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I: 01, 3-17, LNM 39 (1967)

**AVANISSIAN, Vazgain**

Sur l'harmonicité des fonctions séparément harmoniques (Potential theory)

This paper proves a harmonic version of Hartogs' theorem: separately harmonic functions are jointly harmonic (without any boundedness assumption) using a complex extension procedure. The talk is an extract from the author's original work in*Ann. ENS,* **178**, 1961

Comment: This talk was justified by the current interest of the seminar in doubly excessive functions, see Cairoli 102 in the same volume

Keywords: Doubly harmonic functions

Nature: Exposition

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I: 02, 18-33, LNM 39 (1967)

**CAIROLI, Renzo**

Semi-groupes de transition et fonctions excessives (Markov processes, Potential theory)

A study of product kernels, product semi-groups and product Markov processes

Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes

Keywords: Product semigroups

Nature: Original

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I: 03, 34-51, LNM 39 (1967)

**COURRÈGE, Philippe**

Noyaux de convolution singuliers opérant sur les fonctions höldériennes et noyaux de convolution régularisants (Potential theory)

The Poisson equation $ėlta\,(Uf)=-f$, where $U$ is the Newtonian potential is proved to be true in the strictest sense when $f$ is a Hölder function (while it is not for mere continuous functions). This involves an exposition of singular integral kernels on Hölder spaces

Comment: This talk was a by-product of the extensive work of Courrège, Bony and Priouret on Feller semi-groups on manifolds with boundary (*Ann. Inst. Fourier,* **16**, 1968)

Keywords: Newtonian potential

Nature: Exposition

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I: 04, 52-53, LNM 39 (1967)

**DELLACHERIE, Claude**

Un complément au théorème de Weierstrass-Stone (Functional analysis)

An easy but useful remark on the relation between the ``lattice'' and ``algebra'' forms of Stone's theorem, which apparently belongs to the folklore

Keywords: Stone-Weierstrass theorem

Nature: Original

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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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I: 06, 72-162, LNM 39 (1967)

**MEYER, Paul-André**

Intégrales stochastiques I--IV (4 talks) (Martingale theory, Stochastic calculus)

This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (*Nagoya Math. J.* **30**, 1967) on square integrable martingales. The filtration is assumed to be free from fixed times of discontinuity, a restriction lifted in the modern theory. A new feature is the definition of the second increasing process associated with a square integrable martingale (a ``square bracket'' in the modern terminology). In the second talk, stochastic integrals are defined with respect to local martingales (introduced from Ito-Watanabe, *Ann. Inst. Fourier,* **15**, 1965), and the general integration by parts formula is proved. Also a restricted class of semimartingales is defined and an ``Ito formula'' for change of variables is given, different from that of Kunita-Watanabe. The third talk contains the famous Kunita-Watanabe theorem giving the structure of martingale additive functionals of a Hunt process, and a new proof of Lévy's description of the structure of processes with independent increments (in the time homogeneous case). The fourth talk deals mostly with Lévy systems (Motoo-Watanabe, *J. Math. Kyoto Univ.*, **4**, 1965; Watanabe, *Japanese J. Math.*, **36**, 1964)

Comment: This paper was a step in the development of stochastic integration. Practically every detail of it has been reworked since, starting with Doléans-Dade-Meyer 409. Note a few corrections in Meyer 312

Keywords: Square integrable martingales, Angle bracket, Stochastic integrals

Nature: Exposition, Original additions

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I: 07, 163-165, LNM 39 (1967)

**MEYER, Paul-André**

Sur un théorème de Deny (Potential theory, Measure theory)

In the potential theory of a resolvent which satisfies the absolute continuity hypothesis, every sequence of excessive functions contains a subsequence which converges except on a set of potential zero. It is also proved that a sequence which converges weakly in $L^1$ but not strongly must oscillate around its limit

Comment: a version of this result in classical potential theory was proved by Deny,*C.R. Acad. Sci.*, **218**, 1944. The cone of excessive functions possesses good compactness properties, discovered by Mokobodzki. See Dellacherie-Meyer, *Probabilités et Potentiel,* end of chapter XII

Keywords: A.e. convergence, Subsequences

Nature: Original

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I: 08, 166-176, LNM 39 (1967)

**WEIL, Michel**

Retournement du temps dans les processus markoviens (Markov processes)

This talk presents the now classical results of Nagasawa (*Nagoya Math. J.*, **24**, 1964) extending to continuous time the results proved by Hunt in discrete time on time reversal of a Markov process at an ``L-time'' or return time

Comment: See also 202. These results have been essentially the best ones until they were extended to Kuznetsov measures, see Dellacherie-Meyer,*Probabilités et Potentiel,* chapter XIX **14**

Keywords: Time reversal, Dual semigroups

Nature: Exposition

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I: 09, 177-189, LNM 39 (1967)

**WEIL, Michel**

Résolvantes en dualité (Markov processes, Potential theory)

Given two sub-Markov resolvents in duality, whose kernels are absolutely continuous with respect to a given measure, it is shown how to choose their densities to get true Green kernels, excessive in one variable and coexcessive in the other one. It is shown also that coexcessive functions are exactly the densities of excessive measures

Comment: These results, now classical, are due to Kunita-T. Watanabe,*Ill. J. Math.*, **9**, 1965 and *J. Math. Mech.*, **15**, 1966

Keywords: Green potentials, Dual semigroups

Nature: Exposition

Retrieve article from Numdam

Sur l'harmonicité des fonctions séparément harmoniques (Potential theory)

This paper proves a harmonic version of Hartogs' theorem: separately harmonic functions are jointly harmonic (without any boundedness assumption) using a complex extension procedure. The talk is an extract from the author's original work in

Comment: This talk was justified by the current interest of the seminar in doubly excessive functions, see Cairoli 102 in the same volume

Keywords: Doubly harmonic functions

Nature: Exposition

Retrieve article from Numdam

I: 02, 18-33, LNM 39 (1967)

Semi-groupes de transition et fonctions excessives (Markov processes, Potential theory)

A study of product kernels, product semi-groups and product Markov processes

Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes

Keywords: Product semigroups

Nature: Original

Retrieve article from Numdam

I: 03, 34-51, LNM 39 (1967)

Noyaux de convolution singuliers opérant sur les fonctions höldériennes et noyaux de convolution régularisants (Potential theory)

The Poisson equation $ėlta\,(Uf)=-f$, where $U$ is the Newtonian potential is proved to be true in the strictest sense when $f$ is a Hölder function (while it is not for mere continuous functions). This involves an exposition of singular integral kernels on Hölder spaces

Comment: This talk was a by-product of the extensive work of Courrège, Bony and Priouret on Feller semi-groups on manifolds with boundary (

Keywords: Newtonian potential

Nature: Exposition

Retrieve article from Numdam

I: 04, 52-53, LNM 39 (1967)

Un complément au théorème de Weierstrass-Stone (Functional analysis)

An easy but useful remark on the relation between the ``lattice'' and ``algebra'' forms of Stone's theorem, which apparently belongs to the folklore

Keywords: Stone-Weierstrass theorem

Nature: Original

Retrieve article from Numdam

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

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

I: 06, 72-162, LNM 39 (1967)

Intégrales stochastiques I--IV (4 talks) (Martingale theory, Stochastic calculus)

This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (

Comment: This paper was a step in the development of stochastic integration. Practically every detail of it has been reworked since, starting with Doléans-Dade-Meyer 409. Note a few corrections in Meyer 312

Keywords: Square integrable martingales, Angle bracket, Stochastic integrals

Nature: Exposition, Original additions

Retrieve article from Numdam

I: 07, 163-165, LNM 39 (1967)

Sur un théorème de Deny (Potential theory, Measure theory)

In the potential theory of a resolvent which satisfies the absolute continuity hypothesis, every sequence of excessive functions contains a subsequence which converges except on a set of potential zero. It is also proved that a sequence which converges weakly in $L^1$ but not strongly must oscillate around its limit

Comment: a version of this result in classical potential theory was proved by Deny,

Keywords: A.e. convergence, Subsequences

Nature: Original

Retrieve article from Numdam

I: 08, 166-176, LNM 39 (1967)

Retournement du temps dans les processus markoviens (Markov processes)

This talk presents the now classical results of Nagasawa (

Comment: See also 202. These results have been essentially the best ones until they were extended to Kuznetsov measures, see Dellacherie-Meyer,

Keywords: Time reversal, Dual semigroups

Nature: Exposition

Retrieve article from Numdam

I: 09, 177-189, LNM 39 (1967)

Résolvantes en dualité (Markov processes, Potential theory)

Given two sub-Markov resolvents in duality, whose kernels are absolutely continuous with respect to a given measure, it is shown how to choose their densities to get true Green kernels, excessive in one variable and coexcessive in the other one. It is shown also that coexcessive functions are exactly the densities of excessive measures

Comment: These results, now classical, are due to Kunita-T. Watanabe,

Keywords: Green potentials, Dual semigroups

Nature: Exposition

Retrieve article from Numdam