Difference between revisions of "Ray-Knight compactification"
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| − | A powerful tool to study homogeneous | + | A powerful tool to study homogeneous [[Markov_process#The_strong_Markov_property.|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 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> | + | <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> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 18:36, 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
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