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X: 15, 235-239, LNM 511 (1976)

**WILLIAMS, David**

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see*Ann. Prob.* **3**, 1975. For modern references, we should ask Yor

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

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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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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

Retrieve article from Numdam

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

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

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