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I: 01, 3-17, LNM 39 (1967)

**AVANISSIAN, Vazgain**

Sur l'harmonicité des fonctions séparément harmoniques (Potential theory)

This paper proves a harmonic version of Hartogs' theorem: separately harmonic functions are jointly harmonic (without any boundedness assumption) using a complex extension procedure. The talk is an extract from the author's original work in*Ann. ENS,* **178**, 1961

Comment: This talk was justified by the current interest of the seminar in doubly excessive functions, see Cairoli 102 in the same volume

Keywords: Doubly harmonic functions

Nature: Exposition

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I: 03, 34-51, LNM 39 (1967)

**COURRÈGE, Philippe**

Noyaux de convolution singuliers opérant sur les fonctions höldériennes et noyaux de convolution régularisants (Potential theory)

The Poisson equation $ėlta\,(Uf)=-f$, where $U$ is the Newtonian potential is proved to be true in the strictest sense when $f$ is a Hölder function (while it is not for mere continuous functions). This involves an exposition of singular integral kernels on Hölder spaces

Comment: This talk was a by-product of the extensive work of Courrège, Bony and Priouret on Feller semi-groups on manifolds with boundary (*Ann. Inst. Fourier,* **16**, 1968)

Keywords: Newtonian potential

Nature: Exposition

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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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I: 08, 166-176, LNM 39 (1967)

**WEIL, Michel**

Retournement du temps dans les processus markoviens (Markov processes)

This talk presents the now classical results of Nagasawa (*Nagoya Math. J.*, **24**, 1964) extending to continuous time the results proved by Hunt in discrete time on time reversal of a Markov process at an ``L-time'' or return time

Comment: See also 202. These results have been essentially the best ones until they were extended to Kuznetsov measures, see Dellacherie-Meyer,*Probabilités et Potentiel,* chapter XIX **14**

Keywords: Time reversal, Dual semigroups

Nature: Exposition

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I: 09, 177-189, LNM 39 (1967)

**WEIL, Michel**

Résolvantes en dualité (Markov processes, Potential theory)

Given two sub-Markov resolvents in duality, whose kernels are absolutely continuous with respect to a given measure, it is shown how to choose their densities to get true Green kernels, excessive in one variable and coexcessive in the other one. It is shown also that coexcessive functions are exactly the densities of excessive measures

Comment: These results, now classical, are due to Kunita-T. Watanabe,*Ill. J. Math.*, **9**, 1965 and *J. Math. Mech.*, **15**, 1966

Keywords: Green potentials, Dual semigroups

Nature: Exposition

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II: 04, 43-74, LNM 51 (1968)

**DOLÉANS-DADE, Catherine**

Espaces $H^m$ sur les variétés, et applications aux équations aux dérivées partielles sur une variété compacte (Functional analysis)

An attempt to teach to the members of the seminar the basic facts of the analytic theory of diffusion processes

Keywords: Sobolev spaces, Second order elliptic equations

Nature: Exposition

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II: 05, 75-110, LNM 51 (1968)

**GIROUX, Gaston**

Théorie des frontières dans les chaînes de Markov (Markov processes)

A presentation of the theory of Markov chains under the hypothesis that all states are regular

Comment: This is the subject of the short monograph of Chung,*Lectures on Boundary Theory for Markov Chains,* Princeton 1970

Keywords: Markov chains, Boundary theory

Nature: Exposition

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II: 06, 111-122, LNM 51 (1968)

**IGOT, Jean-Pierre**

Un théorème de Linnik (Independence)

The result proved (which implies that of Linnik mentioned in the title) is due to Ostrovskii (*Uspehi Mat. Nauk,* **20**, 1965) and states that the convolution of a normal and a Poisson law decomposes only into factors of the same type

Keywords: Infinitely divisible laws, Characteristic functions

Nature: Exposition

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II: 07, 123-139, LNM 51 (1968)

**SAM LAZARO, José de**

Sur les moments spectraux d'ordre supérieur (Second order processes)

The essential result of the paper (Shiryaev,*Th. Prob. Appl.*, **5**, 1960; Sinai, *Th. Prob. Appl.*, **8**, 1963) is the definition of multiple stochastic integrals with respect to a second order process whose covariance satisfies suitable spectral properties

Keywords: Spectral representation, Multiple stochastic integrals

Nature: Exposition

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II: 08, 140-165, LNM 51 (1968)

**MEYER, Paul-André**

Guide détaillé de la théorie ``générale'' des processus (General theory of processes)

This paper states and comments the essential results of a theory which was considered difficult in those times. New terminology was introduced (for instance, the accessible and previsible $\sigma$-fields) though not quite the definitive one (the word ``optional'' only timidly appears instead of the awkward ``well-measurable''). A few new results on the $\sigma$-fields ${\cal F}_{T-}$ and increasing processes are given at the end, the only ones to be proved

Comment: This paper had pedagogical importance in its time, but is now obsolete

Keywords: Previsible processes, Section theorems

Nature: Exposition

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II: 10, 171-174, LNM 51 (1968)

**MEYER, Paul-André**

Les résolvantes fortement fellériennes d'après Mokobodzki (Potential theory)

On a compact space, a submarkov kernel $N$ has the strong Feller property if it maps Borel bounded functions into continuous functions, and the stronger Feller property if the mapping $x\rightarrow \epsilon_x N$ is continuous in the norm topology of measures. It is proved that the product of two strong Feller kernels is stronger Feller, and as a consequence if the kernels of are resolvent are strong Feller they are automatically stronger Feller

Comment: This follows from a result on weakly compact operators on continuous functions due to Grothendieck (*Canadian J. Math.*, **5**, 1953). Mokobodzki's proof is less general (it uses positivity) but very simple. This result is rather useful

Keywords: Resolvents, Strong Feller properties

Nature: Exposition

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III: 01, 1-23, LNM 88 (1969)

**ARTZNER, Philippe**

Extension du théorème de Sazonov-Minlos d'après L.~Schwartz (Measure theory, Functional analysis)

Exposition of three notes by L.~Schwartz (*CRAS* **265**, 1967 and **266**, 1968) showing that some classes of maps between spaces $\ell^p$ and $\ell^q$ transform Gaussian cylindrical measures into Radon measures. The result turns out to be an extension of Minlos' theorem

Comment: Self-contained and detailed exposition, possibly still useful

Keywords: Radonifying maps

Nature: Exposition

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III: 13, 163-174, LNM 88 (1969)

**MEYER, Paul-André**

Les inégalités de Burkholder en théorie des martingales, d'après Gundy (Martingale theory)

A proof of the famous Burkholder inequalities in discrete time, from Gundy,*Ann. Math. Stat.*, **39**, 1968

Keywords: Burkholder inequalities

Nature: Exposition

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III: 14, 175-189, LNM 88 (1969)

**MEYER, Paul-André**

Processus à accroissements indépendants et positifs (Markov processes, Independent increments)

This is an exposition of the theory of subordinators (Lévy processes with increasing paths), aiming at presenting Chung's conjecture that a certain identity known to hold a.e. actually holds everywhere, also equivalent to the fact that single points are polar sets for subordinators without drift

Comment: The conjecture was proved by Kesten (see 503) who actually knew of the problem through this talk. See also 502

Keywords: Subordinators, Polar sets

Nature: Exposition

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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: 02, 28-36, LNM 124 (1970)

**CARTIER, Pierre**

Sur certaines variables aléatoires associées au réarrangement croissant d'un échantillon (Miscellanea)

Presentation of combinatorial results due to D.~Foata and to Cartier-Foata, which are relevant for the theory of order statistics

Comment: See Cartier-Foata,*Problèmes combinatoires de commutation et réarrangements,* LN **85**, 1969

Keywords: Combinatorics, Generating functions, Order statistics

Nature: Exposition

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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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IV: 04, 47-59, LNM 124 (1970)

**DACUNHA-CASTELLE, Didier**

Principe de dualité pour des espaces de suites associés à une suite de variables aléatoires (Miscellanea)

The ``duality principe'' (too technical to be stated here) is a result of L.~Schwartz on Banach spaces of type (L), i.e., consisting of sequences $(b_n)$ such that $\sum_n b_nX_n$ is bounded in probability, where $(X_n)$ is a given sequence of r.v.'s

Comment: Related to the theory of radonifying maps, then in fast progress. To be completed

Keywords: Banach spaces, Radonifying maps

Nature: Exposition

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IV: 12, 133-150, LNM 124 (1970)

**MEYER, Paul-André**

Ensembles régénératifs, d'après Hoffmann-Jørgensen (Markov processes)

The theory of recurrent events in discrete time was a highlight of the old probability theory. It was extended to continuous time by Kingman (see for instance*Z. für W-theorie,* **2**, 1964), under the very restrictive assumption that the ``event'' has a non-zero probability to occur at fixed times. The general theory is due to Krylov and Yushkevich (*Trans. Moscow Math. Soc.*, **13**, 1965), a deep paper difficult to read and to apply in concrete cases. Hoffmann-Jørgensen (*Math. Scand.*, **24**, 1969) developed the theory under simple and efficient axioms. It is shown that a regenerative set defined axiomatically is the same thing as the set of returns of a strong Markov process to a fixed state, or the range of a subordinator

Comment: This result was expanded to involve a Markovian regeneration property instead of independence. See Maisonneuve-Meyer 813. The subject is related to excursion theory, Lévy systems, semi-Markovian processes (Lévy), F-processes (Neveu), Markov renewal processes (Pyke), and the literature is very extensive. See for instance Dynkin (*Th. Prob. Appl.*, **16**, 1971) and Maisonneuve, *Systèmes Régénératifs,* *Astérisque * **15**, 1974

Keywords: Renewal theory, Regenerative sets, Recurrent events

Nature: Exposition

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IV: 14, 162-169, LNM 124 (1970)

**MEYER, Paul-André**

Quelques inégalités sur les martingales, d'après Dubins et Freedman (Martingale theory)

The original paper appeared in*Ann. Math. Stat.*, **36**, 1965, and the inequalities are extensions to martingales of the Borel-Cantelli lemma and the strong law of large numbers. For martingales with bounded jumps, exponential bounds are given (Neveu, *Martingales à temps discret,* gives a better one)

Comment: Though the proofs are very clever, so much work has been devoted to martingale inequalities since the paper was written that it is probably obsolete

Keywords: Inequalities

Nature: Exposition

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IV: 19, 240-282, LNM 124 (1970)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**; **LETTA, Giorgio**; **MEYER, Paul-André**

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (*Comm. Pure Appl. Math.*, **22**, 1969), which constructs by a probability method a unique semigroup whose generator is an elliptic second order operator with continuous coefficients (the analytic approach either deals with operators in divergence form, or requires some Hölder condition). The contribution of G.~Letta nicely simplified the proof

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,*Multidimensional Diffusion Processes,* Springer 1979

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

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V: 02, 17-20, LNM 191 (1971)

**ASSOUAD, Patrice**

Démonstration de la ``Conjecture de Chung'' par Carleson (Markov processes, Independent increments)

Chung conjectured that singletons are polar sets for driftless subordinators. This paper gives Carleson's (unpublished) analytic proof of it

Comment: See Chung,*C. R. Acad. Sci. *, **260**, 1965, p.4665. For the statement of the problem see Meyer 314. For Kesten's earlier (contrary to a statement in the paper!) probabilistic proof see Bretagnolle 503. See also *Séminaire Bourbaki * 21th year, **361**, June 1969

Keywords: Subordinators, Polar sets

Nature: Exposition

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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: 05, 58-75, LNM 191 (1971)

**CARTIER, Pierre**

Introduction à l'étude des mouvements browniens à plusieurs paramètres (Gaussian processes)

Settles in particular a disagreement between statements of Lévy (*Processus Stochastiques et Mouvement Brownien,* 1948) and McKean (*Teor. Ver. i Prim.* **8**, 1963) on the domain of analyticity of some Gaussian random functions

Comment: More recent work of Cartier on covariances appeared in the L.~Schwartz volume*Mathematical Analysis and Applications, A*, Academic Press 1981

Keywords: Several parameter Brownian motions, Covariance

Nature: New exposition of known results

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V: 06, 76-76, LNM 191 (1971)

**CHUNG, Kai Lai**

A simple proof of Doob's convergence theorem (Potential theory)

Doob's theorem is a version of the main convergence theorem of potential theory: the limit of a decreasing sequence of excessive functions differs of its regularized version on a semi-polar set

Comment: It is also shown that a function $f$ satisfying $f\ge P_Kf$ for all compact sets $K$ differs from its regularized function on a semi-polar set

Keywords: Excessive functions, Semi-polar sets

Nature: New exposition of known results

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V: 10, 87-102, LNM 191 (1971)

**DELLACHERIE, Claude**

Les théorèmes de Mazurkiewicz-Sierpinski et de Lusin (Descriptive set theory)

Synthetic presentation of (then) little known results on the perfect kernels of closed random sets and uniformization of random sets with countable sections

Comment: See Dellacherie-Meyer,*Probabilités et Potentiel,* Chap. XI

Keywords: Analytic sets, Random sets, Section theorems

Nature: New exposition of known results

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V: 16, 170-176, LNM 191 (1971)

**MEYER, Paul-André**

Sur un article de Dubins (Martingale theory)

Description of a Skorohod imbedding procedure for real valued r.v.'s due to Dubins (*Ann. Math. Stat.*, **39,** 1968), using a remarkable discrete approximation of measures. It does not use randomization

Comment: This beautiful method to realize Skorohod's imbedding is related to that of Chacon and Walsh in 1002. For a deeper study see Bretagnolle 802. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Exposition

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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: 19, 196-208, LNM 191 (1971)

**MEYER, Paul-André**

Représentation intégrale des fonctions excessives. Résultats de Mokobodzki (Markov processes, Potential theory)

Main result: the convex cone of excessive functions for a resolvent which satisfies the absolute continuity hypothesis is the union of convex compact metrizable ``hats''\ in a suitable topology, and therefore has the integral representation property. The original proof of Mokobodzki, self-contained and unpublished, is given here

Comment: See Mokobodzki's work on cones of potentials,*Séminaire Bourbaki,* May 1970

Keywords: Minimal excessive functions, Martin boundary, Integral representations

Nature: Exposition

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V: 20, 209-210, LNM 191 (1971)

**MEYER, Paul-André**

Un théorème sur la répartition des temps locaux (Markov processes)

Kesten discovered that the value at a terminal time $T$ of the local time $L$ of a Markov process $X$ at a single point has an exponential distribution, and that $X_T$ and $L_T$ are independent. A short proof is given

Comment: The result can be deduced from excursion theory

Keywords: Local times

Nature: New exposition of known results

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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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V: 25, 270-274, LNM 191 (1971)

**MEYER, Paul-André**

Balayage pour les processus de Markov continus à droite, d'après C.T. Shih (Markov processes, Potential theory)

Hunt's fundamental theorem on balayage gives the probabilistic interpretation of the réduite of an excessive function set on a set. It was proved originally within the special class of Hunt's processes, then extended to ``standard'' processes. Using a method of compactification, Shih (*Ann. Inst. Fourier,* **20-1**, 1970) showed it was quite general

Comment: Shih's paper is the origin of the general definition of ``right processes''

Keywords: Excessive functions, Réduite

Nature: Exposition

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VI: 04, 72-89, LNM 258 (1972)

**CHATTERJI, Shrishti Dhav**

Un principe de sous-suites dans la théorie des probabilités (Measure theory)

This paper is devoted to results of the following kind: any sequence of random variables with a given weak property contains a subsequence which satisfies a stronger property. An example is due to Komlós: any sequence bounded in $L^1$ contains a subsequence which converges a.s. in the Cesaro sense. Several results of this kind, mostly due to the author, are presented without detailed proofs

Comment: See 1302 for extensions to the case of Banach space valued random variables. See also Aldous,*Zeit. für W-theorie,* **40**, 1977

Keywords: Subsequences, Central limit theorem, Law of the iterated logarithm

Nature: Exposition

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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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VI: 13, 151-158, LNM 258 (1972)

**MEYER, Paul-André**

Les résultats récents de Burkholder, Davis et Gundy (Martingale theory)

The well-known norm equivalence between the maximum and the square-function of a martingale in moderate Orlicz spaces is presented following the celebrated papers of Burkholder-Gundy (*Acta Math.,* **124**, 1970), Burkholder-Davis-Gundy (*Proc. 6-th Berkeley Symposium,* **3**, 1972). The technique of proof is now obsolete

Keywords: Burkholder inequalities, Moderate convex functions

Nature: Exposition

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VII: 06, 48-50, LNM 321 (1973)

**DELLACHERIE, Claude**

Une démonstration du théorème de Souslin-Lusin (Descriptive set theory)

The basic fact that the image of a Borel set under an injective Borel mapping is Borel is deduced from a separation theorem concerning countably many disjoint analytic sets

Comment: This is a step in the author's simplification of the proofs of the great theorems on analytic and Borel sets. See*Un cours sur les ensembles analytiques,* in *Analytic Sets,* C.A. Rogers ed., Academic Press 1980

Keywords: Borel sets, Analytic sets, Separation theorem

Nature: New exposition of known results

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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: 15, 146-154, LNM 321 (1973)

**MEYER, Paul-André**

Chirurgie sur un processus de Markov, d'après Knight et Pittenger (Markov processes)

This paper presents a remarkable result of Knight and Pittenger, according to which excising from the sample path of a Markov process all the excursions from a given set $A$ which meet another set $B$ preserves the Markov property

Comment: The original paper appeared in*Zeit. für W-theorie,* **23**, 1972

Keywords: Transformations of Markov processes, Excursions

Nature: Exposition

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VII: 17, 172-179, LNM 321 (1973)

**MEYER, Paul-André**

Application de l'exposé précédent aux processus de Markov (Markov processes)

This paper is devoted to results of J.F. Mertens on optimal stopping and strongly supermedian functions for a right Markov process (*Zeit. für W-theorie,* **26**, 1973), which are shown to be closely related to those of the preceding paper 716 on general gambling houses. An interesting result of Mokobodzki is included, showing that the extreme points of the convex set of all balayées of a given measure $\lambda$ are the balayées of $\lambda$ on sets

Comment: See related papers by Mertens in*Zeit. für W-theorie,* **22**, 1972 and *Invent. Math.*, **23**, 1974. The original result of Mokobodzki appeared in the *Sémin. Théorie du Potentiel,* 1969-70

Keywords: Excessive functions, Supermedian functions, Réduite

Nature: Exposition, Original proofs

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VII: 18, 180-197, LNM 321 (1973)

**MEYER, Paul-André**

Résultats d'Azéma en théorie générale des processus (General theory of processes)

This paper presents several results from a paper of Azéma (*Invent. Math.*, **18**, 1972) which have become (in a slightly extended version) standard tools in the general theory of processes. The problem is that of ``localizing'' a time $L$ which is not a stopping time. With $L$ are associated the supermartingale $c^L_t=P\{L>t|{\cal F}_t\}$ and the previsible increasing processes $p^L$ which generates it (and is the dual previsible projection of the unit mass on the graph of $L$). Then the left support of $dp^L$ is the smallest left-closed previsible set containing the graph of $L$, while the set $\{c^L_-=1\}$ is the greatest previsible set to the left of $L$. Other useful results are the following: given a progressive process $X$, the process $\limsup_{s\rightarrow t} X_s$ is optional, previsible if $s<t$ is added, and a few similar results

Comment: These results have been included (with their optional counterpart, whose interest was discovered later) in Dellacherie-Meyer,*Probabilités et Potentiel*, Vol. E, Chapter XX **12**--17

Keywords: Optimal stopping, Previsible processes

Nature: Exposition

Retrieve article from Numdam

VII: 19, 198-204, LNM 321 (1973)

**MEYER, Paul-André**

Limites médiales d'après Mokobodzki (Measure theory, Functional analysis)

Given a sequence of (classes of) random variables on a probability space which converges in some of the standard ways of measure theory, the problem is to find some universal method (independent from the underlying probability) to identify its limit. For convergence in probability, and thus for all strong $L^p$ topologies, Mokobodzki had discovered the procedure of rapid ultrafilters (see 304). The same problem is now solved for weak convergences, using a special kind of Banach limits

Comment: The paper contains a few annoying misprints, in particular p.199 line 9*s.c;s.* should be deleted and line 17 *atomique * should be *absolument continu.* For a misprint-free version see Dellacherie-Meyer, *Probabiliés et Potentiel,* Volume C, Chapter X, **55**--57

Keywords: Continuum axiom, Weak convergence of r.v.'s, Medial limit

Nature: Exposition

Retrieve article from Numdam

VII: 21, 210-216, LNM 321 (1973)

**MEYER, Paul-André**

Note sur l'interprétation des mesures d'équilibre (Markov processes)

Let $(X_t)$ be a transient Markov process (we omit the detailed assumptions) with a potential density $u(x,y)$. Let $\mu$ be the measure whose potential is the equilibrium potential of a set $A$. Then the distribution of the process at the last exit time from $A$ is given by $$E_x[f\circ X_{L-}, 0<L<\infty]=\int u(x,y)\,f(y)\,\mu(dy)$$ This formula, due to Chung, is deduced under minimal duality hypotheses from a general formula of Azéma, and a well-known theorem on Revuz measures

Keywords: Equilibrium potentials, Last exit time, Revuz measures

Nature: Exposition, Original proofs

Retrieve article from Numdam

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

Retrieve article from Numdam

VII: 23, 223-247, LNM 321 (1973)

**MEYER, Paul-André**

Sur un problème de filtration (General theory of processes)

This is an attempt to present, in the usual language of the seminar, a a simple case from filtering theory (additive white noise in one dimension). The results proved are the Kallianpur-Striebel formula (*Ann. Math. Stat.*, **39**, 1968) and a theorem of Clark

Keywords: Filtering theory, Innovation

Nature: Exposition

Retrieve article from Numdam

VII: 25, 273-283, LNM 321 (1973)

**PINSKY, Mark A.**

Fonctionnelles multiplicatives opératrices (Markov processes)

This paper presents results due to the author (*Advances in Probability,* **3**, 1973). The essential idea is to consider multiplicative functionals of a Markov process taking values in the algebra of bounded operators of a Banach space $L$. Such a functional defines a semi-group acting on bounded $L$-valued functions. This semi-group determines the functional. The structure of functionals is investigated in the case of a finite Markov chain. The case where $L$ is finite dimensional and the Markov process is Browian motion is investigated too. Asymptotic results near $0$ are described

Comment: This paper explores the same idea as Jacod (*Mém. Soc. Math. France,* **35**, 1973), though in a very different way. See 816

Keywords: Multiplicative functionals, Multiplicative kernels

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

VIII: 16, 290-309, LNM 381 (1974)

**MEYER, Paul-André**

Noyaux multiplicatifs (Markov processes)

This paper presents results due to Jacod (*Mém. Soc. Math. France,* **35**, 1973): given a pair $(X,Y)$ which jointly is a Markov process, and whose first component $X$ is a Markov process by itself, describe the conditional distribution of the joint path of $(X,Y)$ over a given path of $X$. These distributions constitute a multiplicative kernel, and attempts are made to regularize it

Comment: Though the paper is fairly technical, it does not improve substantially on Jacod's results

Keywords: Multiplicative kernels, Semimarkovian processes

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

IX: 04, 206-212, LNM 465 (1975)

**CHOU, Ching Sung**

Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)

A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities

Keywords: Inequalities, Burkholder inequalities

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

IX: 15, 336-372, LNM 465 (1975)

**DELLACHERIE, Claude**

Ensembles analytiques, théorèmes de séparation et applications (Descriptive set theory)

According to the standard (``first'') separation theorem, in a compact metric space or any space which is Borel isomorphic to it, two disjoint analytic sets can be separated by Borel sets, and in particular any bianalytic set (analytic and coanalytic i.e., complement of analytic) is Borel. Not so in general metric spaces. That the same statement holds in full generality with ``bianalytic'' instead of ``Borel'' is the second separation theorem, which according to the general opinion was considered much more difficult than the first. This result and many more (on projections of Borel sets with compact sections or countable sections, for instance) are fully proved in this exposition

Comment: See also the next paper 916, the set of lectures by Dellacherie in C.A. Rogers,*Analytic Sets,* Academic Press 1981, and chapter XXIV of Dellacherie-Meyer, *Probabilités et potentiel *

Keywords: Second separation theorem

Nature: Exposition

Retrieve article from Numdam

IX: 21, 425-436, LNM 465 (1975)

**ÉMERY, Michel**

Primitive d'une mesure sur les compacts d'un espace métrique (Measure theory)

It is well known that the set ${\cal K}$ of all compact subsets of a compact metric space has a natural compact metric topology. The ``distribution function'' of a positive measure on ${\cal K}$ associates with every $A\in{\cal K}$ the measure of the subset $\{K\subset A\}$ of ${\cal K}$. It is shown here (following A.~Revuz,*Ann. Inst. Fourier,* **6**, 1955-56) that the distribution functions of measures are characterized by simple algebraic properties and right continuity

Comment: This elegant theorem apparently never had applications

Keywords: Distribution functions on ordered spaces

Nature: Exposition

Retrieve article from Numdam

IX: 27, 486-493, LNM 465 (1975)

**WEIL, Michel**

Surlois d'entrée (Markov processes)

The results presented here are due to T. Leviatan (*Ann. Prob.*, **1**, 1973) and concern the construction of a Markov process with creation of mass, corresponding to a given transition semigroup $(P_t)$ and a given ``super-entrance law'' $(\mu_t)$, consisting of bounded measures such that $\mu_{s+t}\ge\mu_s P_t$. The proof is a clever argument of projective limits. The paper mentions briefly the relation with earlier results of Helms (*Z. für W-theorie,* **7**, 1967)

Comment: This beautiful paper was superseded by the (slightly later) fundamental paper of Kuznetsov (1974)

Keywords: Entrance laws, Creation of mass, Kuznetsov measures

Nature: Exposition

Retrieve article from Numdam

IX: 34, 530-533, LNM 465 (1975)

**MEYER, Paul-André**

Sur la démonstration de prévisibilité de Chung et Walsh (Markov processes)

A new proof of the result that for a Hunt process, the previsible stopping times are exactly those at which the process does not jump was given by Chung-Walsh (*Z. für W-theorie,* **29**, 1974). Their idea is used here in a modified way, using a formula of Dawson which ``explicitly'' computes conditional expectations and projections. Then it is extended to Ray processes

Comment: The contents of this paper became Chapter XIV**44**--47 in Dellacherie-Meyer, *Probabilités et Potentiel*

Keywords: Hunt processes, Previsible times

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

XI: 10, 109-119, LNM 581 (1977)

**MEYER, Paul-André**

Convergence faible de processus, d'après Mokobodzki (General theory of processes)

The following simple question of Benveniste was answered positively by Mokobodzki (*Séminaire de Théorie du Potentiel,* Lect. Notes in M. 563, 1976): given a sequence $(U^n)$ of optional processes such that $U^n_T$ converges weakly in $L^1$ for every stopping time $T$, does there exist an optional process $U$ such that $U^n_T$ converges to $U_T$? The proof is rather elaborate

Keywords: Weak convergence in $L^1$

Nature: Exposition

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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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

XI: 38, 539-565, LNM 581 (1977)

**TORTRAT, Albert**

Désintégration d'une probabilité. Statistiques exhaustives (Measure theory)

This is a detailed review (including proofs) of the problem of existence of conditional distributions

Keywords: Disintegration of measures, Conditional distributions, Sufficient statistics

Nature: Exposition

Retrieve article from Numdam

XI: 39, 566-573, LNM 581 (1977)

**ÉMERY, Michel**

Information associée à un semigroupe (Markov processes)

This paper contains the proof of two important theorems of Donsker and Varadhan (*Comm. Pure and Appl. Math.*, 1975)

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

XII: 34, 468-481, LNM 649 (1978)

**SAINT RAYMOND, Jean**

Quelques remarques sur un article de Donsker et Varadhan (Markov processes)

This paper is a partial exposition of the celebrated papers of Donsker and Varadhan in*Comm. Pure Appl. Math.* 28, 1975 and 29, 1976, aiming at simplifying the proofs and weakening a few technical hypotheses

Keywords: Large deviations

Nature: Exposition

Retrieve article from Numdam

XII: 57, 763-769, LNM 649 (1978)

**MEYER, Paul-André**

La formule d'Ito pour le mouvement brownien, d'après Brosamler (Brownian motion, Stochastic calculus)

This paper presents the results of a paper by Brosamler (*Trans. Amer. Math. Soc.* 149, 1970) on the Ito formula $f(B_t)=...$ for $n$-dimensional Brownian motion, under the weakest possible assumptions: namely up to the first exit time from an open set $W$ and assuming only that $f$ is locally in $L^1$ in $W$, and its Laplacian in the sense of distributions is a measure in $W$

Keywords: Ito formula

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

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: 23, 253-259, LNM 721 (1979)

**SPILIOTIS, Jean**

Sur les intégrales stochastiques de L.C. Young (Stochastic calculus)

This is a partial exposition of a theory of stochastic integration due to L.C. Young (*Advances in Prob.* **3**, 1974)

Keywords: Stochastic integrals

Nature: Exposition

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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

Retrieve article from Numdam

XIII: 31, 371-377, LNM 721 (1979)

**DELLACHERIE, Claude**

Inégalités de convexité pour les processus croissants et les sousmartingales (Martingale theory)

Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,*Probabilités et Potentiels B,* Chapter VI

Keywords: Martingale inequalities, Convex functions

Nature: Exposition, Original proofs

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XIII: 52, 611-613, LNM 721 (1979)

**MEYER, Paul-André**

Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (Martingale theory)

In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (*Ann. Prob.* **8**, 1980) to develop the whole theory of stochastic differential equations

Keywords: Doob's inequality, Stochastic differential equations

Nature: Exposition

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XIII: 54, 620-623, LNM 721 (1979)

**MEYER, Paul-André**

Caractérisation des semimartingales, d'après Dellacherie (Stochastic calculus)

This short paper contains the proof of a very important theorem, due to Dellacherie (with the crucial help of Mokobodzki for the functional analytic part). Namely, semimartingales are exactly the processes which give rise to a nice vector measure on the previsible $\sigma$-field, with values in the (non locally convex) space $L^0$. It is only fair to say that this direction was initiated by Métivier and Pellaumail, and that the main result was independently discovered by Bichteler,*Ann. Prob.* **9**, 1981)

Comment: An important lemma which simplifies the proof and has other applications is given by Yan in 1425

Keywords: Semimartingales, Stochastic integrals

Nature: Exposition

Retrieve article from Numdam

XIII: 55, 624-624, LNM 721 (1979)

**YOR, Marc**

Un exemple de J. Pitman (General theory of processes)

The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale $X$, namely martingales of the form $Z_{g_t}X_t$ where $Z$ is previsible. Taking $X$ to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form

Comment: The general problem of finding all martingales which vanish on the zeros of a given continuous martingale is discussed by Azéma and Yor in 2622

Keywords: Balayage, Balayage formula

Nature: Exposition

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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

Retrieve article from Numdam

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

Retrieve article from Numdam

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: 20, 173-188, LNM 784 (1980)

**MEYER, Paul-André**

Les résultats de Jeulin sur le grossissement des tribus (General theory of processes, Stochastic calculus)

This is an introduction to beautiful results of Jeulin on enlargements, for which see*Zeit. für W-Theorie,* **52**, 1980, and above all the Lecture Notes vol. 833, *Semimartingales et grossissement d'une filtration *

Comment: See also 1329, 1350

Keywords: Enlargement of filtrations, Semimartingales

Nature: Exposition

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XIV: 24, 209-219, LNM 784 (1980)

**PELLAUMAIL, Jean**

Remarques sur l'intégrale stochastique (Stochastic calculus)

This is an exposition of stochastic integrals and stochastic differential equations for Banach space valued processes along the lines of Métivier-Pellaumail*Stochastic Integration* (1980), the class of semimartingales being defined by the Métivier-Pellaumail inequality (1413)

Keywords: Stochastic integrals, Stochastic differential equations, Métivier-Pellaumail inequality

Nature: Exposition

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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

Retrieve article from Numdam

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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XV: 28, 388-398, LNM 850 (1981)

**SPILIOTIS, Jean**

Sur les travaux de Krylov en théorie de l'intégrale stochastique (Martingale theory)

The well-known work of Malliavin deals with the existence of smooth densities for solutions of stochastic differential equations with smooth coefficients satisfying a hypoellipticity condition. N.V.~Krylov's earlier work (among many papers see*Izvestija Akad Nauk,* **38**, 1974, and Krylov's book *Controlled Diffusion processes,* Springer 1980) dealt with the existence of densities for several dimensional stochastic integrals with measurable bounded integrands, satisfying an ellipticity condition. It is a puzzling fact that nobody ever succeeded in unifying these results. Krylov's method depends on results of the Russian school on Monge-Ampère equations (see Pogorelov *The Minkowski Multidimensional Problem,* 1978). This exposition attempts, rather modestly, to explain in the seminar's language what it is all about, and in particular to show the place where a crucial lemma on convex functions is used

Keywords: Stochastic integrals, Existence of densities

Nature: Exposition

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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: 07, 133-133, LNM 920 (1982)

**MEYER, Paul-André**

Appendice : Un résultat de D. Williams (Malliavin's calculus)

This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as ``quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process

Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis

Nature: Exposition

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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: 11, 153-158, LNM 920 (1982)

**MEYER, Paul-André**

Interpolation entre espaces d'Orlicz (Functional analysis)

This is an exposition of Calderon's complex interpolation method, in the case of moderate Orlicz spaces, aiming at its application in 1609

Keywords: Interpolation, Orlicz spaces, Moderate convex functions

Nature: Exposition

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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: 20, 187-193, LNM 986 (1983)

**MEYER, Paul-André**

Le théorème de convergence des martingales dans les variétés riemanniennes, d'après R.W. Darling et W.A. Zheng (Stochastic differential geometry)

Exposition of two results on the asymptotic behaviour of martingales in a Riemannian manifold: First, Darling's theorem says that on the event where the Riemannian quadratic variation $<X,X>_\infty$ of a martingale $X$ is finite, $X_\infty$ exists in the Aleksandrov compactification of $V$. Second, Zheng's theorem asserts that on the event where $X_\infty$ exists in $V$, the Riemannian quadratic variation $<X,X>_\infty$ is finite

Comment: Darling's result is in*Publ. R.I.M.S. Kyoto* **19** (1983) and Zheng's in *Zeit. für W-theorie* **63** (1983). As observed in He-Yan-Zheng 1718, a stronger version of Zheng's theorem holds (with the same argument): On the event where $X_\infty$ exists in $V$, $X$ is a semimartingale up to infinity (so for instance solutions to good SDE's driven by $X$ also have a limit at infinity)

Keywords: Martingales in manifolds

Nature: Exposition

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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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XLIII: 05, 127-186, LNM 2006 (2011)

**LAURENT, Stéphane**

On standardness and I-cosiness (Filtrations)

Keywords: Filtrations, Cosiness

Nature: Original, Exposition

XLIII: 08, 215-219, LNM 2006 (2011)

**PRATELLI, Maurizio**

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition

XLIV: 14, 279-280, LNM 2046 (2012)

**BOGSO, Antoine-Marie**; **PROFETA, Christophe**; **ROYNETTE, Bernard**

On Peacocks: a general introduction to two articles (Theory of processes)

Keywords: Processes increasing in the convex order: peacocks

Nature: Exposition

Sur l'harmonicité des fonctions séparément harmoniques (Potential theory)

This paper proves a harmonic version of Hartogs' theorem: separately harmonic functions are jointly harmonic (without any boundedness assumption) using a complex extension procedure. The talk is an extract from the author's original work in

Comment: This talk was justified by the current interest of the seminar in doubly excessive functions, see Cairoli 102 in the same volume

Keywords: Doubly harmonic functions

Nature: Exposition

Retrieve article from Numdam

I: 03, 34-51, LNM 39 (1967)

Noyaux de convolution singuliers opérant sur les fonctions höldériennes et noyaux de convolution régularisants (Potential theory)

The Poisson equation $ėlta\,(Uf)=-f$, where $U$ is the Newtonian potential is proved to be true in the strictest sense when $f$ is a Hölder function (while it is not for mere continuous functions). This involves an exposition of singular integral kernels on Hölder spaces

Comment: This talk was a by-product of the extensive work of Courrège, Bony and Priouret on Feller semi-groups on manifolds with boundary (

Keywords: Newtonian potential

Nature: Exposition

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

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

Retrieve article from Numdam

I: 08, 166-176, LNM 39 (1967)

Retournement du temps dans les processus markoviens (Markov processes)

This talk presents the now classical results of Nagasawa (

Comment: See also 202. These results have been essentially the best ones until they were extended to Kuznetsov measures, see Dellacherie-Meyer,

Keywords: Time reversal, Dual semigroups

Nature: Exposition

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I: 09, 177-189, LNM 39 (1967)

Résolvantes en dualité (Markov processes, Potential theory)

Given two sub-Markov resolvents in duality, whose kernels are absolutely continuous with respect to a given measure, it is shown how to choose their densities to get true Green kernels, excessive in one variable and coexcessive in the other one. It is shown also that coexcessive functions are exactly the densities of excessive measures

Comment: These results, now classical, are due to Kunita-T. Watanabe,

Keywords: Green potentials, Dual semigroups

Nature: Exposition

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II: 04, 43-74, LNM 51 (1968)

Espaces $H^m$ sur les variétés, et applications aux équations aux dérivées partielles sur une variété compacte (Functional analysis)

An attempt to teach to the members of the seminar the basic facts of the analytic theory of diffusion processes

Keywords: Sobolev spaces, Second order elliptic equations

Nature: Exposition

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II: 05, 75-110, LNM 51 (1968)

Théorie des frontières dans les chaînes de Markov (Markov processes)

A presentation of the theory of Markov chains under the hypothesis that all states are regular

Comment: This is the subject of the short monograph of Chung,

Keywords: Markov chains, Boundary theory

Nature: Exposition

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II: 06, 111-122, LNM 51 (1968)

Un théorème de Linnik (Independence)

The result proved (which implies that of Linnik mentioned in the title) is due to Ostrovskii (

Keywords: Infinitely divisible laws, Characteristic functions

Nature: Exposition

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II: 07, 123-139, LNM 51 (1968)

Sur les moments spectraux d'ordre supérieur (Second order processes)

The essential result of the paper (Shiryaev,

Keywords: Spectral representation, Multiple stochastic integrals

Nature: Exposition

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II: 08, 140-165, LNM 51 (1968)

Guide détaillé de la théorie ``générale'' des processus (General theory of processes)

This paper states and comments the essential results of a theory which was considered difficult in those times. New terminology was introduced (for instance, the accessible and previsible $\sigma$-fields) though not quite the definitive one (the word ``optional'' only timidly appears instead of the awkward ``well-measurable''). A few new results on the $\sigma$-fields ${\cal F}_{T-}$ and increasing processes are given at the end, the only ones to be proved

Comment: This paper had pedagogical importance in its time, but is now obsolete

Keywords: Previsible processes, Section theorems

Nature: Exposition

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II: 10, 171-174, LNM 51 (1968)

Les résolvantes fortement fellériennes d'après Mokobodzki (Potential theory)

On a compact space, a submarkov kernel $N$ has the strong Feller property if it maps Borel bounded functions into continuous functions, and the stronger Feller property if the mapping $x\rightarrow \epsilon_x N$ is continuous in the norm topology of measures. It is proved that the product of two strong Feller kernels is stronger Feller, and as a consequence if the kernels of are resolvent are strong Feller they are automatically stronger Feller

Comment: This follows from a result on weakly compact operators on continuous functions due to Grothendieck (

Keywords: Resolvents, Strong Feller properties

Nature: Exposition

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III: 01, 1-23, LNM 88 (1969)

Extension du théorème de Sazonov-Minlos d'après L.~Schwartz (Measure theory, Functional analysis)

Exposition of three notes by L.~Schwartz (

Comment: Self-contained and detailed exposition, possibly still useful

Keywords: Radonifying maps

Nature: Exposition

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III: 13, 163-174, LNM 88 (1969)

Les inégalités de Burkholder en théorie des martingales, d'après Gundy (Martingale theory)

A proof of the famous Burkholder inequalities in discrete time, from Gundy,

Keywords: Burkholder inequalities

Nature: Exposition

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III: 14, 175-189, LNM 88 (1969)

Processus à accroissements indépendants et positifs (Markov processes, Independent increments)

This is an exposition of the theory of subordinators (Lévy processes with increasing paths), aiming at presenting Chung's conjecture that a certain identity known to hold a.e. actually holds everywhere, also equivalent to the fact that single points are polar sets for subordinators without drift

Comment: The conjecture was proved by Kesten (see 503) who actually knew of the problem through this talk. See also 502

Keywords: Subordinators, Polar sets

Nature: Exposition

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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: 02, 28-36, LNM 124 (1970)

Sur certaines variables aléatoires associées au réarrangement croissant d'un échantillon (Miscellanea)

Presentation of combinatorial results due to D.~Foata and to Cartier-Foata, which are relevant for the theory of order statistics

Comment: See Cartier-Foata,

Keywords: Combinatorics, Generating functions, Order statistics

Nature: Exposition

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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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IV: 04, 47-59, LNM 124 (1970)

Principe de dualité pour des espaces de suites associés à une suite de variables aléatoires (Miscellanea)

The ``duality principe'' (too technical to be stated here) is a result of L.~Schwartz on Banach spaces of type (L), i.e., consisting of sequences $(b_n)$ such that $\sum_n b_nX_n$ is bounded in probability, where $(X_n)$ is a given sequence of r.v.'s

Comment: Related to the theory of radonifying maps, then in fast progress. To be completed

Keywords: Banach spaces, Radonifying maps

Nature: Exposition

Retrieve article from Numdam

IV: 12, 133-150, LNM 124 (1970)

Ensembles régénératifs, d'après Hoffmann-Jørgensen (Markov processes)

The theory of recurrent events in discrete time was a highlight of the old probability theory. It was extended to continuous time by Kingman (see for instance

Comment: This result was expanded to involve a Markovian regeneration property instead of independence. See Maisonneuve-Meyer 813. The subject is related to excursion theory, Lévy systems, semi-Markovian processes (Lévy), F-processes (Neveu), Markov renewal processes (Pyke), and the literature is very extensive. See for instance Dynkin (

Keywords: Renewal theory, Regenerative sets, Recurrent events

Nature: Exposition

Retrieve article from Numdam

IV: 14, 162-169, LNM 124 (1970)

Quelques inégalités sur les martingales, d'après Dubins et Freedman (Martingale theory)

The original paper appeared in

Comment: Though the proofs are very clever, so much work has been devoted to martingale inequalities since the paper was written that it is probably obsolete

Keywords: Inequalities

Nature: Exposition

Retrieve article from Numdam

IV: 19, 240-282, LNM 124 (1970)

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

Retrieve article from Numdam

V: 02, 17-20, LNM 191 (1971)

Démonstration de la ``Conjecture de Chung'' par Carleson (Markov processes, Independent increments)

Chung conjectured that singletons are polar sets for driftless subordinators. This paper gives Carleson's (unpublished) analytic proof of it

Comment: See Chung,

Keywords: Subordinators, Polar sets

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

V: 05, 58-75, LNM 191 (1971)

Introduction à l'étude des mouvements browniens à plusieurs paramètres (Gaussian processes)

Settles in particular a disagreement between statements of Lévy (

Comment: More recent work of Cartier on covariances appeared in the L.~Schwartz volume

Keywords: Several parameter Brownian motions, Covariance

Nature: New exposition of known results

Retrieve article from Numdam

V: 06, 76-76, LNM 191 (1971)

A simple proof of Doob's convergence theorem (Potential theory)

Doob's theorem is a version of the main convergence theorem of potential theory: the limit of a decreasing sequence of excessive functions differs of its regularized version on a semi-polar set

Comment: It is also shown that a function $f$ satisfying $f\ge P_Kf$ for all compact sets $K$ differs from its regularized function on a semi-polar set

Keywords: Excessive functions, Semi-polar sets

Nature: New exposition of known results

Retrieve article from Numdam

V: 10, 87-102, LNM 191 (1971)

Les théorèmes de Mazurkiewicz-Sierpinski et de Lusin (Descriptive set theory)

Synthetic presentation of (then) little known results on the perfect kernels of closed random sets and uniformization of random sets with countable sections

Comment: See Dellacherie-Meyer,

Keywords: Analytic sets, Random sets, Section theorems

Nature: New exposition of known results

Retrieve article from Numdam

V: 16, 170-176, LNM 191 (1971)

Sur un article de Dubins (Martingale theory)

Description of a Skorohod imbedding procedure for real valued r.v.'s due to Dubins (

Comment: This beautiful method to realize Skorohod's imbedding is related to that of Chacon and Walsh in 1002. For a deeper study see Bretagnolle 802. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

V: 19, 196-208, LNM 191 (1971)

Représentation intégrale des fonctions excessives. Résultats de Mokobodzki (Markov processes, Potential theory)

Main result: the convex cone of excessive functions for a resolvent which satisfies the absolute continuity hypothesis is the union of convex compact metrizable ``hats''\ in a suitable topology, and therefore has the integral representation property. The original proof of Mokobodzki, self-contained and unpublished, is given here

Comment: See Mokobodzki's work on cones of potentials,

Keywords: Minimal excessive functions, Martin boundary, Integral representations

Nature: Exposition

Retrieve article from Numdam

V: 20, 209-210, LNM 191 (1971)

Un théorème sur la répartition des temps locaux (Markov processes)

Kesten discovered that the value at a terminal time $T$ of the local time $L$ of a Markov process $X$ at a single point has an exponential distribution, and that $X_T$ and $L_T$ are independent. A short proof is given

Comment: The result can be deduced from excursion theory

Keywords: Local times

Nature: New exposition of known results

Retrieve article from Numdam

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

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: 25, 270-274, LNM 191 (1971)

Balayage pour les processus de Markov continus à droite, d'après C.T. Shih (Markov processes, Potential theory)

Hunt's fundamental theorem on balayage gives the probabilistic interpretation of the réduite of an excessive function set on a set. It was proved originally within the special class of Hunt's processes, then extended to ``standard'' processes. Using a method of compactification, Shih (

Comment: Shih's paper is the origin of the general definition of ``right processes''

Keywords: Excessive functions, Réduite

Nature: Exposition

Retrieve article from Numdam

VI: 04, 72-89, LNM 258 (1972)

Un principe de sous-suites dans la théorie des probabilités (Measure theory)

This paper is devoted to results of the following kind: any sequence of random variables with a given weak property contains a subsequence which satisfies a stronger property. An example is due to Komlós: any sequence bounded in $L^1$ contains a subsequence which converges a.s. in the Cesaro sense. Several results of this kind, mostly due to the author, are presented without detailed proofs

Comment: See 1302 for extensions to the case of Banach space valued random variables. See also Aldous,

Keywords: Subsequences, Central limit theorem, Law of the iterated logarithm

Nature: Exposition

Retrieve article from Numdam

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

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

VI: 13, 151-158, LNM 258 (1972)

Les résultats récents de Burkholder, Davis et Gundy (Martingale theory)

The well-known norm equivalence between the maximum and the square-function of a martingale in moderate Orlicz spaces is presented following the celebrated papers of Burkholder-Gundy (

Keywords: Burkholder inequalities, Moderate convex functions

Nature: Exposition

Retrieve article from Numdam

VII: 06, 48-50, LNM 321 (1973)

Une démonstration du théorème de Souslin-Lusin (Descriptive set theory)

The basic fact that the image of a Borel set under an injective Borel mapping is Borel is deduced from a separation theorem concerning countably many disjoint analytic sets

Comment: This is a step in the author's simplification of the proofs of the great theorems on analytic and Borel sets. See

Keywords: Borel sets, Analytic sets, Separation theorem

Nature: New exposition of known results

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

VII: 15, 146-154, LNM 321 (1973)

Chirurgie sur un processus de Markov, d'après Knight et Pittenger (Markov processes)

This paper presents a remarkable result of Knight and Pittenger, according to which excising from the sample path of a Markov process all the excursions from a given set $A$ which meet another set $B$ preserves the Markov property

Comment: The original paper appeared in

Keywords: Transformations of Markov processes, Excursions

Nature: Exposition

Retrieve article from Numdam

VII: 17, 172-179, LNM 321 (1973)

Application de l'exposé précédent aux processus de Markov (Markov processes)

This paper is devoted to results of J.F. Mertens on optimal stopping and strongly supermedian functions for a right Markov process (

Comment: See related papers by Mertens in

Keywords: Excessive functions, Supermedian functions, Réduite

Nature: Exposition, Original proofs

Retrieve article from Numdam

VII: 18, 180-197, LNM 321 (1973)

Résultats d'Azéma en théorie générale des processus (General theory of processes)

This paper presents several results from a paper of Azéma (

Comment: These results have been included (with their optional counterpart, whose interest was discovered later) in Dellacherie-Meyer,

Keywords: Optimal stopping, Previsible processes

Nature: Exposition

Retrieve article from Numdam

VII: 19, 198-204, LNM 321 (1973)

Limites médiales d'après Mokobodzki (Measure theory, Functional analysis)

Given a sequence of (classes of) random variables on a probability space which converges in some of the standard ways of measure theory, the problem is to find some universal method (independent from the underlying probability) to identify its limit. For convergence in probability, and thus for all strong $L^p$ topologies, Mokobodzki had discovered the procedure of rapid ultrafilters (see 304). The same problem is now solved for weak convergences, using a special kind of Banach limits

Comment: The paper contains a few annoying misprints, in particular p.199 line 9

Keywords: Continuum axiom, Weak convergence of r.v.'s, Medial limit

Nature: Exposition

Retrieve article from Numdam

VII: 21, 210-216, LNM 321 (1973)

Note sur l'interprétation des mesures d'équilibre (Markov processes)

Let $(X_t)$ be a transient Markov process (we omit the detailed assumptions) with a potential density $u(x,y)$. Let $\mu$ be the measure whose potential is the equilibrium potential of a set $A$. Then the distribution of the process at the last exit time from $A$ is given by $$E_x[f\circ X_{L-}, 0<L<\infty]=\int u(x,y)\,f(y)\,\mu(dy)$$ This formula, due to Chung, is deduced under minimal duality hypotheses from a general formula of Azéma, and a well-known theorem on Revuz measures

Keywords: Equilibrium potentials, Last exit time, Revuz measures

Nature: Exposition, Original proofs

Retrieve article from Numdam

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

Retrieve article from Numdam

VII: 23, 223-247, LNM 321 (1973)

Sur un problème de filtration (General theory of processes)

This is an attempt to present, in the usual language of the seminar, a a simple case from filtering theory (additive white noise in one dimension). The results proved are the Kallianpur-Striebel formula (

Keywords: Filtering theory, Innovation

Nature: Exposition

Retrieve article from Numdam

VII: 25, 273-283, LNM 321 (1973)

Fonctionnelles multiplicatives opératrices (Markov processes)

This paper presents results due to the author (

Comment: This paper explores the same idea as Jacod (

Keywords: Multiplicative functionals, Multiplicative kernels

Nature: Exposition

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

VIII: 16, 290-309, LNM 381 (1974)

Noyaux multiplicatifs (Markov processes)

This paper presents results due to Jacod (

Comment: Though the paper is fairly technical, it does not improve substantially on Jacod's results

Keywords: Multiplicative kernels, Semimarkovian processes

Nature: Exposition

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: 04, 206-212, LNM 465 (1975)

Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)

A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities

Keywords: Inequalities, Burkholder inequalities

Nature: Exposition

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

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IX: 15, 336-372, LNM 465 (1975)

Ensembles analytiques, théorèmes de séparation et applications (Descriptive set theory)

According to the standard (``first'') separation theorem, in a compact metric space or any space which is Borel isomorphic to it, two disjoint analytic sets can be separated by Borel sets, and in particular any bianalytic set (analytic and coanalytic i.e., complement of analytic) is Borel. Not so in general metric spaces. That the same statement holds in full generality with ``bianalytic'' instead of ``Borel'' is the second separation theorem, which according to the general opinion was considered much more difficult than the first. This result and many more (on projections of Borel sets with compact sections or countable sections, for instance) are fully proved in this exposition

Comment: See also the next paper 916, the set of lectures by Dellacherie in C.A. Rogers,

Keywords: Second separation theorem

Nature: Exposition

Retrieve article from Numdam

IX: 21, 425-436, LNM 465 (1975)

Primitive d'une mesure sur les compacts d'un espace métrique (Measure theory)

It is well known that the set ${\cal K}$ of all compact subsets of a compact metric space has a natural compact metric topology. The ``distribution function'' of a positive measure on ${\cal K}$ associates with every $A\in{\cal K}$ the measure of the subset $\{K\subset A\}$ of ${\cal K}$. It is shown here (following A.~Revuz,

Comment: This elegant theorem apparently never had applications

Keywords: Distribution functions on ordered spaces

Nature: Exposition

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IX: 27, 486-493, LNM 465 (1975)

Surlois d'entrée (Markov processes)

The results presented here are due to T. Leviatan (

Comment: This beautiful paper was superseded by the (slightly later) fundamental paper of Kuznetsov (1974)

Keywords: Entrance laws, Creation of mass, Kuznetsov measures

Nature: Exposition

Retrieve article from Numdam

IX: 34, 530-533, LNM 465 (1975)

Sur la démonstration de prévisibilité de Chung et Walsh (Markov processes)

A new proof of the result that for a Hunt process, the previsible stopping times are exactly those at which the process does not jump was given by Chung-Walsh (

Comment: The contents of this paper became Chapter XIV

Keywords: Hunt processes, Previsible times

Nature: Exposition

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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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

XI: 10, 109-119, LNM 581 (1977)

Convergence faible de processus, d'après Mokobodzki (General theory of processes)

The following simple question of Benveniste was answered positively by Mokobodzki (

Keywords: Weak convergence in $L^1$

Nature: Exposition

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

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

XI: 38, 539-565, LNM 581 (1977)

Désintégration d'une probabilité. Statistiques exhaustives (Measure theory)

This is a detailed review (including proofs) of the problem of existence of conditional distributions

Keywords: Disintegration of measures, Conditional distributions, Sufficient statistics

Nature: Exposition

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XI: 39, 566-573, LNM 581 (1977)

Information associée à un semigroupe (Markov processes)

This paper contains the proof of two important theorems of Donsker and Varadhan (

Nature: Exposition

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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

Retrieve article from Numdam

XII: 34, 468-481, LNM 649 (1978)

Quelques remarques sur un article de Donsker et Varadhan (Markov processes)

This paper is a partial exposition of the celebrated papers of Donsker and Varadhan in

Keywords: Large deviations

Nature: Exposition

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XII: 57, 763-769, LNM 649 (1978)

La formule d'Ito pour le mouvement brownien, d'après Brosamler (Brownian motion, Stochastic calculus)

This paper presents the results of a paper by Brosamler (

Keywords: Ito formula

Nature: Exposition

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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: 23, 253-259, LNM 721 (1979)

Sur les intégrales stochastiques de L.C. Young (Stochastic calculus)

This is a partial exposition of a theory of stochastic integration due to L.C. Young (

Keywords: Stochastic integrals

Nature: Exposition

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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: 31, 371-377, LNM 721 (1979)

Inégalités de convexité pour les processus croissants et les sousmartingales (Martingale theory)

Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,

Keywords: Martingale inequalities, Convex functions

Nature: Exposition, Original proofs

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XIII: 52, 611-613, LNM 721 (1979)

Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (Martingale theory)

In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (

Keywords: Doob's inequality, Stochastic differential equations

Nature: Exposition

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XIII: 54, 620-623, LNM 721 (1979)

Caractérisation des semimartingales, d'après Dellacherie (Stochastic calculus)

This short paper contains the proof of a very important theorem, due to Dellacherie (with the crucial help of Mokobodzki for the functional analytic part). Namely, semimartingales are exactly the processes which give rise to a nice vector measure on the previsible $\sigma$-field, with values in the (non locally convex) space $L^0$. It is only fair to say that this direction was initiated by Métivier and Pellaumail, and that the main result was independently discovered by Bichteler,

Comment: An important lemma which simplifies the proof and has other applications is given by Yan in 1425

Keywords: Semimartingales, Stochastic integrals

Nature: Exposition

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XIII: 55, 624-624, LNM 721 (1979)

Un exemple de J. Pitman (General theory of processes)

The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale $X$, namely martingales of the form $Z_{g_t}X_t$ where $Z$ is previsible. Taking $X$ to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form

Comment: The general problem of finding all martingales which vanish on the zeros of a given continuous martingale is discussed by Azéma and Yor in 2622

Keywords: Balayage, Balayage formula

Nature: Exposition

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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: 20, 173-188, LNM 784 (1980)

Les résultats de Jeulin sur le grossissement des tribus (General theory of processes, Stochastic calculus)

This is an introduction to beautiful results of Jeulin on enlargements, for which see

Comment: See also 1329, 1350

Keywords: Enlargement of filtrations, Semimartingales

Nature: Exposition

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XIV: 24, 209-219, LNM 784 (1980)

Remarques sur l'intégrale stochastique (Stochastic calculus)

This is an exposition of stochastic integrals and stochastic differential equations for Banach space valued processes along the lines of Métivier-Pellaumail

Keywords: Stochastic integrals, Stochastic differential equations, Métivier-Pellaumail inequality

Nature: Exposition

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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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XV: 28, 388-398, LNM 850 (1981)

Sur les travaux de Krylov en théorie de l'intégrale stochastique (Martingale theory)

The well-known work of Malliavin deals with the existence of smooth densities for solutions of stochastic differential equations with smooth coefficients satisfying a hypoellipticity condition. N.V.~Krylov's earlier work (among many papers see

Keywords: Stochastic integrals, Existence of densities

Nature: Exposition

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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: 07, 133-133, LNM 920 (1982)

Appendice : Un résultat de D. Williams (Malliavin's calculus)

This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as ``quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process

Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis

Nature: Exposition

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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: 11, 153-158, LNM 920 (1982)

Interpolation entre espaces d'Orlicz (Functional analysis)

This is an exposition of Calderon's complex interpolation method, in the case of moderate Orlicz spaces, aiming at its application in 1609

Keywords: Interpolation, Orlicz spaces, Moderate convex functions

Nature: Exposition

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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: 20, 187-193, LNM 986 (1983)

Le théorème de convergence des martingales dans les variétés riemanniennes, d'après R.W. Darling et W.A. Zheng (Stochastic differential geometry)

Exposition of two results on the asymptotic behaviour of martingales in a Riemannian manifold: First, Darling's theorem says that on the event where the Riemannian quadratic variation $<X,X>_\infty$ of a martingale $X$ is finite, $X_\infty$ exists in the Aleksandrov compactification of $V$. Second, Zheng's theorem asserts that on the event where $X_\infty$ exists in $V$, the Riemannian quadratic variation $<X,X>_\infty$ is finite

Comment: Darling's result is in

Keywords: Martingales in manifolds

Nature: Exposition

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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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XLIII: 05, 127-186, LNM 2006 (2011)

On standardness and I-cosiness (Filtrations)

Keywords: Filtrations, Cosiness

Nature: Original, Exposition

XLIII: 08, 215-219, LNM 2006 (2011)

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition

XLIV: 14, 279-280, LNM 2046 (2012)

On Peacocks: a general introduction to two articles (Theory of processes)

Keywords: Processes increasing in the convex order: peacocks

Nature: Exposition