## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 21, Number 3 (1993), 1591-1598.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989132

Digital Object Identifier
doi:10.1214/aop/1176989132

Mathematical Reviews number (MathSciNet)
MR1235430

Zentralblatt MATH identifier
0788.60091

JSTOR
links.jstor.org

Subjects
Primary: 60J55: Local time and additive functionals
Secondary: 60J60: Diffusion processes [See also 58J65]

Keywords
Local time excursions

#### Citation

Eisenbaum, Nathalie; Kaspi, Haya. A Necessary and Sufficient Condition for the Markov Property of the Local Time Process. Ann. Probab. 21 (1993), no. 3, 1591--1598. doi:10.1214/aop/1176989132. https://projecteuclid.org/euclid.aop/1176989132