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I: 02, 18-33, LNM 39 (1967)

**CAIROLI, Renzo**

Semi-groupes de transition et fonctions excessives (Markov processes, Potential theory)

A study of product kernels, product semi-groups and product Markov processes

Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes

Keywords: Product semigroups

Nature: Original

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I: 08, 166-176, LNM 39 (1967)

**WEIL, Michel**

Retournement du temps dans les processus markoviens (Markov processes)

This talk presents the now classical results of Nagasawa (*Nagoya Math. J.*, **24**, 1964) extending to continuous time the results proved by Hunt in discrete time on time reversal of a Markov process at an ``L-time'' or return time

Comment: See also 202. These results have been essentially the best ones until they were extended to Kuznetsov measures, see Dellacherie-Meyer,*Probabilités et Potentiel,* chapter XIX **14**

Keywords: Time reversal, Dual semigroups

Nature: Exposition

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I: 09, 177-189, LNM 39 (1967)

**WEIL, Michel**

Résolvantes en dualité (Markov processes, Potential theory)

Given two sub-Markov resolvents in duality, whose kernels are absolutely continuous with respect to a given measure, it is shown how to choose their densities to get true Green kernels, excessive in one variable and coexcessive in the other one. It is shown also that coexcessive functions are exactly the densities of excessive measures

Comment: These results, now classical, are due to Kunita-T. Watanabe,*Ill. J. Math.*, **9**, 1965 and *J. Math. Mech.*, **15**, 1966

Keywords: Green potentials, Dual semigroups

Nature: Exposition

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II: 01, 1-21, LNM 51 (1968)

**AZÉMA, Jacques**; **DUFLO, Marie**; **REVUZ, Daniel**

Classes récurrentes d'un processus de Markov (Markov processes)

This is an improved version of a paper by the same authors (*Ann. Inst. H. Poincaré,* **2**, 1966). Its aim is a theory of recurrence in continuous time (for a Hunt process). The main point is to use the finely open sets instead of the ordinary ones to define recurrence

Comment: The subject is further investigated by the same authors in 302

Keywords: Recurrent sets, Fine topology

Nature: Original

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II: 02, 22-33, LNM 51 (1968)

**CARTIER, Pierre**; **MEYER, Paul-André**; **WEIL, Michel**

Le retournement du temps~: compléments à l'exposé de M.~Weil (Markov processes)

In 108, M.~Weil had presented the work of Nagasawa on the time reversal of a Markov process at a ``L-time'' or return time. Here the results are improved on three points: a Markovian filtration is given for the reversed process; an analytic condition on the semigroup is lifted; finally, the behaviour of the*coexcessive * functions on the sample functions of the original process is investigated

Comment: The results of this paper have become part of the standard theory of time reversal. See 312 for a correction

Keywords: Time reversal, Dual semigroups

Nature: Original

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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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II: 05, 75-110, LNM 51 (1968)

**GIROUX, Gaston**

Théorie des frontières dans les chaînes de Markov (Markov processes)

A presentation of the theory of Markov chains under the hypothesis that all states are regular

Comment: This is the subject of the short monograph of Chung,*Lectures on Boundary Theory for Markov Chains,* Princeton 1970

Keywords: Markov chains, Boundary theory

Nature: Exposition

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III: 02, 24-33, LNM 88 (1969)

**AZÉMA, Jacques**; **DUFLO, Marie**; **REVUZ, Daniel**

Mesure invariante des processus de Markov récurrents (Markov processes)

A condition similar to the Harris recurrence condition is studied in continuous time. It is shown that it implies the existence (up to a constant factor) of a unique $\sigma$-finite excessive measure, which is invariant. The invariant measure for a time-changed process is described

Comment: This is related to several papers by the same authors on recurrent Markov processes, and in particular to 201

Keywords: Recurrent potential theory, Invariant measures

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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III: 05, 97-114, LNM 88 (1969)

**DELLACHERIE, Claude**

Ensembles aléatoires I (Descriptive set theory, Markov processes, General theory of processes)

A deep theorem of Lusin asserts that a Borel set with countable sections is a countable union of Borel graphs. It is applied here in the general theory of processes to show that an optional set with countable sections is a countable union of graphs of stopping times, and in the theory of Markov processes, that a Borel set which is a.s. hit by the process at countably many times must be semi-polar

Comment: See Dellacherie,*Capacités et Processus Stochastiques,* Springer 1972

Keywords: Sets with countable sections

Nature: Original

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III: 06, 115-136, LNM 88 (1969)

**DELLACHERIE, Claude**

Ensembles aléatoires II (Descriptive set theory, Markov processes)

Among the many proofs that an uncountable Borel set of the line contains a perfect set, a proof of Sierpinski (*Fund. Math.*, **5**, 1924) can be extended to an abstract set-up to show that a non-semi-polar Borel set contains a non-semi-polar compact set

Comment: See Dellacherie,*Capacités et Processus Stochastiques,* Springer 1972. More recent proofs no longer depend on ``rabotages'': Dellacherie-Meyer, *Probabilités et potentiel,* Appendix to Chapter IV

Keywords: Sierpinski's ``rabotages'', Semi-polar sets

Nature: Original

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III: 09, 144-151, LNM 88 (1969)

**MEYER, Paul-André**

Un résultat de théorie du potentiel (Potential theory, Markov processes)

Under strong duality hypotheses, it is shown that a measure which does not charge polar sets is equivalent to a measure whose Green potential is bounded

Comment: See Meyer,*Processus de Markov,* Lecture Notes in M. **26**

Keywords: Green potentials, Dual semigroups

Nature: Original

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III: 12, 160-162, LNM 88 (1969)

**MEYER, Paul-André**

Rectification à des exposés antérieurs (Markov processes, Martingale theory)

Corrections are given to the talk 202 by Cartier, Meyer and Weil and to the talk 106 by Meyer

Comment: This note introduces ``Walsh's fork'', the well-known strong Markov process whose dual is not strong Markov

Keywords: Time reversal, Stochastic integrals

Nature: Correction

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III: 14, 175-189, LNM 88 (1969)

**MEYER, Paul-André**

Processus à accroissements indépendants et positifs (Markov processes, Independent increments)

This is an exposition of the theory of subordinators (Lévy processes with increasing paths), aiming at presenting Chung's conjecture that a certain identity known to hold a.e. actually holds everywhere, also equivalent to the fact that single points are polar sets for subordinators without drift

Comment: The conjecture was proved by Kesten (see 503) who actually knew of the problem through this talk. See also 502

Keywords: Subordinators, Polar sets

Nature: Exposition

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IV: 06, 71-72, LNM 124 (1970)

**DELLACHERIE, Claude**

Au sujet des sauts d'un processus de Hunt (Markov processes)

Two a.s. results on jumps: the process cannot jump*from * a semi-polar set; at the first hitting time of any finely closed set $F$, either the process does not jump, or it jumps from outside $F$

Comment: Both results are improvements of previous results of Meyer and Weil

Keywords: Hunt processes, Semi-polar sets

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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IV: 12, 133-150, LNM 124 (1970)

**MEYER, Paul-André**

Ensembles régénératifs, d'après Hoffmann-Jørgensen (Markov processes)

The theory of recurrent events in discrete time was a highlight of the old probability theory. It was extended to continuous time by Kingman (see for instance*Z. für W-theorie,* **2**, 1964), under the very restrictive assumption that the ``event'' has a non-zero probability to occur at fixed times. The general theory is due to Krylov and Yushkevich (*Trans. Moscow Math. Soc.*, **13**, 1965), a deep paper difficult to read and to apply in concrete cases. Hoffmann-Jørgensen (*Math. Scand.*, **24**, 1969) developed the theory under simple and efficient axioms. It is shown that a regenerative set defined axiomatically is the same thing as the set of returns of a strong Markov process to a fixed state, or the range of a subordinator

Comment: This result was expanded to involve a Markovian regeneration property instead of independence. See Maisonneuve-Meyer 813. The subject is related to excursion theory, Lévy systems, semi-Markovian processes (Lévy), F-processes (Neveu), Markov renewal processes (Pyke), and the literature is very extensive. See for instance Dynkin (*Th. Prob. Appl.*, **16**, 1971) and Maisonneuve, *Systèmes Régénératifs,* *Astérisque * **15**, 1974

Keywords: Renewal theory, Regenerative sets, Recurrent events

Nature: Exposition

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IV: 13, 151-161, LNM 124 (1970)

**MAISONNEUVE, Bernard**; **MORANDO, Philippe**

Temps locaux pour les ensembles régénératifs (Markov processes)

This paper uses the results of the preceding one 412 to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of increase is exactly the given set

Comment: Same references as the preceding paper 412

Keywords: Renewal theory, Regenerative sets, Local times

Nature: Original

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IV: 17, 208-215, LNM 124 (1970)

**REVUZ, Daniel**

Application d'un théorème de Mokobodzki aux opérateurs potentiels dans le cas récurrent (Potential theory, Markov processes)

Mokododzki's theorem asserts that if the kernels of a resolvent are strong Feller, i.e., map bounded functions into continuous functions, then they must satisfy a norm continuity property (see 210). This is used to show the existence for``normal'' recurrent processes of a nice potential operator, defined for suitable functions of zero integral with respect to the invariant measure

Comment: For additional work of Revuz on recurrence, see*Ann. Inst. Fourier,* **21**, 1971

Keywords: Recurrent potential theory

Nature: Original

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IV: 18, 216-239, LNM 124 (1970)

**WEIL, Michel**

Quasi-processus (Markov processes)

Excessive measures which are not potentials of measures were shown by Hunt (*Ill. J. Math.*, **4**, 1960) to be associated with a probabilistic object which is a kind of projective limit of Markov processes. Hunt's construction was performed in discrete time only, and is difficult in continuous time because of measure theoretic difficulties (the standard theorem on projective limits cannot be applied). Here the construction is done in full detail

Comment: Further work by M.~Weil on the same subject in 532; see the references there

Keywords: Hunt quasi-processes

Nature: Original

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IV: 19, 240-282, LNM 124 (1970)

**DELLACHERIE, Claude**; **DOLÉANS-DADE, Catherine**; **LETTA, Giorgio**; **MEYER, Paul-André**

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (*Comm. Pure Appl. Math.*, **22**, 1969), which constructs by a probability method a unique semigroup whose generator is an elliptic second order operator with continuous coefficients (the analytic approach either deals with operators in divergence form, or requires some Hölder condition). The contribution of G.~Letta nicely simplified the proof

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,*Multidimensional Diffusion Processes,* Springer 1979

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

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V: 02, 17-20, LNM 191 (1971)

**ASSOUAD, Patrice**

Démonstration de la ``Conjecture de Chung'' par Carleson (Markov processes, Independent increments)

Chung conjectured that singletons are polar sets for driftless subordinators. This paper gives Carleson's (unpublished) analytic proof of it

Comment: See Chung,*C. R. Acad. Sci. *, **260**, 1965, p.4665. For the statement of the problem see Meyer 314. For Kesten's earlier (contrary to a statement in the paper!) probabilistic proof see Bretagnolle 503. See also *Séminaire Bourbaki * 21th year, **361**, June 1969

Keywords: Subordinators, Polar sets

Nature: Exposition

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V: 03, 21-36, LNM 191 (1971)

**BRETAGNOLLE, Jean**

Résultats de Kesten sur les processus à accroissements indépendants (Markov processes, Independent increments)

The question is to find all Lévy processes for which single points are polar. Kesten's answer (*Mem. Amer. Math. Soc.*, **93**, 1969) is almost complete and in particular proves Chung's conjecture. The proofs in this paper have been considerably reworked

Comment: See also 502 in the same volume

Keywords: Subordinators, Polar sets

Nature: Exposition, Original additions

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V: 17, 177-190, LNM 191 (1971)

**MEYER, Paul-André**

Processus de Poisson ponctuels d'après K. Ito (Markov processes, Point processes)

Presents (a preliminary form of) the celebrated paper of Ito (*Proc. Sixth Berkeley Symposium,* **3**, 1972) on excursion theory, with an extension (the use of possibly unbounded entrance laws instead of initial measures) which has become part of the now classical theory

Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form

Keywords: Poisson point processes, Excursions, Local times

Nature: Exposition, Original additions

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V: 19, 196-208, LNM 191 (1971)

**MEYER, Paul-André**

Représentation intégrale des fonctions excessives. Résultats de Mokobodzki (Markov processes, Potential theory)

Main result: the convex cone of excessive functions for a resolvent which satisfies the absolute continuity hypothesis is the union of convex compact metrizable ``hats''\ in a suitable topology, and therefore has the integral representation property. The original proof of Mokobodzki, self-contained and unpublished, is given here

Comment: See Mokobodzki's work on cones of potentials,*Séminaire Bourbaki,* May 1970

Keywords: Minimal excessive functions, Martin boundary, Integral representations

Nature: Exposition

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V: 20, 209-210, LNM 191 (1971)

**MEYER, Paul-André**

Un théorème sur la répartition des temps locaux (Markov processes)

Kesten discovered that the value at a terminal time $T$ of the local time $L$ of a Markov process $X$ at a single point has an exponential distribution, and that $X_T$ and $L_T$ are independent. A short proof is given

Comment: The result can be deduced from excursion theory

Keywords: Local times

Nature: New exposition of known results

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V: 22, 213-236, LNM 191 (1971)

**MEYER, Paul-André**

Le retournement du temps, d'après Chung et Walsh (Markov processes)

The paper of Chung and Walsh (*Acta Math.*, **134**, 1970) proved that any right continuous strong Markov process had a reversed left continuous moderate Markov process at any $L$-time, with a suitably constructed dual semigroup. Appendix 1 gives a useful characterization of càdlàg processes using stopping times (connected with amarts). Appendix 2 proves (following Mokobodzki) that any excessive function strongly dominated by a potential of function is such a potential

Comment: The theorem of Chung-Walsh remains the deepest on time reversal (to be supplemented by the consideration of Kuznetsov's measures)

Keywords: Time reversal, Dual semigroups

Nature: Exposition, Original additions

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V: 24, 251-269, LNM 191 (1971)

**MEYER, Paul-André**

Solutions de l'équation de Poisson dans le cas récurrent (Potential theory, Markov processes)

The problem is to solve the Poisson equation for measures, $\mu-\mu P=\theta$ for given $\theta$, in the case of a recurrent transition kernel $P$. Here a ``filling scheme'' technique is used

Comment: The paper was motivated by Métivier (*Ann. Math. Stat.*, **40**, 1969) and is completely superseded by one of Revuz (*Ann. Inst. Fourier,* **21**, 1971)

Keywords: Recurrent potential theory, Filling scheme, Harris recurrence, Poisson equation

Nature: Original

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V: 25, 270-274, LNM 191 (1971)

**MEYER, Paul-André**

Balayage pour les processus de Markov continus à droite, d'après C.T. Shih (Markov processes, Potential theory)

Hunt's fundamental theorem on balayage gives the probabilistic interpretation of the réduite of an excessive function set on a set. It was proved originally within the special class of Hunt's processes, then extended to ``standard'' processes. Using a method of compactification, Shih (*Ann. Inst. Fourier,* **20-1**, 1970) showed it was quite general

Comment: Shih's paper is the origin of the general definition of ``right processes''

Keywords: Excessive functions, Réduite

Nature: Exposition

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V: 26, 275-277, LNM 191 (1971)

**REVUZ, Daniel**

Remarque sur les potentiels de mesure (Markov processes, Potential theory)

The standard proof of the equivalence between semi-polar sets being polar and a very precise domination principle (Blumenthal-Getoor,*Markov Processes and Potential Theory,* 1968) uses the assumption that excessive functions are lower semicontinuous. This assumption is weakened

Comment: To be asked

Keywords: Polar sets, Semi-polar sets, Excessive functions

Nature: Original

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V: 28, 283-289, LNM 191 (1971)

**WALSH, John B.**

Two footnotes to a theorem of Ray (Markov processes)

Ray's theorem (*Ann. of Math.*, **70**, 1959) is the construction of a good semigroup (and process) from a Ray resolvent. The first ``footnote'' gives the construction of a second semigroup with nice properties from the left instead of the right side. The second ``footnote'' studies the filtration of a Ray process

Comment: See Meyer-Walsh,*Invent. Math.*, **14**, 1971

Keywords: Ray compactification

Nature: Original

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V: 29, 290-310, LNM 191 (1971)

**WALSH, John B.**

Some topologies connected with Lebesgue measure (Markov processes, General theory of processes, Potential theory)

It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called*essential topology,* used in the paper of Chung and Walsh 522 in the same volume

Comment: See Doob*Bull. Amer. Math. Soc.*, **72**, 1966. An important application in given by Walsh 623 in the next volume. See the paper 1025 of Benveniste. For the use of a different topology see Ito *J. Math. Soc. Japan,* **20**, 1968

Keywords: Essential topology

Nature: Original

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V: 31, 342-346, LNM 191 (1971)

**WEIL, Michel**

Décomposition d'un temps terminal (Markov processes)

It is shown that for a Hunt process, a terminal time can be represented as the infimum of a previsible terminal time, and a totally inaccessible terminal time

Keywords: Terminal times

Nature: Original

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V: 32, 347-361, LNM 191 (1971)

**WEIL, Michel**

Quasi-processus et énergie (Markov processes, Potential theory)

The energy of an excessive function $f$ with respect to an excessive measure $\xi$ has a simple proba\-bi\-listic interpretation if $\xi$ is is the potential of a measure $\mu$ and $f$ is the potential of an additive functional $(A_t)$, as ${1\over2}E_\mu[A_\infty^2]$. If $\xi$ is not a potential, still it can be associated with it a quasi-process (see Weil 418) with a birthtime $b$ and a death time $d$, and the formal expression ${1\over2}E[(A_d-A_b)^2]$ is given a precise meaning and represents the energy

Comment: This subject has been renewed by the introduction of Kuznetsov's measures. See Fitzsimmons*Sem. Stoch. Proc.*, 1987

Keywords: Hunt quasi-processes, Energy

Nature: Original

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V: 33, 362-372, LNM 191 (1971)

**WEIL, Michel**

Conditionnement par rapport au passé strict (Markov processes)

Given a totally inaccessible terminal time $T$, it is shown how to compute conditional expectations of the future with respect to the strict past $\sigma$-field ${\cal F}_{T-}$. The formula involves the Lévy system of the process

Comment: B. Maisonneuve pointed out once that the paper, though essentially correct, has a small mistake somewhere. See Dellacherie-Meyer,*Probabilité et Potentiels,* Chap. XX **46**--48

Keywords: Terminal times, Lévy systems

Nature: Original

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VI: 02, 35-50, LNM 258 (1972)

**AZÉMA, Jacques**

Une remarque sur les temps de retour. Trois applications (Markov processes, General theory of processes)

This paper is the first step in the investigations of Azéma on the ``dual'' form of the general theory of processes (for which see Azéma (*Ann. Sci. ENS,* **6**, 1973, and 814). Here the $\sigma$-fields of cooptional and coprevisible sets are introduced in a Markovian set-up, and without their definitive names. A section theorem by return times is proved, and applications to the theory of Markov processes are given

Keywords: Homogeneous processes, Coprevisible processes, Cooptional processes, Section theorems, Projection theorems, Time reversal

Nature: Original

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VI: 09, 109-112, LNM 258 (1972)

**SAM LAZARO, José de**; **MEYER, Paul-André**

Un gros processus de Markov. Application à certains flots (Markov processes)

In a vague but useful sense, a ``big'' process over a given process consists of random variables whose values are a part of the path of the original process (the best known example is the excursion process). Here it is shown how the past of a Markov process can be turned into a big (homogeneous) Markov process, and how its semigroup is computed using an idea of Dawson (*Trans. Amer. Math. Soc.*, **131**, 1968)

Comment: For a complete account of Dawson's formula, see Dellacherie-Meyer,*Probabilités et Potentiel,* \no XIV.45

Keywords: Prediction theory, Filtered flows

Nature: Original

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VI: 15, 164-167, LNM 258 (1972)

**MEYER, Paul-André**

Une note sur le théorème du balayage de Hunt (Markov processes, Potential theory)

The theorem involved is the characterization of the réduite $P_A u$ of an excessive function $u$ on a set $A$ as equal (except on a well-defined semipolar set) to the infimum of the excessive functions that dominate $u$ on $A$. This theorem is slightly improved under the absolute continuity hypothesis. The proof rests on the following property of the fine topology: every (nearly Borel) finely closed set is contained in a fine $G_\delta$ which differs from it by a polar set

Keywords: Réduite, Fine topology, Absolute continuity hypothesis

Nature: Original

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VI: 16, 168-172, LNM 258 (1972)

**MEYER, Paul-André**; **WALSH, John B.**

Un résultat sur les résolvantes de Ray (Markov processes)

This is a complement to the authors' paper on Ray processes in*Invent. Math.,* **14**, 1971: a lemma is proved on the existence of many martingales which are continuous whenever the process is continuous (a wrong reference for it was given in the paper). Then it is shown that the mapping $x\rightarrow P_x$ is continuous in the weak topology of measures, when the path space is given the topology of convergence in measure. Note that a correction is mentioned on the errata page of vol. VII

Comment: The idea of using the topology of convergence in measure on a path space turned out to be a fruitful idea; see Meyer and Zheng*Ann. Inst. Henri Poincaré* **20**, 1984

Keywords: Ray compactification, Weak convergence of measures

Nature: Original

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VI: 18, 177-197, LNM 258 (1972)

**NAGASAWA, Masao**

Branching property of Markov processes (Markov processes)

To be completed

Keywords: Branching processes

Nature: Original

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VI: 21, 215-232, LNM 258 (1972)

**WALSH, John B.**

Transition functions of Markov processes (Markov processes)

Assume that a cadlag process satisfies the strong Markov property with respect to some family of kernels $P_t$ (not necessarily a semigroup). It is shown that these kernels can be modified into a true strong Markov transition function with a few additional properties. A similar problem is solved for a left continuous, moderate Markov process. The technique involves a Ray compactification which is eliminated at the end, and a useful lemma shows how to construct supermedian functions which separate points

Comment: The problem discussed here has great theoretical importance, but little practical importance except for time reversal. The construction of a nice transition function for a Markov process has been also discussed by Kuznetsov ()

Keywords: Transition functions, Strong Markov property, Moderate Markov property, Ray compactification

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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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: 02, 25-32, LNM 321 (1973)

**MEYER, Paul-André**

Une mise au point sur les systèmes de Lévy. Remarques sur l'exposé de A. Benveniste (Markov processes)

This is an addition to the preceding paper 701, extending the theory to right processes by means of a Ray compactification

Comment: All this material has become classical. See for instance Dellacherie-Meyer,*Probabilités et Potentiel,* vol. D, chapter XV, 31--35

Keywords: Lévy systems, Ray compactification

Nature: Original

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VII: 07, 51-57, LNM 321 (1973)

**DELLACHERIE, Claude**

Une conjecture sur les ensembles semi-polaires (Markov processes)

For a right process satisfying the absolute continuity hypothesis and assuming singletons are semi-polar sets, it is conjectured that a (nearly-)Borel set is semipolar if and only if it does not contain uncountable families of disjoint, non-polar compact sets. This statement implies that two processes which have the same polar sets also have the same semi-polar sets

Comment: The conjecture can be proved, using a general result of Mokobodzki, see 1238

Keywords: Polar sets, Semi-polar sets

Nature: 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 functionals

Nature: Exposition, Original additions

Retrieve article from Numdam

VII: 13, 122-135, LNM 321 (1973)

**KHALILI-FRANÇON, Elisabeth**

Processus de Galton-Watson (Markov processes)

This paper is mostly a survey of previous results with comments and some alternative proofs

Comment: An erroneous statement is corrected in 939

Keywords: Branching processes, Galton-Watson processes

Nature: Exposition, Original additions

Retrieve article from Numdam

VII: 15, 146-154, LNM 321 (1973)

**MEYER, Paul-André**

Chirurgie sur un processus de Markov, d'après Knight et Pittenger (Markov processes)

This paper presents a remarkable result of Knight and Pittenger, according to which excising from the sample path of a Markov process all the excursions from a given set $A$ which meet another set $B$ preserves the Markov property

Comment: The original paper appeared in*Zeit. für W-theorie,* **23**, 1972

Keywords: Transformations of Markov processes, Excursions

Nature: Exposition

Retrieve article from Numdam

VII: 17, 172-179, LNM 321 (1973)

**MEYER, Paul-André**

Application de l'exposé précédent aux processus de Markov (Markov processes)

This paper is devoted to results of J.F. Mertens on optimal stopping and strongly supermedian functions for a right Markov process (*Zeit. für W-theorie,* **26**, 1973), which are shown to be closely related to those of the preceding paper 716 on general gambling houses. An interesting result of Mokobodzki is included, showing that the extreme points of the convex set of all balayées of a given measure $\lambda$ are the balayées of $\lambda$ on sets

Comment: See related papers by Mertens in*Zeit. für W-theorie,* **22**, 1972 and *Invent. Math.*, **23**, 1974. The original result of Mokobodzki appeared in the *Sémin. Théorie du Potentiel,* 1969-70

Keywords: Excessive functions, Supermedian functions, Réduite

Nature: Exposition, Original proofs

Retrieve article from Numdam

VII: 20, 205-209, LNM 321 (1973)

**MEYER, Paul-André**

Remarques sur les hypothèses droites (Markov processes)

The following technical point is discussed: why does one assume among the ``right hypotheses'' that excessive functions are nearly-Borel?

Keywords: Right processes, Excessive functions

Nature: Original

Retrieve article from Numdam

VII: 21, 210-216, LNM 321 (1973)

**MEYER, Paul-André**

Note sur l'interprétation des mesures d'équilibre (Markov processes)

Let $(X_t)$ be a transient Markov process (we omit the detailed assumptions) with a potential density $u(x,y)$. Let $\mu$ be the measure whose potential is the equilibrium potential of a set $A$. Then the distribution of the process at the last exit time from $A$ is given by $$E_x[f\circ X_{L-}, 0<L<\infty]=\int u(x,y)\,f(y)\,\mu(dy)$$ This formula, due to Chung, is deduced under minimal duality hypotheses from a general formula of Azéma, and a well-known theorem on Revuz measures

Keywords: Equilibrium potentials, Last exit time, Revuz measures

Nature: Exposition, Original proofs

Retrieve article from Numdam

VII: 25, 273-283, LNM 321 (1973)

**PINSKY, Mark A.**

Fonctionnelles multiplicatives opératrices (Markov processes)

This paper presents results due to the author (*Advances in Probability,* **3**, 1973). The essential idea is to consider multiplicative functionals of a Markov process taking values in the algebra of bounded operators of a Banach space $L$. Such a functional defines a semi-group acting on bounded $L$-valued functions. This semi-group determines the functional. The structure of functionals is investigated in the case of a finite Markov chain. The case where $L$ is finite dimensional and the Markov process is Browian motion is investigated too. Asymptotic results near $0$ are described

Comment: This paper explores the same idea as Jacod (*Mém. Soc. Math. France,* **35**, 1973), though in a very different way. See 816

Keywords: Multiplicative functionals, Multiplicative kernels

Nature: Exposition

Retrieve article from Numdam

VIII: 01, 1-10, LNM 381 (1974)

**AZÉMA, Jacques**; **MEYER, Paul-André**

Une nouvelle représentation du type de Skorohod (Markov processes)

A Skorohod imbedding theorem for general Markov processes is proved, in which the stopping time is a randomized ``left'' terminal time. A uniqueness result is proved

Comment: The result is deduced from a representation of measures by left additive functionals, due to Azéma (*Invent. Math.* **18**, 1973 and this volume, 814). A general survey on the Skorohod embedding problem is Ob\lój, *Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding, Multiplicative functionals

Nature: Original

Retrieve article from Numdam

VIII: 03, 20-21, LNM 381 (1974)

**CHUNG, Kai Lai**

Note on last exit decomposition (Markov processes)

This is a useful complement to the monograph of Chung*Lectures on Boundary Theory for Markov Chains,* Annals of Math. Studies 65, Princeton 1970

Keywords: Markov chains

Nature: Original

Retrieve article from Numdam

VIII: 06, 27-36, LNM 381 (1974)

**DINGES, Hermann**

Stopping sequences (Markov processes, Potential theory)

Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process

Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see 523, 524, 612. This article contains announcements of further results

Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme

Nature: Original

Retrieve article from Numdam

VIII: 09, 80-133, LNM 381 (1974)

**GEBUHRER, Marc Olivier**

Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)

The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (*Arkiv för Math.*, **6**, 1965-67), which is studied as a Lorentz invariant diffusion process (in the usual sense) on the standard hyperboloid of velocities in special relativity, on which the Lorentz group acts. The Brownian paths themselves are constructed by integration and possess a speed smaller than the velocity of light but no higher derivatives. The second part studies more generally invariant Markov processes on a Riemannian symmetric space of non-compact type, their generators and the corresponding semigroups

Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces

Nature: Exposition, Original additions

Retrieve article from Numdam

VIII: 11, 150-154, LNM 381 (1974)

**HEATH, David C.**

Skorohod stopping via Potential Theory (Potential theory, Markov processes)

The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $**R**^n$ is given. A theorem of Mokobodzki on réduites is used as an important technical tool

Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

VIII: 13, 172-261, LNM 381 (1974)

**MAISONNEUVE, Bernard**; **MEYER, Paul-André**

Ensembles aléatoires markoviens homogènes (5 talks) (Markov processes)

This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see 412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (``left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (*Ann. Prob.* **1**, 1973; *Indiana Math. J.* **23**, 1973). The basic and earlier paper of Dynkin on the same subject (* Teor. Ver. Prim.* **16**, 1971) was not known to the authors.\par Chapter III is devoted to the original work of Maisonneuve on incursions. Roughly, the incursion at time $t$ is trivial if $t\in M$, and if $t\notin M$ it consists of the post-$t$ part of the excursion straddling $t$. Thus the incursion process is a path valued, non adapted process. It is only adapted to the filtration ${\cal F}_{D_t}$ where $D_t$ is the first hitting time of $M$ after $t$. Contrary to the Ito theory of excursions, no change of time using a local time is performed. The main result is the fact that, if a suitable regeneration property is assumed only on the set $M$ then, in a suitable topology on the space of paths, this process is a right-continuous strong Markov process. Considerable effort is devoted to proving that it is even a right process (the technique is heavy and many errors have crept in, some of them corrected in 932-933).\par Chapter IV makes the connection between II and III: the main results of Chapter II are proved anew (without balayage or Laplace transforms): they amount to computing the Lévy system of the incursion process. Finally, Chapter V consists of applications, among which a short discussion of the boundary theory for Markov chains

Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as*Systèmes Régénératifs,* *Astérisque,* **15**, 1974, a much simpler version of his own results, and discovered important improvements later on (some of which are included in Dellacherie-Maisonneuve-Meyer, *Probabilités et Potentiel,* Chapter XX, 1992). Along the slightly different line of Dynkin, see El~Karoui-Reinhard, *Compactification et balayage de processus droits,* *Astérisque 21,* 1975. A recent book on excursion theory is Blumenthal, *Excursions of Markov Processes,* Birkhäuser 1992

Keywords: Regenerative systems, Regenerative sets, Renewal theory, Local times, Excursions, Markov chains, Incursions

Nature: Original

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 functionals

Nature: Exposition, Original additions

Retrieve article from Numdam

VIII: 15, 289-289, LNM 381 (1974)

**MEYER, Paul-André**

Une note sur la compactification de Ray (Markov processes)

This short note shows that (contrary to the belief of Meyer and Walsh in a preceding paper) the state space of a Ray process is universally measurable in its Ray compactification

Comment: This is now considered a standard fact

Keywords: Ray compactification, Right processes

Nature: Original

Retrieve article from Numdam

VIII: 16, 290-309, LNM 381 (1974)

**MEYER, Paul-André**

Noyaux multiplicatifs (Markov processes)

This paper presents results due to Jacod (*Mém. Soc. Math. France,* **35**, 1973): given a pair $(X,Y)$ which jointly is a Markov process, and whose first component $X$ is a Markov process by itself, describe the conditional distribution of the joint path of $(X,Y)$ over a given path of $X$. These distributions constitute a multiplicative kernel, and attempts are made to regularize it

Comment: Though the paper is fairly technical, it does not improve substantially on Jacod's results

Keywords: Multiplicative kernels, Semimarkovian processes

Nature: Exposition

Retrieve article from Numdam

VIII: 18, 316-328, LNM 381 (1974)

**PRIOURET, Pierre**

Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)

The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic

Comment: To be completed

Keywords: Processes with jumps

Nature: Original

Retrieve article from Numdam

VIII: 19, 329-343, LNM 381 (1974)

**SMYTHE, Robert T.**

Remarks on the hypotheses of duality (Markov processes)

To develop the full strength of potential theory, it is helpful to assume that the basic Markov process has a right-continuous, strong Markov dual. The paper investigates what remains if this assumption is not made, i.e., if the process (transient and satisfying the absolute continuity hypothesis (hypothesis (L)) only has a left continuous moderately Markov dual process (Smythe-Walsh ,*Invent. Math.*, **19**, 1973)

Comment: The independent paper Garsia Alvarez-Meyer*Ann. Prob.* **1**, 1973, has some results in common with this one

Keywords: Dual semigroups

Nature: Original

Retrieve article from Numdam

IX: 26, 471-485, LNM 465 (1975)

**NAGASAWA, Masao**

Multiplicative excessive measures and duality between equations of Boltzmann and of branching processes (Markov processes, Statistical mechanics)

The author investigates the connection between the branching Markov processes constructed over some given Markov processes and a non-linear equation close to Boltzmann's equation. A special class of excessive measures for the branching Markov process is described and studied, as well as the corresponding dual processes

Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 1011

Keywords: Boltzmann equation, Branching processes

Nature: Original

Retrieve article from Numdam

IX: 27, 486-493, LNM 465 (1975)

**WEIL, Michel**

Surlois d'entrée (Markov processes)

The results presented here are due to T. Leviatan (*Ann. Prob.*, **1**, 1973) and concern the construction of a Markov process with creation of mass, corresponding to a given transition semigroup $(P_t)$ and a given ``super-entrance law'' $(\mu_t)$, consisting of bounded measures such that $\mu_{s+t}\ge\mu_s P_t$. The proof is a clever argument of projective limits. The paper mentions briefly the relation with earlier results of Helms (*Z. für W-theorie,* **7**, 1967)

Comment: This beautiful paper was superseded by the (slightly later) fundamental paper of Kuznetsov (1974)

Keywords: Entrance laws, Creation of mass, Kuznetsov measures

Nature: Exposition

Retrieve article from Numdam

IX: 29, 495-495, LNM 465 (1975)

**DELLACHERIE, Claude**

Une propriété des ensembles semi-polaires (Markov processes)

It is shown that semi-polar sets are exactly those which have potential 0 for all continuous additive functionals (or for all time-changed processes)

Keywords: Semi-polar sets

Nature: Original

Retrieve article from Numdam

IX: 30, 496-514, LNM 465 (1975)

**SHARPE, Michael J.**

Homogeneous extensions of random measures (Markov processes)

Homogeneous random measures are the appropriate definition of additive functionals which may explode. The problem discussed here is the extension of such a measure given up to a terminal time into a measure defined up to the lifetime

Comment: The subject is taken over in a systematic way in Sharpe,*General Theory of Markov processes,* Academic Press 1988

Keywords: Homogeneous random measures, Terminal times, Subprocesses

Nature: Original

Retrieve article from Numdam

IX: 31, 515-517, LNM 465 (1975)

**HEATH, David C.**

Skorohod stopping in discrete time (Markov processes, Potential theory)

Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times

Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath 811. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

IX: 32, 518-521, LNM 465 (1975)

**MAISONNEUVE, Bernard**; **MEYER, Paul-André**

Ensembles aléatoires markoviens homogènes. Mise au point et compléments (Markov processes)

This paper corrects or simplifies many details in the long paper 713 by the same authors

Comment: See also the next paper 933

Keywords: Regenerative systems, Last-exit decompositions, Excursions

Nature: Original

Retrieve article from Numdam

IX: 33, 522-529, LNM 465 (1975)

**MAISONNEUVE, Bernard**

Le comportement de dernière sortie (Markov processes)

This paper contains improvements to the paper 813 by Maisonneuve-Meyer, whose results are briefly recalled. Incursion processes and Lévy systems are altogether avoided, last-exist decompositions are derived, and the strong Markov property of the analogue of the age process in renewal theory is proved, as well as a non-homogeneous Markov property for some processes starting at last-exit times. The extension of these results to abstractly defined regenerative systems is mentioned

Comment: More detailed versions of these results appear in Maisonneuve,*Ann. Prob.*, **3**, 1975, *Z. für W-theorie,* **80**, 1989, and in Chapter XX of Dellacherie-Maisonneuve-Meyer, *Probabilités et Potentiel,* Hermann 1992

Keywords: Regenerative systems, Last-exit decompositions, Excursions

Nature: Original

Retrieve article from Numdam

IX: 34, 530-533, LNM 465 (1975)

**MEYER, Paul-André**

Sur la démonstration de prévisibilité de Chung et Walsh (Markov processes)

A new proof of the result that for a Hunt process, the previsible stopping times are exactly those at which the process does not jump was given by Chung-Walsh (*Z. für W-theorie,* **29**, 1974). Their idea is used here in a modified way, using a formula of Dawson which ``explicitly'' computes conditional expectations and projections. Then it is extended to Ray processes

Comment: The contents of this paper became Chapter XIV**44**--47 in Dellacherie-Meyer, *Probabilités et Potentiel*

Keywords: Hunt processes, Previsible times

Nature: Exposition

Retrieve article from Numdam

IX: 36, 555-555, LNM 465 (1975)

**MEYER, Paul-André**

Une remarque sur les processus de Markov (Markov processes)

It is shown that, under a fixed measure $**P**^{\mu}$, the optional processes and times relative to the uncompleted filtrations $({\cal F}_{t+}^{\circ})$ and $({\cal F}_{t}^{\circ})$ are undistinguishable from each other

Comment: No applications are known

Keywords: Stopping times

Nature: Original

Retrieve article from Numdam

IX: 37, 556-564, LNM 465 (1975)

**MEYER, Paul-André**

Retour aux retournements (Markov processes, General theory of processes)

The first part of the talk is devoted to an important correction to the theorem on p.285 of 814 (Azéma's theory of cooptional and coprevisible sets and its application to Markov processes): the definition of left additive functionals should allow an a.s. explosion at time 0 for initial points belonging to a polar set. The second part belongs to the general theory of processes: if the index set is not the right half-line as usual but the left half-line, the section and projection theorems must be modified in a not quite trivial way

Comment: See Chapter XXVIII of Dellacherie-Maisonneuve-Meyer*Probabilités et potentiel *

Keywords: Time reversal, Cooptional processes, Coprevisible processes, Homogeneous processes

Nature: Exposition, Original additions

Retrieve article from Numdam

IX: 39, 589-589, LNM 465 (1975)

**KHALILI-FRANÇON, Elisabeth**

Correction à ``Processus de Galton-Watson'' (Markov processes)

Proposition 1(ii) p.126 of 713 is true only in the case $q=0$

Keywords: Branching processes

Nature: Correction

Retrieve article from Numdam

X: 03, 24-39, LNM 511 (1976)

**JACOD, Jean**; **MÉMIN, Jean**

Un théorème de représentation des martingales pour les ensembles régénératifs (Martingale theory, Markov processes, Stochastic calculus)

The natural filtration of a regenerative set $M$ is that of the corresponding ``age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration

Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in 1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second ``Azéma's martingale'' (not the well known one which has the chaotic representation property)

Keywords: Regenerative sets, Renewal theory, Stochastic integrals, Previsible representation

Nature: Original

Retrieve article from Numdam

X: 05, 44-77, LNM 511 (1976)

**KUNITA, Hiroshi**

Absolute continuity for Markov processes (Markov processes)

This paper is devoted to a ``progressive'' Lebesgue decomposition of the laws of a Markov process with respect to a second one in the same filtration, and the structure of the corresponding density. The two processes are assumed to be Hunt processes, and for part of the paper satisfy Hunt's hypothesis (K) (all excessive functions are regular, or semi-polar sets are polar). The topics discussed are the following: Lévy systems and the relation between the Lévy systems of a process and of its transform by a multiplicative functional; structure of exact perfect terminal times, which are shown to be hitting times of sets in space-time, by the process $(X_{t-},X_t)$ (a version of a result of Walsh-Weil,*Ann. Sci. ENS,* **5**, 1972); the ``Lebesgue decomposition'' of a Markov process with respect to another, and the fact that if absolute continuity holds on the germ field it also holds up to some maximal terminal time; a condition for this terminal time to be equal to the lifetime, under hypothesis (K)

Comment: The pasting together of the Lebesgue decompositions of a probability measure with respect to another one, on the $\sigma$-fields of a given filtration, is called the*Kunita decomposition,* and is not restricted to Markov processes. For the general case, see Yoeurp, in LN **1118**, *Grossissements de filtrations,* 1985

Keywords: Absolute continuity of laws, Hunt processes, Terminal times, Kunita decomposition

Nature: Original

Retrieve article from Numdam

X: 10, 125-183, LNM 511 (1976)

**MEYER, Paul-André**

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book*Topics in Harmonic Analysis related to the Littlewood-Paley theory,* Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $**R**^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $**R**^n\times **R**_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $**R**^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (*CRAS Paris,* **278A**, 1974, p.1103) in potential theory and by Kunita (*Nagoya M. J.*, **36**, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false

Comment: This paper was rediscovered by Varopoulos (*J. Funct. Anal.*, **38**, 1980), and was then rewritten by Meyer in 1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in 1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912, and Meyer 1908 reporting on Cowling's extension of Stein's work. An erratum is given in 1253

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

Retrieve article from Numdam

X: 11, 184-193, LNM 511 (1976)

**NAGASAWA, Masao**

A probabilistic approach to non-linear Dirichlet problem (Markov processes)

The theory of branching Markov processes in continuous time developed in particular by Ikeda-Nagasawa-Watanabe (*J. Math. Kyoto Univ.*, **8**, 1968 and **9**, 1969) and Nagasawa (*Kodai Math. Sem. Rep.* **20**, 1968) leads to the probabilistic solution of a non-linear Dirichlet problem

Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 926

Keywords: Branching processes, Dirichlet problem

Nature: Original

Retrieve article from Numdam

X: 12, 194-208, LNM 511 (1976)

**ROST, Hermann**

Skorohod stopping times of minimal variance (Markov processes)

Root's (*Ann. Math. Stat.*, **40**, 1969) solution of the Shorohod imbedding problem for Brownian motion uses the hitting time of a barrier in space-time. Here Root's construction is extended to general Markov processes, an optimality property of Root's stopping times is proved, as well as the uniqueness of such stopping times

Comment: For previous work of the author on Skorohod's imbedding see*Ann. M. Stat.* **40**, 1969 and *Invent. Math.* **14**, 1971, and in this Seminar 523, 613, 806. A general survey on the Skorohod embedding problem is Ob\lój, *Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

X: 14, 216-234, LNM 511 (1976)

**WILLIAMS, David**

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams*Diffusions, Markov Processes and Martingales,* vol. 1 (second edition), Wiley 1994. See also 1024

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

Retrieve article from Numdam

X: 24, 505-520, LNM 511 (1976)

**WILLIAMS, David**

The Q-matrix problem 2: Kolmogorov backward equations (Markov processes)

This is an addition to 1014, the problem being now of constructing a chain whose transition probabilities satisfy the Kolmogorov backward equations, as defined in a precise way in the paper. A different construction is required

Keywords: Markov chains

Nature: Original

Retrieve article from Numdam

X: 26, 532-535, LNM 511 (1976)

**NAGASAWA, Masao**

Note on pasting of two Markov processes (Markov processes)

The pasting or piecing out theorem says roughly that two Markov processes taking values in two open sets and agreeing up to the first exit time of their intersection can be extended into a single Markov process taking values in their union. The word ``roughly'' replaces a precise definition, necessary in particular to handle jumps. Though the result is intuitively obvious, its proof is surprisingly messy. It is due to Courrège-Priouret,*Publ. Inst. Stat. Univ. Paris,* **14**, 1965. Here it is reduced to a ``revival theorem'' of Ikeda-Nagasawa-Watanabe, *J. Math. Kyoto Univ.*, **8**, 1968

Comment: The piecing out theorem is also reduced to a revival theorem in Meyer,*Ann. Inst. Fourier,* **25**,1975

Keywords: Piecing-out theorem

Nature: Original

Retrieve article from Numdam

XI: 14, 257-297, LNM 581 (1977)

**YOR, Marc**

Sur les théories du filtrage et de la prédiction (General theory of processes, Markov processes)

To be completed

Comment: MR 57, 10801

Keywords: Filtering theory, Prediction theory

Nature: Original

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 1513

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

Nature: Original

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XI: 39, 566-573, LNM 581 (1977)

**ÉMERY, Michel**

Information associée à un semigroupe (Markov processes)

This paper contains the proof of two important theorems of Donsker and Varadhan (*Comm. Pure and Appl. Math.*, 1975)

Nature: Exposition

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XII: 22, 310-331, LNM 649 (1978)

**WILLIAMS, David**

The Q-matrix problem 3: The Lévy-kernel problem for chains (Markov processes)

After solving the Q-matrix problem in 1014, the author constructs here a Markov chain from a Q-matrix on a countable space $I$ which satisfies several desirable conditions. Among them, the following: though the process is defined on a (Ray) compactification of $I$, the Q-matrix should describe the full Lévy kernel. Otherwise stated, whenever the process jumps, it does so from a point of $I$ to a point of $I$. The construction is extremely delicate

Keywords: Markov chains

Nature: Original

Retrieve article from Numdam

XII: 34, 468-481, LNM 649 (1978)

**SAINT RAYMOND, Jean**

Quelques remarques sur un article de Donsker et Varadhan (Markov processes)

This paper is a partial exposition of the celebrated papers of Donsker and Varadhan in*Comm. Pure Appl. Math.* 28, 1975 and 29, 1976, aiming at simplifying the proofs and weakening a few technical hypotheses

Keywords: Large deviations

Nature: Exposition

Retrieve article from Numdam

XII: 53, 741-741, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Inégalités de Littlewood-Paley'' (Applications of martingale theory, Markov processes)

This is an erratum to 1010

Keywords: Littlewood-Paley theory, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Correction

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 functionals

Nature: Original

Retrieve article from Numdam

XII: 60, 804-805, LNM 649 (1978)

**LETTA, Giorgio**

Un système de notations pour les processus de Markov (Markov processes)

Instead of notations like $X_T$, $\Theta_T$, etc, the author suggests to use kernel notations---for instance, $X^T$ is the submarkovian kernel $f\mapsto f\circ X_T$ ($0$ on $\{T=\infty\}$). Then the main properties of Markov processes are expressed by simple kernel equalities

Nature: Original

Retrieve article from Numdam

XIII: 33, 385-399, LNM 721 (1979)

**LE JAN, Yves**

Martingales et changement de temps (Martingale theory, Markov processes)

The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to 1108 and 1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary

Keywords: Changes of time, Energy, Douglas formula

Nature: Original

Retrieve article from Numdam

XIII: 43, 490-494, LNM 721 (1979)

**WILLIAMS, David**

Conditional excursion theory (Brownian motion, Markov processes)

To be completed

Keywords: Excursions

Nature: Original

Retrieve article from Numdam

XIII: 44, 495-520, LNM 721 (1979)

**BISMUT, Jean-Michel**

Problèmes à frontière libre et arbres de mesures (Miscellanea, Markov processes)

An optimization problem is discussed, in which one is free to choose at any time among three different transition semi-groups

Keywords: Control theory

Nature: Original

Retrieve article from Numdam

XIII: 50, 574-609, LNM 721 (1979)

**JEULIN, Thierry**

Grossissement d'une filtration et applications (General theory of processes, Markov processes)

This is a sequel to the papers 1209 and 1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths

Keywords: Enlargement of filtrations, Williams decomposition

Nature: 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 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)

**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

Retrieve article from Numdam

XIV: 39, 347-356, LNM 784 (1980)

**CHUNG, Kai Lai**

On stopped Feynman-Kac functionals (Markov processes, Diffusion theory)

Let $(X_t)$ be a strong Markov process with continuous paths on the line, and let $\tau_b$ be the hitting time of the point $b$. It is assumed that $\tau_b$ is $P_a$-a.s. finite for all $a,b$. The purpose of the paper is to study the quantities $u(a,b)=E_a[\,\exp(\int_0^{\tau_b} q(X_s)\,ds)\,]$ where $q$ is bounded. Then (among other results) if $u(a,b)<\infty$ for all $a<b$, we have $u(a,b)\,u(b,a)\le 1$ for all $a,b$

Keywords: Hitting probabilities

Nature: Original

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XIV: 42, 397-409, LNM 784 (1980)

**GETOOR, Ronald K.**

Transience and recurrence of Markov processes (Markov processes)

From the introduction: The purpose of this paper is to present an elementary exposition of some various conditions that have been used to define transience or recurrence of a Markov process... an elementary and unified discussion of these ideas may be worthwhile

Keywords: Recurrent Markov processes

Nature: Exposition, Original additions

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 functionals

Nature: 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 potentials

Nature: Original

Retrieve article from Numdam

XIV: 45, 437-474, LNM 784 (1980)

**TAKSAR, Michael I.**

Regenerative sets on real line (Markov processes, Renewal theory)

From the introduction: A number of papers are devoted to studying regenerative sets on a positive half-line... our objective is to construct translation invariant sets of this type on the entire real line. Besides we start from a weaker definition of regenerativity

Comment: This important paper, if written in recent years, would have merged into the theory of Kuznetsov measures

Keywords: Regenerative sets

Nature: Original

Retrieve article from Numdam

XV: 10, 151-166, LNM 850 (1981)

**MEYER, Paul-André**

Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)

The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (*J. Funct. Anal.*, 38, 1980)

Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ

Nature: Original

Retrieve article from Numdam

XV: 11, 167-188, LNM 850 (1981)

**BOULEAU, Nicolas**

Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov (Markov processes)

The main result of the paper of Kunita (*Nagoya Math. J.*, **36**, 1969) showed that the domain of the extended generator $A$ of a right Markov semigroup is an algebra if and only if the angle brackets of all martingales are absolutely continuous with respect to the measure $dt$. See also 1010. Such semigroups are called here ``semigroups of Lebesgue type''. Kunita's result is sharpened here: it is proved in particular that if some non-affine convex function $f$ operates on the domain, then the semigroup is of Lebesgue type (Kunita's result corresponds to $f(x)=x^2$) and if the second derivative of $f$ is not absolutely continuous, then the semigroup has no diffusion part (i.e., all martingales are purely discontinuous). The second part of the paper is devoted to the behaviour of the extended domain under an absolutely continuous change of probability (arising from a multiplicative functional)

Keywords: Semigroup theory, Carré du champ, Infinitesimal generators

Nature: Original

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XV: 21, 290-306, LNM 850 (1981)

**CHACON, Rafael V.**; **LE JAN, Yves**; **WALSH, John B.**

Spatial trajectories (Markov processes, General theory of processes)

It is well known that Markov processes with the same excessive functions are the same up to a strictly increasing continuous time-change. It is therefore natural to study spatial trajectories, i.e., trajectories up to a strictly increasing continuous time changes, and in particular to provide the space of all spatial trajectories with a reasonable $\sigma$-field so that it may carry measures. It is shown here that the space of right-continuous spatial trajectories with left-hand limits is a Blackwell space. The class of intrinsic stopping times defined on this space is also investigated

Comment: See Chacon-Jamison,*Israel J. of M.*, **33**, 1979

Keywords: Spatial trajectories

Nature: Original

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XV: 22, 307-310, LNM 850 (1981)

**LE JAN, Yves**

Tribus markoviennes et prédiction (Markov processes, General theory of processes)

The problem discussed here is whether a given filtration is generated by a Ray process. The answer is positive under very general conditions. Knight's prediction theory (1007) is used

Keywords: Prediction theory

Nature: Original

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XVII: 28, 243-297, LNM 986 (1983)

**ALDOUS, David J.**

Random walks on finite groups and rapidly mixing Markov chains (Markov processes)

The ``mixing time'' for a Markov chain---how many steps are needed to approximately reach the stationary distribution---is defined here by taking the variation distance between measures and the worst possible starting point, and bounded above by coupling arguments. For the simple random walk on the discrete cube $\{0,1\}^d$ with large $d$, there is a ``cut-off phenomenon'', an abrupt change in variation distance from 1 to 0 around time $1/4\ d\,\log d$. For a natural model of riffle shuffle of an $n$-card deck, there is an analogous cut-off at time $3/2\ \log n$. The relationship between ``rapid mixing'' and approximate exponential distribution for first hitting times on small subsets, is also discussed

Comment: In the 1960s and 1970s, Markov chains were considered by probabilists as rather trite objects. This work was one of several papers that prompted a reassessment and focused attention on the question of mixing time. In 1981, Diaconis-Sashahani (*Z. Wahrsch. Verw. Gebiete* **57**) had established the cut-off phenomenon for a different shuffling scheme. For a random walk on a graph, Alon (*Combinatorica* **6**, 1986) related an eigenvalue-based mixing time to expansion properties of the graph, and parallel work of Lawler-Sokal (*Trans. Amer. Math. Soc.* **309**, 1988) in the broader setting of reversible chains made a connection with models from statistical physics. Jerrum-Sinclair (*Inform. and Comput.* **82**, 1989) gave the first deep use of Markov chain methods in the theory of algorithms, while Geman-Geman (*IEEE Trans. Pattern Anal. Machine Intell.* **6**, 1984) promoted the use of Markov chains in image reconstruction. Such papers brought the attention of probabilists to the Metropolis algorithm in statistical physics, and foreshadowed the development of Markov chain Monte Carlo methods in Bayesian statistics, e.g. Smith (*Philos. Trans. Roy. Soc. London* **337**, 1991)

Keywords: Markov chains, Hitting probabilities

Nature: Original

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XX: 13, 162-185, LNM 1204 (1986)

**BOULEAU, Nicolas**; **LAMBERTON, Damien**

Théorie de Littlewood-Paley et processus stables (Applications of martingale theory, Markov processes)

Meyer' probabilistic approach to Littlewood-Paley inequalities (1010, 1510) is extended by replacing the underlying Brownian motion with a stable process. The following spectral multiplicator theorem is obtained: If $(P_t)_{t\geq 0}$ is a symmetric Markov semigroup with spectral representation $P_t=\int_{[0,\infty)}e^{-t\lambda} dE_{\lambda}$, and if $M$ is a function on $**R**_+$ defined by $M(\lambda)=\lambda\int_0^\infty r(y)e^{-y\lambda}dy,$ where $r(y)$ is bounded and Borel on $**R**_+$, then the operator $T_M=\int_{[0,\infty)}M(\lambda)dE_{\lambda},$ which is obviously bounded on $L^2$, is actually bounded on all $L^p$ spaces of the invariant measure, $1<p<\infty$. The method also leads to new Littlewood-Paley inequalities for semigroups admitting a carré du champ operator

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Stable processes, Inequalities, Singular integrals, Carré du champ

Nature: Original

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XXIV: 30, 448-452, LNM 1426 (1990)

**ÉMERY, Michel**; **LÉANDRE, Rémi**

Sur une formule de Bismut (Markov processes, Stochastic differential geometry)

This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group

Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes

Nature: Exposition, Original additions

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XXIX: 16, 166-180, LNM 1613 (1995)

**APPLEBAUM, David**

A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)

This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.

Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (*Stochastics Stochastics Rep.* **56**, 1996). The same question is addressed by Cohen in the next article 2917

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

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XXIX: 17, 181-193, LNM 1613 (1995)

**COHEN, Serge**

Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)

The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (*Stochastics Stochastics Rep.* **56**, 1996). Here this language is used to study the Markov property of solutions to SDE's with jumps. In particular,two definitions of a Lévy process in a Riemannian manifold are compared: One as the solution to a SDE driven by some Euclidean Lévy process, the other by subordinating some Riemannian Brownian motion. It is shown that in general the former is not of the second kind

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

Retrieve article from Numdam

XLI: 05, 121-135, LNM 1934 (2008)

**KYPRIANOU, Andreas E.**; **PALMOWSKI, Zbigniew**

Fluctuations of spectrally negative Markov additive processes (Theory of Markov processes)

Nature: Original

Semi-groupes de transition et fonctions excessives (Markov processes, Potential theory)

A study of product kernels, product semi-groups and product Markov processes

Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes

Keywords: Product semigroups

Nature: Original

Retrieve article from Numdam

I: 08, 166-176, LNM 39 (1967)

Retournement du temps dans les processus markoviens (Markov processes)

This talk presents the now classical results of Nagasawa (

Comment: See also 202. These results have been essentially the best ones until they were extended to Kuznetsov measures, see Dellacherie-Meyer,

Keywords: Time reversal, Dual semigroups

Nature: Exposition

Retrieve article from Numdam

I: 09, 177-189, LNM 39 (1967)

Résolvantes en dualité (Markov processes, Potential theory)

Given two sub-Markov resolvents in duality, whose kernels are absolutely continuous with respect to a given measure, it is shown how to choose their densities to get true Green kernels, excessive in one variable and coexcessive in the other one. It is shown also that coexcessive functions are exactly the densities of excessive measures

Comment: These results, now classical, are due to Kunita-T. Watanabe,

Keywords: Green potentials, Dual semigroups

Nature: Exposition

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II: 01, 1-21, LNM 51 (1968)

Classes récurrentes d'un processus de Markov (Markov processes)

This is an improved version of a paper by the same authors (

Comment: The subject is further investigated by the same authors in 302

Keywords: Recurrent sets, Fine topology

Nature: Original

Retrieve article from Numdam

II: 02, 22-33, LNM 51 (1968)

Le retournement du temps~: compléments à l'exposé de M.~Weil (Markov processes)

In 108, M.~Weil had presented the work of Nagasawa on the time reversal of a Markov process at a ``L-time'' or return time. Here the results are improved on three points: a Markovian filtration is given for the reversed process; an analytic condition on the semigroup is lifted; finally, the behaviour of the

Comment: The results of this paper have become part of the standard theory of time reversal. See 312 for a correction

Keywords: Time reversal, Dual semigroups

Nature: Original

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

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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II: 05, 75-110, LNM 51 (1968)

Théorie des frontières dans les chaînes de Markov (Markov processes)

A presentation of the theory of Markov chains under the hypothesis that all states are regular

Comment: This is the subject of the short monograph of Chung,

Keywords: Markov chains, Boundary theory

Nature: Exposition

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III: 02, 24-33, LNM 88 (1969)

Mesure invariante des processus de Markov récurrents (Markov processes)

A condition similar to the Harris recurrence condition is studied in continuous time. It is shown that it implies the existence (up to a constant factor) of a unique $\sigma$-finite excessive measure, which is invariant. The invariant measure for a time-changed process is described

Comment: This is related to several papers by the same authors on recurrent Markov processes, and in particular to 201

Keywords: Recurrent potential theory, Invariant measures

Nature: Original

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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

III: 05, 97-114, LNM 88 (1969)

Ensembles aléatoires I (Descriptive set theory, Markov processes, General theory of processes)

A deep theorem of Lusin asserts that a Borel set with countable sections is a countable union of Borel graphs. It is applied here in the general theory of processes to show that an optional set with countable sections is a countable union of graphs of stopping times, and in the theory of Markov processes, that a Borel set which is a.s. hit by the process at countably many times must be semi-polar

Comment: See Dellacherie,

Keywords: Sets with countable sections

Nature: Original

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III: 06, 115-136, LNM 88 (1969)

Ensembles aléatoires II (Descriptive set theory, Markov processes)

Among the many proofs that an uncountable Borel set of the line contains a perfect set, a proof of Sierpinski (

Comment: See Dellacherie,

Keywords: Sierpinski's ``rabotages'', Semi-polar sets

Nature: Original

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III: 09, 144-151, LNM 88 (1969)

Un résultat de théorie du potentiel (Potential theory, Markov processes)

Under strong duality hypotheses, it is shown that a measure which does not charge polar sets is equivalent to a measure whose Green potential is bounded

Comment: See Meyer,

Keywords: Green potentials, Dual semigroups

Nature: Original

Retrieve article from Numdam

III: 12, 160-162, LNM 88 (1969)

Rectification à des exposés antérieurs (Markov processes, Martingale theory)

Corrections are given to the talk 202 by Cartier, Meyer and Weil and to the talk 106 by Meyer

Comment: This note introduces ``Walsh's fork'', the well-known strong Markov process whose dual is not strong Markov

Keywords: Time reversal, Stochastic integrals

Nature: Correction

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III: 14, 175-189, LNM 88 (1969)

Processus à accroissements indépendants et positifs (Markov processes, Independent increments)

This is an exposition of the theory of subordinators (Lévy processes with increasing paths), aiming at presenting Chung's conjecture that a certain identity known to hold a.e. actually holds everywhere, also equivalent to the fact that single points are polar sets for subordinators without drift

Comment: The conjecture was proved by Kesten (see 503) who actually knew of the problem through this talk. See also 502

Keywords: Subordinators, Polar sets

Nature: Exposition

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IV: 06, 71-72, LNM 124 (1970)

Au sujet des sauts d'un processus de Hunt (Markov processes)

Two a.s. results on jumps: the process cannot jump

Comment: Both results are improvements of previous results of Meyer and Weil

Keywords: Hunt processes, Semi-polar sets

Nature: Original

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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

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IV: 12, 133-150, LNM 124 (1970)

Ensembles régénératifs, d'après Hoffmann-Jørgensen (Markov processes)

The theory of recurrent events in discrete time was a highlight of the old probability theory. It was extended to continuous time by Kingman (see for instance

Comment: This result was expanded to involve a Markovian regeneration property instead of independence. See Maisonneuve-Meyer 813. The subject is related to excursion theory, Lévy systems, semi-Markovian processes (Lévy), F-processes (Neveu), Markov renewal processes (Pyke), and the literature is very extensive. See for instance Dynkin (

Keywords: Renewal theory, Regenerative sets, Recurrent events

Nature: Exposition

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IV: 13, 151-161, LNM 124 (1970)

Temps locaux pour les ensembles régénératifs (Markov processes)

This paper uses the results of the preceding one 412 to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of increase is exactly the given set

Comment: Same references as the preceding paper 412

Keywords: Renewal theory, Regenerative sets, Local times

Nature: Original

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IV: 17, 208-215, LNM 124 (1970)

Application d'un théorème de Mokobodzki aux opérateurs potentiels dans le cas récurrent (Potential theory, Markov processes)

Mokododzki's theorem asserts that if the kernels of a resolvent are strong Feller, i.e., map bounded functions into continuous functions, then they must satisfy a norm continuity property (see 210). This is used to show the existence for``normal'' recurrent processes of a nice potential operator, defined for suitable functions of zero integral with respect to the invariant measure

Comment: For additional work of Revuz on recurrence, see

Keywords: Recurrent potential theory

Nature: Original

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IV: 18, 216-239, LNM 124 (1970)

Quasi-processus (Markov processes)

Excessive measures which are not potentials of measures were shown by Hunt (

Comment: Further work by M.~Weil on the same subject in 532; see the references there

Keywords: Hunt quasi-processes

Nature: Original

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IV: 19, 240-282, LNM 124 (1970)

Diffusions à coefficients continus, d'après Stroock et Varadhan (Markov processes, Diffusion theory)

This paper consists of four seminar talks on a celebrated paper of Stroock-Varadhan (

Comment: The results were so definitive that apparently the subject attracted no further work. See Stroock-Varadhan,

Keywords: Elliptic differential operators, Uniqueness in law

Nature: Exposition

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V: 02, 17-20, LNM 191 (1971)

Démonstration de la ``Conjecture de Chung'' par Carleson (Markov processes, Independent increments)

Chung conjectured that singletons are polar sets for driftless subordinators. This paper gives Carleson's (unpublished) analytic proof of it

Comment: See Chung,

Keywords: Subordinators, Polar sets

Nature: Exposition

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V: 03, 21-36, LNM 191 (1971)

Résultats de Kesten sur les processus à accroissements indépendants (Markov processes, Independent increments)

The question is to find all Lévy processes for which single points are polar. Kesten's answer (

Comment: See also 502 in the same volume

Keywords: Subordinators, Polar sets

Nature: Exposition, Original additions

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V: 17, 177-190, LNM 191 (1971)

Processus de Poisson ponctuels d'après K. Ito (Markov processes, Point processes)

Presents (a preliminary form of) the celebrated paper of Ito (

Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form

Keywords: Poisson point processes, Excursions, Local times

Nature: Exposition, Original additions

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V: 19, 196-208, LNM 191 (1971)

Représentation intégrale des fonctions excessives. Résultats de Mokobodzki (Markov processes, Potential theory)

Main result: the convex cone of excessive functions for a resolvent which satisfies the absolute continuity hypothesis is the union of convex compact metrizable ``hats''\ in a suitable topology, and therefore has the integral representation property. The original proof of Mokobodzki, self-contained and unpublished, is given here

Comment: See Mokobodzki's work on cones of potentials,

Keywords: Minimal excessive functions, Martin boundary, Integral representations

Nature: Exposition

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V: 20, 209-210, LNM 191 (1971)

Un théorème sur la répartition des temps locaux (Markov processes)

Kesten discovered that the value at a terminal time $T$ of the local time $L$ of a Markov process $X$ at a single point has an exponential distribution, and that $X_T$ and $L_T$ are independent. A short proof is given

Comment: The result can be deduced from excursion theory

Keywords: Local times

Nature: New exposition of known results

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V: 22, 213-236, LNM 191 (1971)

Le retournement du temps, d'après Chung et Walsh (Markov processes)

The paper of Chung and Walsh (

Comment: The theorem of Chung-Walsh remains the deepest on time reversal (to be supplemented by the consideration of Kuznetsov's measures)

Keywords: Time reversal, Dual semigroups

Nature: Exposition, Original additions

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V: 24, 251-269, LNM 191 (1971)

Solutions de l'équation de Poisson dans le cas récurrent (Potential theory, Markov processes)

The problem is to solve the Poisson equation for measures, $\mu-\mu P=\theta$ for given $\theta$, in the case of a recurrent transition kernel $P$. Here a ``filling scheme'' technique is used

Comment: The paper was motivated by Métivier (

Keywords: Recurrent potential theory, Filling scheme, Harris recurrence, Poisson equation

Nature: Original

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V: 25, 270-274, LNM 191 (1971)

Balayage pour les processus de Markov continus à droite, d'après C.T. Shih (Markov processes, Potential theory)

Hunt's fundamental theorem on balayage gives the probabilistic interpretation of the réduite of an excessive function set on a set. It was proved originally within the special class of Hunt's processes, then extended to ``standard'' processes. Using a method of compactification, Shih (

Comment: Shih's paper is the origin of the general definition of ``right processes''

Keywords: Excessive functions, Réduite

Nature: Exposition

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V: 26, 275-277, LNM 191 (1971)

Remarque sur les potentiels de mesure (Markov processes, Potential theory)

The standard proof of the equivalence between semi-polar sets being polar and a very precise domination principle (Blumenthal-Getoor,

Comment: To be asked

Keywords: Polar sets, Semi-polar sets, Excessive functions

Nature: Original

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V: 28, 283-289, LNM 191 (1971)

Two footnotes to a theorem of Ray (Markov processes)

Ray's theorem (

Comment: See Meyer-Walsh,

Keywords: Ray compactification

Nature: Original

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V: 29, 290-310, LNM 191 (1971)

Some topologies connected with Lebesgue measure (Markov processes, General theory of processes, Potential theory)

It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called

Comment: See Doob

Keywords: Essential topology

Nature: Original

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V: 31, 342-346, LNM 191 (1971)

Décomposition d'un temps terminal (Markov processes)

It is shown that for a Hunt process, a terminal time can be represented as the infimum of a previsible terminal time, and a totally inaccessible terminal time

Keywords: Terminal times

Nature: Original

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V: 32, 347-361, LNM 191 (1971)

Quasi-processus et énergie (Markov processes, Potential theory)

The energy of an excessive function $f$ with respect to an excessive measure $\xi$ has a simple proba\-bi\-listic interpretation if $\xi$ is is the potential of a measure $\mu$ and $f$ is the potential of an additive functional $(A_t)$, as ${1\over2}E_\mu[A_\infty^2]$. If $\xi$ is not a potential, still it can be associated with it a quasi-process (see Weil 418) with a birthtime $b$ and a death time $d$, and the formal expression ${1\over2}E[(A_d-A_b)^2]$ is given a precise meaning and represents the energy

Comment: This subject has been renewed by the introduction of Kuznetsov's measures. See Fitzsimmons

Keywords: Hunt quasi-processes, Energy

Nature: Original

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V: 33, 362-372, LNM 191 (1971)

Conditionnement par rapport au passé strict (Markov processes)

Given a totally inaccessible terminal time $T$, it is shown how to compute conditional expectations of the future with respect to the strict past $\sigma$-field ${\cal F}_{T-}$. The formula involves the Lévy system of the process

Comment: B. Maisonneuve pointed out once that the paper, though essentially correct, has a small mistake somewhere. See Dellacherie-Meyer,

Keywords: Terminal times, Lévy systems

Nature: Original

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VI: 02, 35-50, LNM 258 (1972)

Une remarque sur les temps de retour. Trois applications (Markov processes, General theory of processes)

This paper is the first step in the investigations of Azéma on the ``dual'' form of the general theory of processes (for which see Azéma (

Keywords: Homogeneous processes, Coprevisible processes, Cooptional processes, Section theorems, Projection theorems, Time reversal

Nature: Original

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VI: 09, 109-112, LNM 258 (1972)

Un gros processus de Markov. Application à certains flots (Markov processes)

In a vague but useful sense, a ``big'' process over a given process consists of random variables whose values are a part of the path of the original process (the best known example is the excursion process). Here it is shown how the past of a Markov process can be turned into a big (homogeneous) Markov process, and how its semigroup is computed using an idea of Dawson (

Comment: For a complete account of Dawson's formula, see Dellacherie-Meyer,

Keywords: Prediction theory, Filtered flows

Nature: Original

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VI: 15, 164-167, LNM 258 (1972)

Une note sur le théorème du balayage de Hunt (Markov processes, Potential theory)

The theorem involved is the characterization of the réduite $P_A u$ of an excessive function $u$ on a set $A$ as equal (except on a well-defined semipolar set) to the infimum of the excessive functions that dominate $u$ on $A$. This theorem is slightly improved under the absolute continuity hypothesis. The proof rests on the following property of the fine topology: every (nearly Borel) finely closed set is contained in a fine $G_\delta$ which differs from it by a polar set

Keywords: Réduite, Fine topology, Absolute continuity hypothesis

Nature: Original

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VI: 16, 168-172, LNM 258 (1972)

Un résultat sur les résolvantes de Ray (Markov processes)

This is a complement to the authors' paper on Ray processes in

Comment: The idea of using the topology of convergence in measure on a path space turned out to be a fruitful idea; see Meyer and Zheng

Keywords: Ray compactification, Weak convergence of measures

Nature: Original

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VI: 18, 177-197, LNM 258 (1972)

Branching property of Markov processes (Markov processes)

To be completed

Keywords: Branching processes

Nature: Original

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VI: 21, 215-232, LNM 258 (1972)

Transition functions of Markov processes (Markov processes)

Assume that a cadlag process satisfies the strong Markov property with respect to some family of kernels $P_t$ (not necessarily a semigroup). It is shown that these kernels can be modified into a true strong Markov transition function with a few additional properties. A similar problem is solved for a left continuous, moderate Markov process. The technique involves a Ray compactification which is eliminated at the end, and a useful lemma shows how to construct supermedian functions which separate points

Comment: The problem discussed here has great theoretical importance, but little practical importance except for time reversal. The construction of a nice transition function for a Markov process has been also discussed by Kuznetsov ()

Keywords: Transition functions, Strong Markov property, Moderate Markov property, Ray compactification

Nature: Original

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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

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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

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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

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VII: 02, 25-32, LNM 321 (1973)

Une mise au point sur les systèmes de Lévy. Remarques sur l'exposé de A. Benveniste (Markov processes)

This is an addition to the preceding paper 701, extending the theory to right processes by means of a Ray compactification

Comment: All this material has become classical. See for instance Dellacherie-Meyer,

Keywords: Lévy systems, Ray compactification

Nature: Original

Retrieve article from Numdam

VII: 07, 51-57, LNM 321 (1973)

Une conjecture sur les ensembles semi-polaires (Markov processes)

For a right process satisfying the absolute continuity hypothesis and assuming singletons are semi-polar sets, it is conjectured that a (nearly-)Borel set is semipolar if and only if it does not contain uncountable families of disjoint, non-polar compact sets. This statement implies that two processes which have the same polar sets also have the same semi-polar sets

Comment: The conjecture can be proved, using a general result of Mokobodzki, see 1238

Keywords: Polar sets, Semi-polar sets

Nature: Original

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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

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VII: 13, 122-135, LNM 321 (1973)

Processus de Galton-Watson (Markov processes)

This paper is mostly a survey of previous results with comments and some alternative proofs

Comment: An erroneous statement is corrected in 939

Keywords: Branching processes, Galton-Watson processes

Nature: Exposition, Original additions

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VII: 15, 146-154, LNM 321 (1973)

Chirurgie sur un processus de Markov, d'après Knight et Pittenger (Markov processes)

This paper presents a remarkable result of Knight and Pittenger, according to which excising from the sample path of a Markov process all the excursions from a given set $A$ which meet another set $B$ preserves the Markov property

Comment: The original paper appeared in

Keywords: Transformations of Markov processes, Excursions

Nature: Exposition

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VII: 17, 172-179, LNM 321 (1973)

Application de l'exposé précédent aux processus de Markov (Markov processes)

This paper is devoted to results of J.F. Mertens on optimal stopping and strongly supermedian functions for a right Markov process (

Comment: See related papers by Mertens in

Keywords: Excessive functions, Supermedian functions, Réduite

Nature: Exposition, Original proofs

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VII: 20, 205-209, LNM 321 (1973)

Remarques sur les hypothèses droites (Markov processes)

The following technical point is discussed: why does one assume among the ``right hypotheses'' that excessive functions are nearly-Borel?

Keywords: Right processes, Excessive functions

Nature: Original

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VII: 21, 210-216, LNM 321 (1973)

Note sur l'interprétation des mesures d'équilibre (Markov processes)

Let $(X_t)$ be a transient Markov process (we omit the detailed assumptions) with a potential density $u(x,y)$. Let $\mu$ be the measure whose potential is the equilibrium potential of a set $A$. Then the distribution of the process at the last exit time from $A$ is given by $$E_x[f\circ X_{L-}, 0<L<\infty]=\int u(x,y)\,f(y)\,\mu(dy)$$ This formula, due to Chung, is deduced under minimal duality hypotheses from a general formula of Azéma, and a well-known theorem on Revuz measures

Keywords: Equilibrium potentials, Last exit time, Revuz measures

Nature: Exposition, Original proofs

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VII: 25, 273-283, LNM 321 (1973)

Fonctionnelles multiplicatives opératrices (Markov processes)

This paper presents results due to the author (

Comment: This paper explores the same idea as Jacod (

Keywords: Multiplicative functionals, Multiplicative kernels

Nature: Exposition

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VIII: 01, 1-10, LNM 381 (1974)

Une nouvelle représentation du type de Skorohod (Markov processes)

A Skorohod imbedding theorem for general Markov processes is proved, in which the stopping time is a randomized ``left'' terminal time. A uniqueness result is proved

Comment: The result is deduced from a representation of measures by left additive functionals, due to Azéma (

Keywords: Skorohod imbedding, Multiplicative functionals

Nature: Original

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VIII: 03, 20-21, LNM 381 (1974)

Note on last exit decomposition (Markov processes)

This is a useful complement to the monograph of Chung

Keywords: Markov chains

Nature: Original

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VIII: 06, 27-36, LNM 381 (1974)

Stopping sequences (Markov processes, Potential theory)

Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process

Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see 523, 524, 612. This article contains announcements of further results

Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme

Nature: Original

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VIII: 09, 80-133, LNM 381 (1974)

Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)

The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (

Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces

Nature: Exposition, Original additions

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VIII: 11, 150-154, LNM 381 (1974)

Skorohod stopping via Potential Theory (Potential theory, Markov processes)

The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $

Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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VIII: 13, 172-261, LNM 381 (1974)

Ensembles aléatoires markoviens homogènes (5 talks) (Markov processes)

This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see 412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (``left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (

Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as

Keywords: Regenerative systems, Regenerative sets, Renewal theory, Local times, Excursions, Markov chains, Incursions

Nature: Original

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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

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VIII: 15, 289-289, LNM 381 (1974)

Une note sur la compactification de Ray (Markov processes)

This short note shows that (contrary to the belief of Meyer and Walsh in a preceding paper) the state space of a Ray process is universally measurable in its Ray compactification

Comment: This is now considered a standard fact

Keywords: Ray compactification, Right processes

Nature: Original

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VIII: 16, 290-309, LNM 381 (1974)

Noyaux multiplicatifs (Markov processes)

This paper presents results due to Jacod (

Comment: Though the paper is fairly technical, it does not improve substantially on Jacod's results

Keywords: Multiplicative kernels, Semimarkovian processes

Nature: Exposition

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VIII: 18, 316-328, LNM 381 (1974)

Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)

The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic

Comment: To be completed

Keywords: Processes with jumps

Nature: Original

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VIII: 19, 329-343, LNM 381 (1974)

Remarks on the hypotheses of duality (Markov processes)

To develop the full strength of potential theory, it is helpful to assume that the basic Markov process has a right-continuous, strong Markov dual. The paper investigates what remains if this assumption is not made, i.e., if the process (transient and satisfying the absolute continuity hypothesis (hypothesis (L)) only has a left continuous moderately Markov dual process (Smythe-Walsh ,

Comment: The independent paper Garsia Alvarez-Meyer

Keywords: Dual semigroups

Nature: Original

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IX: 26, 471-485, LNM 465 (1975)

Multiplicative excessive measures and duality between equations of Boltzmann and of branching processes (Markov processes, Statistical mechanics)

The author investigates the connection between the branching Markov processes constructed over some given Markov processes and a non-linear equation close to Boltzmann's equation. A special class of excessive measures for the branching Markov process is described and studied, as well as the corresponding dual processes

Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 1011

Keywords: Boltzmann equation, Branching processes

Nature: Original

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IX: 27, 486-493, LNM 465 (1975)

Surlois d'entrée (Markov processes)

The results presented here are due to T. Leviatan (

Comment: This beautiful paper was superseded by the (slightly later) fundamental paper of Kuznetsov (1974)

Keywords: Entrance laws, Creation of mass, Kuznetsov measures

Nature: Exposition

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IX: 29, 495-495, LNM 465 (1975)

Une propriété des ensembles semi-polaires (Markov processes)

It is shown that semi-polar sets are exactly those which have potential 0 for all continuous additive functionals (or for all time-changed processes)

Keywords: Semi-polar sets

Nature: Original

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IX: 30, 496-514, LNM 465 (1975)

Homogeneous extensions of random measures (Markov processes)

Homogeneous random measures are the appropriate definition of additive functionals which may explode. The problem discussed here is the extension of such a measure given up to a terminal time into a measure defined up to the lifetime

Comment: The subject is taken over in a systematic way in Sharpe,

Keywords: Homogeneous random measures, Terminal times, Subprocesses

Nature: Original

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IX: 31, 515-517, LNM 465 (1975)

Skorohod stopping in discrete time (Markov processes, Potential theory)

Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times

Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath 811. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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IX: 32, 518-521, LNM 465 (1975)

Ensembles aléatoires markoviens homogènes. Mise au point et compléments (Markov processes)

This paper corrects or simplifies many details in the long paper 713 by the same authors

Comment: See also the next paper 933

Keywords: Regenerative systems, Last-exit decompositions, Excursions

Nature: Original

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IX: 33, 522-529, LNM 465 (1975)

Le comportement de dernière sortie (Markov processes)

This paper contains improvements to the paper 813 by Maisonneuve-Meyer, whose results are briefly recalled. Incursion processes and Lévy systems are altogether avoided, last-exist decompositions are derived, and the strong Markov property of the analogue of the age process in renewal theory is proved, as well as a non-homogeneous Markov property for some processes starting at last-exit times. The extension of these results to abstractly defined regenerative systems is mentioned

Comment: More detailed versions of these results appear in Maisonneuve,

Keywords: Regenerative systems, Last-exit decompositions, Excursions

Nature: Original

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IX: 34, 530-533, LNM 465 (1975)

Sur la démonstration de prévisibilité de Chung et Walsh (Markov processes)

A new proof of the result that for a Hunt process, the previsible stopping times are exactly those at which the process does not jump was given by Chung-Walsh (

Comment: The contents of this paper became Chapter XIV

Keywords: Hunt processes, Previsible times

Nature: Exposition

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IX: 36, 555-555, LNM 465 (1975)

Une remarque sur les processus de Markov (Markov processes)

It is shown that, under a fixed measure $

Comment: No applications are known

Keywords: Stopping times

Nature: Original

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IX: 37, 556-564, LNM 465 (1975)

Retour aux retournements (Markov processes, General theory of processes)

The first part of the talk is devoted to an important correction to the theorem on p.285 of 814 (Azéma's theory of cooptional and coprevisible sets and its application to Markov processes): the definition of left additive functionals should allow an a.s. explosion at time 0 for initial points belonging to a polar set. The second part belongs to the general theory of processes: if the index set is not the right half-line as usual but the left half-line, the section and projection theorems must be modified in a not quite trivial way

Comment: See Chapter XXVIII of Dellacherie-Maisonneuve-Meyer

Keywords: Time reversal, Cooptional processes, Coprevisible processes, Homogeneous processes

Nature: Exposition, Original additions

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IX: 39, 589-589, LNM 465 (1975)

Correction à ``Processus de Galton-Watson'' (Markov processes)

Proposition 1(ii) p.126 of 713 is true only in the case $q=0$

Keywords: Branching processes

Nature: Correction

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X: 03, 24-39, LNM 511 (1976)

Un théorème de représentation des martingales pour les ensembles régénératifs (Martingale theory, Markov processes, Stochastic calculus)

The natural filtration of a regenerative set $M$ is that of the corresponding ``age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration

Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in 1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second ``Azéma's martingale'' (not the well known one which has the chaotic representation property)

Keywords: Regenerative sets, Renewal theory, Stochastic integrals, Previsible representation

Nature: Original

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X: 05, 44-77, LNM 511 (1976)

Absolute continuity for Markov processes (Markov processes)

This paper is devoted to a ``progressive'' Lebesgue decomposition of the laws of a Markov process with respect to a second one in the same filtration, and the structure of the corresponding density. The two processes are assumed to be Hunt processes, and for part of the paper satisfy Hunt's hypothesis (K) (all excessive functions are regular, or semi-polar sets are polar). The topics discussed are the following: Lévy systems and the relation between the Lévy systems of a process and of its transform by a multiplicative functional; structure of exact perfect terminal times, which are shown to be hitting times of sets in space-time, by the process $(X_{t-},X_t)$ (a version of a result of Walsh-Weil,

Comment: The pasting together of the Lebesgue decompositions of a probability measure with respect to another one, on the $\sigma$-fields of a given filtration, is called the

Keywords: Absolute continuity of laws, Hunt processes, Terminal times, Kunita decomposition

Nature: Original

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X: 10, 125-183, LNM 511 (1976)

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book

Comment: This paper was rediscovered by Varopoulos (

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

Retrieve article from Numdam

X: 11, 184-193, LNM 511 (1976)

A probabilistic approach to non-linear Dirichlet problem (Markov processes)

The theory of branching Markov processes in continuous time developed in particular by Ikeda-Nagasawa-Watanabe (

Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see 618, 926

Keywords: Branching processes, Dirichlet problem

Nature: Original

Retrieve article from Numdam

X: 12, 194-208, LNM 511 (1976)

Skorohod stopping times of minimal variance (Markov processes)

Root's (

Comment: For previous work of the author on Skorohod's imbedding see

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

X: 14, 216-234, LNM 511 (1976)

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

Retrieve article from Numdam

X: 24, 505-520, LNM 511 (1976)

The Q-matrix problem 2: Kolmogorov backward equations (Markov processes)

This is an addition to 1014, the problem being now of constructing a chain whose transition probabilities satisfy the Kolmogorov backward equations, as defined in a precise way in the paper. A different construction is required

Keywords: Markov chains

Nature: Original

Retrieve article from Numdam

X: 26, 532-535, LNM 511 (1976)

Note on pasting of two Markov processes (Markov processes)

The pasting or piecing out theorem says roughly that two Markov processes taking values in two open sets and agreeing up to the first exit time of their intersection can be extended into a single Markov process taking values in their union. The word ``roughly'' replaces a precise definition, necessary in particular to handle jumps. Though the result is intuitively obvious, its proof is surprisingly messy. It is due to Courrège-Priouret,

Comment: The piecing out theorem is also reduced to a revival theorem in Meyer,

Keywords: Piecing-out theorem

Nature: Original

Retrieve article from Numdam

XI: 14, 257-297, LNM 581 (1977)

Sur les théories du filtrage et de la prédiction (General theory of processes, Markov processes)

To be completed

Comment: MR 57, 10801

Keywords: Filtering theory, Prediction theory

Nature: Original

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

XI: 39, 566-573, LNM 581 (1977)

Information associée à un semigroupe (Markov processes)

This paper contains the proof of two important theorems of Donsker and Varadhan (

Nature: Exposition

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XII: 22, 310-331, LNM 649 (1978)

The Q-matrix problem 3: The Lévy-kernel problem for chains (Markov processes)

After solving the Q-matrix problem in 1014, the author constructs here a Markov chain from a Q-matrix on a countable space $I$ which satisfies several desirable conditions. Among them, the following: though the process is defined on a (Ray) compactification of $I$, the Q-matrix should describe the full Lévy kernel. Otherwise stated, whenever the process jumps, it does so from a point of $I$ to a point of $I$. The construction is extremely delicate

Keywords: Markov chains

Nature: Original

Retrieve article from Numdam

XII: 34, 468-481, LNM 649 (1978)

Quelques remarques sur un article de Donsker et Varadhan (Markov processes)

This paper is a partial exposition of the celebrated papers of Donsker and Varadhan in

Keywords: Large deviations

Nature: Exposition

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XII: 53, 741-741, LNM 649 (1978)

Correction à ``Inégalités de Littlewood-Paley'' (Applications of martingale theory, Markov processes)

This is an erratum to 1010

Keywords: Littlewood-Paley theory, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Correction

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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

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XII: 60, 804-805, LNM 649 (1978)

Un système de notations pour les processus de Markov (Markov processes)

Instead of notations like $X_T$, $\Theta_T$, etc, the author suggests to use kernel notations---for instance, $X^T$ is the submarkovian kernel $f\mapsto f\circ X_T$ ($0$ on $\{T=\infty\}$). Then the main properties of Markov processes are expressed by simple kernel equalities

Nature: Original

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XIII: 33, 385-399, LNM 721 (1979)

Martingales et changement de temps (Martingale theory, Markov processes)

The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to 1108 and 1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary

Keywords: Changes of time, Energy, Douglas formula

Nature: Original

Retrieve article from Numdam

XIII: 43, 490-494, LNM 721 (1979)

Conditional excursion theory (Brownian motion, Markov processes)

To be completed

Keywords: Excursions

Nature: Original

Retrieve article from Numdam

XIII: 44, 495-520, LNM 721 (1979)

Problèmes à frontière libre et arbres de mesures (Miscellanea, Markov processes)

An optimization problem is discussed, in which one is free to choose at any time among three different transition semi-groups

Keywords: Control theory

Nature: Original

Retrieve article from Numdam

XIII: 50, 574-609, LNM 721 (1979)

Grossissement d'une filtration et applications (General theory of processes, Markov processes)

This is a sequel to the papers 1209 and 1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths

Keywords: Enlargement of filtrations, Williams decomposition

Nature: Original

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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

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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: 39, 347-356, LNM 784 (1980)

On stopped Feynman-Kac functionals (Markov processes, Diffusion theory)

Let $(X_t)$ be a strong Markov process with continuous paths on the line, and let $\tau_b$ be the hitting time of the point $b$. It is assumed that $\tau_b$ is $P_a$-a.s. finite for all $a,b$. The purpose of the paper is to study the quantities $u(a,b)=E_a[\,\exp(\int_0^{\tau_b} q(X_s)\,ds)\,]$ where $q$ is bounded. Then (among other results) if $u(a,b)<\infty$ for all $a<b$, we have $u(a,b)\,u(b,a)\le 1$ for all $a,b$

Keywords: Hitting probabilities

Nature: Original

Retrieve article from Numdam

XIV: 42, 397-409, LNM 784 (1980)

Transience and recurrence of Markov processes (Markov processes)

From the introduction: The purpose of this paper is to present an elementary exposition of some various conditions that have been used to define transience or recurrence of a Markov process... an elementary and unified discussion of these ideas may be worthwhile

Keywords: Recurrent Markov processes

Nature: Exposition, Original additions

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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

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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

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XIV: 45, 437-474, LNM 784 (1980)

Regenerative sets on real line (Markov processes, Renewal theory)

From the introduction: A number of papers are devoted to studying regenerative sets on a positive half-line... our objective is to construct translation invariant sets of this type on the entire real line. Besides we start from a weaker definition of regenerativity

Comment: This important paper, if written in recent years, would have merged into the theory of Kuznetsov measures

Keywords: Regenerative sets

Nature: Original

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XV: 10, 151-166, LNM 850 (1981)

Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)

The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (

Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ

Nature: Original

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XV: 11, 167-188, LNM 850 (1981)

Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov (Markov processes)

The main result of the paper of Kunita (

Keywords: Semigroup theory, Carré du champ, Infinitesimal generators

Nature: Original

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XV: 21, 290-306, LNM 850 (1981)

Spatial trajectories (Markov processes, General theory of processes)

It is well known that Markov processes with the same excessive functions are the same up to a strictly increasing continuous time-change. It is therefore natural to study spatial trajectories, i.e., trajectories up to a strictly increasing continuous time changes, and in particular to provide the space of all spatial trajectories with a reasonable $\sigma$-field so that it may carry measures. It is shown here that the space of right-continuous spatial trajectories with left-hand limits is a Blackwell space. The class of intrinsic stopping times defined on this space is also investigated

Comment: See Chacon-Jamison,

Keywords: Spatial trajectories

Nature: Original

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XV: 22, 307-310, LNM 850 (1981)

Tribus markoviennes et prédiction (Markov processes, General theory of processes)

The problem discussed here is whether a given filtration is generated by a Ray process. The answer is positive under very general conditions. Knight's prediction theory (1007) is used

Keywords: Prediction theory

Nature: Original

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XVII: 28, 243-297, LNM 986 (1983)

Random walks on finite groups and rapidly mixing Markov chains (Markov processes)

The ``mixing time'' for a Markov chain---how many steps are needed to approximately reach the stationary distribution---is defined here by taking the variation distance between measures and the worst possible starting point, and bounded above by coupling arguments. For the simple random walk on the discrete cube $\{0,1\}^d$ with large $d$, there is a ``cut-off phenomenon'', an abrupt change in variation distance from 1 to 0 around time $1/4\ d\,\log d$. For a natural model of riffle shuffle of an $n$-card deck, there is an analogous cut-off at time $3/2\ \log n$. The relationship between ``rapid mixing'' and approximate exponential distribution for first hitting times on small subsets, is also discussed

Comment: In the 1960s and 1970s, Markov chains were considered by probabilists as rather trite objects. This work was one of several papers that prompted a reassessment and focused attention on the question of mixing time. In 1981, Diaconis-Sashahani (

Keywords: Markov chains, Hitting probabilities

Nature: Original

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XX: 13, 162-185, LNM 1204 (1986)

Théorie de Littlewood-Paley et processus stables (Applications of martingale theory, Markov processes)

Meyer' probabilistic approach to Littlewood-Paley inequalities (1010, 1510) is extended by replacing the underlying Brownian motion with a stable process. The following spectral multiplicator theorem is obtained: If $(P_t)_{t\geq 0}$ is a symmetric Markov semigroup with spectral representation $P_t=\int_{[0,\infty)}e^{-t\lambda} dE_{\lambda}$, and if $M$ is a function on $

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Stable processes, Inequalities, Singular integrals, Carré du champ

Nature: Original

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XXIV: 30, 448-452, LNM 1426 (1990)

Sur une formule de Bismut (Markov processes, Stochastic differential geometry)

This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group

Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes

Nature: Exposition, Original additions

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XXIX: 16, 166-180, LNM 1613 (1995)

A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)

This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.

Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

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XXIX: 17, 181-193, LNM 1613 (1995)

Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)

The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

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XLI: 05, 121-135, LNM 1934 (2008)

Fluctuations of spectrally negative Markov additive processes (Theory of Markov processes)

Nature: Original