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
1247Keywords: Classical potential theory,
Barrier,
Regular pointsNature: New proof of known results
Retrieve article from Numdam
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 processesNature: New proof of known results Retrieve article from Numdam
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 inequalityNature: New proof of known results Retrieve article from Numdam
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 lawNature: New proof of known results Retrieve article from Numdam
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 processesNature: New proof of known results Retrieve article from Numdam
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 Retrieve article from Numdam
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 intervalNature: New proof of known results Retrieve article from Numdam
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 inequalitiesNature: New proof of known results Retrieve article from Numdam
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
3219Keywords: Filtrations,
Spider martingalesNature: New proof of known results Retrieve article from Numdam
