# Multiplicative functionals of dual processes

• Volume: 21, Issue: 2, page 43-83
• ISSN: 0373-0956

top

## Abstract

top
Let $X$ and $\stackrel{^}{X}$ be a pair of dual standard Markov processes. We associate to each exact multiplicative function $\left(MF\right)$, $M$ of $X$ a unique exact $MF$, $\stackrel{^}{M}$ of $\stackrel{^}{X}$ in a natural manner. Any $MF$, $M$ is assumed to satisfy $0\le {M}_{t}\le 1$. The map $M\to \stackrel{^}{M}$ is bijective and multiplicative in the sense that: $\left(MN{\right)}^{\wedge }=\stackrel{^}{M}\stackrel{^}{N}$.This correspondence is studied in some detail and several important examples are discussed.These results are then applied to study additive functionals.

## How to cite

top

Getoor, Ronald K.. "Multiplicative functionals of dual processes." Annales de l'institut Fourier 21.2 (1971): 43-83. <http://eudml.org/doc/74039>.

@article{Getoor1971,
abstract = {Let $X$ and $\widehat\{X\}$ be a pair of dual standard Markov processes. We associate to each exact multiplicative function $(MF)$, $M$ of $X$ a unique exact $MF$, $\widehat\{M\}$ of $\widehat\{X\}$ in a natural manner. Any $MF$, $M$ is assumed to satisfy $0\le M_t \le 1$. The map $M \rightarrow \widehat\{M\}$ is bijective and multiplicative in the sense that: $(MN) ^\wedge = \widehat\{M\} \widehat\{N\}$.This correspondence is studied in some detail and several important examples are discussed.These results are then applied to study additive functionals.},
author = {Getoor, Ronald K.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {43-83},
publisher = {Association des Annales de l'Institut Fourier},
title = {Multiplicative functionals of dual processes},
url = {http://eudml.org/doc/74039},
volume = {21},
year = {1971},
}

TY - JOUR
AU - Getoor, Ronald K.
TI - Multiplicative functionals of dual processes
JO - Annales de l'institut Fourier
PY - 1971
PB - Association des Annales de l'Institut Fourier
VL - 21
IS - 2
SP - 43
EP - 83
AB - Let $X$ and $\widehat{X}$ be a pair of dual standard Markov processes. We associate to each exact multiplicative function $(MF)$, $M$ of $X$ a unique exact $MF$, $\widehat{M}$ of $\widehat{X}$ in a natural manner. Any $MF$, $M$ is assumed to satisfy $0\le M_t \le 1$. The map $M \rightarrow \widehat{M}$ is bijective and multiplicative in the sense that: $(MN) ^\wedge = \widehat{M} \widehat{N}$.This correspondence is studied in some detail and several important examples are discussed.These results are then applied to study additive functionals.
LA - eng
UR - http://eudml.org/doc/74039
ER -

## References

top
1. [1] R. M. BLUMENTHAL and R. K. GETOOR, Markov Processes and Potential Theory, Academic Press, New York and London, (1968). Zbl0169.49204MR41 #9348
2. [2] R. M. BLUMENTHAL and R. K. GETOOR, Additive functionals of Markov processes in duality, Trans. Amer. Math. Soc. 112, 131-163 (1964). Zbl0133.40904MR28 #3483
3. [3] R. K. GETOOR, Duality of multiplicative functionals, Bull. Amer. Math. Soc. 76, 1053-1056 (1970). Zbl0234.60088
4. [4] G. A. HUNT, Markoff processes and potentials III, III, Ill. J. Math. 2, 151-213 (1958). Zbl0100.13804MR21 #5824
5. [5] P. A. MEYER, Probability and Potentials, Ginn (Blaisdell). Boston. 1966. Zbl0138.10401MR34 #5119
6. [6] P. A. MEYER, Semi-groupes en dualité, Séminaire de théorie du potentiel (Sem. Brelot, Choquet, Deny). Paris, 5th year. 1960/1961.
7. [7] P. A. MEYER, Quelques résultats sur les processus, Invent. Math. 1, 101-115 (1966). Zbl0178.53403MR34 #862
8. [8] P. A. MEYER, Intégrales stochastiques IV, Séminaire de Probabilités I, Lecture Notes in Math. 39. Springer-Verlag. 1967. Zbl0157.25001MR37 #7000
9. [9] D. REVUZ, Mesures associées aux fonctionnelles additives de Markov, Trans. Amer. Math. Soc. 148, 501-531 (1970). Zbl0266.60053MR43 #5611
10. [10] M. WEIL, Propriétés de continuité fine des fonctions coexcessives. Zeit. f. Wahrscheinlichkeitstheorie, 12, 75-86 (1969). Zbl0165.52502MR41 #1122

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.