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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: 05, 25-26, LNM 381 (1974)

**DELLACHERIE, Claude**

Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (General theory of processes)

This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems

Comment: A gap in the proof is filled in 928 and 2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,*Z. für W-theorie,* **38**, 1977 and Yor 1221. For another approach to the restricted case considered here, see Ruiz de Chavez 1821. The previsible representation property of Brownian motion and compensated Poisson process was know by Itô; it is a consequence of the (stronger) chaotic representation property, established by Wiener in 1938. The converse was also known by Itô: among the martingales which are also Lévy processes, only Brownian motions and compensated Poisson processes have the previsible representation property

Keywords: Brownian motion, Poisson processes, Previsible representation

Nature: Original

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VIII: 09, 80-133, LNM 381 (1974)

**GEBUHRER, Marc Olivier**

Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)

The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (*Arkiv för Math.*, **6**, 1965-67), which is studied as a Lorentz invariant diffusion process (in the usual sense) on the standard hyperboloid of velocities in special relativity, on which the Lorentz group acts. The Brownian paths themselves are constructed by integration and possess a speed smaller than the velocity of light but no higher derivatives. The second part studies more generally invariant Markov processes on a Riemannian symmetric space of non-compact type, their generators and the corresponding semigroups

Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces

Nature: Exposition, Original additions

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X: 10, 125-183, LNM 511 (1976)

**MEYER, Paul-André**

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book*Topics in Harmonic Analysis related to the Littlewood-Paley theory,* Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $**R**^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $**R**^n\times **R**_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $**R**^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (*CRAS Paris,* **278A**, 1974, p.1103) in potential theory and by Kunita (*Nagoya M. J.*, **36**, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false

Comment: This paper was rediscovered by Varopoulos (*J. Funct. Anal.*, **38**, 1980), and was then rewritten by Meyer in 1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in 1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912, and Meyer 1908 reporting on Cowling's extension of Stein's work. An erratum is given in 1253

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

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XV: 10, 151-166, LNM 850 (1981)

**MEYER, Paul-André**

Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)

The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (*J. Funct. Anal.*, 38, 1980)

Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ

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: 21, 194-197, LNM 986 (1983)

**PRICE, Gareth C.**; **WILLIAMS, David**

Rolling with `slipping': I (Stochastic calculus, Stochastic differential geometry)

If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$

Keywords: Brownian motion in a manifold, Previsible representation

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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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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XXIV: 30, 448-452, LNM 1426 (1990)

**ÉMERY, Michel**; **LÉANDRE, Rémi**

Sur une formule de Bismut (Markov processes, Stochastic differential geometry)

This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group

Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes

Nature: Exposition, Original additions

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XXV: 33, 407-424, LNM 1485 (1991)

**ROSEN, Jay S.**

Second order limit laws for the local times of stable processes (Limit theorems)

Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable

Comment: Another proof due to Eisenbaum 3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi*Ann. Prob.* **28** (2000). A good reference on this subject is Marcus-Rosen, *Markov Processes, Gaussian Processes, and Local Times*, Cambridge University Press (2006)

Keywords: Local times, Stable processes, Method of moments, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXVI: 24, 322-347, LNM 1526 (1992)

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

Une décomposition non-canonique du drap brownien (Brownian sheet, Gaussian processes)

In 2415, the authors have introduced a transform of Brownian motion. Here, a similar transform is defined on the Brownian sheet; this transform is shown to be strongly mixing

Comment: This work was motivated by Föllmer's article on Martin boundaries on Wiener space (in*Diffusion processes and related problems in analysis*, vol.~I, Birkhäuser 1990)

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XXIX: 26, 266-289, LNM 1613 (1995)

**EISENBAUM, Nathalie**

Une version sans conditionnement du théorème d'isomorphisme de Dynkin (Limit theorems)

After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen 2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (*Proc. Conf. Probability in Banach Spaces~8*, Birkhäuser 1992) on Laplace transforms of the increments of local time are extended

Comment: A general reference on the subject is Marcus-Rosen,*Markov Processes, Gaussian Processes, and Local Times*, Cambridge University Press (2006)

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXXI: 05, 54-61, LNM 1655 (1997)

**FANG, Shizan**; **FRANCHI, Jacques**

A differentiable isomorphism between Wiener space and path group (Malliavin's calculus)

The Itô map $I$ is known to realize a measurable isomorphism between Wiener space $W$ and the group ${\cal P}$ of paths with values in a Riemannian manifold. Here, the pullback $I^{*}$ is shown to be a diffeomorphism (in the sense of Malliavin derivatives) between the exterior algebras $\Lambda (W)$ and $\Lambda ({\cal P})$. This allows to transfer the Weitzenböck-Shigekawa identity from $\Lambda (W)$ to $\Lambda ({\cal P})$, yielding for example the de~Rham-Hodge-Kodaira decomposition on ${\cal P}$

Keywords: Wiener space, Path group, Brownian motion in a manifold, Differential forms

Nature: Original

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XXXI: 20, 216-224, LNM 1655 (1997)

**EISENBAUM, Nathalie**

Théorèmes limites pour les temps locaux d'un processus stable symétrique (Limit theorems)

Using Dynkin's isomorphism, a central-limit type theorem is derived for the local times of a stable symmetric process of index $\beta$ at a finite number $n$ of levels. The limiting process is expressed in terms of a fractional, $n$-dimensional Brownian sheet with Hurst index $\beta-1$. The case when $n=1$ is due to Rosen 2533, and, for Brownian local times, to Yor 1709

Comment: This kind of result is now understood as a weak form of theorems à la Ray-Knight, describing the local times of a stable symmetric process: see Eisenbaum-Kaspi-Marcus-Rosen-Shi*Ann. Prob.* **28** (2000) for a Ray-Knight theorem involving fractional Brownian motion. Marcus-Rosen, *Markov Processes, Gaussian Processes, and Local Times*, Cambridge University Press (2006) is a general reference on the subject

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

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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XLIII: 01, 3-70, LNM 2006 (2011)

**PICARD, Jean**

Representation formulae for the fractional Brownian motion (Theory of processes)

Keywords: Fractional Brownian motion, Brownian motion

Nature: Original, Survey

XLIII: 02, 73-94, LNM 2006 (2011)

**ARNAUDON, Marc**; **COULIBALY, Koléhè Abdoulaye**; **THALMAIER, Anton**

Horizontal diffusion in $C^1$ path space (Theory of processes)

Keywords: Brownian motion, Damped parallel transport, Horizontal diffusion, Monge-Kantorovich problem, Ricci curvature

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: 08, 215-219, LNM 2006 (2011)

**PRATELLI, Maurizio**

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition

XLIII: 09, 221-239, LNM 2006 (2011)

**MAROUBY, Matthieu**

Simulation of a Local Time Fractional Stable Motion (Theory of processes)

Keywords: Stable processes, Self-similar processes, Shot noise series, Local times, Fractional Brownian motion, Simulation

Nature: Original

XLIV: 08, 167-190, LNM 2046 (2012)

**HAJRI, Hatem**

Discrete approximation to solution flows of Tanaka's SDE related to Walsh Brownian motion (Stochastic calculus, Limit theorems)

Keywords: Walsh's Brownian motion, Tanaka's SDE, Local times

Nature: Original

XLIV: 09, 191-206, LNM 2046 (2012)

**DEMNI, Nizar**; **HMIDI, Taoufik**

Spectral Distribution of the Free unitary Brownian motion: another approach (Non commutative probability theory)

Keywords: Free unitary Brownian motion, Spectral distribution

Nature: Original

XLIV: 12, 247-269, LNM 2046 (2012)

**MARTY, Renaud**; **SØLNA, Knut**

Asymptotic behavior of oscillatory fractional processes (Theory of processes, Limit theorems)

Keywords: Fractional processes, Brownian motion, Waves in random media

Nature: Original

XLIV: 17, 375-399, LNM 2046 (2012)

**HARRIS, Simon C.**; **ROBERTS, Matthew I.**

Branching Brownian motion: Almost sure growth along scaled paths (Limit theorems, Theory of processes)

Keywords: Branching Brownian motion

Nature: Original

XLIV: 21, 467-467, LNM 2046 (2012)

**ÉMERY, Michel**; **YOR, Marc**

Erratum to Séminaire XXVII

Comment: This is an erratum to 2714.

Keywords: Brownian motion, Continuous martingale

Nature: Correction

XLV: 10, 277-300, LNM 2078 (2013)

**DONEY, R. A.**; **VAKEROUDIS, S.**

Windings of Planar Stable Processes (Theory of processes)

Keywords: Stable processes, Lévy processes, Brownian motion, windings, exit time from a cone, Spitzer's Theorem, skew-product representation, Lamperti's relation, Law of the Iterated Logarithm for small times

Nature: Original

XLV: 15, 365-400, LNM 2078 (2013)

**PAGÈS, Gilles**

Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options (Theory of processes)

Keywords: Co-monotony, antithetic simulation method, processes with independent increments, Liouville processes, fractional Brownian motion, Asian options, sensitivity, barrier options

Nature: Original

XLVI: 06, 125-193, LNM 2123 (2014)

**GENG, Xi**; **QIAN, Zhongmin**; **YANG, Danyu**

$G$-Brownian Motion as Rough Paths and Differential Equations Driven by $G$-Brownian Motion (Stochastic analysis)

This article studies stochastic differential equations driven by the $G$-Brownian motion in the context of rough paths theory

Keywords: rough path, $G$-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: 05, 25-26, LNM 381 (1974)

Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (General theory of processes)

This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems

Comment: A gap in the proof is filled in 928 and 2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,

Keywords: Brownian motion, Poisson processes, Previsible representation

Nature: Original

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VIII: 09, 80-133, LNM 381 (1974)

Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (Markov processes)

The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (

Keywords: Relativistic Brownian motion, Invariant Markov processes, Symmetric spaces

Nature: Exposition, Original additions

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X: 10, 125-183, LNM 511 (1976)

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book

Comment: This paper was rediscovered by Varopoulos (

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

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XV: 10, 151-166, LNM 850 (1981)

Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)

The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (

Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ

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

Retrieve article from Numdam

XVII: 21, 194-197, LNM 986 (1983)

Rolling with `slipping': I (Stochastic calculus, Stochastic differential geometry)

If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$

Keywords: Brownian motion in a manifold, Previsible representation

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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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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XXIV: 30, 448-452, LNM 1426 (1990)

Sur une formule de Bismut (Markov processes, Stochastic differential geometry)

This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group

Keywords: Brownian bridge, Brownian motion in a manifold, Transformations of Markov processes

Nature: Exposition, Original additions

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XXV: 33, 407-424, LNM 1485 (1991)

Second order limit laws for the local times of stable processes (Limit theorems)

Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable

Comment: Another proof due to Eisenbaum 3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi

Keywords: Local times, Stable processes, Method of moments, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXVI: 24, 322-347, LNM 1526 (1992)

Une décomposition non-canonique du drap brownien (Brownian sheet, Gaussian processes)

In 2415, the authors have introduced a transform of Brownian motion. Here, a similar transform is defined on the Brownian sheet; this transform is shown to be strongly mixing

Comment: This work was motivated by Föllmer's article on Martin boundaries on Wiener space (in

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XXIX: 26, 266-289, LNM 1613 (1995)

Une version sans conditionnement du théorème d'isomorphisme de Dynkin (Limit theorems)

After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen 2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (

Comment: A general reference on the subject is Marcus-Rosen,

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

Nature: Original

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XXXI: 05, 54-61, LNM 1655 (1997)

A differentiable isomorphism between Wiener space and path group (Malliavin's calculus)

The Itô map $I$ is known to realize a measurable isomorphism between Wiener space $W$ and the group ${\cal P}$ of paths with values in a Riemannian manifold. Here, the pullback $I^{*}$ is shown to be a diffeomorphism (in the sense of Malliavin derivatives) between the exterior algebras $\Lambda (W)$ and $\Lambda ({\cal P})$. This allows to transfer the Weitzenböck-Shigekawa identity from $\Lambda (W)$ to $\Lambda ({\cal P})$, yielding for example the de~Rham-Hodge-Kodaira decomposition on ${\cal P}$

Keywords: Wiener space, Path group, Brownian motion in a manifold, Differential forms

Nature: Original

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XXXI: 20, 216-224, LNM 1655 (1997)

Théorèmes limites pour les temps locaux d'un processus stable symétrique (Limit theorems)

Using Dynkin's isomorphism, a central-limit type theorem is derived for the local times of a stable symmetric process of index $\beta$ at a finite number $n$ of levels. The limiting process is expressed in terms of a fractional, $n$-dimensional Brownian sheet with Hurst index $\beta-1$. The case when $n=1$ is due to Rosen 2533, and, for Brownian local times, to Yor 1709

Comment: This kind of result is now understood as a weak form of theorems à la Ray-Knight, describing the local times of a stable symmetric process: see Eisenbaum-Kaspi-Marcus-Rosen-Shi

Keywords: Stable processes, Local times, Central limit theorem, Dynkin isomorphism, Fractional Brownian motion, Brownian sheet

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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XLIII: 01, 3-70, LNM 2006 (2011)

Representation formulae for the fractional Brownian motion (Theory of processes)

Keywords: Fractional Brownian motion, Brownian motion

Nature: Original, Survey

XLIII: 02, 73-94, LNM 2006 (2011)

Horizontal diffusion in $C^1$ path space (Theory of processes)

Keywords: Brownian motion, Damped parallel transport, Horizontal diffusion, Monge-Kantorovich problem, Ricci curvature

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: 08, 215-219, LNM 2006 (2011)

A Remark on the $1/H$-variation of the Fractional Brownian Motion (Theory of processes)

Keywords: Fractional Brownian motion, $p$-variation, Ergodic theorem

Nature: Exposition

XLIII: 09, 221-239, LNM 2006 (2011)

Simulation of a Local Time Fractional Stable Motion (Theory of processes)

Keywords: Stable processes, Self-similar processes, Shot noise series, Local times, Fractional Brownian motion, Simulation

Nature: Original

XLIV: 08, 167-190, LNM 2046 (2012)

Discrete approximation to solution flows of Tanaka's SDE related to Walsh Brownian motion (Stochastic calculus, Limit theorems)

Keywords: Walsh's Brownian motion, Tanaka's SDE, Local times

Nature: Original

XLIV: 09, 191-206, LNM 2046 (2012)

Spectral Distribution of the Free unitary Brownian motion: another approach (Non commutative probability theory)

Keywords: Free unitary Brownian motion, Spectral distribution

Nature: Original

XLIV: 12, 247-269, LNM 2046 (2012)

Asymptotic behavior of oscillatory fractional processes (Theory of processes, Limit theorems)

Keywords: Fractional processes, Brownian motion, Waves in random media

Nature: Original

XLIV: 17, 375-399, LNM 2046 (2012)

Branching Brownian motion: Almost sure growth along scaled paths (Limit theorems, Theory of processes)

Keywords: Branching Brownian motion

Nature: Original

XLIV: 21, 467-467, LNM 2046 (2012)

Erratum to Séminaire XXVII

Comment: This is an erratum to 2714.

Keywords: Brownian motion, Continuous martingale

Nature: Correction

XLV: 10, 277-300, LNM 2078 (2013)

Windings of Planar Stable Processes (Theory of processes)

Keywords: Stable processes, Lévy processes, Brownian motion, windings, exit time from a cone, Spitzer's Theorem, skew-product representation, Lamperti's relation, Law of the Iterated Logarithm for small times

Nature: Original

XLV: 15, 365-400, LNM 2078 (2013)

Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options (Theory of processes)

Keywords: Co-monotony, antithetic simulation method, processes with independent increments, Liouville processes, fractional Brownian motion, Asian options, sensitivity, barrier options

Nature: Original

XLVI: 06, 125-193, LNM 2123 (2014)

$G$-Brownian Motion as Rough Paths and Differential Equations Driven by $G$-Brownian Motion (Stochastic analysis)

This article studies stochastic differential equations driven by the $G$-Brownian motion in the context of rough paths theory

Keywords: rough path, $G$-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