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I: 05, 54-71, LNM 39 (1967)

**FERNIQUE, Xavier**

Séries de distributions aléatoires indépendantes (2 talks) (Miscellanea)

This is part of X.~Fernique's research on random distributions (probability measures on ${\cal D}'$, and more generally on the dual space $E'$ of a nuclear LF space $E$) and their characteristic functions, which are exactly, according to Minlos' theorem, the continuous positive definite functions on $E$ assuming the value $1$ at $0$. Here it is proved that a series of independent random distributions converges a.s. if and only if the product of their characteristic functions converges pointwise to a continuous limit, and converges a.s. after centering if and only if the product of absolute values converges

Comment: See for further results*Ann. Inst. Fourier,* **17-1**, 1967; *Invent. Math.*, **3**, 1967, and *C.R. Acad. Sc.*, **266**, 1968 for the extension of Lévy's continuity theorem (also presented at Séminaire Bourbaki, June 1966, **311**)

Keywords: Random distributions, Minlos theorem

Nature: Original

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IV: 02, 28-36, LNM 124 (1970)

**CARTIER, Pierre**

Sur certaines variables aléatoires associées au réarrangement croissant d'un échantillon (Miscellanea)

Presentation of combinatorial results due to D.~Foata and to Cartier-Foata, which are relevant for the theory of order statistics

Comment: See Cartier-Foata,*Problèmes combinatoires de commutation et réarrangements,* LN **85**, 1969

Keywords: Combinatorics, Generating functions, Order statistics

Nature: Exposition

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IV: 04, 47-59, LNM 124 (1970)

**DACUNHA-CASTELLE, Didier**

Principe de dualité pour des espaces de suites associés à une suite de variables aléatoires (Miscellanea)

The ``duality principe'' (too technical to be stated here) is a result of L.~Schwartz on Banach spaces of type (L), i.e., consisting of sequences $(b_n)$ such that $\sum_n b_nX_n$ is bounded in probability, where $(X_n)$ is a given sequence of r.v.'s

Comment: Related to the theory of radonifying maps, then in fast progress. To be completed

Keywords: Banach spaces, Radonifying maps

Nature: Exposition

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V: 01, 1-16, LNM 191 (1971)

**ARTZNER, Philippe**

Fonctions caractéristiques et mesures planes invariantes par rotation (Miscellanea)

A study of the class of probability measures on the line which are projections of a measure on the plane invariant by rotation

Keywords: Characteristic functions

Nature: Original

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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: 28, 301-318, LNM 321 (1973)

**WALDENFELS, Wilhelm von**

Some remarks on Burkhardt's model for pressure broadening of spectral lines (Miscellanea)

A model proposed by Burkhardt in 1940 for the deformation of the radiation emitted by an atom due to the surrounding atoms is transformed into probabilistic language and exactly solved

Nature: Original

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VIII: 08, 78-79, LNM 381 (1974)

**FERNIQUE, Xavier**

Une démonstration simple du théorème de R.M.~Dudley et M.~Kanter sur les lois 0-1 pour les mesures stables (Miscellanea)

The theorem concerns stable laws on a linear space, and asserts that every measurable linear subspace has probability 0 or 1. The title describes accurately the paper

Keywords: Stable measures

Nature: Original

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VIII: 20, 344-354, LNM 381 (1974)

**WALDENFELS, Wilhelm von**

Taylor expansion of a Poisson measure (Miscellanea)

To be completed

Comment: To be completed

Keywords: Poisson point processes

Nature: Original

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IX: 03, 154-205, LNM 465 (1975)

**NANOPOULOS, Photius**

Mesures d'information et représentation de semi-groupes associés (Miscellanea)

Several attempts have been made to define the information content of an event on a measurable space independently from its probability. This paper develops a point of view of Kampé de Fériet-Forte,*C.R.A.S. Paris,* **265A**, 1969

Keywords: Information theory

Nature: Original

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IX: 09, 246-267, LNM 465 (1975)

**DACUNHA-CASTELLE, Didier**

Processus et espaces de Banach invariants par réarrangement (Miscellanea)

The first topic of this paper is the class of processes on an interval with exchangeable increments, which includes all processes with independent and stationary increments, but also the standard Brownian bridge (for several results quoted here, see Kallenberg,*Z. für W-theorie,* **27**, 1973). The results are applied to problems in the classification of Banach spaces, in the spirit of Dacunha-Castelle-Schreiber, *Ann. Inst. H. Poincaré,* **10**, 1974

Keywords: Banach spaces, Exchangeable random variables

Nature: Original

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IX: 10, 268-284, LNM 465 (1975)

**DACUNHA-CASTELLE, Didier**

Sous-espaces symétriques des espaces d'Orlicz (Miscellanea)

Will be asked from the author

Comment: To be asked from the author. See Dacunha-Castelle-Schreiber,*Ann. Inst. H. Poincaré,* **10**, 1974

Keywords: Banach spaces

Nature: Original

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IX: 11, 285-293, LNM 465 (1975)

**ARTZNER, Philippe**

Quelques résultats de décomposabilité en algèbre linéaire et en algèbre quadratique aléatoires (Miscellanea)

It is shown that if a random quadratic form on $**R**^4$ is a.s. of rank 3 and has an absolutely continuous law, then its law is indecomposable. To prove this, it is shown that the sum of two independent planes in $**R**^4$ cannot at the same time be a.s. a hyperplane and have an absolutely continuous law (with respect to the natural measure for hyperplanes)

Keywords: Independent random subspaces, Independent quadratic forms

Nature: Original

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IX: 13, 305-317, LNM 465 (1975)

**FÖLLMER, Hans**

Phase transition and Martin boundary (Miscellanea)

To be completed

Comment: To be completed

Keywords: Random fields, Martin boundary

Nature: Original

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IX: 38, 565-588, LNM 465 (1975)

**WALDENFELS, Wilhelm von**

Interval partitions and pair interactions (Miscellanea)

To be completed

Comment: To be completed

Nature: Original

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X: 06, 78-85, LNM 511 (1976)

**MANDREKAR, Vidyadhar**

Germ-field Markov property for multiparameter processes (Miscellanea)

The paper studies the relations between several Markov properties of a process indexed by an open set of $**R**^n$

Comment: To be completed

Keywords: Several parameter Brownian motions, Several parameter processes

Nature: Original

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XI: 01, 1-20, LNM 581 (1977)

**AVANISSIAN, Vazgain**

Fonctions harmoniques d'ordre infini et l'harmonicité réelle liée à l'opérateur laplacien itéré (Potential theory, Miscellanea)

This paper studies two classes of functions in (an open set of) $**R**^n$, $n\ge1$: 1) Harmonic functions of infinite order (see Avanissian and Fernique, *Ann. Inst. Fourier,* **18-2**, 1968), which are $C^\infty$ functions satisfying a growth condition on their iterated laplacians, and are shown to be real analytic. 2) Infinitely differentiable functions (or distributions) similar to completely monotonic functions on the line, i.e., whose iterated laplacians are alternatively positive and negative (they were introduced by Lelong). Among the results is the fact that the second class is included in the first

Keywords: Harmonic functions, Real analytic functions, Completely monotonic functions

Nature: Original

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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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XIII: 44, 495-520, LNM 721 (1979)

**BISMUT, Jean-Michel**

Problèmes à frontière libre et arbres de mesures (Miscellanea, Markov processes)

An optimization problem is discussed, in which one is free to choose at any time among three different transition semi-groups

Keywords: Control theory

Nature: Original

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XIII: 46, 533-547, LNM 721 (1979)

**NANOPOULOS, Photius**

Mesures de probabilités sur les entiers et ensembles progressions (Miscellanea)

To be completed

Nature: Original

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XIII: 47, 548-556, LNM 721 (1979)

**FUJISAKI, Masatoshi**

On the uniqueness of optimal controls (Miscellanea)

``In section 1 we can give simple criteria for the uniqueness of the optimal controls whose existence is proved by Ikeda-Watanabe in the completely observable case,*Osaka Math. J.*, **14**, 1977. In section 2 we consider the same problem in the partially observable case.'' (From the author's summary)

Keywords: Control theory

Nature: Original

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XIII: 49, 570-573, LNM 721 (1979)

**CARMONA, René**

Opérateur de Schrödinger à résolvante compacte (Miscellanea)

A sufficient condition for a Schrödinger operator $(-1/2)ėlta+V$ to have a compact resolvent is proved, using standard properties of Brownian paths

Keywords: Schrödinger operators

Nature: Original

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XLVI: 10, 195-205, LNM 2123 (2014)

**BAILLEUL, Ismaël**

Flows driven by Banach space-valued rough paths (Miscellanea)

This article gives a quick proof of the existence of a solution to a rough differential equations by constructing directly its associated flow. Bound on the solutions, which are valid in an infinite dimensional setting, are also given

Keywords: rough paths

Nature: Original

Séries de distributions aléatoires indépendantes (2 talks) (Miscellanea)

This is part of X.~Fernique's research on random distributions (probability measures on ${\cal D}'$, and more generally on the dual space $E'$ of a nuclear LF space $E$) and their characteristic functions, which are exactly, according to Minlos' theorem, the continuous positive definite functions on $E$ assuming the value $1$ at $0$. Here it is proved that a series of independent random distributions converges a.s. if and only if the product of their characteristic functions converges pointwise to a continuous limit, and converges a.s. after centering if and only if the product of absolute values converges

Comment: See for further results

Keywords: Random distributions, Minlos theorem

Nature: Original

Retrieve article from Numdam

IV: 02, 28-36, LNM 124 (1970)

Sur certaines variables aléatoires associées au réarrangement croissant d'un échantillon (Miscellanea)

Presentation of combinatorial results due to D.~Foata and to Cartier-Foata, which are relevant for the theory of order statistics

Comment: See Cartier-Foata,

Keywords: Combinatorics, Generating functions, Order statistics

Nature: Exposition

Retrieve article from Numdam

IV: 04, 47-59, LNM 124 (1970)

Principe de dualité pour des espaces de suites associés à une suite de variables aléatoires (Miscellanea)

The ``duality principe'' (too technical to be stated here) is a result of L.~Schwartz on Banach spaces of type (L), i.e., consisting of sequences $(b_n)$ such that $\sum_n b_nX_n$ is bounded in probability, where $(X_n)$ is a given sequence of r.v.'s

Comment: Related to the theory of radonifying maps, then in fast progress. To be completed

Keywords: Banach spaces, Radonifying maps

Nature: Exposition

Retrieve article from Numdam

V: 01, 1-16, LNM 191 (1971)

Fonctions caractéristiques et mesures planes invariantes par rotation (Miscellanea)

A study of the class of probability measures on the line which are projections of a measure on the plane invariant by rotation

Keywords: Characteristic functions

Nature: Original

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: 28, 301-318, LNM 321 (1973)

Some remarks on Burkhardt's model for pressure broadening of spectral lines (Miscellanea)

A model proposed by Burkhardt in 1940 for the deformation of the radiation emitted by an atom due to the surrounding atoms is transformed into probabilistic language and exactly solved

Nature: Original

Retrieve article from Numdam

VIII: 08, 78-79, LNM 381 (1974)

Une démonstration simple du théorème de R.M.~Dudley et M.~Kanter sur les lois 0-1 pour les mesures stables (Miscellanea)

The theorem concerns stable laws on a linear space, and asserts that every measurable linear subspace has probability 0 or 1. The title describes accurately the paper

Keywords: Stable measures

Nature: Original

Retrieve article from Numdam

VIII: 20, 344-354, LNM 381 (1974)

Taylor expansion of a Poisson measure (Miscellanea)

To be completed

Comment: To be completed

Keywords: Poisson point processes

Nature: Original

Retrieve article from Numdam

IX: 03, 154-205, LNM 465 (1975)

Mesures d'information et représentation de semi-groupes associés (Miscellanea)

Several attempts have been made to define the information content of an event on a measurable space independently from its probability. This paper develops a point of view of Kampé de Fériet-Forte,

Keywords: Information theory

Nature: Original

Retrieve article from Numdam

IX: 09, 246-267, LNM 465 (1975)

Processus et espaces de Banach invariants par réarrangement (Miscellanea)

The first topic of this paper is the class of processes on an interval with exchangeable increments, which includes all processes with independent and stationary increments, but also the standard Brownian bridge (for several results quoted here, see Kallenberg,

Keywords: Banach spaces, Exchangeable random variables

Nature: Original

Retrieve article from Numdam

IX: 10, 268-284, LNM 465 (1975)

Sous-espaces symétriques des espaces d'Orlicz (Miscellanea)

Will be asked from the author

Comment: To be asked from the author. See Dacunha-Castelle-Schreiber,

Keywords: Banach spaces

Nature: Original

Retrieve article from Numdam

IX: 11, 285-293, LNM 465 (1975)

Quelques résultats de décomposabilité en algèbre linéaire et en algèbre quadratique aléatoires (Miscellanea)

It is shown that if a random quadratic form on $

Keywords: Independent random subspaces, Independent quadratic forms

Nature: Original

Retrieve article from Numdam

IX: 13, 305-317, LNM 465 (1975)

Phase transition and Martin boundary (Miscellanea)

To be completed

Comment: To be completed

Keywords: Random fields, Martin boundary

Nature: Original

Retrieve article from Numdam

IX: 38, 565-588, LNM 465 (1975)

Interval partitions and pair interactions (Miscellanea)

To be completed

Comment: To be completed

Nature: Original

Retrieve article from Numdam

X: 06, 78-85, LNM 511 (1976)

Germ-field Markov property for multiparameter processes (Miscellanea)

The paper studies the relations between several Markov properties of a process indexed by an open set of $

Comment: To be completed

Keywords: Several parameter Brownian motions, Several parameter processes

Nature: Original

Retrieve article from Numdam

XI: 01, 1-20, LNM 581 (1977)

Fonctions harmoniques d'ordre infini et l'harmonicité réelle liée à l'opérateur laplacien itéré (Potential theory, Miscellanea)

This paper studies two classes of functions in (an open set of) $

Keywords: Harmonic functions, Real analytic functions, Completely monotonic functions

Nature: Original

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

XIII: 44, 495-520, LNM 721 (1979)

Problèmes à frontière libre et arbres de mesures (Miscellanea, Markov processes)

An optimization problem is discussed, in which one is free to choose at any time among three different transition semi-groups

Keywords: Control theory

Nature: Original

Retrieve article from Numdam

XIII: 46, 533-547, LNM 721 (1979)

Mesures de probabilités sur les entiers et ensembles progressions (Miscellanea)

To be completed

Nature: Original

Retrieve article from Numdam

XIII: 47, 548-556, LNM 721 (1979)

On the uniqueness of optimal controls (Miscellanea)

``In section 1 we can give simple criteria for the uniqueness of the optimal controls whose existence is proved by Ikeda-Watanabe in the completely observable case,

Keywords: Control theory

Nature: Original

Retrieve article from Numdam

XIII: 49, 570-573, LNM 721 (1979)

Opérateur de Schrödinger à résolvante compacte (Miscellanea)

A sufficient condition for a Schrödinger operator $(-1/2)ėlta+V$ to have a compact resolvent is proved, using standard properties of Brownian paths

Keywords: Schrödinger operators

Nature: Original

Retrieve article from Numdam

XLVI: 10, 195-205, LNM 2123 (2014)

Flows driven by Banach space-valued rough paths (Miscellanea)

This article gives a quick proof of the existence of a solution to a rough differential equations by constructing directly its associated flow. Bound on the solutions, which are valid in an infinite dimensional setting, are also given

Keywords: rough paths

Nature: Original