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V: 01, 1-16, LNM 191 (1971)

**ARTZNER, Philippe**

Fonctions caractéristiques et mesures planes invariantes par rotation (Miscellanea)

A study of the class of probability measures on the line which are projections of a measure on the plane invariant by rotation

Keywords: Characteristic functions

Nature: Original

Retrieve article from Numdam

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 This paper gives Carleson's (unpublished) analytic proof of it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

Retrieve article from Numdam

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

Retrieve article from Numdam

V: 04, 37-57, LNM 191 (1971)

**CAIROLI, Renzo**

Décomposition de processus à indices doubles (Several parameter processes)

A discrete submartingale is decomposed into an increasing process and three different kinds of ``martingales''. Extension to continuous time. Earlier than the fundamental paper of Cairoli-Walsh (*Acta Math.*, **134**, 1975)

Comment: See Cairoli 401

Keywords: Two-parameter martingales

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

V: 07, 77-81, LNM 191 (1971)

**DELLACHERIE, Claude**

Quelques commentaires sur les prolongements de capacités (Descriptive set theory)

Remarks on the extension of capacities from sets to functions. Probably superseded by the work of Mokobodzki on functional capacities

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

Keywords: Capacities

Nature: Original

Retrieve article from Numdam

V: 08, 82-85, LNM 191 (1971)

**DELLACHERIE, Claude**

Une démonstration du théorème de séparation des ensembles analytiques (Descriptive set theory)

The first separation theorem can be deduced from Choquet's capacity theorem

Comment: Starting point in Sion,*Ann. Inst. Fourier,* **13**, 1963. This proof has become standard, see Dellacherie-Meyer, *Probabilités et Potentiel,* Chap. III

Keywords: Analytic sets, Capacities, Separation theorem

Nature: Original

Retrieve article from Numdam

V: 09, 86-86, LNM 191 (1971)

**DELLACHERIE, Claude**

Correction à ``Ensembles Aléatoires II'' (Descriptive set theory)

Correction to Dellacherie 306

Comment: See Dellacherie 511

Keywords: Sierpinski's ``rabotages''

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

V: 11, 103-126, LNM 191 (1971)

**DELLACHERIE, Claude**

Ensembles pavés et rabotages (Descriptive set theory)

A systematic study of the ``rabotages de Sierpinski'', used in Dellacherie 306 to solve several problems in probabilistic potential theory. The main paper on this subject

Comment: See Dellacherie,*Capacités et Processus Stochastiques,* 1970. Author should be consulted on recent developments (see 1526)

Keywords: Analytic sets, Capacities, Sierpinski's ``rabotages''

Nature: Original

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V: 12, 127-137, LNM 191 (1971)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**

Un contre-exemple au problème des laplaciens approchés (Martingale theory)

The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years

Comment: Problem originated in Meyer,*Ill. J. Math.*, **7**, 1963

Keywords: Submartingales, Supermartingales

Nature: Original

Retrieve article from Numdam

V: 13, 138-140, LNM 191 (1971)

**DOLÉANS-DADE, Catherine**

Une martingale uniformément intégrable, non localement de carré intégrable (Martingale theory)

Now well known! This paper helped to set the basic notions of the theory

Keywords: Square integrable martingales

Nature: Original

Retrieve article from Numdam

V: 14, 141-146, LNM 191 (1971)

**DOLÉANS-DADE, Catherine**

Intégrales stochastiques par rapport à une famille de probabilités (Stochastic calculus)

Given a family of probability laws on the same space, construct versions of stochastic integrals which do not depend on the law

Comment: Expanded by Stricker-Yor, Calcul stochastique dépendant d'un paramètre,*Z. für W-theorie,* **45**, 1978

Keywords: Stochastic integrals

Nature: Original

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V: 15, 147-169, LNM 191 (1971)

**MAISONNEUVE, Bernard**

Ensembles régénératifs, temps locaux et subordinateurs (General theory of processes, Renewal theory)

New approach to the theory of regenerative sets (Kingman; Krylov-Yushkevic 1965, Hoffmann-Jørgensen,*Math. Scand.*, **24**, 1969), including a general definition of local time of a random set

Comment: See Meyer 412, Morando-Maisonneuve 413, later work of Maisonneuve in 813 and later

Keywords: Local times, Subordinators, Renewal theory

Nature: Original

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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

Retrieve article from Numdam

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

Retrieve article from Numdam

V: 21, 211-212, LNM 191 (1971)

**MEYER, Paul-André**

Deux petits résultats de théorie du potentiel (Potential theory)

Excessive functions are characterized by their domination property over potentials. The strong ordering relation between two functions is carried over to their réduites

Comment: See Dellacherie-Meyer*Probability and Potentials,* Chapter XII, \S2

Keywords: Excessive functions, Réduite, Strong ordering

Nature: Original

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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

Retrieve article from Numdam

V: 24, 251-269, LNM 191 (1971)

**MEYER, Paul-André**

Solutions de l'équation de Poisson dans le cas récurrent (Potential theory, Markov processes)

The problem is to solve the Poisson equation for measures, $\mu-\mu P=\theta$ for given $\theta$, in the case of a recurrent transition kernel $P$. Here a ``filling scheme'' technique is used

Comment: The paper was motivated by Métivier (*Ann. Math. Stat.*, **40**, 1969) and is completely superseded by one of Revuz (*Ann. Inst. Fourier,* **21**, 1971)

Keywords: Recurrent potential theory, Filling scheme, Harris recurrence, Poisson equation

Nature: Original

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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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V: 26, 275-277, LNM 191 (1971)

**REVUZ, Daniel**

Remarque sur les potentiels de mesure (Markov processes, Potential theory)

The standard proof of the equivalence between semi-polar sets being polar and a very precise domination principle (Blumenthal-Getoor,*Markov Processes and Potential Theory,* 1968) uses the assumption that excessive functions are lower semicontinuous. This assumption is weakened

Comment: To be asked

Keywords: Polar sets, Semi-polar sets, Excessive functions

Nature: Original

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V: 27, 278-282, LNM 191 (1971)

**SAM LAZARO, José de**; **MEYER, Paul-André**

Une remarque sur le flot du mouvement brownien (Brownian motion, Ergodic theory)

It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense

Comment: See Sam Lazaro-Meyer,*Z. für W-theorie,* **18**, 1971

Keywords: Brownian motion, Wiener chaos, Screw-lines

Nature: Original

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V: 28, 283-289, LNM 191 (1971)

**WALSH, John B.**

Two footnotes to a theorem of Ray (Markov processes)

Ray's theorem (*Ann. of Math.*, **70**, 1959) is the construction of a good semigroup (and process) from a Ray resolvent. The first ``footnote'' gives the construction of a second semigroup with nice properties from the left instead of the right side. The second ``footnote'' studies the filtration of a Ray process

Comment: See Meyer-Walsh,*Invent. Math.*, **14**, 1971

Keywords: Ray compactification

Nature: Original

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V: 29, 290-310, LNM 191 (1971)

**WALSH, John B.**

Some topologies connected with Lebesgue measure (Markov processes, General theory of processes, Potential theory)

It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called*essential topology,* used in the paper of Chung and Walsh 522 in the same volume

Comment: See Doob*Bull. Amer. Math. Soc.*, **72**, 1966. An important application in given by Walsh 623 in the next volume. See the paper 1025 of Benveniste. For the use of a different topology see Ito *J. Math. Soc. Japan,* **20**, 1968

Keywords: Essential topology

Nature: Original

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V: 30, 311-341, LNM 191 (1971)

**WATANABE, Takesi**

On balayées of excessive measures and functions with respect to resolvents (Potential theory)

A general study of balayage of excessive measures as dual to réduite of excessive functions, first for a single kernel, then for a resolvent on a measurable space, and finally for a standard process

Comment: See Kunita and T. Watanabe,*Ill. J. Math.*, **9**, 1965. For the modern theory of balayage of measures (using Kuznetsov's processes) see Getoor, *Excessive Measures,* 1990, Chapter 4

Keywords: Excessive measures, Balayage

Nature: Original

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V: 31, 342-346, LNM 191 (1971)

**WEIL, Michel**

Décomposition d'un temps terminal (Markov processes)

It is shown that for a Hunt process, a terminal time can be represented as the infimum of a previsible terminal time, and a totally inaccessible terminal time

Keywords: Terminal times

Nature: Original

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V: 32, 347-361, LNM 191 (1971)

**WEIL, Michel**

Quasi-processus et énergie (Markov processes, Potential theory)

The energy of an excessive function $f$ with respect to an excessive measure $\xi$ has a simple proba\-bi\-listic interpretation if $\xi$ is is the potential of a measure $\mu$ and $f$ is the potential of an additive functional $(A_t)$, as ${1\over2}E_\mu[A_\infty^2]$. If $\xi$ is not a potential, still it can be associated with it a quasi-process (see Weil 418) with a birthtime $b$ and a death time $d$, and the formal expression ${1\over2}E[(A_d-A_b)^2]$ is given a precise meaning and represents the energy

Comment: This subject has been renewed by the introduction of Kuznetsov's measures. See Fitzsimmons*Sem. Stoch. Proc.*, 1987

Keywords: Hunt quasi-processes, Energy

Nature: Original

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V: 33, 362-372, LNM 191 (1971)

**WEIL, Michel**

Conditionnement par rapport au passé strict (Markov processes)

Given a totally inaccessible terminal time $T$, it is shown how to compute conditional expectations of the future with respect to the strict past $\sigma$-field ${\cal F}_{T-}$. The formula involves the Lévy system of the process

Comment: B. Maisonneuve pointed out once that the paper, though essentially correct, has a small mistake somewhere. See Dellacherie-Meyer,*Probabilité et Potentiels,* Chap. XX **46**--48

Keywords: Terminal times, Lévy systems

Nature: Original

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Fonctions caractéristiques et mesures planes invariantes par rotation (Miscellanea)

A study of the class of probability measures on the line which are projections of a measure on the plane invariant by rotation

Keywords: Characteristic functions

Nature: Original

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: 04, 37-57, LNM 191 (1971)

Décomposition de processus à indices doubles (Several parameter processes)

A discrete submartingale is decomposed into an increasing process and three different kinds of ``martingales''. Extension to continuous time. Earlier than the fundamental paper of Cairoli-Walsh (

Comment: See Cairoli 401

Keywords: Two-parameter martingales

Nature: Original

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: 07, 77-81, LNM 191 (1971)

Quelques commentaires sur les prolongements de capacités (Descriptive set theory)

Remarks on the extension of capacities from sets to functions. Probably superseded by the work of Mokobodzki on functional capacities

Comment: See Dellacherie-Meyer,

Keywords: Capacities

Nature: Original

Retrieve article from Numdam

V: 08, 82-85, LNM 191 (1971)

Une démonstration du théorème de séparation des ensembles analytiques (Descriptive set theory)

The first separation theorem can be deduced from Choquet's capacity theorem

Comment: Starting point in Sion,

Keywords: Analytic sets, Capacities, Separation theorem

Nature: Original

Retrieve article from Numdam

V: 09, 86-86, LNM 191 (1971)

Correction à ``Ensembles Aléatoires II'' (Descriptive set theory)

Correction to Dellacherie 306

Comment: See Dellacherie 511

Keywords: Sierpinski's ``rabotages''

Nature: Original

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: 11, 103-126, LNM 191 (1971)

Ensembles pavés et rabotages (Descriptive set theory)

A systematic study of the ``rabotages de Sierpinski'', used in Dellacherie 306 to solve several problems in probabilistic potential theory. The main paper on this subject

Comment: See Dellacherie,

Keywords: Analytic sets, Capacities, Sierpinski's ``rabotages''

Nature: Original

Retrieve article from Numdam

V: 12, 127-137, LNM 191 (1971)

Un contre-exemple au problème des laplaciens approchés (Martingale theory)

The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years

Comment: Problem originated in Meyer,

Keywords: Submartingales, Supermartingales

Nature: Original

Retrieve article from Numdam

V: 13, 138-140, LNM 191 (1971)

Une martingale uniformément intégrable, non localement de carré intégrable (Martingale theory)

Now well known! This paper helped to set the basic notions of the theory

Keywords: Square integrable martingales

Nature: Original

Retrieve article from Numdam

V: 14, 141-146, LNM 191 (1971)

Intégrales stochastiques par rapport à une famille de probabilités (Stochastic calculus)

Given a family of probability laws on the same space, construct versions of stochastic integrals which do not depend on the law

Comment: Expanded by Stricker-Yor, Calcul stochastique dépendant d'un paramètre,

Keywords: Stochastic integrals

Nature: Original

Retrieve article from Numdam

V: 15, 147-169, LNM 191 (1971)

Ensembles régénératifs, temps locaux et subordinateurs (General theory of processes, Renewal theory)

New approach to the theory of regenerative sets (Kingman; Krylov-Yushkevic 1965, Hoffmann-Jørgensen,

Comment: See Meyer 412, Morando-Maisonneuve 413, later work of Maisonneuve in 813 and later

Keywords: Local times, Subordinators, Renewal theory

Nature: Original

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: 21, 211-212, LNM 191 (1971)

Deux petits résultats de théorie du potentiel (Potential theory)

Excessive functions are characterized by their domination property over potentials. The strong ordering relation between two functions is carried over to their réduites

Comment: See Dellacherie-Meyer

Keywords: Excessive functions, Réduite, Strong ordering

Nature: Original

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: 24, 251-269, LNM 191 (1971)

Solutions de l'équation de Poisson dans le cas récurrent (Potential theory, Markov processes)

The problem is to solve the Poisson equation for measures, $\mu-\mu P=\theta$ for given $\theta$, in the case of a recurrent transition kernel $P$. Here a ``filling scheme'' technique is used

Comment: The paper was motivated by Métivier (

Keywords: Recurrent potential theory, Filling scheme, Harris recurrence, Poisson equation

Nature: Original

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

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V: 26, 275-277, LNM 191 (1971)

Remarque sur les potentiels de mesure (Markov processes, Potential theory)

The standard proof of the equivalence between semi-polar sets being polar and a very precise domination principle (Blumenthal-Getoor,

Comment: To be asked

Keywords: Polar sets, Semi-polar sets, Excessive functions

Nature: Original

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V: 27, 278-282, LNM 191 (1971)

Une remarque sur le flot du mouvement brownien (Brownian motion, Ergodic theory)

It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense

Comment: See Sam Lazaro-Meyer,

Keywords: Brownian motion, Wiener chaos, Screw-lines

Nature: Original

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V: 28, 283-289, LNM 191 (1971)

Two footnotes to a theorem of Ray (Markov processes)

Ray's theorem (

Comment: See Meyer-Walsh,

Keywords: Ray compactification

Nature: Original

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V: 29, 290-310, LNM 191 (1971)

Some topologies connected with Lebesgue measure (Markov processes, General theory of processes, Potential theory)

It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called

Comment: See Doob

Keywords: Essential topology

Nature: Original

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V: 30, 311-341, LNM 191 (1971)

On balayées of excessive measures and functions with respect to resolvents (Potential theory)

A general study of balayage of excessive measures as dual to réduite of excessive functions, first for a single kernel, then for a resolvent on a measurable space, and finally for a standard process

Comment: See Kunita and T. Watanabe,

Keywords: Excessive measures, Balayage

Nature: Original

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V: 31, 342-346, LNM 191 (1971)

Décomposition d'un temps terminal (Markov processes)

It is shown that for a Hunt process, a terminal time can be represented as the infimum of a previsible terminal time, and a totally inaccessible terminal time

Keywords: Terminal times

Nature: Original

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V: 32, 347-361, LNM 191 (1971)

Quasi-processus et énergie (Markov processes, Potential theory)

The energy of an excessive function $f$ with respect to an excessive measure $\xi$ has a simple proba\-bi\-listic interpretation if $\xi$ is is the potential of a measure $\mu$ and $f$ is the potential of an additive functional $(A_t)$, as ${1\over2}E_\mu[A_\infty^2]$. If $\xi$ is not a potential, still it can be associated with it a quasi-process (see Weil 418) with a birthtime $b$ and a death time $d$, and the formal expression ${1\over2}E[(A_d-A_b)^2]$ is given a precise meaning and represents the energy

Comment: This subject has been renewed by the introduction of Kuznetsov's measures. See Fitzsimmons

Keywords: Hunt quasi-processes, Energy

Nature: Original

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V: 33, 362-372, LNM 191 (1971)

Conditionnement par rapport au passé strict (Markov processes)

Given a totally inaccessible terminal time $T$, it is shown how to compute conditional expectations of the future with respect to the strict past $\sigma$-field ${\cal F}_{T-}$. The formula involves the Lévy system of the process

Comment: B. Maisonneuve pointed out once that the paper, though essentially correct, has a small mistake somewhere. See Dellacherie-Meyer,

Keywords: Terminal times, Lévy systems

Nature: Original

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