Let $p_t, t \geqslant 0$ be the probability transition semigroup of a continuous one-dimensional diffusion. We examine continuous Markov processes $\xi_s$, defined for $-\infty < s < \infty$, which are governed by $p_t$. It is shown that the class of such processes, modulo convex combinations and translations, can consist of at most three elements. In addition, it is shown that the first passage times for these processes are related to a previously known existence condition.
"Further Results on One-dimensional Diffusions with Time Parameter set S(- \infty, \infty)$." Ann. Probab. 7 (3) 537 - 542, June, 1979. https://doi.org/10.1214/aop/1176995054