On Markov Random Sets

A Markov random set is a time-homogeneous random closed set on the half-line $t\geqq 0$, satisfying the Markov property of independence between the future and the past when the present is known. Such sets are introduced as a special class of Markov processes. They may be described by a non-increasing right-continuous positive function $g(x)$, $x>0$, integrable near 0 and a non-negative number $\alpha$, determined up to an arbitrary positive constant factor. If $y(t)$ is a continuous strong Markov process, the $t$-set $\{y(t)=\mathrm{const}\}$ is a Markov random set. The most interesting Markov sets are obtained by simple transformations from the Brownien motion process.