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-linesNature: Original
Retrieve article from Numdam
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 representationNature: Original Retrieve article from Numdam
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 spacesNature: Exposition,
Original additions Retrieve article from Numdam
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
1253Keywords: Littlewood-Paley theory,
Riesz transforms,
Brownian motion,
Inequalities,
Harmonic functions,
Singular integrals,
Carré du champ,
Infinitesimal generators,
Semigroup theoryNature: Original Retrieve article from Numdam
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 champNature: Original Retrieve article from Numdam
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
2926Keywords: Brownian motion,
Several parameter processesNature: Original Retrieve article from Numdam
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 representationNature: Original Retrieve article from Numdam
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
2612Keywords: Changes of measure,
Brownian motion in a manifold,
Lie groupNature: Original Retrieve article from Numdam
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 theoremsNature: Original Retrieve article from Numdam
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-intersectionNature: Original proofs Retrieve article from Numdam
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
1928Comment: 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-intersectionNature: Original proofs Retrieve article from Numdam
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-intersectionNature: Original Retrieve article from Numdam
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 processesNature: Exposition,
Original additions Retrieve article from Numdam
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 sheetNature: Original Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
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 sheetNature: Original Retrieve article from Numdam
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 formsNature: Original Retrieve article from Numdam
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
1709Comment: 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 sheetNature: Original Retrieve article from Numdam
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
3325Keywords: Filtrations,
Spider martingales,
Walsh's Brownian motion,
Cosiness,
Slutsky's lemmaNature: New exposition of known results,
Original additions Retrieve article from Numdam
XLIII: 01, 3-70, LNM 2006 (2011)
PICARD, Jean
Representation formulae for the fractional Brownian motion (
Theory of processes)
Keywords: Fractional Brownian motion,
Brownian motionNature: 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 curvatureNature: 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 motionNature: 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 theoremNature: 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,
SimulationNature: 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 timesNature: 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 distributionNature: 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 mediaNature: 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 motionNature: Original
XLIV: 21, 467-467, LNM 2046 (2012)
ÉMERY, Michel;
YOR, Marc
Erratum to Séminaire XXVIIComment: This is an erratum to
2714.
Keywords: Brownian motion,
Continuous martingaleNature: 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 timesNature: 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 optionsNature: 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 motionNature: 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 transformNature: Original
