Multiplicative functionals of dual processes

Ronald K. Getoor

Annales de l'institut Fourier (1971)

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

Abstract

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Let X and X ^ be a pair of dual standard Markov processes. We associate to each exact multiplicative function ( M F ) , M of X a unique exact M F , M ^ of X ^ in a natural manner. Any M F , M is assumed to satisfy 0 ≤ M t ≤ 1 . The map M → M ^ is bijective and multiplicative in the sense that: ( M N ) ∧ = M ^ 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

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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

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  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

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