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

**ASSOUAD, Patrice**

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

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 This paper gives Carleson's (unpublished) analytic proof of it

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

Keywords: Subordinators, Polar sets

Nature: Exposition

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

**BRETAGNOLLE, Jean**

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

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

Comment: See also 502 in the same volume

Keywords: Subordinators, Polar sets

Nature: Exposition, Original additions

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V: 06, 76-76, LNM 191 (1971)

**CHUNG, Kai Lai**

A simple proof of Doob's convergence theorem (Potential theory)

Doob's theorem is a version of the main convergence theorem of potential theory: the limit of a decreasing sequence of excessive functions differs of its regularized version on a semi-polar set

Comment: It is also shown that a function $f$ satisfying $f\ge P_Kf$ for all compact sets $K$ differs from its regularized function on a semi-polar set

Keywords: Excessive functions, Semi-polar sets

Nature: New exposition of known results

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

**REVUZ, Daniel**

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

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

Comment: To be asked

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

Nature: Original

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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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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

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X: 29, 544-544, LNM 511 (1976)

**DELLACHERIE, Claude**

Correction à des exposés de 1973/74 (Descriptive set theory)

Corrections to 915 and 918

Keywords: Analytic sets, Semi-polar sets, Suslin spaces

Nature: Original

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XII: 38, 509-511, LNM 649 (1978)

**DELLACHERIE, Claude**

Appendice à l'exposé de Mokobodzki (Measure theory, General theory of processes)

Some comments on 1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from 707

Keywords: Sets with countable sections, Semi-polar sets

Nature: Original

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XII: 43, 564-566, LNM 649 (1978)

**DELLACHERIE, Claude**; **MOKOBODZKI, Gabriel**

Deux propriétés des ensembles minces (abstraits) (Descriptive set theory)

Given a class ${\cal S}$ of Borel sets understood as ``small'' sets, the class ${\cal L}$ consisting of their conplements understood as ``large'' sets, a set $A$ is said to be ${\cal S}$-thin if does not contain uncountably many disjoint ``large'' sets. For instance, if ${\cal S}$ is the class of polar sets, then thin sets are the same as semi-polar sets. Two general theorems are proved here on thin sets

Keywords: Thin sets, Semi-polar sets, Essential suprema

Nature: Original

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XVI: 07, 133-133, LNM 920 (1982)

**MEYER, Paul-André**

Appendice : Un résultat de D. Williams (Malliavin's calculus)

This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as ``quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process

Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis

Nature: Exposition

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Ensembles aléatoires II (Descriptive set theory, Markov processes)

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

Comment: See Dellacherie,

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

Nature: Original

Retrieve article from Numdam

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

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

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

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

Keywords: Subordinators, Polar sets

Nature: Exposition

Retrieve article from Numdam

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

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

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

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

Keywords: Hunt processes, Semi-polar sets

Nature: Original

Retrieve article from Numdam

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

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

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

Comment: See Chung,

Keywords: Subordinators, Polar sets

Nature: Exposition

Retrieve article from Numdam

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

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

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

Comment: See also 502 in the same volume

Keywords: Subordinators, Polar sets

Nature: Exposition, Original additions

Retrieve article from Numdam

V: 06, 76-76, LNM 191 (1971)

A simple proof of Doob's convergence theorem (Potential theory)

Doob's theorem is a version of the main convergence theorem of potential theory: the limit of a decreasing sequence of excessive functions differs of its regularized version on a semi-polar set

Comment: It is also shown that a function $f$ satisfying $f\ge P_Kf$ for all compact sets $K$ differs from its regularized function on a semi-polar set

Keywords: Excessive functions, Semi-polar sets

Nature: New exposition of known results

Retrieve article from Numdam

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

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

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

Comment: To be asked

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

Nature: Original

Retrieve article from Numdam

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

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

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

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

Keywords: Polar sets, Semi-polar sets

Nature: Original

Retrieve article from Numdam

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

Une propriété des ensembles semi-polaires (Markov processes)

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

Keywords: Semi-polar sets

Nature: Original

Retrieve article from Numdam

X: 29, 544-544, LNM 511 (1976)

Correction à des exposés de 1973/74 (Descriptive set theory)

Corrections to 915 and 918

Keywords: Analytic sets, Semi-polar sets, Suslin spaces

Nature: Original

Retrieve article from Numdam

XII: 38, 509-511, LNM 649 (1978)

Appendice à l'exposé de Mokobodzki (Measure theory, General theory of processes)

Some comments on 1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from 707

Keywords: Sets with countable sections, Semi-polar sets

Nature: Original

Retrieve article from Numdam

XII: 43, 564-566, LNM 649 (1978)

Deux propriétés des ensembles minces (abstraits) (Descriptive set theory)

Given a class ${\cal S}$ of Borel sets understood as ``small'' sets, the class ${\cal L}$ consisting of their conplements understood as ``large'' sets, a set $A$ is said to be ${\cal S}$-thin if does not contain uncountably many disjoint ``large'' sets. For instance, if ${\cal S}$ is the class of polar sets, then thin sets are the same as semi-polar sets. Two general theorems are proved here on thin sets

Keywords: Thin sets, Semi-polar sets, Essential suprema

Nature: Original

Retrieve article from Numdam

XVI: 07, 133-133, LNM 920 (1982)

Appendice : Un résultat de D. Williams (Malliavin's calculus)

This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as ``quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process

Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis

Nature: Exposition

Retrieve article from Numdam