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V: 27, 278-282, LNM 191 (1971)

**SAM LAZARO, José de**; **MEYER, Paul-André**

Une remarque sur le flot du mouvement brownien (Brownian motion, Ergodic theory)

It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense

Comment: See Sam Lazaro-Meyer,*Z. für W-theorie,* **18**, 1971

Keywords: Brownian motion, Wiener chaos, Screw-lines

Nature: Original

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VIII: 02, 11-19, LNM 381 (1974)

**BRETAGNOLLE, Jean**

Une remarque sur le problème de Skorohod (Brownian motion)

The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see 516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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VIII: 10, 134-149, LNM 381 (1974)

**KNIGHT, Frank B.**

Existence of small oscillations at zeros of brownian motion (Brownian motion)

The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an ``iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros

Comment: This result may be compared to Kahane,*C.R. Acad. Sci.* **248**, 1974

Keywords: Law of the iterated logarithm, Local times

Nature: Original

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X: 02, 19-23, LNM 511 (1976)

**CHACON, Rafael V.**; **WALSH, John B.**

One-dimensional potential imbedding (Brownian motion)

The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool

Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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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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XII: 31, 428-445, LNM 649 (1978)

**KNIGHT, Frank B.**

On the sojourn times of killed Brownian motion (Brownian motion)

To be completed

Keywords: Sojourn times, Laplace transforms

Nature: Original

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XII: 57, 763-769, LNM 649 (1978)

**MEYER, Paul-André**

La formule d'Ito pour le mouvement brownien, d'après Brosamler (Brownian motion, Stochastic calculus)

This paper presents the results of a paper by Brosamler (*Trans. Amer. Math. Soc.* 149, 1970) on the Ito formula $f(B_t)=...$ for $n$-dimensional Brownian motion, under the weakest possible assumptions: namely up to the first exit time from an open set $W$ and assuming only that $f$ is locally in $L^1$ in $W$, and its Laplacian in the sense of distributions is a measure in $W$

Keywords: Ito formula

Nature: Exposition

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XIII: 06, 90-115, LNM 721 (1979)

**AZÉMA, Jacques**; **YOR, Marc**

Une solution simple au problème de Skorokhod (Brownian motion)

An explicit solution is given to Skorohod's problem: given a distribution $\mu$ with mean $0$ and finite second moment $\sigma^2$, find a (non randomized) stopping time $T$ of a Brownian motion $(X_t)$ such that $X_T$ has the distribution $\mu$ and $E[T]=\sigma^2$. It is shown that if $S_t$ is the one-sided supremum of $X$ at time $t$, $T=\inf\{t:S_t\ge\psi(X_t)\}$ solves the problem, where $\psi(x)$ is the barycenter of $\mu$ restricted to $[x,\infty[$. The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray-Knight theorem on local times

Comment: The subject is further investigated in 1356 and 1441. See also 1515. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XIII: 36, 427-440, LNM 721 (1979)

**YOR, Marc**

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (Brownian motion)

The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property

Comment: Additional results on the same subject appear in 1545 and in Malric*Ann. Inst. H. Poincaré * **26** (1990)

Keywords: Stochastic integrals

Nature: Original

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XIII: 43, 490-494, LNM 721 (1979)

**WILLIAMS, David**

Conditional excursion theory (Brownian motion, Markov processes)

To be completed

Keywords: Excursions

Nature: Original

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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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XIII: 56, 625-633, LNM 721 (1979)

**AZÉMA, Jacques**; **YOR, Marc**

Le problème de Skorokhod~: compléments à l'exposé précédent (Brownian motion)

What the title calls ``the preceding talk'' is 1306. The method is extended to (centered) measures possessing a moment of order one instead of two, preserving the uniform integrability of the stopped martingale

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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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: 21, 189-199, LNM 784 (1980)

**YOR, Marc**

Application d'un lemme de Jeulin au grossissement de la filtration brownienne (General theory of processes, Brownian motion)

The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment

Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')

Keywords: Enlargement of filtrations

Nature: Original

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XIV: 38, 343-346, LNM 784 (1980)

**YOR, Marc**

Remarques sur une formule de Paul Lévy (Brownian motion)

Given a two-dimensional Brownian motion $(X_t,Y_t)$, Lévy's area integral formula gives the characteristic function $E[\,\exp(iu\int_0^1 X_s\,dY_s-Y_s\,dX_s)\,\,|\,\, X_0=x, Y_0=y]$. A short proof of this formula is given, and it is shown how to deduce from it the apparently more general $E[\exp(iu\int_0^1 X_sdY_s+iv\int_0^1 Y_sdX_s)\,]$ computed by Berthuet

Keywords: Area integral formula

Nature: Original

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XIV: 40, 357-391, LNM 784 (1980)

**FALKNER, Neil**

On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times (Brownian motion, Potential theory)

The problem discussed here is the Skorohod representation of a measure $\nu$ as the distribution of $B_T$, where $(B_t)$ is Brownian motion in $**R**^n$ with the initial measure $\mu$, and $T$ is a *non-randomized * stopping time. The conditions given are sufficient in all cases, necessary if $\mu$ does not charge polar sets

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XIV: 41, 392-396, LNM 784 (1980)

**PIERRE, Michel**

Le problème de Skorohod~: une remarque sur la démonstration d'Azéma-Yor (Brownian motion)

This is an addition to 1306 and 1356, showing how the proof can be reduced to that of a regular case, where it becomes simpler

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XV: 01, 1-5, LNM 850 (1981)

**FERNIQUE, Xavier**

Sur les lois de certaines intégrales associées à des mouvements browniens (Brownian motion)

Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)

Keywords: Iterated stochastic integrals

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: 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: 15, 210-226, LNM 850 (1981)

**JEULIN, Thierry**; **YOR, Marc**

Sur les distributions de certaines fonctionnelles du mouvement brownien (Brownian motion)

This paper gives new proofs and extensions of results due to Knight, concerning occupation times by the process $(S_t,B_t)$ up to time $T_a$, where $(B_t)$ is Brownian motion, $T_a$ the hitting time of $a$, and $(S_t)$ is $\sup_{s\le t} B_s$. The method uses enlargement of filtrations, and martingales similar to those of 1306. Theorem 3.7 is a decomposition of Brownian paths akin to Williams' decomposition

Comment: See also 1516

Keywords: Explicit laws, Occupation times, Enlargement of filtrations, Williams decomposition

Nature: Original

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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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XV: 29, 399-412, LNM 850 (1981)

**YOEURP, Chantha**

Sur la dérivation stochastique au sens de Davis (Stochastic calculus, Brownian motion)

The problem is that of estimating $H_t$ from a knowledge of the process $\int H_sdX_s$, where $X_t$ is a continuous (semi)martingale, as a limit of ratios of the form $\int_t^{t+\alpha} H_sdX_s/(X_{t+\alpha}-X_t)$, or replacing $t+\alpha$ by suitable stopping times

Comment: The problem was suggested and partially solved by Mark H.A. Davis (*Teor. Ver. Prim.*, **20**, 1975, 887--892). See also 1428

Keywords: Stochastic integrals

Nature: Original

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XV: 45, 643-668, LNM 850 (1981)

**AUERHAN, J.**; **LÉPINGLE, Dominique**

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (General theory of processes, Brownian motion, Martingale theory)

This is a sequel to 1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in 1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given

Comment: Further results are given by Malric*Ann. Inst. H. Poincaré * **26** (1990)

Nature: Original

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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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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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XVI: 20, 234-237, LNM 920 (1982)

**YOEURP, Chantha**

Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (Brownian motion, Stochastic calculus)

A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$

Comment: See 1023, 1321

Keywords: Multiplicative decomposition, Change of variable formula, Local times

Nature: Original

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XVI: 21, 238-247, LNM 920 (1982)

**YOR, Marc**

Sur la transformée de Hilbert des temps locaux browniens et une extension de la formule d'Itô (Brownian motion)

This paper is about the application to the function $(x-a)\log|x-a|-(x-a)$ (whose second derivative is $1/x-a$) of the Ito-Tanaka formula; the last term then involves a formal Hilbert transform $\tilde L^a_t$ of the local time process $L^a_t$. Such processes had been defined by Ito and McKean, and studied by Yamada as examples of Fukushima's ``additive functionals of zero energy''. Here it is proved, as a consequence of a general theorem, that this process has a jointly continuous version---more precisely, Hölder continuous of all orders $<1/2$ in $a$ and in $t$

Comment: For a modern version with references see Yor,*Some Aspects of Brownian Motion II*, Birkhäuser 1997

Keywords: Local times, Hilbert transform, Ito formula

Nature: Original

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XVII: 09, 89-105, LNM 986 (1983)

**YOR, Marc**

Le drap brownien comme limite en loi des temps locaux linéaires (Brownian motion, Local time, Brownian sheet)

A central limit theorem is obtained for the increments $L^x_t-L^0_t$ of Brownian local times. The limiting process is expressed in terms of a Brownian sheet, independent of the initial Brownian motion

Comment: This type of result is closely related to the Ray-Knight theorems, which describe the law of Brownian local times considered at certain random times. This has been extended first by Rosen in 2533, where Brownian motion is replaced with a symmetric stable process, then by Eisenbaum 2926

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XVII: 22, 198-204, LNM 986 (1983)

**KARANDIKAR, Rajeeva L.**

Girsanov type formula for a Lie group valued Brownian motion (Brownian motion, Stochastic differential geometry)

A formula for the change of measure of a Lie group valued Brownian motion is stated and proved. It needs a Borel correspondence between paths in the Lie algebra and paths in the group, that transforms all (continuous) semimartingales in the algebra into their stochastic exponential

Comment: For more on stochastic exponentials in Lie groups, see Hakim-Dowek-Lépingle 2023 and Arnaudon 2612

Keywords: Changes of measure, Brownian motion in a manifold, Lie group

Nature: Original

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XVII: 24, 221-224, LNM 986 (1983)

**BASS, Richard F.**

Skorohod imbedding via stochastic integrals (Brownian motion)

A centered probability $\mu$ on $\bf R$ is the law of $g(X_1)$, for a suitable function $g$ and $(X_t,\ t\le 1)$ a Brownian motion. The martingale with terminal value $g(X_1)$ is a time change $(T(t), \ t\le1)$ of a Brownian motion $\beta$; it is shown that $T(1)$ is a stopping time for $\beta$, thus showing the Skorohod embedding for $\mu$

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XVII: 25, 225-226, LNM 986 (1983)

**MEILIJSON, Isaac**

On the Azéma-Yor stopping time (Brownian motion)

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XVII: 26, 227-239, LNM 986 (1983)

**VALLOIS, Pierre**

Le problème de Skorokhod sur $\bf R$ : une approche avec le temps local (Brownian motion)

A solution to Skorohod's embedding problem is given, using the first hiting time of a set by the 2-dimensional process which consists of Brownian motion and its local time at zero. The author's aim is to ``correct'' the asymmetry inherent to the Azéma-Yor construction

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding, Local times

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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XIX: 28, 314-331, LNM 1123 (1985)

**LE GALL, Jean-François**

Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan (Brownian motion)

The normalized self-intersection local time of planar Brownian motion was shown to exist by Varadhan (Appendix to*Euclidean quantum field theory*, by K.~Symanzik, in *Local Quantum Theory*, Academic Press, 1969). This is established anew here by a completely different method, using the intersection local time of two independent planar Brownian motions (whose existence was established by Geman, Horowitz and Rosen, *Ann. Prob.* **12**, 1984) and a sequence of dyadic decompositions of the triangle $\{0<s<t\le1\}$

Comment: Later, Dynkin, Rosen, Le Gall and others have shown existence of a renormalized local time for the multiple self-intersection of arbitrary order $n$ of planar Brownian motion. A good reference is Le Gall,*École d'Été de Saint-Flour XX*, Springer LNM 1527

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original proofs

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XIX: 29, 332-349, LNM 1123 (1985)

**YOR, Marc**

Compléments aux formules de Tanaka-Rosen (Brownian motion)

Several variants of Rosen's works (*Comm. Math. Phys.* **88** (1983), *Ann. Proba.* **13** (1985), *Ann. Proba.* **14** (1986)) are presented. They yield Tanaka-type formulae for the self-intersection local times of Brownian motion in dimension 2 and beyond, establishing again Varadhan's normalization result (Appendix to *Euclidean quantum field theory*, by K.~Symanzik, in *Local Quantum Theory*, Academic Press, 1969). The methods involve stochastic calculus, which was not needed in 1928

Comment: Examples of further work on this subject, using stochastic calculus or not, are Werner,*Ann. I.H.P.* **29** (1993) who gives many references, Khoshnevisan-Bass, *Ann. I.H.P.* **29** (1993), Rosen-Yor *Ann. Proba.* **19** (1991)

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original proofs

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XIX: 30, 350-365, LNM 1123 (1985)

**YOR, Marc**

Renormalisation et convergence en loi pour des temps locaux d'intersection du mouvement brownien dans ${\bf R}^3$ (Brownian motion)

It is shown that no renormalization à la Varadhan occurs for the self-intersection local times of 3-dimensional Brownian motion; but a weaker result is established: when the point $y\in**R**^3$ tends to $0$, the self-intersection local time at $y$, on the triangle $\{0<s<u\le t\},\ t\ge0$, centered and divided by $(-\log|y|)^{1/2}$, converges in law to a Brownian motion. Several variants of this theorem are established

Comment: This result was used by Le Gall in his work on fluctuations of the Wiener sausage:*Ann. Prob.* **16** (1988). Many results by Rosen have the same flavour

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original

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XX: 31, 465-502, LNM 1204 (1986)

**McGILL, Paul**

Integral representation of martingales in the Brownian excursion filtration (Brownian motion, Stochastic calculus)

An integral representation is obtained of all square integrable martingales in the filtration $({\cal E}^x,\ x\in**R**)$, where ${\cal E}^x$ denotes the Brownian excursion $\sigma$-field below $x$ introduced by D. Williams 1343, who also showed that every $({\cal E}^x)$ martingale is continuous

Comment: Another filtration $(\tilde{\cal E}^x,\ x\in**R**)$ of Brownian excursions below $x$ has been proposed by Azéma; the structure of martingales is quite diffferent: they are discontinuous. See Y. Hu's thesis (Paris VI, 1996), and chap.~16 of Yor, *Some Aspects of Brownian Motion, Part~II*, Birkhäuser, 1997

Keywords: Previsible representation, Martingales, Filtrations

Nature: Original

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XX: 33, 515-531, LNM 1204 (1986)

**ROSEN, Jay S.**

A renormalized local time for multiple intersections of planar Brownian motion (Brownian motion)

Using Fourier techniques, the existence of a renormalized local time for $n$-fold self-intersections of planar Brownian motion is obtained, thus extending the case $n=2$, obtained in the pioneering work of Varadhan (Appendix to*Euclidean quantum field theory*, by K.~Symanzik, in *Local Quantum Theory*, Academic Press, 1969)

Comment: Closely related to 2036. A general reference is Le Gall,*École d'Été de Saint-Flour XX*, Springer LNM 1527

Keywords: Local times, Self-intersection

Nature: Original

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XX: 35, 543-552, LNM 1204 (1986)

**YOR, Marc**

Sur la représentation comme intégrales stochastiques des temps d'occupation du mouvement brownien dans ${\bf R}^d$ (Brownian motion)

Varadhan's renormalization result (Appendix to*Euclidean quantum field theory*, by K.~Symanzik, in *Local Quantum Theory* consists in centering certain sequences of Brownian functionals and showing $L^2$-convergence. The same results are obtained here by writing these centered functionals as stochastic integrals

Comment: One of mny applications of stochastic calculus to the existence and regularity of self-intersection local times. See Rosen's papers on this topic in general, and page 196 of Le Gall,*École d'Été de Saint-Flour XX*, Springer LNM 1527

Keywords: Local times, Self-intersection, Previsible representation

Nature: Original proofs

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XXXI: 29, 306-314, LNM 1655 (1997)

**YOR, Marc**

Some remarks about the joint law of Brownian motion and its supremum (Brownian motion)

Seshadri's identity says that if $S_1$ denotes the maximum of a Brownian motion $B$ on the interval $[0,1]$, the r.v. $2S_1(S_1-B_1)$ is independent of $B_1$ and exponentially distributed. Several variants of this are obtained

Comment: See also 3320

Keywords: Maximal process, Seshadri's identity

Nature: Original

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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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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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XXXIII: 20, 388-394, LNM 1709 (1999)

**PITMAN, James W.**

The distribution of local times of a Brownian bridge (Brownian motion)

Several useful identities for the one-dimensional marginals of local times of Brownian bridges are derived. This is a variation and extension on the well-known joint law of the maximum and the value of Brownian motion at a given time

Comment: Useful references are Borodin,*Russian Math. Surveys* (1989) and the book *Brownian motion and stochastic calculus* by Karatzas-Shrieve (Springer, 1991)

Keywords: Local times, Brownian bridge

Nature: Original

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XLI: 07, 161-179, LNM 1934 (2008)

**BARAKA, Driss**; **MOUNTFORD, Thomas**

A law of the iterated logarithm for fractional Brownian motions (Theory of fractional Brownian motion)

Nature: Original

XLI: 08, 181-197, LNM 1934 (2008)

**NOURDIN, Ivan**

A simple theory for the study of SDEs driven by a fractional Brownian motion in dimension one (Theory of fractional Brownian motion)

Nature: Original

XLII: 14, 383-396, LNM 1978 (2009)

**ÉMERY, Michel**

Recognising whether a filtration is Brownian: a case study (Theory of Brownian motion)

Keywords: Brownian filtration

Nature: Original

XLIII: 03, 95-104, LNM 2006 (2011)

**ROSEN, Jay**

A stochastic calculus proof of the CLT for the $L^{2}$ modulus of continuity of local time (Theory of Brownian motion)

Keywords: Central Limit Theorem, Moduli of continuity, Local times, Brownian motion

Nature: Original

XLIII: 10, 241-268, LNM 2006 (2011)

**BÉRARD BERGERY, Blandine**; **VALLOIS, Pierre**

Convergence at first and second order of some approximations of stochastic integrals (Theory of Brownian motion, Theory of stochastic integrals)

Keywords: Stochastic integration by regularization, Quadratic variation, First and second order convergence, Stochastic Fubini's theorem

Nature: Original

XLIII: 19, 437-439, LNM 2006 (2011)

**BAKER, David**; **YOR, Marc**

On martingales with given marginals and the scaling property (Martingale theory, Theory of Brownian motion)

Nature: Original

XLVI: 14, 359-375, LNM 2123 (2014)

**ROSENBAUM, Mathieu**; **YOR, Marc**

On the law of a triplet associated with the pseudo-Brownian bridge (Theory of Brownian motion)

This article gives a remarkable identity in law which relates the Brownian motion, its local time, and the the inverse of its local time

Keywords: Brownian motion, pseudo-Brownian bridge, Bessel process, local time, hitting times, scaling, uniform sampling, Mellin transform

Nature: Original

Une remarque sur le flot du mouvement brownien (Brownian motion, Ergodic theory)

It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense

Comment: See Sam Lazaro-Meyer,

Keywords: Brownian motion, Wiener chaos, Screw-lines

Nature: Original

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VIII: 02, 11-19, LNM 381 (1974)

Une remarque sur le problème de Skorohod (Brownian motion)

The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see 516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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VIII: 10, 134-149, LNM 381 (1974)

Existence of small oscillations at zeros of brownian motion (Brownian motion)

The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an ``iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros

Comment: This result may be compared to Kahane,

Keywords: Law of the iterated logarithm, Local times

Nature: Original

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X: 02, 19-23, LNM 511 (1976)

One-dimensional potential imbedding (Brownian motion)

The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool

Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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

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

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XII: 31, 428-445, LNM 649 (1978)

On the sojourn times of killed Brownian motion (Brownian motion)

To be completed

Keywords: Sojourn times, Laplace transforms

Nature: Original

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XII: 57, 763-769, LNM 649 (1978)

La formule d'Ito pour le mouvement brownien, d'après Brosamler (Brownian motion, Stochastic calculus)

This paper presents the results of a paper by Brosamler (

Keywords: Ito formula

Nature: Exposition

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XIII: 06, 90-115, LNM 721 (1979)

Une solution simple au problème de Skorokhod (Brownian motion)

An explicit solution is given to Skorohod's problem: given a distribution $\mu$ with mean $0$ and finite second moment $\sigma^2$, find a (non randomized) stopping time $T$ of a Brownian motion $(X_t)$ such that $X_T$ has the distribution $\mu$ and $E[T]=\sigma^2$. It is shown that if $S_t$ is the one-sided supremum of $X$ at time $t$, $T=\inf\{t:S_t\ge\psi(X_t)\}$ solves the problem, where $\psi(x)$ is the barycenter of $\mu$ restricted to $[x,\infty[$. The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray-Knight theorem on local times

Comment: The subject is further investigated in 1356 and 1441. See also 1515. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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XIII: 36, 427-440, LNM 721 (1979)

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (Brownian motion)

The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property

Comment: Additional results on the same subject appear in 1545 and in Malric

Keywords: Stochastic integrals

Nature: Original

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XIII: 43, 490-494, LNM 721 (1979)

Conditional excursion theory (Brownian motion, Markov processes)

To be completed

Keywords: Excursions

Nature: Original

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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

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XIII: 56, 625-633, LNM 721 (1979)

Le problème de Skorokhod~: compléments à l'exposé précédent (Brownian motion)

What the title calls ``the preceding talk'' is 1306. The method is extended to (centered) measures possessing a moment of order one instead of two, preserving the uniform integrability of the stopped martingale

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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

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: 21, 189-199, LNM 784 (1980)

Application d'un lemme de Jeulin au grossissement de la filtration brownienne (General theory of processes, Brownian motion)

The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment

Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')

Keywords: Enlargement of filtrations

Nature: Original

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XIV: 38, 343-346, LNM 784 (1980)

Remarques sur une formule de Paul Lévy (Brownian motion)

Given a two-dimensional Brownian motion $(X_t,Y_t)$, Lévy's area integral formula gives the characteristic function $E[\,\exp(iu\int_0^1 X_s\,dY_s-Y_s\,dX_s)\,\,|\,\, X_0=x, Y_0=y]$. A short proof of this formula is given, and it is shown how to deduce from it the apparently more general $E[\exp(iu\int_0^1 X_sdY_s+iv\int_0^1 Y_sdX_s)\,]$ computed by Berthuet

Keywords: Area integral formula

Nature: Original

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XIV: 40, 357-391, LNM 784 (1980)

On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times (Brownian motion, Potential theory)

The problem discussed here is the Skorohod representation of a measure $\nu$ as the distribution of $B_T$, where $(B_t)$ is Brownian motion in $

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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XIV: 41, 392-396, LNM 784 (1980)

Le problème de Skorohod~: une remarque sur la démonstration d'Azéma-Yor (Brownian motion)

This is an addition to 1306 and 1356, showing how the proof can be reduced to that of a regular case, where it becomes simpler

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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XV: 01, 1-5, LNM 850 (1981)

Sur les lois de certaines intégrales associées à des mouvements browniens (Brownian motion)

Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)

Keywords: Iterated stochastic integrals

Nature: Original

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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

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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

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XV: 15, 210-226, LNM 850 (1981)

Sur les distributions de certaines fonctionnelles du mouvement brownien (Brownian motion)

This paper gives new proofs and extensions of results due to Knight, concerning occupation times by the process $(S_t,B_t)$ up to time $T_a$, where $(B_t)$ is Brownian motion, $T_a$ the hitting time of $a$, and $(S_t)$ is $\sup_{s\le t} B_s$. The method uses enlargement of filtrations, and martingales similar to those of 1306. Theorem 3.7 is a decomposition of Brownian paths akin to Williams' decomposition

Comment: See also 1516

Keywords: Explicit laws, Occupation times, Enlargement of filtrations, Williams decomposition

Nature: Original

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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

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XV: 29, 399-412, LNM 850 (1981)

Sur la dérivation stochastique au sens de Davis (Stochastic calculus, Brownian motion)

The problem is that of estimating $H_t$ from a knowledge of the process $\int H_sdX_s$, where $X_t$ is a continuous (semi)martingale, as a limit of ratios of the form $\int_t^{t+\alpha} H_sdX_s/(X_{t+\alpha}-X_t)$, or replacing $t+\alpha$ by suitable stopping times

Comment: The problem was suggested and partially solved by Mark H.A. Davis (

Keywords: Stochastic integrals

Nature: Original

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XV: 45, 643-668, LNM 850 (1981)

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (General theory of processes, Brownian motion, Martingale theory)

This is a sequel to 1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in 1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given

Comment: Further results are given by Malric

Nature: Original

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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

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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

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XVI: 20, 234-237, LNM 920 (1982)

Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (Brownian motion, Stochastic calculus)

A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$

Comment: See 1023, 1321

Keywords: Multiplicative decomposition, Change of variable formula, Local times

Nature: Original

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XVI: 21, 238-247, LNM 920 (1982)

Sur la transformée de Hilbert des temps locaux browniens et une extension de la formule d'Itô (Brownian motion)

This paper is about the application to the function $(x-a)\log|x-a|-(x-a)$ (whose second derivative is $1/x-a$) of the Ito-Tanaka formula; the last term then involves a formal Hilbert transform $\tilde L^a_t$ of the local time process $L^a_t$. Such processes had been defined by Ito and McKean, and studied by Yamada as examples of Fukushima's ``additive functionals of zero energy''. Here it is proved, as a consequence of a general theorem, that this process has a jointly continuous version---more precisely, Hölder continuous of all orders $<1/2$ in $a$ and in $t$

Comment: For a modern version with references see Yor,

Keywords: Local times, Hilbert transform, Ito formula

Nature: Original

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XVII: 09, 89-105, LNM 986 (1983)

Le drap brownien comme limite en loi des temps locaux linéaires (Brownian motion, Local time, Brownian sheet)

A central limit theorem is obtained for the increments $L^x_t-L^0_t$ of Brownian local times. The limiting process is expressed in terms of a Brownian sheet, independent of the initial Brownian motion

Comment: This type of result is closely related to the Ray-Knight theorems, which describe the law of Brownian local times considered at certain random times. This has been extended first by Rosen in 2533, where Brownian motion is replaced with a symmetric stable process, then by Eisenbaum 2926

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XVII: 22, 198-204, LNM 986 (1983)

Girsanov type formula for a Lie group valued Brownian motion (Brownian motion, Stochastic differential geometry)

A formula for the change of measure of a Lie group valued Brownian motion is stated and proved. It needs a Borel correspondence between paths in the Lie algebra and paths in the group, that transforms all (continuous) semimartingales in the algebra into their stochastic exponential

Comment: For more on stochastic exponentials in Lie groups, see Hakim-Dowek-Lépingle 2023 and Arnaudon 2612

Keywords: Changes of measure, Brownian motion in a manifold, Lie group

Nature: Original

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XVII: 24, 221-224, LNM 986 (1983)

Skorohod imbedding via stochastic integrals (Brownian motion)

A centered probability $\mu$ on $\bf R$ is the law of $g(X_1)$, for a suitable function $g$ and $(X_t,\ t\le 1)$ a Brownian motion. The martingale with terminal value $g(X_1)$ is a time change $(T(t), \ t\le1)$ of a Brownian motion $\beta$; it is shown that $T(1)$ is a stopping time for $\beta$, thus showing the Skorohod embedding for $\mu$

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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XVII: 25, 225-226, LNM 986 (1983)

On the Azéma-Yor stopping time (Brownian motion)

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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XVII: 26, 227-239, LNM 986 (1983)

Le problème de Skorokhod sur $\bf R$ : une approche avec le temps local (Brownian motion)

A solution to Skorohod's embedding problem is given, using the first hiting time of a set by the 2-dimensional process which consists of Brownian motion and its local time at zero. The author's aim is to ``correct'' the asymmetry inherent to the Azéma-Yor construction

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding, Local times

Nature: Original

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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

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XIX: 28, 314-331, LNM 1123 (1985)

Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan (Brownian motion)

The normalized self-intersection local time of planar Brownian motion was shown to exist by Varadhan (Appendix to

Comment: Later, Dynkin, Rosen, Le Gall and others have shown existence of a renormalized local time for the multiple self-intersection of arbitrary order $n$ of planar Brownian motion. A good reference is Le Gall,

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original proofs

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XIX: 29, 332-349, LNM 1123 (1985)

Compléments aux formules de Tanaka-Rosen (Brownian motion)

Several variants of Rosen's works (

Comment: Examples of further work on this subject, using stochastic calculus or not, are Werner,

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original proofs

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XIX: 30, 350-365, LNM 1123 (1985)

Renormalisation et convergence en loi pour des temps locaux d'intersection du mouvement brownien dans ${\bf R}^3$ (Brownian motion)

It is shown that no renormalization à la Varadhan occurs for the self-intersection local times of 3-dimensional Brownian motion; but a weaker result is established: when the point $y\in

Comment: This result was used by Le Gall in his work on fluctuations of the Wiener sausage:

Keywords: Brownian motion, Local times, Self-intersection

Nature: Original

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XX: 31, 465-502, LNM 1204 (1986)

Integral representation of martingales in the Brownian excursion filtration (Brownian motion, Stochastic calculus)

An integral representation is obtained of all square integrable martingales in the filtration $({\cal E}^x,\ x\in

Comment: Another filtration $(\tilde{\cal E}^x,\ x\in

Keywords: Previsible representation, Martingales, Filtrations

Nature: Original

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XX: 33, 515-531, LNM 1204 (1986)

A renormalized local time for multiple intersections of planar Brownian motion (Brownian motion)

Using Fourier techniques, the existence of a renormalized local time for $n$-fold self-intersections of planar Brownian motion is obtained, thus extending the case $n=2$, obtained in the pioneering work of Varadhan (Appendix to

Comment: Closely related to 2036. A general reference is Le Gall,

Keywords: Local times, Self-intersection

Nature: Original

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XX: 35, 543-552, LNM 1204 (1986)

Sur la représentation comme intégrales stochastiques des temps d'occupation du mouvement brownien dans ${\bf R}^d$ (Brownian motion)

Varadhan's renormalization result (Appendix to

Comment: One of mny applications of stochastic calculus to the existence and regularity of self-intersection local times. See Rosen's papers on this topic in general, and page 196 of Le Gall,

Keywords: Local times, Self-intersection, Previsible representation

Nature: Original proofs

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XXXI: 29, 306-314, LNM 1655 (1997)

Some remarks about the joint law of Brownian motion and its supremum (Brownian motion)

Seshadri's identity says that if $S_1$ denotes the maximum of a Brownian motion $B$ on the interval $[0,1]$, the r.v. $2S_1(S_1-B_1)$ is independent of $B_1$ and exponentially distributed. Several variants of this are obtained

Comment: See also 3320

Keywords: Maximal process, Seshadri's identity

Nature: Original

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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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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

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XXXIII: 20, 388-394, LNM 1709 (1999)

The distribution of local times of a Brownian bridge (Brownian motion)

Several useful identities for the one-dimensional marginals of local times of Brownian bridges are derived. This is a variation and extension on the well-known joint law of the maximum and the value of Brownian motion at a given time

Comment: Useful references are Borodin,

Keywords: Local times, Brownian bridge

Nature: Original

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XLI: 07, 161-179, LNM 1934 (2008)

A law of the iterated logarithm for fractional Brownian motions (Theory of fractional Brownian motion)

Nature: Original

XLI: 08, 181-197, LNM 1934 (2008)

A simple theory for the study of SDEs driven by a fractional Brownian motion in dimension one (Theory of fractional Brownian motion)

Nature: Original

XLII: 14, 383-396, LNM 1978 (2009)

Recognising whether a filtration is Brownian: a case study (Theory of Brownian motion)

Keywords: Brownian filtration

Nature: Original

XLIII: 03, 95-104, LNM 2006 (2011)

A stochastic calculus proof of the CLT for the $L^{2}$ modulus of continuity of local time (Theory of Brownian motion)

Keywords: Central Limit Theorem, Moduli of continuity, Local times, Brownian motion

Nature: Original

XLIII: 10, 241-268, LNM 2006 (2011)

Convergence at first and second order of some approximations of stochastic integrals (Theory of Brownian motion, Theory of stochastic integrals)

Keywords: Stochastic integration by regularization, Quadratic variation, First and second order convergence, Stochastic Fubini's theorem

Nature: Original

XLIII: 19, 437-439, LNM 2006 (2011)

On martingales with given marginals and the scaling property (Martingale theory, Theory of Brownian motion)

Nature: Original

XLVI: 14, 359-375, LNM 2123 (2014)

On the law of a triplet associated with the pseudo-Brownian bridge (Theory of Brownian motion)

This article gives a remarkable identity in law which relates the Brownian motion, its local time, and the the inverse of its local time

Keywords: Brownian motion, pseudo-Brownian bridge, Bessel process, local time, hitting times, scaling, uniform sampling, Mellin transform

Nature: Original