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II: 01, 1-21, LNM 51 (1968)

**AZÉMA, Jacques**; **DUFLO, Marie**; **REVUZ, Daniel**

Classes récurrentes d'un processus de Markov (Markov processes)

This is an improved version of a paper by the same authors (*Ann. Inst. H. Poincaré,* **2**, 1966). Its aim is a theory of recurrence in continuous time (for a Hunt process). The main point is to use the finely open sets instead of the ordinary ones to define recurrence

Comment: The subject is further investigated by the same authors in 302

Keywords: Recurrent sets, Fine topology

Nature: Original

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II: 02, 22-33, LNM 51 (1968)

**CARTIER, Pierre**; **MEYER, Paul-André**; **WEIL, Michel**

Le retournement du temps~: compléments à l'exposé de M.~Weil (Markov processes)

In 108, M.~Weil had presented the work of Nagasawa on the time reversal of a Markov process at a ``L-time'' or return time. Here the results are improved on three points: a Markovian filtration is given for the reversed process; an analytic condition on the semigroup is lifted; finally, the behaviour of the*coexcessive * functions on the sample functions of the original process is investigated

Comment: The results of this paper have become part of the standard theory of time reversal. See 312 for a correction

Keywords: Time reversal, Dual semigroups

Nature: Original

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II: 03, 34-42, LNM 51 (1968)

**DOLÉANS-DADE, Catherine**

Fonctionnelles additives parfaites (Markov processes)

The identity defining additive (or multiplicative) functionals involves an exceptional set depending on a continuous time $t$. If the exceptional set can be chosen independently of $t$, the functional is perfect. It is shown that every additive functional of a Hunt process admitting a reference measure has a perfect version

Comment: The existence of a reference measure was lifted by Dellacherie in 304. However, the whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

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II: 04, 43-74, LNM 51 (1968)

**DOLÉANS-DADE, Catherine**

Espaces $H^m$ sur les variétés, et applications aux équations aux dérivées partielles sur une variété compacte (Functional analysis)

An attempt to teach to the members of the seminar the basic facts of the analytic theory of diffusion processes

Keywords: Sobolev spaces, Second order elliptic equations

Nature: Exposition

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II: 05, 75-110, LNM 51 (1968)

**GIROUX, Gaston**

Théorie des frontières dans les chaînes de Markov (Markov processes)

A presentation of the theory of Markov chains under the hypothesis that all states are regular

Comment: This is the subject of the short monograph of Chung,*Lectures on Boundary Theory for Markov Chains,* Princeton 1970

Keywords: Markov chains, Boundary theory

Nature: Exposition

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II: 06, 111-122, LNM 51 (1968)

**IGOT, Jean-Pierre**

Un théorème de Linnik (Independence)

The result proved (which implies that of Linnik mentioned in the title) is due to Ostrovskii (*Uspehi Mat. Nauk,* **20**, 1965) and states that the convolution of a normal and a Poisson law decomposes only into factors of the same type

Keywords: Infinitely divisible laws, Characteristic functions

Nature: Exposition

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II: 07, 123-139, LNM 51 (1968)

**SAM LAZARO, José de**

Sur les moments spectraux d'ordre supérieur (Second order processes)

The essential result of the paper (Shiryaev,*Th. Prob. Appl.*, **5**, 1960; Sinai, *Th. Prob. Appl.*, **8**, 1963) is the definition of multiple stochastic integrals with respect to a second order process whose covariance satisfies suitable spectral properties

Keywords: Spectral representation, Multiple stochastic integrals

Nature: Exposition

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II: 08, 140-165, LNM 51 (1968)

**MEYER, Paul-André**

Guide détaillé de la théorie ``générale'' des processus (General theory of processes)

This paper states and comments the essential results of a theory which was considered difficult in those times. New terminology was introduced (for instance, the accessible and previsible $\sigma$-fields) though not quite the definitive one (the word ``optional'' only timidly appears instead of the awkward ``well-measurable''). A few new results on the $\sigma$-fields ${\cal F}_{T-}$ and increasing processes are given at the end, the only ones to be proved

Comment: This paper had pedagogical importance in its time, but is now obsolete

Keywords: Previsible processes, Section theorems

Nature: Exposition

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II: 09, 166-170, LNM 51 (1968)

**MEYER, Paul-André**

Une majoration du processus croissant associé à une surmartingale (Martingale theory)

Let $(X_t)$ be the potential generated by a previsible increasing process $(A_t)$. Then a norm equivalence in $L^p,\ 1<p<\infty$ is given between the random variables $X^\ast$ and $A_\infty$

Comment: This paper became obsolete after the $H^1$-$BMO$ theory

Keywords: Inequalities, Potential of an increasing process

Nature: Original

Retrieve article from Numdam

II: 10, 171-174, LNM 51 (1968)

**MEYER, Paul-André**

Les résolvantes fortement fellériennes d'après Mokobodzki (Potential theory)

On a compact space, a submarkov kernel $N$ has the strong Feller property if it maps Borel bounded functions into continuous functions, and the stronger Feller property if the mapping $x\rightarrow \epsilon_x N$ is continuous in the norm topology of measures. It is proved that the product of two strong Feller kernels is stronger Feller, and as a consequence if the kernels of are resolvent are strong Feller they are automatically stronger Feller

Comment: This follows from a result on weakly compact operators on continuous functions due to Grothendieck (*Canadian J. Math.*, **5**, 1953). Mokobodzki's proof is less general (it uses positivity) but very simple. This result is rather useful

Keywords: Resolvents, Strong Feller properties

Nature: Exposition

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II: 11, 175-199, LNM 51 (1968)

**MEYER, Paul-André**

Compactifications associées à une résolvante (Potential theory)

Let $E$ be a locally compact space, $(U_p)$ be a submarkovian resolvent, with a potential kernel $U=U_0$ which maps $C_k$ (the continuous functions with compact support) into continuous bounded functions. Let $F$ be a compact space containing $E$ as a dense subset, but inducing possibly a coarser topology. It is assumed that all potentials $Uf$ with $f\in C_k$ extend to continuous functions on $F$, and that points of $F$ are separated by continuous functions on $F$ whose restriction to $E$ is supermedian. Then it is shown how to extend the resolvent to $F$ and imitate the construction of a Ray semigroup and a strong Markov process. This was an attempt to compactify the space using only supermedian functions, not $p$-supermedian for all $p>0$. An application to Markov chains is given

Comment: This method of compactification suggested by Chung's boundary theory for Markov chains (similarly Doob,*Trans. Amer. Math. Soc.*, **149**, 1970) never superseded the standard Ray-Knight approach

Keywords: Resolvents, Ray compactification, Martin boundary, Boundary theory

Nature: Original

Retrieve article from Numdam

Classes récurrentes d'un processus de Markov (Markov processes)

This is an improved version of a paper by the same authors (

Comment: The subject is further investigated by the same authors in 302

Keywords: Recurrent sets, Fine topology

Nature: Original

Retrieve article from Numdam

II: 02, 22-33, LNM 51 (1968)

Le retournement du temps~: compléments à l'exposé de M.~Weil (Markov processes)

In 108, M.~Weil had presented the work of Nagasawa on the time reversal of a Markov process at a ``L-time'' or return time. Here the results are improved on three points: a Markovian filtration is given for the reversed process; an analytic condition on the semigroup is lifted; finally, the behaviour of the

Comment: The results of this paper have become part of the standard theory of time reversal. See 312 for a correction

Keywords: Time reversal, Dual semigroups

Nature: Original

Retrieve article from Numdam

II: 03, 34-42, LNM 51 (1968)

Fonctionnelles additives parfaites (Markov processes)

The identity defining additive (or multiplicative) functionals involves an exceptional set depending on a continuous time $t$. If the exceptional set can be chosen independently of $t$, the functional is perfect. It is shown that every additive functional of a Hunt process admitting a reference measure has a perfect version

Comment: The existence of a reference measure was lifted by Dellacherie in 304. However, the whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see 623

Keywords: Additive functionals, Perfection

Nature: Original

Retrieve article from Numdam

II: 04, 43-74, LNM 51 (1968)

Espaces $H^m$ sur les variétés, et applications aux équations aux dérivées partielles sur une variété compacte (Functional analysis)

An attempt to teach to the members of the seminar the basic facts of the analytic theory of diffusion processes

Keywords: Sobolev spaces, Second order elliptic equations

Nature: Exposition

Retrieve article from Numdam

II: 05, 75-110, LNM 51 (1968)

Théorie des frontières dans les chaînes de Markov (Markov processes)

A presentation of the theory of Markov chains under the hypothesis that all states are regular

Comment: This is the subject of the short monograph of Chung,

Keywords: Markov chains, Boundary theory

Nature: Exposition

Retrieve article from Numdam

II: 06, 111-122, LNM 51 (1968)

Un théorème de Linnik (Independence)

The result proved (which implies that of Linnik mentioned in the title) is due to Ostrovskii (

Keywords: Infinitely divisible laws, Characteristic functions

Nature: Exposition

Retrieve article from Numdam

II: 07, 123-139, LNM 51 (1968)

Sur les moments spectraux d'ordre supérieur (Second order processes)

The essential result of the paper (Shiryaev,

Keywords: Spectral representation, Multiple stochastic integrals

Nature: Exposition

Retrieve article from Numdam

II: 08, 140-165, LNM 51 (1968)

Guide détaillé de la théorie ``générale'' des processus (General theory of processes)

This paper states and comments the essential results of a theory which was considered difficult in those times. New terminology was introduced (for instance, the accessible and previsible $\sigma$-fields) though not quite the definitive one (the word ``optional'' only timidly appears instead of the awkward ``well-measurable''). A few new results on the $\sigma$-fields ${\cal F}_{T-}$ and increasing processes are given at the end, the only ones to be proved

Comment: This paper had pedagogical importance in its time, but is now obsolete

Keywords: Previsible processes, Section theorems

Nature: Exposition

Retrieve article from Numdam

II: 09, 166-170, LNM 51 (1968)

Une majoration du processus croissant associé à une surmartingale (Martingale theory)

Let $(X_t)$ be the potential generated by a previsible increasing process $(A_t)$. Then a norm equivalence in $L^p,\ 1<p<\infty$ is given between the random variables $X^\ast$ and $A_\infty$

Comment: This paper became obsolete after the $H^1$-$BMO$ theory

Keywords: Inequalities, Potential of an increasing process

Nature: Original

Retrieve article from Numdam

II: 10, 171-174, LNM 51 (1968)

Les résolvantes fortement fellériennes d'après Mokobodzki (Potential theory)

On a compact space, a submarkov kernel $N$ has the strong Feller property if it maps Borel bounded functions into continuous functions, and the stronger Feller property if the mapping $x\rightarrow \epsilon_x N$ is continuous in the norm topology of measures. It is proved that the product of two strong Feller kernels is stronger Feller, and as a consequence if the kernels of are resolvent are strong Feller they are automatically stronger Feller

Comment: This follows from a result on weakly compact operators on continuous functions due to Grothendieck (

Keywords: Resolvents, Strong Feller properties

Nature: Exposition

Retrieve article from Numdam

II: 11, 175-199, LNM 51 (1968)

Compactifications associées à une résolvante (Potential theory)

Let $E$ be a locally compact space, $(U_p)$ be a submarkovian resolvent, with a potential kernel $U=U_0$ which maps $C_k$ (the continuous functions with compact support) into continuous bounded functions. Let $F$ be a compact space containing $E$ as a dense subset, but inducing possibly a coarser topology. It is assumed that all potentials $Uf$ with $f\in C_k$ extend to continuous functions on $F$, and that points of $F$ are separated by continuous functions on $F$ whose restriction to $E$ is supermedian. Then it is shown how to extend the resolvent to $F$ and imitate the construction of a Ray semigroup and a strong Markov process. This was an attempt to compactify the space using only supermedian functions, not $p$-supermedian for all $p>0$. An application to Markov chains is given

Comment: This method of compactification suggested by Chung's boundary theory for Markov chains (similarly Doob,

Keywords: Resolvents, Ray compactification, Martin boundary, Boundary theory

Nature: Original

Retrieve article from Numdam