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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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IV: 07, 73-75, LNM 124 (1970)

**DELLACHERIE, Claude**

Potentiels de Green et fonctionnelles additives (Markov processes, Potential theory)

Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$

Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces

Keywords: Green potentials, Additive functionals

Nature: Original

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VI: 23, 243-252, LNM 258 (1972)

**MEYER, Paul-André**

Quelques autres applications de la méthode de Walsh (``La perfection en probabilités'') (Markov processes)

This is but an exercise on using the method of the preceding paper 622 to reduce the exceptional sets in other situations: additive functionals, cooptional times and processes, etc

Comment: A correction to this paper is mentioned on the errata list of vol. VII

Keywords: Additive functionals, Return times, Essential topology

Nature: Original

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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: 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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VIII: 14, 262-288, LNM 381 (1974)

**MEYER, Paul-André**

Les travaux d'Azéma sur le retournement du temps (General theory of processes, Markov processes)

This paper is an exposition of a paper by Azéma (*Ann. Sci. ENS,* **6**, 1973) in which the theory ``dual'' to the general theory of processes was developed. It is shown first how the general theory itself can be developed from a family of killing operators, and then how the dual theory follows from a family of shift operators $\theta_t$. A transience hypothesis involving the existence of many ``return times'' permits the construction of a theory completely similar to the usual one. Then some of Azéma's applications to the theory of Markov processes are given, particularly the representation of a measure not charging $\mu$-polar sets as expectation under the initial measure $\mu$ of a left additive functional

Comment: This paper follows (with considerable progress) the line of 602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer,*Probabilités et Potentiel,* Chapter XVIII, 1992

Keywords: Time reversal, Shift operators, Killing operators, Cooptional processes, Coprevisible processes, Additive functionals, Left additive functionals

Nature: Exposition, Original additions

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XI: 37, 529-538, LNM 581 (1977)

**MAISONNEUVE, Bernard**

Changement de temps d'un processus markovien additif (Markov processes)

A Markov additive process $(X_t,S_t)$ (Cinlar,*Z. für W-theorie,* **24**, 1972) is a generalisation of a pair $(X,S)$ where $X$ is a Markov process with arbitrary state space, and $S$ is an additive functional of $X$: in the general situation $S$ is positive real valued, $X$ is a Markov process in itself, and the pair $(X,S)$ is a Markov processes, while $S$ is an additive functional *of the pair.* For instance, subordinators are Markov additive processes with trivial $X$. A simpler proof of a basic formula of Cinlar is given, and it is shown also that a Markov additive process gives rise to a regenerative system in a slightly extended sense

Comment: See also 1513

Keywords: Markov additive processes, Additive functionals, Regenerative sets, Lévy systems

Nature: Original

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XII: 59, 775-803, LNM 649 (1978)

**MEYER, Paul-André**

Martingales locales fonctionnelles additives (two talks) (Markov processes)

The purpose of the paper is to specialize the standard theory of Hardy spaces of martingales to the subspaces of additive martingales of a Markov process. The theory is not complete: the dual of (additive) $H^1$ seems to be different from (additive) $BMO$

Keywords: Hardy spaces, Additive functionals

Nature: Original

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XIV: 36, 324-331, LNM 784 (1980)

**BARLOW, Martin T.**; **ROGERS, L.C.G.**; **WILLIAMS, David**

Wiener-Hopf factorization for matrices (Markov processes)

Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix

Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in 1437

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time, Markov chains

Nature: Original

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XIV: 37, 332-342, LNM 784 (1980)

**ROGERS, L.C.G.**; **WILLIAMS, David**

Time-substitution based on fluctuating additive functionals (Wiener-Hopf factorization for infinitesimal generators) (Markov processes)

This is a first step towards the extension of 1436 to Markov processes with a general state space

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time

Nature: Original

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XIV: 43, 410-417, LNM 784 (1980)

**JACOD, Jean**; **MAISONNEUVE, Bernard**

Remarque sur les fonctionnelles additives non adaptées des processus de Markov (Markov processes)

It occurs sometimes that a Markov process $(X_t)$ satisfies in a filtration ${\cal H}_t$ a Markov property of the form $E[f\circ \theta_t \,|\,{\cal H}_t]= E_{X_t}[f]$, where $f$ is not restricted to be ${\cal H}_t$-measurable. For instance, situations in renewal theory where one is given a Markov pair $(X_t,Y_t)$, and ${\cal H}_t$ describes the path of $X$ up to time $t$, and the whole path of $Y$. In such cases, the authors show that additive functionals which are previsible in the larger filtration are in fact previsible in the filtration of $X$ alone

Keywords: Additive functionals

Nature: Original

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XIV: 44, 418-436, LNM 784 (1980)

**RAO, Murali**

A note on Revuz measure (Markov processes, Potential theory)

The problem is to weaken the hypotheses of Chung (*Ann. Inst. Fourier,* **23**, 1973) implying the representation of the equilibrium potential of a compact set as a Green potential. To this order, Revuz measure techniques are used, and interesting auxiliary results are proved concerning the Revuz measures of natural additive functionals of a Hunt process

Keywords: Revuz measures, Additive functionals, Hunt processes, Equilibrium potentials

Nature: Original

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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

IV: 07, 73-75, LNM 124 (1970)

Potentiels de Green et fonctionnelles additives (Markov processes, Potential theory)

Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$

Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces

Keywords: Green potentials, Additive functionals

Nature: Original

Retrieve article from Numdam

VI: 23, 243-252, LNM 258 (1972)

Quelques autres applications de la méthode de Walsh (``La perfection en probabilités'') (Markov processes)

This is but an exercise on using the method of the preceding paper 622 to reduce the exceptional sets in other situations: additive functionals, cooptional times and processes, etc

Comment: A correction to this paper is mentioned on the errata list of vol. VII

Keywords: Additive functionals, Return times, Essential topology

Nature: Original

Retrieve article from Numdam

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

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

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

VIII: 14, 262-288, LNM 381 (1974)

Les travaux d'Azéma sur le retournement du temps (General theory of processes, Markov processes)

This paper is an exposition of a paper by Azéma (

Comment: This paper follows (with considerable progress) the line of 602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer,

Keywords: Time reversal, Shift operators, Killing operators, Cooptional processes, Coprevisible processes, Additive functionals, Left additive functionals

Nature: Exposition, Original additions

Retrieve article from Numdam

XI: 37, 529-538, LNM 581 (1977)

Changement de temps d'un processus markovien additif (Markov processes)

A Markov additive process $(X_t,S_t)$ (Cinlar,

Comment: See also 1513

Keywords: Markov additive processes, Additive functionals, Regenerative sets, Lévy systems

Nature: Original

Retrieve article from Numdam

XII: 59, 775-803, LNM 649 (1978)

Martingales locales fonctionnelles additives (two talks) (Markov processes)

The purpose of the paper is to specialize the standard theory of Hardy spaces of martingales to the subspaces of additive martingales of a Markov process. The theory is not complete: the dual of (additive) $H^1$ seems to be different from (additive) $BMO$

Keywords: Hardy spaces, Additive functionals

Nature: Original

Retrieve article from Numdam

XIV: 36, 324-331, LNM 784 (1980)

Wiener-Hopf factorization for matrices (Markov processes)

Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix

Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in 1437

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time, Markov chains

Nature: Original

Retrieve article from Numdam

XIV: 37, 332-342, LNM 784 (1980)

Time-substitution based on fluctuating additive functionals (Wiener-Hopf factorization for infinitesimal generators) (Markov processes)

This is a first step towards the extension of 1436 to Markov processes with a general state space

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time

Nature: Original

Retrieve article from Numdam

XIV: 43, 410-417, LNM 784 (1980)

Remarque sur les fonctionnelles additives non adaptées des processus de Markov (Markov processes)

It occurs sometimes that a Markov process $(X_t)$ satisfies in a filtration ${\cal H}_t$ a Markov property of the form $E[f\circ \theta_t \,|\,{\cal H}_t]= E_{X_t}[f]$, where $f$ is not restricted to be ${\cal H}_t$-measurable. For instance, situations in renewal theory where one is given a Markov pair $(X_t,Y_t)$, and ${\cal H}_t$ describes the path of $X$ up to time $t$, and the whole path of $Y$. In such cases, the authors show that additive functionals which are previsible in the larger filtration are in fact previsible in the filtration of $X$ alone

Keywords: Additive functionals

Nature: Original

Retrieve article from Numdam

XIV: 44, 418-436, LNM 784 (1980)

A note on Revuz measure (Markov processes, Potential theory)

The problem is to weaken the hypotheses of Chung (

Keywords: Revuz measures, Additive functionals, Hunt processes, Equilibrium potentials

Nature: Original

Retrieve article from Numdam