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 functionsNature: 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 This paper gives Carleson's (unpublished) analytic proof of it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 setsNature: 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 setsNature: 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
401Keywords: Two-parameter martingalesNature: 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,
CovarianceNature: New exposition of known results Retrieve article from Numdam
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 setsNature: 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: CapacitiesNature: 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 theoremNature: 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
306Comment: See Dellacherie
511Keywords: 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 theoremsNature: 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 Retrieve article from Numdam
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,
SupermartingalesNature: 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 martingalesNature: 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 integralsNature: Original Retrieve article from Numdam
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 theoryNature: Original Retrieve article from Numdam
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 imbeddingNature: Exposition Retrieve article from Numdam
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 timesNature: 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 timeNature: Exposition,
Original additions Retrieve article from Numdam
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 representationsNature: Exposition Retrieve article from Numdam
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 timesNature: 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 orderingNature: Original Retrieve article from Numdam
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 semigroupsNature: Exposition,
Original additions Retrieve article from Numdam
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 imbeddingNature: 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 equationNature: Original Retrieve article from Numdam
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,
RéduiteNature: Exposition Retrieve article from Numdam
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 functionsNature: Original Retrieve article from Numdam
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-linesNature: Original Retrieve article from Numdam
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 compactificationNature: Original Retrieve article from Numdam
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 topologyNature: Original Retrieve article from Numdam
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,
BalayageNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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,
EnergyNature: Original Retrieve article from Numdam
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 systemsNature: Original Retrieve article from Numdam
