The Ray space of a right process

Ronald K. Getoor; Michael J. Sharpe

Annales de l'institut Fourier (1975)

  • Volume: 25, Issue: 3-4, page 207-233
  • ISSN: 0373-0956

Abstract

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Let X be a process with state space E satisfying (a somewhat relaxed version of) Meyer’s “hypothèses droites”. Then by introducing a new topology (called the Ray topology) on E and a compactification F of E in the Ray topology one can regard X as a Ray process. However, this construction depends on the choice of an arbitrary uniformity on E and not just the topology of E . We show that the Ray topology is independent of the choice of this uniformity. We then introduce a space R (the Ray space) which contains E in the Ray topology and which has all of the properties of F which are relevant for the study of X . Although R is not compact it is independent of the choice of the original uniformity on E .

How to cite

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Getoor, Ronald K., and Sharpe, Michael J.. "The Ray space of a right process." Annales de l'institut Fourier 25.3-4 (1975): 207-233. <http://eudml.org/doc/74243>.

@article{Getoor1975,
abstract = {Let $X$ be a process with state space $E$ satisfying (a somewhat relaxed version of) Meyer’s “hypothèses droites”. Then by introducing a new topology (called the Ray topology) on $E$ and a compactification $F$ of $E$ in the Ray topology one can regard $X$ as a Ray process. However, this construction depends on the choice of an arbitrary uniformity on $E$ and not just the topology of $E$. We show that the Ray topology is independent of the choice of this uniformity. We then introduce a space $R$ (the Ray space) which contains $E$ in the Ray topology and which has all of the properties of $F$ which are relevant for the study of $X$. Although $R$ is not compact it is independent of the choice of the original uniformity on $E$.},
author = {Getoor, Ronald K., Sharpe, Michael J.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3-4},
pages = {207-233},
publisher = {Association des Annales de l'Institut Fourier},
title = {The Ray space of a right process},
url = {http://eudml.org/doc/74243},
volume = {25},
year = {1975},
}

TY - JOUR
AU - Getoor, Ronald K.
AU - Sharpe, Michael J.
TI - The Ray space of a right process
JO - Annales de l'institut Fourier
PY - 1975
PB - Association des Annales de l'Institut Fourier
VL - 25
IS - 3-4
SP - 207
EP - 233
AB - Let $X$ be a process with state space $E$ satisfying (a somewhat relaxed version of) Meyer’s “hypothèses droites”. Then by introducing a new topology (called the Ray topology) on $E$ and a compactification $F$ of $E$ in the Ray topology one can regard $X$ as a Ray process. However, this construction depends on the choice of an arbitrary uniformity on $E$ and not just the topology of $E$. We show that the Ray topology is independent of the choice of this uniformity. We then introduce a space $R$ (the Ray space) which contains $E$ in the Ray topology and which has all of the properties of $F$ which are relevant for the study of $X$. Although $R$ is not compact it is independent of the choice of the original uniformity on $E$.
LA - eng
UR - http://eudml.org/doc/74243
ER -

References

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  1. [1] N. BOURBAKI, General Topology, Part 2, Hermann, Paris (1966). 
  2. [2] C. DELLACHERIE, Capacités et Processus Stochastiques, Springer-Verlag, Heidelberg (1972). Zbl0246.60032MR56 #6810
  3. [3] R. K. GETOOR, Lectures on Markov Processes : Ray Processes and Right Processes, Preliminary Version Univ. of Calif. San Diego (1974). To appear Springer Lecture Notes in Mathematics. 
  4. [4] F. KNIGHT, Note on regularization of Markov processes, Ill. Journ. Math., 9 (1965), 548-552. Zbl0143.20002MR31 #1713
  5. [5] P. A. MEYER, Probability and Potentials, Ginn. Boston (1966). Zbl0138.10401MR34 #5119
  6. [6] P. A. MEYER and J. B. WALSH, Quelques applications des résolvantes de Ray, Invent. Math., 14 (1971), 143-166. Zbl0224.60037MR45 #4502
  7. [7] M. OHTSUKA, Dirichlet Problem, Extremal Length, and Prime Ends, Van Nostrand Reinhold Math. Studies, 22 (1970). Zbl0197.08404
  8. [8] K. P. PARTHASARATHY, Probability Measures on Metric Spaces, Academic Press, New York (1967). Zbl0153.19101MR37 #2271
  9. [9] D. B. RAY, Resolvents, transition functions, and strongly Markovian processes, Ann. Math., 70 (1959), 43-72. Zbl0092.34501
  10. [10] C. T. SHIH, On extending potential theory to all strong Markov processes, Ann. Instit. Fourier, 20 (1970), 303-315. Zbl0193.46201MR44 #6040

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