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V: 05, 58-75, LNM 191 (1971)

**CARTIER, Pierre**

Introduction à l'étude des mouvements browniens à plusieurs paramètres (Gaussian processes)

Settles in particular a disagreement between statements of Lévy (*Processus Stochastiques et Mouvement Brownien,* 1948) and McKean (*Teor. Ver. i Prim.* **8**, 1963) on the domain of analyticity of some Gaussian random functions

Comment: More recent work of Cartier on covariances appeared in the L.~Schwartz volume*Mathematical Analysis and Applications, A*, Academic Press 1981

Keywords: Several parameter Brownian motions, Covariance

Nature: New exposition of known results

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VII: 09, 61-76, LNM 321 (1973)

**FARAUT, Jacques**

Fonction brownienne sur une variété riemannienne (Miscellanea, Gaussian processes)

As defined originally by Lévy in the case of spheres and euclidian spaces, a Brownian motion indexed by a point of a metric space $E$ is a centered Gaussian process $(X_t)_{t\in E}$ such that $E[(X_t-X_s)^2]=d(s,t)$, the distance. In a Riemannian manifold $d$ is understood to be the geodesic distance. The results of this paper imply that Brownian motions exist on spheres and Euclidean spaces (Lévy's original result), on real hyperbolic spaces, but not on quaternionic hyperbolic spaces

Comment: This article contains joint work with K. Harzallah

Keywords: Covariance, Riemannian manifold, Riemannian distance, Lévy Brownian motions, Several parameter Brownian motions

Nature: Original

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VII: 10, 77-94, LNM 321 (1973)

**HEINKEL, Bernard**

Une condition suffisante pour la continuité presque sûre des trajectoires de certains processus gaussiens (Gaussian processes)

It is shown that a continuity criterion due to Preston (1972) can be deduced from a theorem of Dudley (1967)

Comment: To be completed

Keywords: Continuity of paths of Gaussian processes

Nature: Original

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VIII: 12, 155-171, LNM 381 (1974)

**HEINKEL, Bernard**

Théorèmes de dérivation du type de Lebesgue et continuité presque sûre de certains processus gaussiens (Gaussian processes)

To be completed

Comment: To be completed

Keywords: Continuity of paths of Gaussian processes

Nature: Original

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IX: 14, 318-335, LNM 465 (1975)

**FERNIQUE, Xavier**

Des résultats nouveaux sur les processus gaussiens (Gaussian processes)

Given a centered Gaussian process indexed by an arbitrary set~$T$, a major problem has been to find conditions implying that the sample functions are a.s. bounded, or a.s. continuous in the natural metric associated with the covariance. Here new necessary conditions for boundedness are given, which turn out to be sufficient in the case of stationary processes on $**R**^n$. The conditions given here involve the existence of a majorizing measure, an idea which became crucial in the theory

Comment: For a systematic account of the theory around the time this paper was written, see Fernique's lectures in*École d'Été de Saint-Four~IV*, LNM **480**, 1974. For the definitive solution, see chapter 11 of Ledoux-Talagrand *Probability in Banach spaces,* Springer 1991

Keywords: Gaussian processes, Sample path regularity

Nature: Original

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XI: 13, 196-256, LNM 581 (1977)

**WEBER, Michel**

Classes uniformes de processus gaussiens stationnaires (Gaussian processes)

To be completed

Comment: See the long and interesting review by Berman in*Math. Reviews,* **56**, **13343**

Nature: Exposition, Original additions

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XII: 44, 567-690, LNM 649 (1978)

**NANOPOULOS, Constantin**; **NOBELIS, Photis**

Régularité et propriétés limites des fonctions aléatoires (Miscellanea, Gaussian processes)

This paper extends to the non-Gaussian case methods to study the regularity of sample paths which have proved useful in the Gaussian case, notably that of majorizing measures (to be completed)

Comment: To be completed

Keywords: Sample path regularity, Majorizing measures

Nature: Original

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XII: 45, 691-706, LNM 649 (1978)

**FERNIQUE, Xavier**

Caractérisation de processus à trajectoires majorées ou continues (Miscellanea, Gaussian processes)

The methods which lead the author to necessary and sufficient conditions for boundedness or continuity of stationary Gaussian processes are extended and applied to non-stationary Gaussian processes and non-Gaussian processes

Keywords: Sample path regularity

Nature: Original

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XIV: 01, 1-16, LNM 784 (1980)

**HEINKEL, Bernard**

Deux exemples d'utilisation de mesures majorantes (Gaussian processes)

From the introduction: ``our purpose is to study in detail two examples to which the method of majorizing measures, but not the entropy method, can be applied''

Nature: Original

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XIV: 02, 17-17, LNM 784 (1980)

**GINÉ, Evarist**

Corrections to ``Domains of attraction in Banach spaces'' (Gaussian processes)

Contains three minor corrections to 1303

Nature: Correction

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XIV: 46, 475-488, LNM 784 (1980)

**WEBER, Michel**

Sur un théorème de Maruyama (Gaussian processes, Ergodic theory)

Given a stationary centered Gaussian process $X$ with spectral measure $\mu$, a new proof is given of the fact that if $\mu$ is continuous, the flow of $X$ is weakly mixing

Keywords: Stationary processes

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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XLII: 13, 365-381, LNM 1978 (2009)

**ERRAOUI, M.**; **ESSAKY, E. H.**

Canonical representation for Gaussian processes (Theory of Gaussian processes)

Nature: Original

Introduction à l'étude des mouvements browniens à plusieurs paramètres (Gaussian processes)

Settles in particular a disagreement between statements of Lévy (

Comment: More recent work of Cartier on covariances appeared in the L.~Schwartz volume

Keywords: Several parameter Brownian motions, Covariance

Nature: New exposition of known results

Retrieve article from Numdam

VII: 09, 61-76, LNM 321 (1973)

Fonction brownienne sur une variété riemannienne (Miscellanea, Gaussian processes)

As defined originally by Lévy in the case of spheres and euclidian spaces, a Brownian motion indexed by a point of a metric space $E$ is a centered Gaussian process $(X_t)_{t\in E}$ such that $E[(X_t-X_s)^2]=d(s,t)$, the distance. In a Riemannian manifold $d$ is understood to be the geodesic distance. The results of this paper imply that Brownian motions exist on spheres and Euclidean spaces (Lévy's original result), on real hyperbolic spaces, but not on quaternionic hyperbolic spaces

Comment: This article contains joint work with K. Harzallah

Keywords: Covariance, Riemannian manifold, Riemannian distance, Lévy Brownian motions, Several parameter Brownian motions

Nature: Original

Retrieve article from Numdam

VII: 10, 77-94, LNM 321 (1973)

Une condition suffisante pour la continuité presque sûre des trajectoires de certains processus gaussiens (Gaussian processes)

It is shown that a continuity criterion due to Preston (1972) can be deduced from a theorem of Dudley (1967)

Comment: To be completed

Keywords: Continuity of paths of Gaussian processes

Nature: Original

Retrieve article from Numdam

VIII: 12, 155-171, LNM 381 (1974)

Théorèmes de dérivation du type de Lebesgue et continuité presque sûre de certains processus gaussiens (Gaussian processes)

To be completed

Comment: To be completed

Keywords: Continuity of paths of Gaussian processes

Nature: Original

Retrieve article from Numdam

IX: 14, 318-335, LNM 465 (1975)

Des résultats nouveaux sur les processus gaussiens (Gaussian processes)

Given a centered Gaussian process indexed by an arbitrary set~$T$, a major problem has been to find conditions implying that the sample functions are a.s. bounded, or a.s. continuous in the natural metric associated with the covariance. Here new necessary conditions for boundedness are given, which turn out to be sufficient in the case of stationary processes on $

Comment: For a systematic account of the theory around the time this paper was written, see Fernique's lectures in

Keywords: Gaussian processes, Sample path regularity

Nature: Original

Retrieve article from Numdam

XI: 13, 196-256, LNM 581 (1977)

Classes uniformes de processus gaussiens stationnaires (Gaussian processes)

To be completed

Comment: See the long and interesting review by Berman in

Nature: Exposition, Original additions

Retrieve article from Numdam

XII: 44, 567-690, LNM 649 (1978)

Régularité et propriétés limites des fonctions aléatoires (Miscellanea, Gaussian processes)

This paper extends to the non-Gaussian case methods to study the regularity of sample paths which have proved useful in the Gaussian case, notably that of majorizing measures (to be completed)

Comment: To be completed

Keywords: Sample path regularity, Majorizing measures

Nature: Original

Retrieve article from Numdam

XII: 45, 691-706, LNM 649 (1978)

Caractérisation de processus à trajectoires majorées ou continues (Miscellanea, Gaussian processes)

The methods which lead the author to necessary and sufficient conditions for boundedness or continuity of stationary Gaussian processes are extended and applied to non-stationary Gaussian processes and non-Gaussian processes

Keywords: Sample path regularity

Nature: Original

Retrieve article from Numdam

XIV: 01, 1-16, LNM 784 (1980)

Deux exemples d'utilisation de mesures majorantes (Gaussian processes)

From the introduction: ``our purpose is to study in detail two examples to which the method of majorizing measures, but not the entropy method, can be applied''

Nature: Original

Retrieve article from Numdam

XIV: 02, 17-17, LNM 784 (1980)

Corrections to ``Domains of attraction in Banach spaces'' (Gaussian processes)

Contains three minor corrections to 1303

Nature: Correction

Retrieve article from Numdam

XIV: 46, 475-488, LNM 784 (1980)

Sur un théorème de Maruyama (Gaussian processes, Ergodic theory)

Given a stationary centered Gaussian process $X$ with spectral measure $\mu$, a new proof is given of the fact that if $\mu$ is continuous, the flow of $X$ is weakly mixing

Keywords: Stationary processes

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

XLII: 13, 365-381, LNM 1978 (2009)

Canonical representation for Gaussian processes (Theory of Gaussian processes)

Nature: Original