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Ray-Knight compactification

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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://www.encyclopediaofmath.org/index.php?title=Ray%E2%80%93Knight_compactification&oldid=22968