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8 matches found
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 816
Keywords: Multiplicative functionals, Multiplicative kernels
Nature: Exposition
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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 processes
Nature: Exposition
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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 41
Keywords: Pseudo-kernels, Regularization
Nature: Original
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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 1030
Keywords: Pseudo-kernels, Regularization
Nature: Original
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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, Regularization
Nature: Original
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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, 1030
Keywords: Pseudo-kernels, Regularization
Nature: Original
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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 theorem
Nature: Original
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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 theorem
Nature: Original
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