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.
"On One-Dimensional Diffusions with Time Parameter Set $(-\infty, \infty)$." Ann. Probab. 5 (5) 807 - 813, October, 1977. https://doi.org/10.1214/aop/1176995724