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II: 03, 34-42, LNM 51 (1968)

**DOLÉANS-DADE, Catherine**

Fonctionnelles additives parfaites (Markov processes)

The identity defining additive (or multiplicative) functionals involves an exceptional set depending on a continuous time $t$. If the exceptional set can be chosen independently of $t$, the functional is perfect. It is shown that every additive functional of a Hunt process admitting a reference measure has a perfect version

Comment: The existence of a reference measure was lifted by Dellacherie in 304. However, the whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

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III: 04, 93-96, LNM 88 (1969)

**DELLACHERIE, Claude**

Une application aux fonctionnelles additives d'un théorème de Mokobodzki (Markov processes)

Mokobodzki showed the existence of ``rapid ultrafilters'' on the integers, with the property that applied to a sequence that converges in probability they converge a.s. (see for instance Dellacherie-Meyer,*Probabilité et potentiels,* Chap. II, **27**). They are used here to prove that every continuous additive functional of a Markov process has a ``perfect'' version

Comment: See also 203. The whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

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VI: 22, 233-242, LNM 258 (1972)

**WALSH, John B.**

The perfection of multiplicative functionals (Markov processes)

In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see 203) that every functional has a perfect modification, see also Dellacherie 304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure

Keywords: Multiplicative functionals, Perfection, Essential topology

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

Retrieve article from Numdam

Fonctionnelles additives parfaites (Markov processes)

The identity defining additive (or multiplicative) functionals involves an exceptional set depending on a continuous time $t$. If the exceptional set can be chosen independently of $t$, the functional is perfect. It is shown that every additive functional of a Hunt process admitting a reference measure has a perfect version

Comment: The existence of a reference measure was lifted by Dellacherie in 304. However, the whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

Retrieve article from Numdam

III: 04, 93-96, LNM 88 (1969)

Une application aux fonctionnelles additives d'un théorème de Mokobodzki (Markov processes)

Mokobodzki showed the existence of ``rapid ultrafilters'' on the integers, with the property that applied to a sequence that converges in probability they converge a.s. (see for instance Dellacherie-Meyer,

Comment: See also 203. The whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

Retrieve article from Numdam

VI: 22, 233-242, LNM 258 (1972)

The perfection of multiplicative functionals (Markov processes)

In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see 203) that every functional has a perfect modification, see also Dellacherie 304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure

Keywords: Multiplicative functionals, Perfection, Essential topology

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