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 setsNature: Original
Retrieve article from Numdam
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
502Keywords: Subordinators,
Polar setsNature: Exposition Retrieve article from Numdam
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 setsNature: Original Retrieve article from Numdam
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 This paper gives Carleson's (unpublished) analytic proof of it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 setsNature: Exposition Retrieve article from Numdam
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 setsNature: Exposition,
Original additions Retrieve article from Numdam
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 setsNature: New exposition of known results Retrieve article from Numdam
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 functionsNature: Original Retrieve article from Numdam
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
1238Keywords: Polar sets,
Semi-polar setsNature: Original 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 setsNature: Original Retrieve article from Numdam
X: 29, 544-544, LNM 511 (1976)
DELLACHERIE, Claude
Correction à des exposés de 1973/74 (
Descriptive set theory)
Corrections to
915 and
918Keywords: Analytic sets,
Semi-polar sets,
Suslin spacesNature: Original Retrieve article from Numdam
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
707Keywords: Sets with countable sections,
Semi-polar setsNature: Original Retrieve article from Numdam
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 supremaNature: Original Retrieve article from Numdam
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 analysisNature: Exposition Retrieve article from Numdam
