VII: 25, 273-283, LNM 321 (1973)
PINSKY, Mark A.
Fonctionnelles multiplicatives opératrices (
Markov processes)
This paper presents results due to the author (
Advances in Probability, 3, 1973). The essential idea is to consider multiplicative functionals of a Markov process taking values in the algebra of bounded operators of a Banach space $L$. Such a functional defines a semi-group acting on bounded $L$-valued functions. This semi-group determines the functional. The structure of functionals is investigated in the case of a finite Markov chain. The case where $L$ is finite dimensional and the Markov process is Browian motion is investigated too. Asymptotic results near $0$ are described
Comment: This paper explores the same idea as Jacod (
Mém. Soc. Math. France, 35, 1973), though in a very different way. See
816Keywords: Multiplicative functionals,
Multiplicative kernelsNature: Exposition
Retrieve article from Numdam
VIII: 16, 290-309, LNM 381 (1974)
MEYER, Paul-André
Noyaux multiplicatifs (
Markov processes)
This paper presents results due to Jacod (
Mém. Soc. Math. France, 35, 1973): given a pair $(X,Y)$ which jointly is a Markov process, and whose first component $X$ is a Markov process by itself, describe the conditional distribution of the joint path of $(X,Y)$ over a given path of $X$. These distributions constitute a multiplicative kernel, and attempts are made to regularize it
Comment: Though the paper is fairly technical, it does not improve substantially on Jacod's results
Keywords: Multiplicative kernels,
Semimarkovian processesNature: Exposition Retrieve article from Numdam
IX: 23, 443-463, LNM 465 (1975)
GETOOR, Ronald K.
On the construction of kernels (
Measure theory)
Given two measurable spaces $(E, {\cal E})$ $(F, {\cal F})$ and a family ${\cal N}\subset{\cal E}$ of negligible sets, a pseudo-kernel $T$ is a mapping from bounded measurable functions on $F$ to classes mod.${\cal N}$ of bounded measurable functions on $E$, which has all a.e. the properties (positivity, countable additivity) of a kernel. Regularizing $T$ consists in finding a true kernel $\hat T$ such that $\hat Tf$ belongs to the class $Tf$ for every measurable bounded $f$ on $F$. The regularization is easy whenever $F$ is compact metric. Then the result is extended to the case of a Lusin space, and to the case of a U-space (Radon space) assuming ${\cal N}$ consists of the negligible sets for a family of measures on $E$. An application is given to densities of continuous additive functionals of a Markov process
Comment: The author states that his paper is purely expository. This is not true, though the proof is a standard one in the theory of conditional distributions. For a deeper result, see Dellacherie
1030. For a presentation in book form, see Dellacherie-Meyer,
Probabilités et Potentiel C, chapter XI
41Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
IX: 24, 464-465, LNM 465 (1975)
MEYER, Paul-André
Une remarque sur la construction de noyaux (
Measure theory)
With the notation of the preceding report
923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity
Comment: See Dellacherie
1030Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
X: 30, 545-577, LNM 511 (1976)
DELLACHERIE, Claude
Sur la construction de noyaux boréliens (
Measure theory)
This answers questions of Getoor
923 and Meyer
924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest
Comment: For a presentation in book form, see Dellacherie-Meyer,
Probabilités et Potentiel C, chapter XI
41. The hypothesis that the space is compact is sometimes troublesome for the applications
Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
XI: 15, 298-302, LNM 581 (1977)
ZANZOTTO, Pio Andrea
Sur l'existence d'un noyau induisant un opérateur sous markovien donné (
Measure theory)
The problem is whether a positive, norm-decreasing operator $L^\infty(\mu)\rightarrow L^\infty(\lambda)$ (of classes, not functions) is induced by a submarkov kernel. No ``countable additivity'' condition is assumed, but completeness of $\lambda$ and tightness of $\mu$
Comment: See
923,
924,
1030Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
XII: 35, 482-488, LNM 649 (1978)
YOR, Marc;
MEYER, Paul-André
Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (
Measure theory)
Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability
Comment: The subject is discussed further in
1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the
weak topology of $L[\infty$, or with the topology of $L^0$
Keywords: Kernels,
Radon-Nikodym theoremNature: Original Retrieve article from Numdam
XV: 27, 371-387, LNM 850 (1981)
DELLACHERIE, Claude
Sur les noyaux $\sigma$-finis (
Measure theory)
This paper is an improvement of
1235. Assume $(X,{\cal X})$ and $(Y,{\cal Y})$ are measurable spaces and $m(x,A)$ is a kernel, i.e., is measurable in $x\in X$ for $A\in{\cal Y}$, and is a $\sigma$-finite measure in $A$ for $x\in X$. Then the problem is to represent the measures $m(x,dy)$ as $g(x,y)\,N(x,dy)$ where $g$ is a jointly measurable function and $N$ is a Markov kernel---possibly enlarging the $\sigma$-field ${\cal X}$ to include analytic sets. The crucial hypothesis (called
measurability of $m$) is the following: for every auxiliary space $(Z, {\cal Z})$, the mapping $(x,z)\mapsto m_x\otimes \epsilon_z$ is again a kernel (in fact, the auxiliary space $
R$ is all one needs). The case of ``basic'' kernels, considered in
1235, is thoroughly discussed
Keywords: Kernels,
Radon-Nikodym theoremNature: Original Retrieve article from Numdam
