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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
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
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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
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
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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
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
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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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