A necessary and sufficient condition for the Markov property of the local time process (Q688065)

This note is devoted to a reciprocal of the Ray-Knight theorem: given a transient Hunt process \(X\) taking values in an interval \(E\) of \(\mathbb{R}\) with lifetime \(\zeta\), if the points of \(E\) are regular and there exists a probability measure \(\mu\) on \(E\) such that the accumulated local time process \((L^ x_ \zeta; x\in E)\) is a Markov process, then a.s. \(X\) has continuous paths, \(\mu\) is a Dirac measure, and \(X\) dies at a fixed point.