Browse by: Author name - Classification - Keywords - Nature

112 matches found
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)
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
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
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
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 fornormal'' 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)
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
Keywords: Subordinators, Polar sets
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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
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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
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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
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
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
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VII: 08, 58-60, LNM 321 (1973)
DELLACHERIE, Claude
Potentiels de fonctionnelles additives. Un contre-exemple de Knight (Markov processes)
An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point
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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
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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
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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
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
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
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
Nature: Original
Retrieve article from Numdam
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
Retrieve article from Numdam
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
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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
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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
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XII: 59, 775-803, LNM 649 (1978)
MEYER, Paul-André
Martingales locales fonctionnelles additives (two talks) (Markov processes)
The purpose of the paper is to specialize the standard theory of Hardy spaces of martingales to the subspaces of additive martingales of a Markov process. The theory is not complete: the dual of (additive) $H^1$ seems to be different from (additive) $BMO$
Nature: Original
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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
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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
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XIII: 43, 490-494, LNM 721 (1979)
WILLIAMS, David
Conditional excursion theory (Brownian motion, Markov processes)
To be completed
Keywords: Excursions
Nature: Original
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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
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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
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XIV: 36, 324-331, LNM 784 (1980)
BARLOW, Martin T.; ROGERS, L.C.G.; WILLIAMS, David
Wiener-Hopf factorization for matrices (Markov processes)
Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix
Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in 1437
Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time, Markov chains
Nature: Original
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XIV: 37, 332-342, LNM 784 (1980)
ROGERS, L.C.G.; WILLIAMS, David
Time-substitution based on fluctuating additive functionals (Wiener-Hopf factorization for infinitesimal generators) (Markov processes)
This is a first step towards the extension of 1436 to Markov processes with a general state space
Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time
Nature: Original
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XIV: 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
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XIV: 43, 410-417, LNM 784 (1980)
JACOD, Jean; MAISONNEUVE, Bernard
Remarque sur les fonctionnelles additives non adaptées des processus de Markov (Markov processes)
It occurs sometimes that a Markov process $(X_t)$ satisfies in a filtration ${\cal H}_t$ a Markov property of the form $E[f\circ \theta_t \,|\,{\cal H}_t]= E_{X_t}[f]$, where $f$ is not restricted to be ${\cal H}_t$-measurable. For instance, situations in renewal theory where one is given a Markov pair $(X_t,Y_t)$, and ${\cal H}_t$ describes the path of $X$ up to time $t$, and the whole path of $Y$. In such cases, the authors show that additive functionals which are previsible in the larger filtration are in fact previsible in the filtration of $X$ alone
Nature: Original
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XIV: 44, 418-436, LNM 784 (1980)
RAO, Murali
A note on Revuz measure (Markov processes, Potential theory)
The problem is to weaken the hypotheses of Chung (Ann. Inst. Fourier, 23, 1973) implying the representation of the equilibrium potential of a compact set as a Green potential. To this order, Revuz measure techniques are used, and interesting auxiliary results are proved concerning the Revuz measures of natural additive functionals of a Hunt process
Keywords: Revuz measures, Additive functionals, Hunt processes, Equilibrium potentials
Nature: Original
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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
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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
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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
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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
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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