A Necessary and Sufficient Condition for the Markov Property of the Local Time Process

Abstract

Let $X$ be a Markov process on an interval $E$ of $\mathbb{R}$, with lifetime $\zeta$, admitting a local time at each point and such that $P_x(X$ hits $y) > 0$ for all $x,y$ in $E$. We prove here that the local times process $(L^x_\zeta, x \in E)$ is a Markov process if and only if $X$ has fixed birth and death points and $X$ has continuous paths. The sufficiency of this condition has been established by Ray, Knight and Walsh. The necessity is proved using arguments based on excursion theory. This result has been proved before in Eisenbaum and Kaspi for symmetric processes using the existence of a zero mean Gaussian process with the Green function as covariance.