Browse by: Author name - Classification - Keywords - Nature

13 matches found
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
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 502
Keywords: Subordinators, Polar sets
Nature: 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 sets
Nature: 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 This paper gives Carleson's (unpublished) analytic proof of it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 sets
Nature: 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 sets
Nature: 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 sets
Nature: 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 functions
Nature: 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 1238
Keywords: Polar sets, Semi-polar sets
Nature: 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 sets
Nature: 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 918
Keywords: Analytic sets, Semi-polar sets, Suslin spaces
Nature: 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 707
Keywords: Sets with countable sections, Semi-polar sets
Nature: 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 suprema
Nature: 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 analysis
Nature: Exposition
Retrieve article from Numdam