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 reversalNature: Original
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VIII: 14, 262-288, LNM 381 (1974)
MEYER, Paul-André
Les travaux d'Azéma sur le retournement du temps (
General theory of processes,
Markov processes)
This paper is an exposition of a paper by Azéma (
Ann. Sci. ENS, 6, 1973) in which the theory ``dual'' to the general theory of processes was developed. It is shown first how the general theory itself can be developed from a family of killing operators, and then how the dual theory follows from a family of shift operators $\theta_t$. A transience hypothesis involving the existence of many ``return times'' permits the construction of a theory completely similar to the usual one. Then some of Azéma's applications to the theory of Markov processes are given, particularly the representation of a measure not charging $\mu$-polar sets as expectation under the initial measure $\mu$ of a left additive functional
Comment: This paper follows (with considerable progress) the line of
602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer,
Probabilités et Potentiel, Chapter XVIII, 1992
Keywords: Time reversal,
Shift operators,
Killing operators,
Cooptional processes,
Coprevisible processes,
Additive functionals,
Left additive functionalsNature: Exposition,
Original additions Retrieve article from Numdam
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 processesNature: Exposition,
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