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170 matches found
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
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I: 07, 163-165, LNM 39 (1967)
MEYER, Paul-André
Sur un théorème de Deny (Potential theory, Measure theory)
In the potential theory of a resolvent which satisfies the absolute continuity hypothesis, every sequence of excessive functions contains a subsequence which converges except on a set of potential zero. It is also proved that a sequence which converges weakly in $L^1$ but not strongly must oscillate around its limit
Comment: a version of this result in classical potential theory was proved by Deny, C.R. Acad. Sci., 218, 1944. The cone of excessive functions possesses good compactness properties, discovered by Mokobodzki. See Dellacherie-Meyer, Probabilités et Potentiel, end of chapter XII
Keywords: A.e. convergence, Subsequences
Nature: Original
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II: 02, 22-33, LNM 51 (1968)
CARTIER, Pierre; MEYER, Paul-André; WEIL, Michel
Le retournement du temps~: compléments à l'exposé de M.~Weil (Markov processes)
In 108, M.~Weil had presented the work of Nagasawa on the time reversal of a Markov process at a L-time'' or return time. Here the results are improved on three points: a Markovian filtration is given for the reversed process; an analytic condition on the semigroup is lifted; finally, the behaviour of the coexcessive functions on the sample functions of the original process is investigated
Comment: The results of this paper have become part of the standard theory of time reversal. See 312 for a correction
Keywords: Time reversal, Dual semigroups
Nature: Original
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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: 09, 166-170, LNM 51 (1968)
MEYER, Paul-André
Une majoration du processus croissant associé à une surmartingale (Martingale theory)
Let $(X_t)$ be the potential generated by a previsible increasing process $(A_t)$. Then a norm equivalence in $L^p,\ 1<p<\infty$ is given between the random variables $X^\ast$ and $A_\infty$
Comment: This paper became obsolete after the $H^1$-$BMO$ theory
Keywords: Inequalities, Potential of an increasing process
Nature: Original
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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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II: 11, 175-199, LNM 51 (1968)
MEYER, Paul-André
Compactifications associées à une résolvante (Potential theory)
Let $E$ be a locally compact space, $(U_p)$ be a submarkovian resolvent, with a potential kernel $U=U_0$ which maps $C_k$ (the continuous functions with compact support) into continuous bounded functions. Let $F$ be a compact space containing $E$ as a dense subset, but inducing possibly a coarser topology. It is assumed that all potentials $Uf$ with $f\in C_k$ extend to continuous functions on $F$, and that points of $F$ are separated by continuous functions on $F$ whose restriction to $E$ is supermedian. Then it is shown how to extend the resolvent to $F$ and imitate the construction of a Ray semigroup and a strong Markov process. This was an attempt to compactify the space using only supermedian functions, not $p$-supermedian for all $p>0$. An application to Markov chains is given
Comment: This method of compactification suggested by Chung's boundary theory for Markov chains (similarly Doob, Trans. Amer. Math. Soc., 149, 1970) never superseded the standard Ray-Knight approach
Keywords: Resolvents, Ray compactification, Martin boundary, Boundary theory
Nature: Original
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III: 08, 143-143, LNM 88 (1969)
MEYER, Paul-André
Un lemme de théorie des martingales (Martingale theory)
The author apparently believed that this classical and useful remark was new (it is often called Hunt's lemma'', see Hunt, Martingales et Processus de Markov, Masson 1966, p.47)
Keywords: Almost sure convergence
Nature: Well-known
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III: 09, 144-151, LNM 88 (1969)
MEYER, Paul-André
Un résultat de théorie du potentiel (Potential theory, Markov processes)
Under strong duality hypotheses, it is shown that a measure which does not charge polar sets is equivalent to a measure whose Green potential is bounded
Comment: See Meyer, Processus de Markov, Lecture Notes in M. 26
Keywords: Green potentials, Dual semigroups
Nature: Original
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III: 10, 152-154, LNM 88 (1969)
MEYER, Paul-André
Un résultat élémentaire sur les temps d'arrêt (General theory of processes)
This useful result asserts that a stopping time is accessible if and only if its graph is contained in a countable union of graphs of previsible stopping times
Comment: Before this was noticed, accessible stopping times were considered important. After this remark, previsible stopping times came to the forefront
Keywords: Stopping times, Accessible times, Previsible times
Nature: Original
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III: 11, 155-159, LNM 88 (1969)
MEYER, Paul-André
Une nouvelle démonstration des théorèmes de section (General theory of processes)
The proof of the section theorems has improved over the years, from complicated-false to complicated-true, and finally to easy-true. This was a step on the way, due to Dellacherie (inspired by Cornea-Licea, Z. für W-theorie, 10, 1968)
Comment: This is essentially the definitive proof, using a general section theorem instead of capacity theory
Keywords: Section theorems, Optional processes, Previsible processes
Nature: Original
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III: 12, 160-162, LNM 88 (1969)
MEYER, Paul-André
Rectification à des exposés antérieurs (Markov processes, Martingale theory)
Corrections are given to the talk 202 by Cartier, Meyer and Weil and to the talk 106 by Meyer
Comment: This note introduces Walsh's fork'', the well-known strong Markov process whose dual is not strong Markov
Keywords: Time reversal, Stochastic integrals
Nature: Correction
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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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IV: 09, 77-107, LNM 124 (1970)
Intégrales stochastiques par rapport aux martingales locales (Martingale theory, Stochastic calculus)
This is a continuation of Meyer 106, with a new complete exposition of the theory, and two substantial improvements: the filtration is general (while in 106 it was assumed free of fixed times of discontinuity) and the definition of semimartingales is the modern one (while in 106 they were the special semimartingales of nowadays). The change of variables formula is given in its full generality
Comment: The results of this paper have become classical, and are reproduced almost literally in Meyer 1017
Keywords: Local martingales, Stochastic integrals, Change of variable formula
Nature: Original
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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: 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
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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
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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
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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
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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: 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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VI: 09, 109-112, LNM 258 (1972)
SAM LAZARO, José de; MEYER, Paul-André
Un gros processus de Markov. Application à certains flots (Markov processes)
In a vague but useful sense, a big'' process over a given process consists of random variables whose values are a part of the path of the original process (the best known example is the excursion process). Here it is shown how the past of a Markov process can be turned into a big (homogeneous) Markov process, and how its semigroup is computed using an idea of Dawson (Trans. Amer. Math. Soc., 131, 1968)
Comment: For a complete account of Dawson's formula, see Dellacherie-Meyer, Probabilités et Potentiel, \no XIV.45
Keywords: Prediction theory, Filtered flows
Nature: Original
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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
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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
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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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VI: 14, 159-163, LNM 258 (1972)
MEYER, Paul-André
Temps d'arrêt algébriquement prévisibles (General theory of processes)
The main results concern the natural filtration of a right continuous process taking values in a Polish spaces, and defined on a Blackwell space $\Omega$. Conditions are given on a process or a random variable on $\Omega$ which insure that it will be previsible or optional under any probability law on $\Omega$
Comment: The subject has been kept alive by Azéma, who used similar techniques in several papers
Keywords: Stopping times, Previsible processes
Nature: Original
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VI: 15, 164-167, LNM 258 (1972)
MEYER, Paul-André
Une note sur le théorème du balayage de Hunt (Markov processes, Potential theory)
The theorem involved is the characterization of the réduite $P_A u$ of an excessive function $u$ on a set $A$ as equal (except on a well-defined semipolar set) to the infimum of the excessive functions that dominate $u$ on $A$. This theorem is slightly improved under the absolute continuity hypothesis. The proof rests on the following property of the fine topology: every (nearly Borel) finely closed set is contained in a fine $G_\delta$ which differs from it by a polar set
Keywords: Réduite, Fine topology, Absolute continuity hypothesis
Nature: Original
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VI: 16, 168-172, LNM 258 (1972)
MEYER, Paul-André; WALSH, John B.
Un résultat sur les résolvantes de Ray (Markov processes)
This is a complement to the authors' paper on Ray processes in Invent. Math., 14, 1971: a lemma is proved on the existence of many martingales which are continuous whenever the process is continuous (a wrong reference for it was given in the paper). Then it is shown that the mapping $x\rightarrow P_x$ is continuous in the weak topology of measures, when the path space is given the topology of convergence in measure. Note that a correction is mentioned on the errata page of vol. VII
Comment: The idea of using the topology of convergence in measure on a path space turned out to be a fruitful idea; see Meyer and Zheng Ann. Inst. Henri Poincaré 20, 1984
Keywords: Ray compactification, Weak convergence of measures
Nature: Original
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VI: 23, 243-252, LNM 258 (1972)
MEYER, Paul-André
Quelques autres applications de la méthode de Walsh (La perfection en probabilités'') (Markov processes)
This is but an exercise on using the method of the preceding paper 622 to reduce the exceptional sets in other situations: additive functionals, cooptional times and processes, etc
Comment: A correction to this paper is mentioned on the errata list of vol. VII
Keywords: Additive functionals, Return times, Essential topology
Nature: Original
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VII: 02, 25-32, LNM 321 (1973)
MEYER, Paul-André
Une mise au point sur les systèmes de Lévy. Remarques sur l'exposé de A. Benveniste (Markov processes)
This is an addition to the preceding paper 701, extending the theory to right processes by means of a Ray compactification
Comment: All this material has become classical. See for instance Dellacherie-Meyer, Probabilités et Potentiel, vol. D, chapter XV, 31--35
Keywords: Lévy systems, Ray compactification
Nature: Original
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VII: 14, 136-145, LNM 321 (1973)
MEYER, Paul-André
Le dual de $H^1$ est $BMO$ (cas continu) (Martingale theory)
The basic results of Fefferman and Fefferman-Stein on functions of bounded mean oscillation in $R$ and $R^n$ and the duality between $BMO$ and $H^1$ were almost immediately translated into discrete martingale theory by Herz and Garsia. The next step, due to Getoor-Sharpe ({\sl Invent. Math.} 16, 1972), delt with continuous martingales. The extension to right continuous martingales, a good exercise in martingale theory, is given here
Comment: See 907 for a correction. This material has been published in book form, see for instance Dellacherie-Meyer, Probabilités et Potentiel, Vol. B, Chapter VII
Keywords: $BMO$, Hardy spaces, Fefferman inequality
Nature: Original
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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: 16, 155-171, LNM 321 (1973)
MEYER, Paul-André; TRAKI, Mohammed
Réduites et jeux de hasard (Potential theory)
This paper arose from an attempt (by the second author) to rewrite the results of Dubins-Savage How to Gamble if you Must in the language of standard (countably additive) measure theory, using the methods of descriptive set theory (analytic sets, section theorems, etc). The attempt is successful, since all general theorems can be proved in this set-up. More recent results in the same line, due to Strauch and Sudderth, are extended too. An appendix includes useful comments by Mokobodzki on the case of a gambling house consisting of a single kernel (discrete potential theory)
Comment: This material is reworked in Dellacherie-Meyer, Probabilités et Potentiel, Vol. C, Chapter X
Keywords: Balayage, Gambling house, Réduite, Optimal strategy
Nature: Original
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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
Nature: Exposition
Retrieve article from Numdam
VII: 20, 205-209, LNM 321 (1973)
MEYER, Paul-André
Remarques sur les hypothèses droites (Markov processes)
The following technical point is discussed: why does one assume among the right hypotheses'' that excessive functions are nearly-Borel?
Keywords: Right processes, Excessive functions
Nature: Original
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
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
VIII: 01, 1-10, LNM 381 (1974)
AZÉMA, Jacques; MEYER, Paul-André
Une nouvelle représentation du type de Skorohod (Markov processes)
A Skorohod imbedding theorem for general Markov processes is proved, in which the stopping time is a randomized left'' terminal time. A uniqueness result is proved
Comment: The result is deduced from a representation of measures by left additive functionals, due to Azéma (Invent. Math. 18, 1973 and this volume, 814). A general survey on the Skorohod embedding problem is Ob\lój, Probab. Surv. 1, 2004
Keywords: Skorohod imbedding, Multiplicative functionals
Nature: Original
Retrieve article from Numdam
VIII: 13, 172-261, LNM 381 (1974)
MAISONNEUVE, Bernard; MEYER, Paul-André
Ensembles aléatoires markoviens homogènes (5 talks) (Markov processes)
This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see 412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (Ann. Prob. 1, 1973; Indiana Math. J. 23, 1973). The basic and earlier paper of Dynkin on the same subject ( Teor. Ver. Prim. 16, 1971) was not known to the authors.\par Chapter III is devoted to the original work of Maisonneuve on incursions. Roughly, the incursion at time $t$ is trivial if $t\in M$, and if $t\notin M$ it consists of the post-$t$ part of the excursion straddling $t$. Thus the incursion process is a path valued, non adapted process. It is only adapted to the filtration ${\cal F}_{D_t}$ where $D_t$ is the first hitting time of $M$ after $t$. Contrary to the Ito theory of excursions, no change of time using a local time is performed. The main result is the fact that, if a suitable regeneration property is assumed only on the set $M$ then, in a suitable topology on the space of paths, this process is a right-continuous strong Markov process. Considerable effort is devoted to proving that it is even a right process (the technique is heavy and many errors have crept in, some of them corrected in 932-933).\par Chapter IV makes the connection between II and III: the main results of Chapter II are proved anew (without balayage or Laplace transforms): they amount to computing the Lévy system of the incursion process. Finally, Chapter V consists of applications, among which a short discussion of the boundary theory for Markov chains
Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as Systèmes Régénératifs, Astérisque, 15, 1974, a much simpler version of his own results, and discovered important improvements later on (some of which are included in Dellacherie-Maisonneuve-Meyer, Probabilités et Potentiel, Chapter XX, 1992). Along the slightly different line of Dynkin, see El~Karoui-Reinhard, Compactification et balayage de processus droits, Astérisque 21, 1975. A recent book on excursion theory is Blumenthal, Excursions of Markov Processes, Birkhäuser 1992
Keywords: Regenerative systems, Regenerative sets, Renewal theory, Local times, Excursions, Markov chains, Incursions
Nature: Original
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
Retrieve article from Numdam
VIII: 15, 289-289, LNM 381 (1974)
MEYER, Paul-André
Une note sur la compactification de Ray (Markov processes)
This short note shows that (contrary to the belief of Meyer and Walsh in a preceding paper) the state space of a Ray process is universally measurable in its Ray compactification
Comment: This is now considered a standard fact
Keywords: Ray compactification, Right processes
Nature: Original
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
VIII: 17, 310-315, LNM 381 (1974)
MEYER, Paul-André
Une représentation de surmartingales (Martingale theory)
Garsia asked whether every right continuous positive supermartingale $(X_t)$ bounded by $1$ is the optional projection of a (non-adapted) decreasing process $(D_t)$, also bounded by $1$. This problem is solved by an explicit formula, and a proof is sketched showing that, if boundedness is not assumed, the proper condition is $D_t\le X^{*}$
Comment: The exponential formula'' appearing in this paper was suggested by a more concrete problem in the theory of Markov processes, using a terminal time. Similar looking formulas occurs in multiplicative decompositions and in 801. For the much more difficult case of positive submartingales, see 1023 and above all Azéma, Z. für W-theorie, 45, 1978 and its exposition 1321
Keywords: Supermartingales, Multiplicative decomposition
Nature: Original
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
Retrieve article from Numdam
IX: 06, 226-236, LNM 465 (1975)
CHOU, Ching Sung; MEYER, Paul-André
Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels (General theory of processes)
Dellacherie has studied in 405 the filtration generated by a point process with one single jump. His study is extended here to the filtration generated by a discrete point process. It is shown in particular how to construct a martingale which has the previsible representation property
Comment: In spite or because of its simplicity, this paper has become a standard reference in the field. For a general account of the subject, see He-Wang-Yan, Semimartingale Theory and Stochastic Calculus, CRC~Press 1992
Keywords: Point processes, Previsible representation
Nature: Original
Retrieve article from Numdam
IX: 07, 237-238, LNM 465 (1975)
MEYER, Paul-André
Complément sur la dualité entre $H^1$ et $BMO$ (Martingale theory)
Fills a gap in the proof of the duality theorem in 714
Keywords: $BMO$
Nature: Correction
Retrieve article from Numdam
IX: 08, 239-245, LNM 465 (1975)
DELLACHERIE, Claude; MEYER, Paul-André
Un nouveau théorème de projection et de section (General theory of processes)
Optional section and projection theorems are proved without assuming the usual conditions'' on the filtration
Comment: This paper is obsolete. As stated at the end by the authors, the result could have been deduced from the general theorem in Dellacherie 705. The result takes its definitive form in Dellacherie-Meyer, Probabilités et Potentiel, theorems IV.84 of vol. A and App.1, \no~6
Keywords: Section theorems, Optional processes, Projection theorems
Nature: Original
Retrieve article from Numdam
IX: 16, 373-389, LNM 465 (1975)
DELLACHERIE, Claude; MEYER, Paul-André
Ensembles analytiques et temps d'arrêt (Descriptive set theory)
This is a sequel to the preceding paper 915. Instead of using the language of trees to prove the second separation theorem, a language more familiar to probabilists is used, in which the space of stopping times on $N^N$ is given a compact metric topology and the space of non-finite stopping times appears as the universal analytic, non-Borel set, from which all analytic sets can be constructed. Many proofs become very natural in this language
Comment: See also the next paper 917, 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, Stopping times
Nature: Original
Retrieve article from Numdam
IX: 24, 464-465, LNM 465 (1975)
MEYER, Paul-André
Une remarque sur la construction de noyaux (Measure theory)
With the notation of the preceding report 923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity
Comment: See Dellacherie 1030
Keywords: Pseudo-kernels, Regularization
Nature: Original
Retrieve article from Numdam
IX: 25, 466-470, LNM 465 (1975)
MEYER, Paul-André; YAN, Jia-An
Génération d'une famille de tribus par un processus croissant (General theory of processes)
The previsible and optional $\sigma$-fields of a filtration $({\cal F}_t)$ on $\Omega$ are studied without the usual hypotheses: no measure is involved, and the filtration is not right continuous. It is proved that if the $\sigma$-fields ${\cal F}_{t-}$ are separable, then so is the previsible $\sigma$-field, and the filtration is the natural one for a continuous strictly increasing process. A similar result is proved for the optional $\sigma$-field assuming $\Omega$ is a Blackwell space, and then every measurable adapted process is optional
Comment: Making the filtration right continuous generally destroys the separability of the optional $\sigma$-field
Keywords: Previsible processes, Optional processes
Nature: Original
Retrieve article from Numdam
IX: 32, 518-521, LNM 465 (1975)
MAISONNEUVE, Bernard; MEYER, Paul-André
Ensembles aléatoires markoviens homogènes. Mise au point et compléments (Markov processes)
This paper corrects or simplifies many details in the long paper 713 by the same authors
Keywords: Regenerative systems, Last-exit decompositions, Excursions
Nature: Original
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: 36, 555-555, LNM 465 (1975)
MEYER, Paul-André
Une remarque sur les processus de Markov (Markov processes)
It is shown that, under a fixed measure $P^{\mu}$, the optional processes and times relative to the uncompleted filtrations $({\cal F}_{t+}^{\circ})$ and $({\cal F}_{t}^{\circ})$ are undistinguishable from each other
Comment: No applications are known
Keywords: Stopping times
Nature: Original
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
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
Retrieve article from Numdam
X: 08, 104-117, LNM 511 (1976)
MEYER, Paul-André; YOR, Marc
Sur la théorie de la prédiction, et le problème de décomposition des tribus ${\cal F}^{\circ}_{t+}$ (General theory of processes)
This paper contains another version of Knight's theory (preceding paper 1007) for cadlag process instead of measurable processes. These results then are applied to the pathology of germ fields: a natural measurability conjecture does not hold, and an example is given of a process $X_t$ such that its natural $\sigma$-field ${\cal F}_{1+}$ is not generated by ${\cal F}_{1}$ and the germ-field at $0$ of the process $(X_{1+s})$
Comment: On the pathology of germ fields, see H. von Weizsäcker, Ann. Inst. Henri Poincaré, 19, 1983
Keywords: Prediction theory, Germ fields
Nature: Original
Retrieve article from Numdam
X: 09, 118-124, LNM 511 (1976)
MEYER, Paul-André
Generation of $\sigma$-fields by step processes (General theory of processes)
On a Blackwell measurable space, let ${\cal F}_t$ be a right continuous filtration, such that for any stopping time $T$ the $\sigma$-field ${\cal F}_T$ is countably generated. Then (discarding possibly one single null set), this filtration is the natural filtration of a right-continuous step process
Comment: This answers a question of Knight, Ann. Math. Stat., 43, 1972
Keywords: Point processes
Nature: Original
Retrieve article from Numdam
X: 10, 125-183, LNM 511 (1976)
MEYER, Paul-André
Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)
This long paper consists of four talks, suggested by E.M.~Stein's book Topics in Harmonic Analysis related to the Littlewood-Paley theory, Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $R^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $R^n\times R_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $R^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (CRAS Paris, 278A, 1974, p.1103) in potential theory and by Kunita (Nagoya M. J., 36, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false
Comment: This paper was rediscovered by Varopoulos (J. Funct. Anal., 38, 1980), and was then rewritten by Meyer in 1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in 1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912, and Meyer 1908 reporting on Cowling's extension of Stein's work. An erratum is given in 1253
Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory
Nature: Original
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
Retrieve article from Numdam
X: 23, 501-504, LNM 511 (1976)
MEYER, Paul-André; YOEURP, Chantha
Sur la décomposition multiplicative des sousmartingales positives (Martingale theory)
This paper expands part of Yoeurp's paper 1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$
Comment: See the comments on 1021 for the general case. The latter result is related to Meyer 817. For a related paper, see 1203. Further study in 1620
Keywords: Multiplicative decomposition
Nature: Original
Retrieve article from Numdam
X: 31, 578-578, LNM 511 (1976)
MEYER, Paul-André
Un point de priorité (Ergodic theory)
An important remark in Sam Lazaro-Meyer 901, on the relation between Palm measures and the Ambrose-Kakutani theorem that any flow can be interpreted as a flow under a function, was made earlier by F.~Papangelou (1970)
Keywords: Palm measures, Flow under a function, Ambrose-Kakutani theorem
Nature: Acknowledgment
Retrieve article from Numdam
XI: 08, 65-78, LNM 581 (1977)
EL KAROUI, Nicole; MEYER, Paul-André
Les changements de temps en théorie générale des processus (General theory of processes)
Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care
Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many I''s remained in spite of the final joint autorship. See the next paper 1109 for the discontinuous case
Keywords: Changes of time
Nature: Original
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
Retrieve article from Numdam
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
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 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
Retrieve article from Numdam
XI: 25, 383-389, LNM 581 (1977)
Une caractérisation de $BMO$ (Martingale theory)
Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention
Comment: Related subjects occur in 1328. The reference to note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible
Keywords: $BMO$, Stochastic exponentials, Martingale inequalities
Nature: Original
Retrieve article from Numdam
XI: 31, 446-481, LNM 581 (1977)
MEYER, Paul-André
Notes sur les intégrales stochastiques (Martingale theory)
This paper contains six additions to 1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times
Comment: Three errors are corrected in 1248 and 1249
Keywords: Stochastic integrals, Hilbert space valued martingales, Operator stochastic integrals, $BMO$
Nature: Original
Retrieve article from Numdam
XI: 32, 482-489, LNM 581 (1977)
MEYER, Paul-André
Sur un théorème de C. Stricker (Martingale theory)
Some emphasis is put on a technical lemma used by Stricker to prove the well-known result that semimartingales remain so under restriction of filtrations (provided they are still adapted). The result is that a semimartingale up to infinity can be sent into the Hardy space $H^1$ by a suitable choice of an equivalent measure. This leads also to a simple proof and an extension of Jacod's theorem that the set of semimartingale laws is convex
Comment: A gap in a proof is filled in 1251
Keywords: Hardy spaces, Changes of measure
Nature: Original
Retrieve article from Numdam
XII: 08, 57-60, LNM 649 (1978)
MEYER, Paul-André
Sur un théorème de J. Jacod (General theory of processes)
Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals
Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration
Keywords: Semimartingales, Enlargement of filtrations, Laws of semimartingales
Nature: Original
Retrieve article from Numdam
XII: 10, 70-77, LNM 649 (1978)
DELLACHERIE, Claude; MEYER, Paul-André
A propos du travail de Yor sur le grossissement des tribus (General theory of processes)
This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$
Keywords: Enlargement of filtrations, Honest times
Nature: Original
Retrieve article from Numdam
XII: 12, 98-113, LNM 649 (1978)
DELLACHERIE, Claude; MEYER, Paul-André; YOR, Marc
Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (Martingale theory, Functional analysis)
The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see 1519; see also 3021 and 3316)
Keywords: Hardy spaces, $BMO$
Nature: Original
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
Retrieve article from Numdam
XII: 30, 425-427, LNM 649 (1978)
DELLACHERIE, Claude; MEYER, Paul-André
Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (General theory of processes)
Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$
Keywords: Stopping times
Nature: Original
Retrieve article from Numdam
XII: 35, 482-488, LNM 649 (1978)
YOR, Marc; MEYER, Paul-André
Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (Measure theory)
Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability
Comment: The subject is discussed further in 1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the weak topology of $L[\infty$, or with the topology of $L^0$
Nature: Original
Retrieve article from Numdam
XII: 48, 739-739, LNM 649 (1978)
MEYER, Paul-André
Correction à Retour sur la représentation de $BMO$'' (Martingale theory)
Two errors in 1131 are corrected
Keywords: Stochastic integrals, $BMO$
Nature: Correction
Retrieve article from Numdam
XII: 49, 739-739, LNM 649 (1978)
MEYER, Paul-André
Correction à Caractérisation de $BMO$ par un opérateur maximal'' (Martingale theory)
Corrects an error in 1131
Keywords: Stochastic integrals, $BMO$
Nature: Correction
Retrieve article from Numdam
XII: 51, 740-740, LNM 649 (1978)
MEYER, Paul-André
Correction à Sur un théorème de C. Stricker'' (Stochastic calculus)
Fills a gap in a proof in 1132
Keywords: Stochastic integrals
Nature: Correction
Retrieve article from Numdam
XII: 53, 741-741, LNM 649 (1978)
MEYER, Paul-André
Correction à Inégalités de Littlewood-Paley'' (Applications of martingale theory, Markov processes)
This is an erratum to 1010
Keywords: Littlewood-Paley theory, Carré du champ, Infinitesimal generators, Semigroup theory
Nature: Correction
Retrieve article from Numdam
XII: 56, 757-762, LNM 649 (1978)
MEYER, Paul-André
Inégalités de normes pour les intégrales stochastiques (Stochastic calculus)
Inequalities of the following kind were introduced by Émery: $$\|X.M\|_{H^p}\le c_p \| X\|_{S^p}\,\| M\|$$ where the left hand side is a stochastic integral of the previsible process $X$ w.r.t. the semimartingale $M$, $S^p$ is a supremum norm, and the norm $H^p$ for semimartingales takes into account the Hardy space norm for the martingale part and the $L^p$ norm of the total variation for the finite variation part. On the right hand side, Émery had used a norm called $H^{\infty}$. Here a weaker $BMO$-like norm for semimartingales is suggested
Keywords: Stochastic integrals, Hardy spaces
Nature: Original
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
Retrieve article from Numdam
XII: 59, 775-803, LNM 649 (1978)
MEYER, Paul-André
Martingales locales fonctionnelles additives (two talks) (Markov processes)
The purpose of the paper is to specialize the standard theory of Hardy spaces of martingales to the subspaces of additive martingales of a Markov process. The theory is not complete: the dual of (additive) $H^1$ seems to be different from (additive) $BMO$
Nature: Original
Retrieve article from Numdam
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
Retrieve article from Numdam
XIII: 16, 204-215, LNM 721 (1979)
Un petit théorème de projection pour processus à deux indices (Several parameter processes)
This paper proves a previsible projection theorem in the case of two-parameter processes, with a two-parameter filtration satisfying the standard commutation property of Cairoli-Walsh. The idea is to apply successively the projection operation of 1315 to the two coordinates
Keywords: Previsible processes (several parameters), Previsible projections, Random measures
Nature: Original
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
Retrieve article from Numdam
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
Retrieve article from Numdam
XIII: 41, 478-487, LNM 721 (1979)
MEYER, Paul-André; STRICKER, Christophe; YOR, Marc
Sur une formule de la théorie du balayage (General theory of processes)
For the notation, see the review of 1340. It is shown here that under the same hypotheses, the semimartingale $Z_{g_t}X_t$ is a sum of three terms: the stochastic integral $\int_0^t \zeta_s dX_s$, where $\zeta$ is the previsible projection of $Z$, an explicit sum of jumps involving $Z-\zeta$, and a mysterious continuous process with finite variation $(R_t)$ such that $dR_t$ is carried by $H$, equal to $0$ if $Z$ was optional
Comment: See 1351, 1357
Keywords: Balayage, Balayage formula
Nature: Original
Retrieve article from Numdam
XIII: 42, 488-489, LNM 721 (1979)
MEYER, Paul-André
Construction de semimartingales s'annulant sur un ensemble donné (General theory of processes)
The title states exactly the subject of this short report, whose conclusion is: in the Brownian filtration, every closed optional set is the set of zeros of a continuous semimartingale
Keywords: Semimartingales
Nature: Original
Retrieve article from Numdam
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
Retrieve article from Numdam
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
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
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
Retrieve article from Numdam
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
Keywords: Enlargement of filtrations, Semimartingales
Nature: Exposition
Retrieve article from Numdam
XV: 05, 44-102, LNM 850 (1981)
MEYER, Paul-André
Géométrie stochastique sans larmes (Stochastic differential geometry)
Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is continuous semimartingales in manifolds, following L.~Schwartz (LN 780, 1980), but with additional features: an indication of J.M.~Bismut hinting to a definition of continuous martingales in a manifold, and the author's own interest on the forgotten intrinsic definition of the second differential $d^2f$ of a function. All this fits together into a geometric approach to semimartingales, and a probabilistic approach to such geometric topics as torsion-free connexions
Comment: A short introduction by the same author can be found in Stochastic Integrals, Springer LNM 851. The same ideas are expanded and presented in the supplement to Volume XVI and the book by Émery, Stochastic Calculus on Manifolds
Keywords: Semimartingales in manifolds, Martingales in manifolds, Transfer principle, Stochastic differential equations, Stochastic integrals, Stratonovich integrals
Nature: Original
Retrieve article from Numdam
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.
Retrieve article from Numdam
XV: 08, 142-142, LNM 850 (1981)
MEYER, Paul-André
Une question de théorie des processus (Stochastic calculus)
It is remarked that the stochastic integrals that appear in stochastic differential geometry are of a particular kind, and asked whether the theory could be developed for processes belonging to a larger class than semimartingales
Comment: For recent work in this area, see T. Lyons' article in Rev. Math. Iberoamericana 14 (1998) on differential equations driven by non-smooth functions
Keywords: Semimartingales
Nature: Open question
Retrieve article from Numdam
XV: 10, 151-166, LNM 850 (1981)
MEYER, Paul-André
Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)
The word original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (J. Funct. Anal., 38, 1980)
Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)
Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ
Nature: Original
Retrieve article from Numdam
XV: 41, 604-617, LNM 850 (1981)
LÉPINGLE, Dominique; MEYER, Paul-André; YOR, Marc
Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)
This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case
Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps
Nature: Original
Retrieve article from Numdam
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
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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: 09, 138-150, LNM 920 (1982)
BAKRY, Dominique; MEYER, Paul-André
Sur les inégalités de Sobolev logarithmiques (two parts) (Applications of martingale theory)
These two papers are variations on a paper of G.F. Feissner (Trans. Amer Math. Soc., 210, 1965). Let $\mu$ be a Gaussian measure, $P_t$ be the corresponding Ornstein-Uhlenbeck semigroup. Nelson's hypercontractivity theorem states (roughly) that $P_t$ is bounded from $L^p(\mu)$ to some $L^q(\mu)$ with $q\ge p$. In another celebrated paper, Gross showed this to be equivalent to a logarithmic Sobolev inequality, meaning that if a function $f$ is in $L^2$ as well as $Af$, where $A$ is the Ornstein-Uhlenbeck generator, then $f$ belongs to the Orlicz space $L^2Log_+L$. The starting point of Feissner was to translate this again as a result on the Riesz potentials'' of the semi-group (defined whenever $f\in L^2$ has integral $0$) $$R^{\alpha}={1\over \Gamma(\alpha)}\int_0^\infty t^{\alpha-1}P_t\,dt\;.$$ Note that $R^{\alpha}R^{\beta}=R^{\alpha+\beta}$. Then the theorem of Gross implies that $R^{1/2}$ is bounded from $L^2$ to $L^2Log_+L$. This suggests the following question: which are in general the smoothing properties of $R^\alpha$? (Feissner in fact considers a slightly different family of potentials).\par The complete result then is the following : for $\alpha$ complex, with real part $\ge0$, $R^\alpha$ is bounded from $L^pLog^r_+L$ to $L^pLog^{r+p\alpha}_+L$. The method uses complex interpolation between two cases: a generalization to Orlicz spaces of a result of Stein, when $\alpha$ is purely imaginary, and the case already known where $\alpha$ has real part $1/2$. The first of these two results, proved by martingale theory, is of a quite general nature
Keywords: Logarithmic Sobolev inequalities, Hypercontractivity, Gaussian measures, Riesz potentials
Nature: Original
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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
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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: 44, 503-508, LNM 920 (1982)
MEYER, Paul-André
Résultats d'Atkinson sur les processus de Markov
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XVI: 53, 623-623, LNM 920 (1982)
MEYER, Paul-André
Correction au Séminaire XV
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XVI: 54, 623-623, LNM 920 (1982)
MEYER, Paul-André
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XVI-S: 56, 151-164, LNM 921 (1982)
MEYER, Paul-André
Variation des solutions d'une e.d.s., d'après J.M. Bismut
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XVI-S: 57, 165-207, LNM 921 (1982)
MEYER, Paul-André
Géométrie différentielle stochastique (bis) (Stochastic differential geometry)
A sequel to 1505. The main theme is that an ordinary differential equation has a non unique extension as a stochastic differential equation: besides the Stratonovich one, given by the transfer principle'', there are other possibilities: choosing among them requires some additional, connection-like, structure. The most striking application is the Dohrn-Guerra correction to the parallel transport along a semimartingale
Comment: For complements, see Émery 1658, Hakim-Dowek-Lépingle 2023, Émery's monography Stochastic Calculus in Manifolds (Springer, 1989) and article 2428, and Arnaudon-Thalmaier 3214
Keywords: Semimartingales in manifolds, Stochastic differential equations, Local characteristics, Nelson's stochastic mechanics, Transfer principle
Nature: Original
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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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XVII: 46, 512-512, LNM 986 (1983)
MEYER, Paul-André
Correction au volume XVI (supplément)
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XVIII: 13, 154-171, LNM 1059 (1984)
MEYER, Paul-André; ZHENG, Wei-An
Intégrales stochastiques non monotones
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XVIII: 16, 179-193, LNM 1059 (1984)
MEYER, Paul-André
Transformations de Riesz pour les lois gaussiennes
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XVIII: 20, 223-244, LNM 1059 (1984)
ZHENG, Wei-An; MEYER, Paul-André
Quelques résultats de mécanique stochastique''
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XVIII: 23, 268-270, LNM 1059 (1984)
MEYER, Paul-André
Un résultat d'approximation
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XVIII: 31, 499-499, LNM 1059 (1984)
MEYER, Paul-André
Rectification à un exposé antérieur
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XIX: 02, 12-26, LNM 1123 (1985)
MEYER, Paul-André; ZHENG, Wei-An
Construction de processus de Nelson réversibles
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XIX: 08, 113-129, LNM 1123 (1985)
MEYER, Paul-André
Sur la théorie de Littlewood-Paley-Stein, d'après Coifman-Rochberg-Weiss et Cowling
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XIX: 11, 176-176, LNM 1123 (1985)
MEYER, Paul-André
Une remarque sur la topologie fine
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XX: 03, 30-33, LNM 1204 (1986)
MEYER, Paul-André
Sur l'existence de l'opérateur carré du champ
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XX: 14, 186-312, LNM 1204 (1986)
MEYER, Paul-André
Élements de probabilités quantiques (chapters I to V)
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XX: 15, 313-316, LNM 1204 (1986)
JOURNÉ, Jean-Lin; MEYER, Paul-André
Une martingale d'opérateurs bornés, non représentable en intégrale stochastique
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XX: 17, 321-330, LNM 1204 (1986)
MEYER, Paul-André
Quelques remarques au sujet du calcul stochastique sur l'espace de Fock
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XX: 19, 334-337, LNM 1204 (1986)
MEYER, Paul-André; ZHENG, Wei-An
Sur la construction de certaines diffusions
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XX: 39, 614-614, LNM 1204 (1986)
MEYER, Paul-André
Correction au Séminaire XVIII
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XX: 41, 614-614, LNM 1204 (1986)
MEYER, Paul-André
Correction au Séminaire XIX
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XX: 42, 614-614, LNM 1204 (1986)
MEYER, Paul-André
Correction au Séminaire XV
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XX: 43, 614-614, LNM 1204 (1986)
MEYER, Paul-André
Correction au Séminaire XVI
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XXI: 02, 8-26, LNM 1247 (1987)
MEYER, Paul-André; YAN, Jia-An
À propos des distributions sur l'espace de Wiener
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XXI: 04, 34-80, LNM 1247 (1987)
MEYER, Paul-André
Élements de probabilités quantiques (chapters VI to VIII)
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XXII: 02, 51-71, LNM 1321 (1988)
HU, Yao-Zhong; MEYER, Paul-André
Chaos de Wiener et intégrales de Feynman
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XXII: 03, 72-81, LNM 1321 (1988)
HU, Yao-Zhong; MEYER, Paul-André
Sur les intégrales multiples de Stratonovitch
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XXII: 06, 86-88, LNM 1321 (1988)
MEYER, Paul-André
Quasimartingales hilbertiennes, d'après Enchev
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XXII: 09, 101-128, LNM 1321 (1988)
MEYER, Paul-André
Élements de probabilités quantiques (chapters IX and X)
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XXII: 11, 138-140, LNM 1321 (1988)
MEYER, Paul-André
Une surmartingale limite de martingales continues
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XXII: 12, 141-143, LNM 1321 (1988)
MEYER, Paul-André
Sur un théorème de B. Rajeev
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XXII: 35, 467-476, LNM 1321 (1988)
MEYER, Paul-André
Distributions, noyaux, symboles, d'après Kree
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XXII: 41, 600-600, LNM 1321 (1988)
MEYER, Paul-André
Erratum au Séminaire XX
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XXIII: 09, 139-141, LNM 1372 (1989)
MEYER, Paul-André
Équations de structure des martingales et probabilités quantiques
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XXIII: 10, 142-145, LNM 1372 (1989)
MEYER, Paul-André
Construction de solutions d'équations de structure''
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XXIII: 11, 146-146, LNM 1372 (1989)
MEYER, Paul-André
Un cas de représentation chaotique discrète
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XXIII: 13, 161-164, LNM 1372 (1989)
LÉANDRE, Rémi; MEYER, Paul-André
Sur le développement d'une diffusion en chaos de Wiener
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XXIII: 16, 175-185, LNM 1372 (1989)
MEYER, Paul-André
Élements de probabilités quantiques (chapters X and XI)
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XXIII: 31, 382-392, LNM 1372 (1989)
MEYER, Paul-André; YAN, Jia-An
Distributions sur l'espace de Wiener (suite), d'après Kubo et Yokoi
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XXIV: 25, 370-396, LNM 1426 (1990)
MEYER, Paul-André
Diffusions quantiques (three parts)
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XXIV: 32, 461-465, LNM 1426 (1990)
RUIZ DE CHAVEZ, Juan; MEYER, Paul-André
Positivité sur l'espace de Fock
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XXIV: 37, 486-487, LNM 1426 (1990)
MEYER, Paul-André
Une remarque sur les lois échangeables
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XXV: 07, 52-60, LNM 1485 (1991)
MEYER, Paul-André
Application du bébé Fock'' au modèle d'Ising
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XXV: 08, 61-78, LNM 1485 (1991)
MEYER, Paul-André; YAN, Jia-An
Les fonctions caractéristiques'' des distributions sur l'espace de Wiener
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XXV: 11, 108-112, LNM 1485 (1991)
MEYER, Paul-André
Sur la méthode de L.~Schwartz pour les e.d.s.
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XXV: 34, 425-426, LNM 1485 (1991)
MEYER, Paul-André
Sur deux estimations d'intégrales multiples
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XXV: 35, 427-427, LNM 1485 (1991)
MEYER, Paul-André
Correction au Séminaire XXII
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XXVI: 23, 307-321, LNM 1526 (1992)
AZÉMA, Jacques; MEYER, Paul-André; YOR, Marc
Martingales relatives
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XXVII: 11, 97-105, LNM 1557 (1993)
MEYER, Paul-André
Représentation de martingales d'opérateurs, d'après Parthasarathy-Sinha
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XXVII: 12, 106-113, LNM 1557 (1993)
MEYER, Paul-André
Les systèmes-produits et l'espace de Fock, d'après W.~Arveson
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XXVII: 13, 114-121, LNM 1557 (1993)
MEYER, Paul-André
Représentation des fonctions conditionnellement de type positif, d'après V.P. Belavkin
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XXVII: 28, 312-327, LNM 1557 (1993)
ATTAL, Stéphane; MEYER, Paul-André
Interprétation probabiliste et extension des intégrales stochastiques non commutatives
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XXVIII: 07, 98-101, LNM 1583 (1994)
MEYER, Paul-André
Sur une transformation du mouvement brownien due à Jeulin et Yor
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XXXI: 24, 252-255, LNM 1655 (1997)
MEYER, Paul-André
Formule d'Itô généralisée pour le mouvement brownien linéaire, d'après Föllmer, Protter, Shiryaev
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