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 This paper gives Carleson's (unpublished) analytic proof of it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