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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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IX: 02, 97-153, LNM 465 (1975)

**BENVENISTE, Albert**

Processus stationnaires et mesures de Palm du flot spécial sous une fonction (Ergodic theory, General theory of processes)

This paper takes over several topics of 901, with important new results and often with simpler proofs. It contains results on the existence of ``perfect'' versions of helixes and stationary processes, a better (uncompleted) version of the filtration itself, a more complete and elegant exposition of the Ambrose-Kakutani theorem, taking the filtration into account (the fundamental counter is adapted). The general theory of processes (projection and section theorems) is developed for a filtered flow, taking into account the fact that the filtrations are uncompleted. It is shown that any bounded measure that does not charge ``polar sets'' is the Palm measure of some increasing helix (see also Geman-Horowitz (*Ann. Inst. H. Poincaré,* **9**, 1973). Then a deeper study of flows under a function is performed, leading to section theorems of optional or previsible homogeneous sets by optional or previsible counters. The last section (written in collaboration with J.~Jacod) concerns a stationary counter (discrete point process) in its natural filtration, and its stochastic intensity: here it is shown (contrary to the case of processes indexed by a half-line) that the stochastic intensity does not determine the law of the counter

Keywords: Filtered flows, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures, Perfection, Point processes

Nature: Original

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X: 25, 521-531, LNM 511 (1976)

**BENVENISTE, Albert**

Séparabilité optionnelle, d'après Doob (General theory of processes)

A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given

Comment: Doob's original paper appeared in*Ann. Inst. Fourier,* **25**, 1975. See also 1105

Keywords: Optional processes, Separability, Section theorems

Nature: Exposition, Original additions

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XI: 02, 21-26, LNM 581 (1977)

**BENVENISTE, Albert**

Application d'un théorème de G. Mokobodzki à la théorie des flots (Ergodic theory, General theory of processes)

The purpose of this paper is to extend to the theory of filtered flows (for which see 901 and 902) the dual version of the general theory of processes due to Azéma (for which see 814 and 937), in particular the association with any measurable process of suitable projections which are homogeneous processes. An important difference here is the fact that the time set is the whole line. Here the class of measurable processes which can be projected is reduced to a (not very explicit) class, and a commutation theorem similar to Azéma's is proved. The proof uses the technique of*medial limits * due to Mokobodzki (see 719), which in fact was developed precisely at the author's request to solve this problem

Keywords: Filtered flows, Stationary processes, Projection theorems, Medial limits

Nature: Original

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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

Retrieve article from Numdam

IX: 02, 97-153, LNM 465 (1975)

Processus stationnaires et mesures de Palm du flot spécial sous une fonction (Ergodic theory, General theory of processes)

This paper takes over several topics of 901, with important new results and often with simpler proofs. It contains results on the existence of ``perfect'' versions of helixes and stationary processes, a better (uncompleted) version of the filtration itself, a more complete and elegant exposition of the Ambrose-Kakutani theorem, taking the filtration into account (the fundamental counter is adapted). The general theory of processes (projection and section theorems) is developed for a filtered flow, taking into account the fact that the filtrations are uncompleted. It is shown that any bounded measure that does not charge ``polar sets'' is the Palm measure of some increasing helix (see also Geman-Horowitz (

Keywords: Filtered flows, Flow under a function, Ambrose-Kakutani theorem, Helix, Palm measures, Perfection, Point processes

Nature: Original

Retrieve article from Numdam

X: 25, 521-531, LNM 511 (1976)

Séparabilité optionnelle, d'après Doob (General theory of processes)

A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given

Comment: Doob's original paper appeared in

Keywords: Optional processes, Separability, Section theorems

Nature: Exposition, Original additions

Retrieve article from Numdam

XI: 02, 21-26, LNM 581 (1977)

Application d'un théorème de G. Mokobodzki à la théorie des flots (Ergodic theory, General theory of processes)

The purpose of this paper is to extend to the theory of filtered flows (for which see 901 and 902) the dual version of the general theory of processes due to Azéma (for which see 814 and 937), in particular the association with any measurable process of suitable projections which are homogeneous processes. An important difference here is the fact that the time set is the whole line. Here the class of measurable processes which can be projected is reduced to a (not very explicit) class, and a commutation theorem similar to Azéma's is proved. The proof uses the technique of

Keywords: Filtered flows, Stationary processes, Projection theorems, Medial limits

Nature: Original

Retrieve article from Numdam