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XIII: 59, 646-646, LNM 721 (1979)

**BARLOW, Martin T.**

On the left endpoints of Brownian excursions (Brownian motion, Excursion theory)

It is shown that no expansion of the Brownian filtration can be found such that $B_t$ remains a semimartingale, and the set of left endpoints of Brownian excursions becomes optional

Keywords: Progressive sets

Nature: Original

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XIV: 07, 62-75, LNM 784 (1980)

**BARLOW, Martin T.**; **YOR, Marc**

Sur la construction d'une martingale continue de valeur absolue donnée (Martingale theory)

This paper consists of two notes on Gilat's theorem (*Ann. Prob.* **5**, 1977, See also 1358). The problem consists in constructing, given a continuous positive submartingale $Y$, a *continuous * martingale $X$ (possibly on a different space) such that $|X|$ has the same law as $Y$. Let $A$ be the increasing process associated with $Y$; it is necessary for the existence of $X$ that $dA$ should be carried by $\{Y=0\}$. This is shown by the first note (Yor's) to be also sufficient---more precisely, in this case the solutions of Gilat's problem are all continuous. The second note (Barlow's) shows how to construct a Gilat martingale by ``putting a random $\pm$ sign in front of each excursion of $Y$'', a simple intuitive idea and a delicate proof

Keywords: Gilat's theorem

Nature: Original

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XIV: 36, 324-331, LNM 784 (1980)

**BARLOW, Martin T.**; **ROGERS, L.C.G.**; **WILLIAMS, David**

Wiener-Hopf factorization for matrices (Markov processes)

Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix

Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in 1437

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time, Markov chains

Nature: Original

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XV: 12, 189-190, LNM 850 (1981)

**BARLOW, Martin T.**

On Brownian local time (Brownian motion)

Let $(L^a_t)$ be the standard (jointly continuous) version of Brownian local times. Perkins has shown that for fixed $t$ $a\mapsto L^a_t$ is a semimartingale relative to the excusrion fields. An example is given of a stopping time $T$ for which $a\mapsto L^a_T$ is not a semimartingale

Keywords: Local times

Nature: Original

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XV: 23, 311-319, LNM 850 (1981)

**ALDOUS, David J.**; **BARLOW, Martin T.**

On countable dense random sets (General theory of processes, Point processes)

This paper is devoted to random sets $B$ which are countable, optional (i.e., can be represented as the union of countably many graphs of stopping times $T_n$) and dense. The main result is that whenever the increasing processes $I_{t\ge T_n}$ have absolutely continuous compensators (in which case the same property holds for any stopping time $T$ whose graph is contained in $B$), then the random set $B$ can be represented as the union of all the points of countably many independent standard Poisson processes (intuitively, a Poisson measure whose rate is $+\infty$ times Lebesgue measure). This may require, however, an innocuous enlargement of filtration. Another characterization of such random sets is roughly that they do not intersect previsible sets of zero Lebesgue measure. Note also an interesting example of a set optional w.r.t. two filtrations, but not w.r.t. their intersection

Keywords: Poisson point processes

Nature: Original

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XVI: 15, 209-211, LNM 920 (1982)

**BARLOW, Martin T.**

$L(B_t,t)$ is not a semimartingale (Brownian motion)

The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale

Keywords: Local times, Semimartingales

Nature: Original

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XVII: 03, 32-61, LNM 986 (1983)

**BARLOW, Martin T.**; **PERKINS, Edwin A.**

Strong existence, uniqueness and non-uniqueness in an equation involving local time

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XVII: 47, 512-512, LNM 986 (1983)

**BARLOW, Martin T.**

Correction to ``$L(B_t,t)$ is not a semimartingale''

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XVIII: 01, 1-28, LNM 1059 (1984)

**BARLOW, Martin T.**; **PERKINS, Edwin A.**

Levels at which every Brownian excursion is exceptional

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XXIII: 23, 275-293, LNM 1372 (1989)

**BARLOW, Martin T.**; **PITMAN, James W.**; **YOR, Marc**

On Walsh's Brownian motions

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XXIII: 24, 294-314, LNM 1372 (1989)

**BARLOW, Martin T.**; **PITMAN, James W.**; **YOR, Marc**

Une extension multidimensionnelle de la loi de l'arc sinus

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XXIV: 12, 188-193, LNM 1426 (1990)

**BARLOW, Martin T.**; **PROTTER, Philip**

On convergence of semimartingales

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XXIV: 13, 194-209, LNM 1426 (1990)

**BARLOW, Martin T.**; **PERKINS, Edwin A.**

On pathwise uniqueness and expansion of filtrations

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XXVI: 06, 70-80, LNM 1526 (1992)

**BARLOW, Martin T.**; **IMKELLER, Peter**

On some sample path properties of Skorohod integral processes

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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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XXXV: 15, 202-205, LNM 1755 (2001)

**BARLOW, Martin T.**; **BURDZY, Krzysztof**; **KASPI, Haya**; **MANDELBAUM, Avi**

Coalescence of skew Brownian motions

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On the left endpoints of Brownian excursions (Brownian motion, Excursion theory)

It is shown that no expansion of the Brownian filtration can be found such that $B_t$ remains a semimartingale, and the set of left endpoints of Brownian excursions becomes optional

Keywords: Progressive sets

Nature: Original

Retrieve article from Numdam

XIV: 07, 62-75, LNM 784 (1980)

Sur la construction d'une martingale continue de valeur absolue donnée (Martingale theory)

This paper consists of two notes on Gilat's theorem (

Keywords: Gilat's theorem

Nature: Original

Retrieve article from Numdam

XIV: 36, 324-331, LNM 784 (1980)

Wiener-Hopf factorization for matrices (Markov processes)

Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix

Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in 1437

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time, Markov chains

Nature: Original

Retrieve article from Numdam

XV: 12, 189-190, LNM 850 (1981)

On Brownian local time (Brownian motion)

Let $(L^a_t)$ be the standard (jointly continuous) version of Brownian local times. Perkins has shown that for fixed $t$ $a\mapsto L^a_t$ is a semimartingale relative to the excusrion fields. An example is given of a stopping time $T$ for which $a\mapsto L^a_T$ is not a semimartingale

Keywords: Local times

Nature: Original

Retrieve article from Numdam

XV: 23, 311-319, LNM 850 (1981)

On countable dense random sets (General theory of processes, Point processes)

This paper is devoted to random sets $B$ which are countable, optional (i.e., can be represented as the union of countably many graphs of stopping times $T_n$) and dense. The main result is that whenever the increasing processes $I_{t\ge T_n}$ have absolutely continuous compensators (in which case the same property holds for any stopping time $T$ whose graph is contained in $B$), then the random set $B$ can be represented as the union of all the points of countably many independent standard Poisson processes (intuitively, a Poisson measure whose rate is $+\infty$ times Lebesgue measure). This may require, however, an innocuous enlargement of filtration. Another characterization of such random sets is roughly that they do not intersect previsible sets of zero Lebesgue measure. Note also an interesting example of a set optional w.r.t. two filtrations, but not w.r.t. their intersection

Keywords: Poisson point processes

Nature: Original

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XVI: 15, 209-211, LNM 920 (1982)

$L(B_t,t)$ is not a semimartingale (Brownian motion)

The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale

Keywords: Local times, Semimartingales

Nature: Original

Retrieve article from Numdam

XVII: 03, 32-61, LNM 986 (1983)

Strong existence, uniqueness and non-uniqueness in an equation involving local time

Retrieve article from Numdam

XVII: 47, 512-512, LNM 986 (1983)

Correction to ``$L(B_t,t)$ is not a semimartingale''

Retrieve article from Numdam

XVIII: 01, 1-28, LNM 1059 (1984)

Levels at which every Brownian excursion is exceptional

Retrieve article from Numdam

XXIII: 23, 275-293, LNM 1372 (1989)

On Walsh's Brownian motions

Retrieve article from Numdam

XXIII: 24, 294-314, LNM 1372 (1989)

Une extension multidimensionnelle de la loi de l'arc sinus

Retrieve article from Numdam

XXIV: 12, 188-193, LNM 1426 (1990)

On convergence of semimartingales

Retrieve article from Numdam

XXIV: 13, 194-209, LNM 1426 (1990)

On pathwise uniqueness and expansion of filtrations

Retrieve article from Numdam

XXVI: 06, 70-80, LNM 1526 (1992)

On some sample path properties of Skorohod integral processes

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

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XXXV: 15, 202-205, LNM 1755 (2001)

Coalescence of skew Brownian motions

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