On One-Dimensional Diffusions with Time Parameter Set $(-\infty, \infty)$

Abstract

Let $p_t, t \geqq 0$ be the probability transition semigroup for a continuous one-dimensional diffusion. We examine continuous Markov processes $\xi_s$, defined for all $-\infty < s < \infty$, which are governed by $p_t$. We determine necessary and sufficient conditions for the set of such processes governed by $p_t$ to be nontrivial, and give an example where these conditions are satisfied.