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
302Keywords: Recurrent sets,
Fine topologyNature: Original
Retrieve article from Numdam
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 semigroupsNature: Original Retrieve article from Numdam
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
623Keywords: Additive functionals,
PerfectionNature: Original Retrieve article from Numdam
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 equationsNature: Exposition Retrieve article from Numdam
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 theoryNature: Exposition Retrieve article from Numdam
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 functionsNature: Exposition Retrieve article from Numdam
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 integralsNature: Exposition Retrieve article from Numdam
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 theoremsNature: Exposition Retrieve article from Numdam
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 processNature: 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 propertiesNature: Exposition Retrieve article from Numdam
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 theoryNature: Original Retrieve article from Numdam
