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VII: 01, 1-24, LNM 321 (1973)

**BENVENISTE, Albert**

Application de deux théorèmes de G.~Mokobodzki à l'étude du noyau de Lévy d'un processus de Hunt sans hypothèse (L) (Markov processes)

The object of the theory of Lévy systems is to compute the previsible compensator of sums $\sum_{s\le t} f(X_{s-},X_s)$ extended to the jump times of a Markov process~$X$, i.e., the times $s$ at which $X_s\not=X_{s-}$. The theory was created by Lévy in the case of a process with independent increments, and the classical results for Markov processes are due to Ikeda-Watanabe,*J. Math. Kyoto Univ.*, **2**, 1962 and Watanabe, *Japan J. Math.*, **34**, 1964. An exposition of their results can be found in the Seminar, 106. The standard assumptions were: 1) $X$ is a Hunt process, implying that jumps occur at totally inaccessible stopping times and the compensator is continuous, 2) Hypothesis (L) (absolute continuity of the resolvent) is satisfied. Here using two results of Mokobodzki: 1) every excessive function dominated in the strong sense in a potential. 2) The existence of medial limits (this volume, 719), Hypothesis (L) is shown to be unnecessary

Comment: Mokobodzki's second result depends on additional axioms in set theory, the continuum hypothesis or Martin's axiom. See also Benveniste-Jacod,*Invent. Math.* **21**, 1973, which no longer uses medial limits

Keywords: Lévy systems, Additive functionals

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: 03, 33-35, LNM 321 (1973)

**DELLACHERIE, Claude**

Un crible généralisé (Descriptive set theory)

Given a Borel set $A$ in the product $E\times F$ of two compact metric sets, the set of all $x\in E$ such that the section $A(x)\subset Y$ is of second category is analytic

Comment: The authour discovered later that the main result is in fact due to Novikov: two references are given in 1252

Keywords: Analytic sets

Nature: Original

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VII: 04, 36-37, LNM 321 (1973)

**DELLACHERIE, Claude**

Temps d'arrêt totalement inaccessibles (General theory of processes)

Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left

Keywords: Stopping times, Accessible sets, Totally inaccessible stopping times

Nature: Original

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VII: 05, 38-47, LNM 321 (1973)

**DELLACHERIE, Claude**

Sur les théorèmes fondamentaux de la théorie générale des processus (General theory of processes)

This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved

Comment: The idea of this paper has proved fruitful. See for instance Lenglart, 1449; Le Jan,*Z. für W-Theorie * **44**, 1978

Keywords: Stopping times, Section theorems

Nature: Original

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VII: 06, 48-50, LNM 321 (1973)

**DELLACHERIE, Claude**

Une démonstration du théorème de Souslin-Lusin (Descriptive set theory)

The basic fact that the image of a Borel set under an injective Borel mapping is Borel is deduced from a separation theorem concerning countably many disjoint analytic sets

Comment: This is a step in the author's simplification of the proofs of the great theorems on analytic and Borel sets. See*Un cours sur les ensembles analytiques,* in *Analytic Sets,* C.A. Rogers ed., Academic Press 1980

Keywords: Borel sets, Analytic sets, Separation theorem

Nature: New exposition of known results

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VII: 07, 51-57, LNM 321 (1973)

**DELLACHERIE, Claude**

Une conjecture sur les ensembles semi-polaires (Markov processes)

For a right process satisfying the absolute continuity hypothesis and assuming singletons are semi-polar sets, it is conjectured that a (nearly-)Borel set is semipolar if and only if it does not contain uncountable families of disjoint, non-polar compact sets. This statement implies that two processes which have the same polar sets also have the same semi-polar sets

Comment: The conjecture can be proved, using a general result of Mokobodzki, see 1238

Keywords: Polar sets, Semi-polar sets

Nature: Original

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VII: 08, 58-60, LNM 321 (1973)

**DELLACHERIE, Claude**

Potentiels de fonctionnelles additives. Un contre-exemple de Knight (Markov processes)

An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point

Keywords: Additive functionals

Nature: Exposition, Original additions

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VII: 09, 61-76, LNM 321 (1973)

**FARAUT, Jacques**

Fonction brownienne sur une variété riemannienne (Miscellanea, Gaussian processes)

As defined originally by Lévy in the case of spheres and euclidian spaces, a Brownian motion indexed by a point of a metric space $E$ is a centered Gaussian process $(X_t)_{t\in E}$ such that $E[(X_t-X_s)^2]=d(s,t)$, the distance. In a Riemannian manifold $d$ is understood to be the geodesic distance. The results of this paper imply that Brownian motions exist on spheres and Euclidean spaces (Lévy's original result), on real hyperbolic spaces, but not on quaternionic hyperbolic spaces

Comment: This article contains joint work with K. Harzallah

Keywords: Covariance, Riemannian manifold, Riemannian distance, Lévy Brownian motions, Several parameter Brownian motions

Nature: Original

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VII: 10, 77-94, LNM 321 (1973)

**HEINKEL, Bernard**

Une condition suffisante pour la continuité presque sûre des trajectoires de certains processus gaussiens (Gaussian processes)

It is shown that a continuity criterion due to Preston (1972) can be deduced from a theorem of Dudley (1967)

Comment: To be completed

Keywords: Continuity of paths of Gaussian processes

Nature: Original

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VII: 11, 95-117, LNM 321 (1973)

**EL KAROUI, Nicole**; **REINHARD, Hervé**

Processus de diffusion dans ${\bf R}^n$ (Diffusion theory)

This paper concerns diffusions (without boundaries) whose generators have Borel bounded coefficients. It consists of two parts. The first one is devoted to the equivalence between the existence and uniqueness of the diffusion semigroup and the uniqueness in law of the solution of the corresponding Ito stochastic differential equation. This allows the authors to use in the elliptic case the deep results of Krylov on s.d.e.'s. The second part concerns mostly the Lipschitz case, and discusses several properties of the diffusion process in itself: the representation of additive functional martingales; the relations between the number of martingales necessary for the representation and the rank of the generator (locally); the existence of a dual diffusion; the support and absolute continuity properties of the semi-group

Comment: This paper is in part an improved version of a paper on degenerate diffusions by Bonami, El-Karoui, Reinhard and Roynette (*Ann. Inst. H. Poincaré,* **7**, 1971)

Keywords: Construction of diffusions, Diffusions with measurable coefficients, Degenerate diffusions

Nature: Original

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VII: 12, 118-121, LNM 321 (1973)

**KAZAMAKI, Norihiko**

Une note sur les martingales faibles (Martingale theory)

Métivier has distinguished in the general theory of processes localization from prelocalization: a process $X$ is a local martingale if there exist stopping times $T_n$ increasing to infinity and martingales $M_n$ such that $X=M_n$ on the closed interval $[0,T_n]$ (omitting for simplicity the convention about time $0$). Replacing the closed intervals by open intervals $[0,T_n[$ defines prelocal martingales or*weak martingales.* It is shown that in the filtration generated by one single stopping time, processes which are prelocally martingales (square integrable martingales) are so globally. It follows that prelocal martingales may not be prelocally square integrable

Comment: The interest of weak martingales arises from their invariance by (possibly discontinuous) changes of time, see Kazamaki,*Zeit. für W-theorie,* **22**, 1972

Keywords: Weak martingales

Nature: Original

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VII: 13, 122-135, LNM 321 (1973)

**KHALILI-FRANÇON, Elisabeth**

Processus de Galton-Watson (Markov processes)

This paper is mostly a survey of previous results with comments and some alternative proofs

Comment: An erroneous statement is corrected in 939

Keywords: Branching processes, Galton-Watson processes

Nature: Exposition, Original additions

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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

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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, Medial limit

Nature: Exposition

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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

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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

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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

Nature: Exposition, Original additions

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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

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VII: 24, 248-272, LNM 321 (1973)

**MÜRMANN, Michael G.**

A semi-Markovian model for the Brownian motion (Statistical mechanics)

A model for physical Brownian motion (the effect on a heavy particle of many interactions with light particles), originally proposed by Spitzer and Holley, in dimension 1, is studied in detail. The resulting process, whose construction is delicate, is non-Markovian

Comment: The last two pages of the manuscript (proof of Proposition 5 and References) were omitted at the production stage, and added as a loose sheet to vol. VIII, while another loose sheet contains an example. These sheets are not mentioned in the table of contents of vol. VIII

Keywords: Infinite particle systems

Nature: Original

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VII: 25, 273-283, LNM 321 (1973)

**PINSKY, Mark A.**

Fonctionnelles multiplicatives opératrices (Markov processes)

This paper presents results due to the author (*Advances in Probability,* **3**, 1973). The essential idea is to consider multiplicative functionals of a Markov process taking values in the algebra of bounded operators of a Banach space $L$. Such a functional defines a semi-group acting on bounded $L$-valued functions. This semi-group determines the functional. The structure of functionals is investigated in the case of a finite Markov chain. The case where $L$ is finite dimensional and the Markov process is Browian motion is investigated too. Asymptotic results near $0$ are described

Comment: This paper explores the same idea as Jacod (*Mém. Soc. Math. France,* **35**, 1973), though in a very different way. See 816

Keywords: Multiplicative functionals, Multiplicative kernels

Nature: Exposition

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VII: 26, 284-290, LNM 321 (1973)

**ROST, Hermann**

Relaxation in infinite spin systems (Statistical mechanics)

The existence of a stochastic process describing an infinitely many interacting particle system is proved

Comment: This is related to Sullivan,*Zeit. für W-theorie,* **31**, 1974

Keywords: Interacting particle systems

Nature: Original

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VII: 27, 291-300, LNM 321 (1973)

**TAYLOR, John C.**

On the existence of resolvents (Potential theory)

Since the basic results of Hunt, a kernel satisfying the complete maximum principle is expected to be the potential kernel of a sub-Markov resolvent. This is not always the case, however, and one should also express that, so to speak, ``potentials vanish at the boundary''. Such a condition is given here on an abstract space, which supersedes an earlier result of the author (*Invent. Math.* **17**, 1972) and a result of Hirsch (*Ann. Inst. Fourier,* **22-1**, 1972)

Comment: The definitive paper of Taylor on this subject appeared in*Ann. Prob.*, **3**, 1975

Keywords: Complete maximum principle, Resolvents

Nature: Original

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VII: 28, 301-318, LNM 321 (1973)

**WALDENFELS, Wilhelm von**

Some remarks on Burkhardt's model for pressure broadening of spectral lines (Miscellanea)

A model proposed by Burkhardt in 1940 for the deformation of the radiation emitted by an atom due to the surrounding atoms is transformed into probabilistic language and exactly solved

Nature: Original

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VII: 29, 319-321, LNM 321 (1973)

**MOKOBODZKI, Gabriel**

Pseudo-quotient de deux mesures, application à la dualité (Potential theory)

Contains the four last pages of 617 omitted from Volume VI

Nature: Correction

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Application de deux théorèmes de G.~Mokobodzki à l'étude du noyau de Lévy d'un processus de Hunt sans hypothèse (L) (Markov processes)

The object of the theory of Lévy systems is to compute the previsible compensator of sums $\sum_{s\le t} f(X_{s-},X_s)$ extended to the jump times of a Markov process~$X$, i.e., the times $s$ at which $X_s\not=X_{s-}$. The theory was created by Lévy in the case of a process with independent increments, and the classical results for Markov processes are due to Ikeda-Watanabe,

Comment: Mokobodzki's second result depends on additional axioms in set theory, the continuum hypothesis or Martin's axiom. See also Benveniste-Jacod,

Keywords: Lévy systems, Additive functionals

Nature: Original

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VII: 02, 25-32, LNM 321 (1973)

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,

Keywords: Lévy systems, Ray compactification

Nature: Original

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VII: 03, 33-35, LNM 321 (1973)

Un crible généralisé (Descriptive set theory)

Given a Borel set $A$ in the product $E\times F$ of two compact metric sets, the set of all $x\in E$ such that the section $A(x)\subset Y$ is of second category is analytic

Comment: The authour discovered later that the main result is in fact due to Novikov: two references are given in 1252

Keywords: Analytic sets

Nature: Original

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VII: 04, 36-37, LNM 321 (1973)

Temps d'arrêt totalement inaccessibles (General theory of processes)

Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left

Keywords: Stopping times, Accessible sets, Totally inaccessible stopping times

Nature: Original

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VII: 05, 38-47, LNM 321 (1973)

Sur les théorèmes fondamentaux de la théorie générale des processus (General theory of processes)

This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved

Comment: The idea of this paper has proved fruitful. See for instance Lenglart, 1449; Le Jan,

Keywords: Stopping times, Section theorems

Nature: Original

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VII: 06, 48-50, LNM 321 (1973)

Une démonstration du théorème de Souslin-Lusin (Descriptive set theory)

The basic fact that the image of a Borel set under an injective Borel mapping is Borel is deduced from a separation theorem concerning countably many disjoint analytic sets

Comment: This is a step in the author's simplification of the proofs of the great theorems on analytic and Borel sets. See

Keywords: Borel sets, Analytic sets, Separation theorem

Nature: New exposition of known results

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VII: 07, 51-57, LNM 321 (1973)

Une conjecture sur les ensembles semi-polaires (Markov processes)

For a right process satisfying the absolute continuity hypothesis and assuming singletons are semi-polar sets, it is conjectured that a (nearly-)Borel set is semipolar if and only if it does not contain uncountable families of disjoint, non-polar compact sets. This statement implies that two processes which have the same polar sets also have the same semi-polar sets

Comment: The conjecture can be proved, using a general result of Mokobodzki, see 1238

Keywords: Polar sets, Semi-polar sets

Nature: Original

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VII: 08, 58-60, LNM 321 (1973)

Potentiels de fonctionnelles additives. Un contre-exemple de Knight (Markov processes)

An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point

Keywords: Additive functionals

Nature: Exposition, Original additions

Retrieve article from Numdam

VII: 09, 61-76, LNM 321 (1973)

Fonction brownienne sur une variété riemannienne (Miscellanea, Gaussian processes)

As defined originally by Lévy in the case of spheres and euclidian spaces, a Brownian motion indexed by a point of a metric space $E$ is a centered Gaussian process $(X_t)_{t\in E}$ such that $E[(X_t-X_s)^2]=d(s,t)$, the distance. In a Riemannian manifold $d$ is understood to be the geodesic distance. The results of this paper imply that Brownian motions exist on spheres and Euclidean spaces (Lévy's original result), on real hyperbolic spaces, but not on quaternionic hyperbolic spaces

Comment: This article contains joint work with K. Harzallah

Keywords: Covariance, Riemannian manifold, Riemannian distance, Lévy Brownian motions, Several parameter Brownian motions

Nature: Original

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VII: 10, 77-94, LNM 321 (1973)

Une condition suffisante pour la continuité presque sûre des trajectoires de certains processus gaussiens (Gaussian processes)

It is shown that a continuity criterion due to Preston (1972) can be deduced from a theorem of Dudley (1967)

Comment: To be completed

Keywords: Continuity of paths of Gaussian processes

Nature: Original

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VII: 11, 95-117, LNM 321 (1973)

Processus de diffusion dans ${\bf R}^n$ (Diffusion theory)

This paper concerns diffusions (without boundaries) whose generators have Borel bounded coefficients. It consists of two parts. The first one is devoted to the equivalence between the existence and uniqueness of the diffusion semigroup and the uniqueness in law of the solution of the corresponding Ito stochastic differential equation. This allows the authors to use in the elliptic case the deep results of Krylov on s.d.e.'s. The second part concerns mostly the Lipschitz case, and discusses several properties of the diffusion process in itself: the representation of additive functional martingales; the relations between the number of martingales necessary for the representation and the rank of the generator (locally); the existence of a dual diffusion; the support and absolute continuity properties of the semi-group

Comment: This paper is in part an improved version of a paper on degenerate diffusions by Bonami, El-Karoui, Reinhard and Roynette (

Keywords: Construction of diffusions, Diffusions with measurable coefficients, Degenerate diffusions

Nature: Original

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VII: 12, 118-121, LNM 321 (1973)

Une note sur les martingales faibles (Martingale theory)

Métivier has distinguished in the general theory of processes localization from prelocalization: a process $X$ is a local martingale if there exist stopping times $T_n$ increasing to infinity and martingales $M_n$ such that $X=M_n$ on the closed interval $[0,T_n]$ (omitting for simplicity the convention about time $0$). Replacing the closed intervals by open intervals $[0,T_n[$ defines prelocal martingales or

Comment: The interest of weak martingales arises from their invariance by (possibly discontinuous) changes of time, see Kazamaki,

Keywords: Weak martingales

Nature: Original

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VII: 13, 122-135, LNM 321 (1973)

Processus de Galton-Watson (Markov processes)

This paper is mostly a survey of previous results with comments and some alternative proofs

Comment: An erroneous statement is corrected in 939

Keywords: Branching processes, Galton-Watson processes

Nature: Exposition, Original additions

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VII: 14, 136-145, LNM 321 (1973)

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 $

Comment: See 907 for a correction. This material has been published in book form, see for instance Dellacherie-Meyer,

Keywords: $BMO$, Hardy spaces, Fefferman inequality

Nature: Original

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VII: 15, 146-154, LNM 321 (1973)

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

Keywords: Transformations of Markov processes, Excursions

Nature: Exposition

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VII: 16, 155-171, LNM 321 (1973)

Réduites et jeux de hasard (Potential theory)

This paper arose from an attempt (by the second author) to rewrite the results of Dubins-Savage

Comment: This material is reworked in Dellacherie-Meyer,

Keywords: Balayage, Gambling house, Réduite, Optimal strategy

Nature: Original

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VII: 17, 172-179, LNM 321 (1973)

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 (

Comment: See related papers by Mertens in

Keywords: Excessive functions, Supermedian functions, Réduite

Nature: Exposition, Original proofs

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VII: 18, 180-197, LNM 321 (1973)

Résultats d'Azéma en théorie générale des processus (General theory of processes)

This paper presents several results from a paper of Azéma (

Comment: These results have been included (with their optional counterpart, whose interest was discovered later) in Dellacherie-Meyer,

Keywords: Optimal stopping, Previsible processes

Nature: Exposition

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VII: 19, 198-204, LNM 321 (1973)

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

Keywords: Continuum axiom, Weak convergence of r.v.'s, Medial limit

Nature: Exposition

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VII: 20, 205-209, LNM 321 (1973)

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

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VII: 21, 210-216, LNM 321 (1973)

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

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VII: 22, 217-222, LNM 321 (1973)

Sur les désintégrations régulières de L. Schwartz (General theory of processes)

This paper presents a small part of an important article of L.~Schwartz (

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

Nature: Exposition, Original additions

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VII: 23, 223-247, LNM 321 (1973)

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 (

Keywords: Filtering theory, Innovation

Nature: Exposition

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VII: 24, 248-272, LNM 321 (1973)

A semi-Markovian model for the Brownian motion (Statistical mechanics)

A model for physical Brownian motion (the effect on a heavy particle of many interactions with light particles), originally proposed by Spitzer and Holley, in dimension 1, is studied in detail. The resulting process, whose construction is delicate, is non-Markovian

Comment: The last two pages of the manuscript (proof of Proposition 5 and References) were omitted at the production stage, and added as a loose sheet to vol. VIII, while another loose sheet contains an example. These sheets are not mentioned in the table of contents of vol. VIII

Keywords: Infinite particle systems

Nature: Original

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VII: 25, 273-283, LNM 321 (1973)

Fonctionnelles multiplicatives opératrices (Markov processes)

This paper presents results due to the author (

Comment: This paper explores the same idea as Jacod (

Keywords: Multiplicative functionals, Multiplicative kernels

Nature: Exposition

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VII: 26, 284-290, LNM 321 (1973)

Relaxation in infinite spin systems (Statistical mechanics)

The existence of a stochastic process describing an infinitely many interacting particle system is proved

Comment: This is related to Sullivan,

Keywords: Interacting particle systems

Nature: Original

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VII: 27, 291-300, LNM 321 (1973)

On the existence of resolvents (Potential theory)

Since the basic results of Hunt, a kernel satisfying the complete maximum principle is expected to be the potential kernel of a sub-Markov resolvent. This is not always the case, however, and one should also express that, so to speak, ``potentials vanish at the boundary''. Such a condition is given here on an abstract space, which supersedes an earlier result of the author (

Comment: The definitive paper of Taylor on this subject appeared in

Keywords: Complete maximum principle, Resolvents

Nature: Original

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VII: 28, 301-318, LNM 321 (1973)

Some remarks on Burkhardt's model for pressure broadening of spectral lines (Miscellanea)

A model proposed by Burkhardt in 1940 for the deformation of the radiation emitted by an atom due to the surrounding atoms is transformed into probabilistic language and exactly solved

Nature: Original

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VII: 29, 319-321, LNM 321 (1973)

Pseudo-quotient de deux mesures, application à la dualité (Potential theory)

Contains the four last pages of 617 omitted from Volume VI

Nature: Correction

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