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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

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VIII: 02, 11-19, LNM 381 (1974)

**BRETAGNOLLE, Jean**

Une remarque sur le problème de Skorohod (Brownian motion)

The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see 516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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VIII: 03, 20-21, LNM 381 (1974)

**CHUNG, Kai Lai**

Note on last exit decomposition (Markov processes)

This is a useful complement to the monograph of Chung*Lectures on Boundary Theory for Markov Chains,* Annals of Math. Studies 65, Princeton 1970

Keywords: Markov chains

Nature: Original

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VIII: 04, 22-24, LNM 381 (1974)

**DELLACHERIE, Claude**

Un ensemble progressivement mesurable... (General theory of processes)

The set of starting times of Brownian excursions from $0$ is a well-known example of a progressive set which does not contain any graph of stopping time. Here it is shown that considering the same set for the excursions from any $a$ and taking the union of all $a$, the corresponding set has the same property and has uncountable sections

Comment: Other such examples are known, such as the set of times at which the law of the iterated logarithm fails

Keywords: Progressive sets, Section theorems

Nature: Original

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VIII: 05, 25-26, LNM 381 (1974)

**DELLACHERIE, Claude**

Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (General theory of processes)

This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems

Comment: A gap in the proof is filled in 928 and 2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,*Z. für W-theorie,* **38**, 1977 and Yor 1221. For another approach to the restricted case considered here, see Ruiz de Chavez 1821. The previsible representation property of Brownian motion and compensated Poisson process was know by Itô; it is a consequence of the (stronger) chaotic representation property, established by Wiener in 1938. The converse was also known by Itô: among the martingales which are also Lévy processes, only Brownian motions and compensated Poisson processes have the previsible representation property

Keywords: Brownian motion, Poisson processes, Previsible representation

Nature: Original

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VIII: 06, 27-36, LNM 381 (1974)

**DINGES, Hermann**

Stopping sequences (Markov processes, Potential theory)

Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process

Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see 523, 524, 612. This article contains announcements of further results

Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme

Nature: Original

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VIII: 07, 37-77, LNM 381 (1974)

**DUPUIS, Claire**

Mesure de Hausdorff de la trajectoire de certains processus à accroissements indépendants et stationnaires (Independent increments)

The problem is to show that, for symmetric Lévy processes with small jumps and without a Brownian part, there exists a natural Hausdorff measure for which almost every path up to time $t$ has ``length'' exactly $t$. The case of Brownian motion had been known for a long time, the case of stable processes was settled by S.J.~Taylor (*J. Math. Mech.*, **16**, 1967) whose methods are generalized here

Keywords: Hausdorff measures, Lévy processes

Nature: Original

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VIII: 08, 78-79, LNM 381 (1974)

**FERNIQUE, Xavier**

Une démonstration simple du théorème de R.M.~Dudley et M.~Kanter sur les lois 0-1 pour les mesures stables (Miscellanea)

The theorem concerns stable laws on a linear space, and asserts that every measurable linear subspace has probability 0 or 1. The title describes accurately the paper

Keywords: Stable measures

Nature: Original

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VIII: 09, 80-133, LNM 381 (1974)

**GEBUHRER, Marc Olivier**

Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)

The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (*Arkiv för Math.*, **6**, 1965-67), which is studied as a Lorentz invariant diffusion process (in the usual sense) on the standard hyperboloid of velocities in special relativity, on which the Lorentz group acts. The Brownian paths themselves are constructed by integration and possess a speed smaller than the velocity of light but no higher derivatives. The second part studies more generally invariant Markov processes on a Riemannian symmetric space of non-compact type, their generators and the corresponding semigroups

Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces

Nature: Exposition, Original additions

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VIII: 10, 134-149, LNM 381 (1974)

**KNIGHT, Frank B.**

Existence of small oscillations at zeros of brownian motion (Brownian motion)

The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an ``iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros

Comment: This result may be compared to Kahane,*C.R. Acad. Sci.* **248**, 1974

Keywords: Law of the iterated logarithm, Local times

Nature: Original

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VIII: 11, 150-154, LNM 381 (1974)

**HEATH, David C.**

Skorohod stopping via Potential Theory (Potential theory, Markov processes)

The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $**R**^n$ is given. A theorem of Mokobodzki on réduites is used as an important technical tool

Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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VIII: 12, 155-171, LNM 381 (1974)

**HEINKEL, Bernard**

Théorèmes de dérivation du type de Lebesgue et continuité presque sûre de certains processus gaussiens (Gaussian processes)

To be completed

Comment: To be completed

Keywords: Continuity of paths of Gaussian processes

Nature: Original

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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

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VIII: 14, 262-288, LNM 381 (1974)

**MEYER, Paul-André**

Les travaux d'Azéma sur le retournement du temps (General theory of processes, Markov processes)

This paper is an exposition of a paper by Azéma (*Ann. Sci. ENS,* **6**, 1973) in which the theory ``dual'' to the general theory of processes was developed. It is shown first how the general theory itself can be developed from a family of killing operators, and then how the dual theory follows from a family of shift operators $\theta_t$. A transience hypothesis involving the existence of many ``return times'' permits the construction of a theory completely similar to the usual one. Then some of Azéma's applications to the theory of Markov processes are given, particularly the representation of a measure not charging $\mu$-polar sets as expectation under the initial measure $\mu$ of a left additive functional

Comment: This paper follows (with considerable progress) the line of 602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer,*Probabilités et Potentiel,* Chapter XVIII, 1992

Keywords: Time reversal, Shift operators, Killing operators, Cooptional processes, Coprevisible processes, Additive functionals, Left additive functionals

Nature: Exposition, Original additions

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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

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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

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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

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VIII: 18, 316-328, LNM 381 (1974)

**PRIOURET, Pierre**

Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)

The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic

Comment: To be completed

Keywords: Processes with jumps

Nature: Original

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VIII: 19, 329-343, LNM 381 (1974)

**SMYTHE, Robert T.**

Remarks on the hypotheses of duality (Markov processes)

To develop the full strength of potential theory, it is helpful to assume that the basic Markov process has a right-continuous, strong Markov dual. The paper investigates what remains if this assumption is not made, i.e., if the process (transient and satisfying the absolute continuity hypothesis (hypothesis (L)) only has a left continuous moderately Markov dual process (Smythe-Walsh ,*Invent. Math.*, **19**, 1973)

Comment: The independent paper Garsia Alvarez-Meyer*Ann. Prob.* **1**, 1973, has some results in common with this one

Keywords: Dual semigroups

Nature: Original

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VIII: 20, 344-354, LNM 381 (1974)

**WALDENFELS, Wilhelm von**

Taylor expansion of a Poisson measure (Miscellanea)

To be completed

Comment: To be completed

Keywords: Poisson point processes

Nature: Original

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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 (

Keywords: Skorohod imbedding, Multiplicative functionals

Nature: Original

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VIII: 02, 11-19, LNM 381 (1974)

Une remarque sur le problème de Skorohod (Brownian motion)

The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see 516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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VIII: 03, 20-21, LNM 381 (1974)

Note on last exit decomposition (Markov processes)

This is a useful complement to the monograph of Chung

Keywords: Markov chains

Nature: Original

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VIII: 04, 22-24, LNM 381 (1974)

Un ensemble progressivement mesurable... (General theory of processes)

The set of starting times of Brownian excursions from $0$ is a well-known example of a progressive set which does not contain any graph of stopping time. Here it is shown that considering the same set for the excursions from any $a$ and taking the union of all $a$, the corresponding set has the same property and has uncountable sections

Comment: Other such examples are known, such as the set of times at which the law of the iterated logarithm fails

Keywords: Progressive sets, Section theorems

Nature: Original

Retrieve article from Numdam

VIII: 05, 25-26, LNM 381 (1974)

Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (General theory of processes)

This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems

Comment: A gap in the proof is filled in 928 and 2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,

Keywords: Brownian motion, Poisson processes, Previsible representation

Nature: Original

Retrieve article from Numdam

VIII: 06, 27-36, LNM 381 (1974)

Stopping sequences (Markov processes, Potential theory)

Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process

Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see 523, 524, 612. This article contains announcements of further results

Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme

Nature: Original

Retrieve article from Numdam

VIII: 07, 37-77, LNM 381 (1974)

Mesure de Hausdorff de la trajectoire de certains processus à accroissements indépendants et stationnaires (Independent increments)

The problem is to show that, for symmetric Lévy processes with small jumps and without a Brownian part, there exists a natural Hausdorff measure for which almost every path up to time $t$ has ``length'' exactly $t$. The case of Brownian motion had been known for a long time, the case of stable processes was settled by S.J.~Taylor (

Keywords: Hausdorff measures, Lévy processes

Nature: Original

Retrieve article from Numdam

VIII: 08, 78-79, LNM 381 (1974)

Une démonstration simple du théorème de R.M.~Dudley et M.~Kanter sur les lois 0-1 pour les mesures stables (Miscellanea)

The theorem concerns stable laws on a linear space, and asserts that every measurable linear subspace has probability 0 or 1. The title describes accurately the paper

Keywords: Stable measures

Nature: Original

Retrieve article from Numdam

VIII: 09, 80-133, LNM 381 (1974)

Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)

The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (

Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces

Nature: Exposition, Original additions

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VIII: 10, 134-149, LNM 381 (1974)

Existence of small oscillations at zeros of brownian motion (Brownian motion)

The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an ``iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros

Comment: This result may be compared to Kahane,

Keywords: Law of the iterated logarithm, Local times

Nature: Original

Retrieve article from Numdam

VIII: 11, 150-154, LNM 381 (1974)

Skorohod stopping via Potential Theory (Potential theory, Markov processes)

The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $

Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

VIII: 12, 155-171, LNM 381 (1974)

Théorèmes de dérivation du type de Lebesgue et continuité presque sûre de certains processus gaussiens (Gaussian processes)

To be completed

Comment: To be completed

Keywords: Continuity of paths of Gaussian processes

Nature: Original

Retrieve article from Numdam

VIII: 13, 172-261, LNM 381 (1974)

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 (

Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as

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)

Les travaux d'Azéma sur le retournement du temps (General theory of processes, Markov processes)

This paper is an exposition of a paper by Azéma (

Comment: This paper follows (with considerable progress) the line of 602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer,

Keywords: Time reversal, Shift operators, Killing operators, Cooptional processes, Coprevisible processes, Additive functionals, Left additive functionals

Nature: Exposition, Original additions

Retrieve article from Numdam

VIII: 15, 289-289, LNM 381 (1974)

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)

Noyaux multiplicatifs (Markov processes)

This paper presents results due to Jacod (

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

Keywords: Multiplicative kernels, Semimarkovian processes

Nature: Exposition

Retrieve article from Numdam

VIII: 17, 310-315, LNM 381 (1974)

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,

Keywords: Supermartingales, Multiplicative decomposition

Nature: Original

Retrieve article from Numdam

VIII: 18, 316-328, LNM 381 (1974)

Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)

The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic

Comment: To be completed

Keywords: Processes with jumps

Nature: Original

Retrieve article from Numdam

VIII: 19, 329-343, LNM 381 (1974)

Remarks on the hypotheses of duality (Markov processes)

To develop the full strength of potential theory, it is helpful to assume that the basic Markov process has a right-continuous, strong Markov dual. The paper investigates what remains if this assumption is not made, i.e., if the process (transient and satisfying the absolute continuity hypothesis (hypothesis (L)) only has a left continuous moderately Markov dual process (Smythe-Walsh ,

Comment: The independent paper Garsia Alvarez-Meyer

Keywords: Dual semigroups

Nature: Original

Retrieve article from Numdam

VIII: 20, 344-354, LNM 381 (1974)

Taylor expansion of a Poisson measure (Miscellanea)

To be completed

Comment: To be completed

Keywords: Poisson point processes

Nature: Original

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