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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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XII: 27, 398-410, LNM 649 (1978)

**GETOOR, Ronald K.**

Homogeneous potentials (General theory of processes)

This is a development in Knight's prediction theory as described in 1007, 1008. Let $(Z_t^\mu)$ be the prediction process associated with a given measure $\mu$. Then it is shown that a bounded homogeneous right continuous supermartingale (or potential) under $\mu$ remains so under the measures $Z_t^\mu$

Keywords: Prediction theory

Nature: Original

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XIV: 42, 397-409, LNM 784 (1980)

**GETOOR, Ronald K.**

Transience and recurrence of Markov processes (Markov processes)

From the introduction: The purpose of this paper is to present an elementary exposition of some various conditions that have been used to define transience or recurrence of a Markov process... an elementary and unified discussion of these ideas may be worthwhile

Keywords: Recurrent Markov processes

Nature: Exposition, Original additions

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XXVI: 34, 485-497, LNM 1526 (1992)

**FITZSIMMONS, Patrick J.**; **GETOOR, Ronald K.**

Some applications of quasi-boundedness for excessive measures

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XLII: 09, 229-259, LNM 1978 (2009)

**FITZSIMMONS, P. J.**; **GETOOR, R. K.**

Lévy systems and time changes (Theory of Lévy processes)

Nature: Original

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,

Keywords: Pseudo-kernels, Regularization

Nature: Original

Retrieve article from Numdam

XII: 27, 398-410, LNM 649 (1978)

Homogeneous potentials (General theory of processes)

This is a development in Knight's prediction theory as described in 1007, 1008. Let $(Z_t^\mu)$ be the prediction process associated with a given measure $\mu$. Then it is shown that a bounded homogeneous right continuous supermartingale (or potential) under $\mu$ remains so under the measures $Z_t^\mu$

Keywords: Prediction theory

Nature: Original

Retrieve article from Numdam

XIV: 42, 397-409, LNM 784 (1980)

Transience and recurrence of Markov processes (Markov processes)

From the introduction: The purpose of this paper is to present an elementary exposition of some various conditions that have been used to define transience or recurrence of a Markov process... an elementary and unified discussion of these ideas may be worthwhile

Keywords: Recurrent Markov processes

Nature: Exposition, Original additions

Retrieve article from Numdam

XXVI: 34, 485-497, LNM 1526 (1992)

Some applications of quasi-boundedness for excessive measures

Retrieve article from Numdam

XLII: 09, 229-259, LNM 1978 (2009)

Lévy systems and time changes (Theory of Lévy processes)

Nature: Original