A Note on Continuous Parameter Zero-Two Law

Abstract

Let $\{X_t\}, 0 \leqq t < \infty$, be a Markov process with state space $(E, \mathscr{E})$. Let $m$ be a $\sigma$-finite measure on $(E, \mathscr{E})$ and let the $L_\infty(E, \mathscr{E}, m)$ operator induced by the transition probability $P_t(x, A), x\in E, A\in \mathscr{E}$, be conservative and ergodic for all $t > 0$. Let $(m)$ abbreviate $m$ modulo 0. For fixed $\alpha > 0$, set $h^\alpha(x) = \lim_{t \rightarrow \infty} \|P_t(x, \bullet) - P_{t + \alpha}(x, \bullet)\|$, where $\|\bullet\|$ is the total variation. THEOREM. Either $h^\alpha(x) = 0(m)$ for $\operatorname{a.e} \alpha\in\mathbb{R}_+$ or $h^\alpha(x) = 2 (m)$ for $\operatorname{a.e} \alpha\in\mathbb{R}_+$. In particular, if $\{X_t\}, 0 \leqq t < \infty$, is a Markov process satisfying a Harris type recurrence condition, then $h^\alpha(x) = 0 (m)$ for $\operatorname{a.e} \alpha\in\mathbb{R}_+$.