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XI: 03, 27-33, LNM 581 (1977)

**CHUNG, Kai Lai**

Pedagogic notes on the barrier theorem (Potential theory)

Let $D$ a bounded open set in $**R**^n$, and let $z$ be a boundary point. Then a barrier at $z$ is a superharmonic function in $D$, strictly positive and with a limit equal to $0$ at $z$. The barrier theorem asserts that if there is a barrier at $z$, then $z$ is regular. An elegant proof of this is given using Brownian motion. Then it is shown that the expectation of $S$, the hitting time of $D^c$, is bounded, upper semi-continuous in $R^n$ and continuous in $D$, and is a barrier at every regular point

Comment: An error is corrected in 1247

Keywords: Classical potential theory, Barrier, Regular points

Nature: New proof of known results

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XIII: 45, 521-532, LNM 721 (1979)

**JEULIN, Thierry**

Un théorème de J.W. Pitman (Brownian motion, Diffusion theory)

This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$

Keywords: Bessel processes

Nature: New proof of known results

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XIV: 14, 125-127, LNM 784 (1980)

**LENGLART, Érik**

Sur l'inégalité de Métivier-Pellaumail (Stochastic calculus)

A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality

Keywords: Doob's inequality, Métivier-Pellaumail inequality

Nature: New proof of known results

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XV: 02, 6-10, LNM 850 (1981)

**FERNIQUE, Xavier**

Sur le théorème de Kantorovitch-Rubinstein dans les espaces polonais (Measure theory)

The theorem asserts the existence, given two probability measures $\mu,\nu$ on a complete separable metric space $(S,d)$, of a measure $\pi$ on $S\times S$ with marginals $\mu$ and $\nu$ such that $\int d(x,y)\,\pi(dx,dy)$ realizes a suitable distance between $\mu$ and $\nu$. An elementary proof is given here by reduction to the compact case

Keywords: Convergence in law

Nature: New proof of known results

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XV: 14, 206-209, LNM 850 (1981)

**McGILL, Paul**

A direct proof of the Ray-Knight theorem (Brownian motion)

The (first) Ray-Knight theorem describes the law of the process $(L_T^{1-a})_{0\le a\le 1}$ where $(L^a_t)$ is the family of local times of Brownian motion starting from $0$ and $T$ is the hitting time of $1$. A direct proof is given indeed. It is reproduced in Revuz-Yor,*Continuous Martingales and Brownian Motion,* Chapter XI, exercice (2.7)

Keywords: Local times, Ray-Knight theorems, Bessel processes

Nature: New proof of known results

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XV: 44, 632-642, LNM 850 (1981)

**SIDIBÉ, Ramatoulaye**

Mesures à accroissements indépendants et P.A.I. non homogènes (Independent increments)

This is an improved version of 1310: the classical theorem of Lévy on the structure of processes with independent increments is elegantly proved by martingale methods, in the non-homogeneous case, and it is proved that the process is a special semimartingale if and only if it is integrable

Nature: New proof of known results

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XV: 46, 669-670, LNM 850 (1981)

**LÉPINGLE, Dominique**

Une remarque sur les lois de certains temps d'atteinte (Brownian motion)

Let $T$ be the exit time of the interval $[-d,c]$ for a Brownian motion starting at $0$. A classical formula giving the Laplace transform of the law of $T$ can be extended by analytical continuation to the positive axis. It is shown here that this extension has a purely probabilistic proof. The same method gives two other formulas

Keywords: Exit time from an interval

Nature: New proof of known results

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XXI: 09, 173-175, LNM 1247 (1987)

**ÉMERY, Michel**; **YUKICH, Joseph E.**

A simple proof of the logarithmic Sobolev inequality on the circle (Real analysis)

The same kind of semi-group argument as in Bakry-Émery 1912 gives an elementary proof of the logarithmic Sobolev inequality on the circle

Keywords: Logarithmic Sobolev inequalities

Nature: New proof of known results

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XXXIII: 04, 217-220, LNM 1709 (1999)

**DE MEYER, Bernard**

Une simplification de l'argument de Tsirelson sur le caractère non-brownien des processus de Walsh (Brownian motion, Filtrations)

Barlow's conjecture is proved with a simpler argument than in 3219

Keywords: Filtrations, Spider martingales

Nature: New proof of known results

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Pedagogic notes on the barrier theorem (Potential theory)

Let $D$ a bounded open set in $

Comment: An error is corrected in 1247

Keywords: Classical potential theory, Barrier, Regular points

Nature: New proof of known results

Retrieve article from Numdam

XIII: 45, 521-532, LNM 721 (1979)

Un théorème de J.W. Pitman (Brownian motion, Diffusion theory)

This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$

Keywords: Bessel processes

Nature: New proof of known results

Retrieve article from Numdam

XIV: 14, 125-127, LNM 784 (1980)

Sur l'inégalité de Métivier-Pellaumail (Stochastic calculus)

A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality

Keywords: Doob's inequality, Métivier-Pellaumail inequality

Nature: New proof of known results

Retrieve article from Numdam

XV: 02, 6-10, LNM 850 (1981)

Sur le théorème de Kantorovitch-Rubinstein dans les espaces polonais (Measure theory)

The theorem asserts the existence, given two probability measures $\mu,\nu$ on a complete separable metric space $(S,d)$, of a measure $\pi$ on $S\times S$ with marginals $\mu$ and $\nu$ such that $\int d(x,y)\,\pi(dx,dy)$ realizes a suitable distance between $\mu$ and $\nu$. An elementary proof is given here by reduction to the compact case

Keywords: Convergence in law

Nature: New proof of known results

Retrieve article from Numdam

XV: 14, 206-209, LNM 850 (1981)

A direct proof of the Ray-Knight theorem (Brownian motion)

The (first) Ray-Knight theorem describes the law of the process $(L_T^{1-a})_{0\le a\le 1}$ where $(L^a_t)$ is the family of local times of Brownian motion starting from $0$ and $T$ is the hitting time of $1$. A direct proof is given indeed. It is reproduced in Revuz-Yor,

Keywords: Local times, Ray-Knight theorems, Bessel processes

Nature: New proof of known results

Retrieve article from Numdam

XV: 44, 632-642, LNM 850 (1981)

Mesures à accroissements indépendants et P.A.I. non homogènes (Independent increments)

This is an improved version of 1310: the classical theorem of Lévy on the structure of processes with independent increments is elegantly proved by martingale methods, in the non-homogeneous case, and it is proved that the process is a special semimartingale if and only if it is integrable

Nature: New proof of known results

Retrieve article from Numdam

XV: 46, 669-670, LNM 850 (1981)

Une remarque sur les lois de certains temps d'atteinte (Brownian motion)

Let $T$ be the exit time of the interval $[-d,c]$ for a Brownian motion starting at $0$. A classical formula giving the Laplace transform of the law of $T$ can be extended by analytical continuation to the positive axis. It is shown here that this extension has a purely probabilistic proof. The same method gives two other formulas

Keywords: Exit time from an interval

Nature: New proof of known results

Retrieve article from Numdam

XXI: 09, 173-175, LNM 1247 (1987)

A simple proof of the logarithmic Sobolev inequality on the circle (Real analysis)

The same kind of semi-group argument as in Bakry-Émery 1912 gives an elementary proof of the logarithmic Sobolev inequality on the circle

Keywords: Logarithmic Sobolev inequalities

Nature: New proof of known results

Retrieve article from Numdam

XXXIII: 04, 217-220, LNM 1709 (1999)

Une simplification de l'argument de Tsirelson sur le caractère non-brownien des processus de Walsh (Brownian motion, Filtrations)

Barlow's conjecture is proved with a simpler argument than in 3219

Keywords: Filtrations, Spider martingales

Nature: New proof of known results

Retrieve article from Numdam