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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: 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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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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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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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

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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

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

Retrieve article from Numdam

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

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

Retrieve article from Numdam

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

Retrieve article from Numdam