Homogeneity and the Strong Markov Property

Abstract

The strong Markov property of a process $X$ at a stopping time $\tau$ may be split into a conditional independence part (CI) and a homogeneity part (H). However, it turns out that (H) often implies at least some version of (CI). In the present paper, we shall assume that (H) holds on the set $\{X_\tau \in B\}$, for all stopping times $\tau$ such that $X_\tau \in F$ a.s., where $F$ is a closed recurrent subset of the state space $S$, while $B \subset F$. If $F = S$, then (CI) will hold on $\{X_\tau \in B\}$ for every stopping time $\tau$, so in this case $X$ is regenerative in $B$. In the general case, the same statement is conditionally true in a suitable sense, given some shift invariant $\sigma$-field.