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
623Keywords: Additive functionals,
PerfectionNature: Original
Retrieve article from Numdam
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
623Keywords: Additive functionals,
PerfectionNature: Original Retrieve article from Numdam
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 topologyNature: Original Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
