# Ray-Knight compactification

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.