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Difference between revisions of "Ray-Knight compactification"

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A powerful tool to study homogeneous strong Markov processes (cf. [[Markov process|Markov process]]) under some general hypotheses. The idea is to imbed as a set the state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777101.png" /> of the process in a compact metrizable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777102.png" /> such that the [[Resolvent|resolvent]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777103.png" /> of the transition semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777104.png" /> (cf. [[Transition-operator semi-group|Transition-operator semi-group]]) has a unique extension to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777105.png" /> as a resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777106.png" /> with nice analytical properties. This Ray resolvent is associated to a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777107.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777108.png" /> need not be the identity: existence of branching points), quite indistinguishable from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r0777109.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077710/r07771010.png" />. The Ray–Knight compactification allows one to extend easily numerous important results for Feller processes (cf. [[Feller process|Feller process]]) to strong Markov processes, to define entrance boundaries, etc.
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A powerful tool to study homogeneous strong [[Markov process]]es under some general hypotheses. The idea is to imbed as a set the state space $E$ of the process in a compact metrizable space $\hat E$ such that the [[resolvent]] $(U_p)_{p\ge 0}$ of the [[transition-operator semi-group]] $(P_t)_{t\ge0}$> has a unique extension to $\hat E$ as a resolvent $(\hat U_p)$ with nice analytical properties. This Ray resolvent is associated to a semi-group $(\hat P_t)$ (note that $\hat P_0$ need not be the identity: existence of branching points), quite indistinguishable from $(P_t)$ on $E$. The Ray–Knight compactification allows one to extend easily numerous important results for [[Feller process]]es to strong Markov processes, to define entrance boundaries, etc.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,  P.A. Meyer,  "Probabilities and potential" , '''C''' , North-Holland  (1988)  pp. Chapt. XII  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.K. Getoor,  "Markov processes: Ray processes and right processes" , ''Lect. notes in math.'' , '''440''' , Springer  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.J. Sharpe,  "General theory of Markov processes" , Acad. Press  (1988)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,  P.A. Meyer,  "Probabilities and potential" , '''C''' , North-Holland  (1988)  pp. Chapt. XII  (Translated from French)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.K. Getoor,  "Markov processes: Ray processes and right processes" , ''Lect. notes in math.'' , '''440''' , Springer  (1975)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  M.J. Sharpe,  "General theory of Markov processes" , Acad. Press  (1988)</TD></TR>
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</table>
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Revision as of 16:01, 14 October 2017

A powerful tool to study homogeneous strong Markov processes under some general hypotheses. The idea is to imbed as a set the state space $E$ of the process in a compact metrizable space $\hat E$ such that the resolvent $(U_p)_{p\ge 0}$ of the transition-operator semi-group $(P_t)_{t\ge0}$> has a unique extension to $\hat E$ as a resolvent $(\hat U_p)$ with nice analytical properties. This Ray resolvent is associated to a semi-group $(\hat P_t)$ (note that $\hat P_0$ need not be the identity: existence of branching points), quite indistinguishable from $(P_t)$ on $E$. The Ray–Knight compactification allows one to extend easily numerous important results for Feller processes to strong Markov processes, to define entrance boundaries, etc.

References

[a1] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , C , North-Holland (1988) pp. Chapt. XII (Translated from French)
[a2] R.K. Getoor, "Markov processes: Ray processes and right processes" , Lect. notes in math. , 440 , Springer (1975)
[a3] M.J. Sharpe, "General theory of Markov processes" , Acad. Press (1988)
How to Cite This Entry:
Ray-Knight compactification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ray-Knight_compactification&oldid=22967
This article was adapted from an original article by C. Dellacherie (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article