modulo negligibility) event where it is a semimartingale up to infinity

Keywords: Semimartingales

Nature: Original additions

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XXIV: 30, 448-452, LNM 1426 (1990)

Sur une formule de Bismut (Markov processes, Stochastic differential geometry)

This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group

Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes

Nature: Exposition, Original additions

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XXXII: 19, 264-305, LNM 1686 (1998)

Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)

Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA

Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,

Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma

Nature: New exposition of known results, Original additions

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