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V: 05, 58-75, LNM 191 (1971)

**CARTIER, Pierre**

Introduction à l'étude des mouvements browniens à plusieurs paramètres (Gaussian processes)

Settles in particular a disagreement between statements of Lévy (*Processus Stochastiques et Mouvement Brownien,* 1948) and McKean (*Teor. Ver. i Prim.* **8**, 1963) on the domain of analyticity of some Gaussian random functions

Comment: More recent work of Cartier on covariances appeared in the L.~Schwartz volume*Mathematical Analysis and Applications, A*, Academic Press 1981

Keywords: Several parameter Brownian motions, Covariance

Nature: New exposition of known results

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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: 10, 87-102, LNM 191 (1971)

**DELLACHERIE, Claude**

Les théorèmes de Mazurkiewicz-Sierpinski et de Lusin (Descriptive set theory)

Synthetic presentation of (then) little known results on the perfect kernels of closed random sets and uniformization of random sets with countable sections

Comment: See Dellacherie-Meyer,*Probabilités et Potentiel,* Chap. XI

Keywords: Analytic sets, Random sets, Section theorems

Nature: New exposition of known results

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V: 20, 209-210, LNM 191 (1971)

**MEYER, Paul-André**

Un théorème sur la répartition des temps locaux (Markov processes)

Kesten discovered that the value at a terminal time $T$ of the local time $L$ of a Markov process $X$ at a single point has an exponential distribution, and that $X_T$ and $L_T$ are independent. A short proof is given

Comment: The result can be deduced from excursion theory

Keywords: Local times

Nature: New exposition of known results

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VII: 06, 48-50, LNM 321 (1973)

**DELLACHERIE, Claude**

Une démonstration du théorème de Souslin-Lusin (Descriptive set theory)

The basic fact that the image of a Borel set under an injective Borel mapping is Borel is deduced from a separation theorem concerning countably many disjoint analytic sets

Comment: This is a step in the author's simplification of the proofs of the great theorems on analytic and Borel sets. See*Un cours sur les ensembles analytiques,* in *Analytic Sets,* C.A. Rogers ed., Academic Press 1980

Keywords: Borel sets, Analytic sets, Separation theorem

Nature: New exposition of known results

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XXXII: 19, 264-305, LNM 1686 (1998)

**BARLOW, Martin T.**; **ÉMERY, Michel**; **KNIGHT, Frank B.**; **SONG, Shiqi**; **YOR, Marc**

Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)

Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA**7**, 1997). Using his methods, the result is extended to spider martingales. A conjecture of M. Barlow is also proved: if $L$ is an honest time in a (possibly multidimensional) Brownian filtration, then ${\cal F}_{L+}$ is generated by ${\cal F}_{L}$ and at most one event. Last, it is shown that a Walsh's Brownian motion can live in the filtration generated by another Walsh's Brownian motion only if the former is obtained from the latter by aggregating rays

Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,*Astérisque* **282** (2002). A simplified proof of Barlow's conjecture is given in 3304. For more on Théorème 1 (Slutsky's lemma), see 3221 and 3325

Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma

Nature: New exposition of known results, Original additions

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Introduction à l'étude des mouvements browniens à plusieurs paramètres (Gaussian processes)

Settles in particular a disagreement between statements of Lévy (

Comment: More recent work of Cartier on covariances appeared in the L.~Schwartz volume

Keywords: Several parameter Brownian motions, Covariance

Nature: New exposition of known results

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: 10, 87-102, LNM 191 (1971)

Les théorèmes de Mazurkiewicz-Sierpinski et de Lusin (Descriptive set theory)

Synthetic presentation of (then) little known results on the perfect kernels of closed random sets and uniformization of random sets with countable sections

Comment: See Dellacherie-Meyer,

Keywords: Analytic sets, Random sets, Section theorems

Nature: New exposition of known results

Retrieve article from Numdam

V: 20, 209-210, LNM 191 (1971)

Un théorème sur la répartition des temps locaux (Markov processes)

Kesten discovered that the value at a terminal time $T$ of the local time $L$ of a Markov process $X$ at a single point has an exponential distribution, and that $X_T$ and $L_T$ are independent. A short proof is given

Comment: The result can be deduced from excursion theory

Keywords: Local times

Nature: New exposition of known results

Retrieve article from Numdam

VII: 06, 48-50, LNM 321 (1973)

Une démonstration du théorème de Souslin-Lusin (Descriptive set theory)

The basic fact that the image of a Borel set under an injective Borel mapping is Borel is deduced from a separation theorem concerning countably many disjoint analytic sets

Comment: This is a step in the author's simplification of the proofs of the great theorems on analytic and Borel sets. See

Keywords: Borel sets, Analytic sets, Separation theorem

Nature: New exposition of known results

Retrieve article from Numdam

XXXII: 19, 264-305, LNM 1686 (1998)

Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (Brownian motion, Filtrations)

Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA

Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,

Keywords: Filtrations, Spider martingales, Walsh's Brownian motion, Cosiness, Slutsky's lemma

Nature: New exposition of known results, Original additions

Retrieve article from Numdam