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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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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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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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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

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

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam