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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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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: 16, 227-250, LNM 850 (1981)

**ROGERS, L.C.G.**

Williams' characterization of the Brownian excursion law: proof and applications (Brownian motion)

In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams*Diffusions, Markov Processes and Martingales,* Wiley 1979, II.67), and it collects a number of applications, including the Azéma-Yor approach to Skorohod's imbedding theorem (1306)

Keywords: Excursions, Explicit laws, Bessel processes, Skorohod imbedding

Nature: Original

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XIX: 27, 297-313, LNM 1123 (1985)

**LE GALL, Jean-François**

Sur la mesure de Hausdorff de la courbe brownienne (Brownian motion)

Previous results on the $h$-measure of the Brownian curve in $**R**^2$ or $**R**^3$ indexed by $t\in[0,1]$, by Cisielski-Taylor *Trans. Amer. Math. Soc.* **103** (1962) and Taylor *Proc. Cambridge Philos. Soc.* **60** (1964) are sharpened. The method uses the description à la Ray-Knight of the local times of Bessel processes

Comment: These Ray-Knight descriptions are useful ; they were later used in questions not related to Hausdorff measures. See for instance Biane-Yor,*Ann. I.H.P.* **23** (1987), Yor, *Ann. I.H.P.* **27** (1991)

Keywords: Hausdorff measures, Brownian motion, Bessel processes, Ray-Knight theorems

Nature: Original

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XXXI: 25, 256-265, LNM 1655 (1997)

**TAKAOKA, Koichiro**

On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman's theorem (Stochastic calculus)

Martingales involving the future minimum of a transient Bessel process are studied, and shown to satisfy a non Markovian SDE. In dimension $>3$, uniqueness in law does not hold for this SDE. This generalizes Saisho-Tanemura*Tokyo J. Math.* **13** (1990)

Comment: Extended to more general diffusions in the next article 3126

Keywords: Continuous martingales, Bessel processes, Pitman's theorem

Nature: Original

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XXXI: 26, 266-271, LNM 1655 (1997)

**RAUSCHER, Bernhard**

Some remarks on Pitman's theorem (Stochastic calculus)

For certain transient diffusions $X$, local martingales which are functins of $X_t$ and the future infimum $\inf_{u\ge t}X_u$ are constructed. This extends the preceding article 3125

Comment: See also chap. 12 of Yor,*Some Aspects of Brownian Motion Part~II*, Birkhäuser (1997)

Keywords: Continuous martingales, Bessel processes, Diffusion processes, Pitman's theorem

Nature: Original

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XLIV: 13, 271-277, LNM 2046 (2012)

**VUOLLE-APIALA, Juha**

Time inversion property for rotation invariant self-similar diffusion processes (Theory of processes)

Keywords: Time inversion, Self-similar, Bessel processes, Diffusion processes, Rotation invariant, Skew product, Radial process

Nature: Original

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

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: 16, 227-250, LNM 850 (1981)

Williams' characterization of the Brownian excursion law: proof and applications (Brownian motion)

In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams

Keywords: Excursions, Explicit laws, Bessel processes, Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XIX: 27, 297-313, LNM 1123 (1985)

Sur la mesure de Hausdorff de la courbe brownienne (Brownian motion)

Previous results on the $h$-measure of the Brownian curve in $

Comment: These Ray-Knight descriptions are useful ; they were later used in questions not related to Hausdorff measures. See for instance Biane-Yor,

Keywords: Hausdorff measures, Brownian motion, Bessel processes, Ray-Knight theorems

Nature: Original

Retrieve article from Numdam

XXXI: 25, 256-265, LNM 1655 (1997)

On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman's theorem (Stochastic calculus)

Martingales involving the future minimum of a transient Bessel process are studied, and shown to satisfy a non Markovian SDE. In dimension $>3$, uniqueness in law does not hold for this SDE. This generalizes Saisho-Tanemura

Comment: Extended to more general diffusions in the next article 3126

Keywords: Continuous martingales, Bessel processes, Pitman's theorem

Nature: Original

Retrieve article from Numdam

XXXI: 26, 266-271, LNM 1655 (1997)

Some remarks on Pitman's theorem (Stochastic calculus)

For certain transient diffusions $X$, local martingales which are functins of $X_t$ and the future infimum $\inf_{u\ge t}X_u$ are constructed. This extends the preceding article 3125

Comment: See also chap. 12 of Yor,

Keywords: Continuous martingales, Bessel processes, Diffusion processes, Pitman's theorem

Nature: Original

Retrieve article from Numdam

XLIV: 13, 271-277, LNM 2046 (2012)

Time inversion property for rotation invariant self-similar diffusion processes (Theory of processes)

Keywords: Time inversion, Self-similar, Bessel processes, Diffusion processes, Rotation invariant, Skew product, Radial process

Nature: Original