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
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 functionalsNature: Original Retrieve article from Numdam
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 topologyNature: Original Retrieve article from Numdam
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 functionalsNature: Original Retrieve article from Numdam
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 functionalsNature: Exposition,
Original additions Retrieve article from Numdam
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 functionalsNature: Exposition,
Original additions Retrieve article from Numdam
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
1513Keywords: Markov additive processes,
Additive functionals,
Regenerative sets,
Lévy systemsNature: Original Retrieve article from Numdam
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 functionalsNature: Original Retrieve article from Numdam
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
1437Keywords: Wiener-Hopf factorizations,
Additive functionals,
Changes of time,
Markov chainsNature: Original Retrieve article from Numdam
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 timeNature: Original Retrieve article from Numdam
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 functionalsNature: Original Retrieve article from Numdam
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 potentialsNature: Original Retrieve article from Numdam
