We present a deterministic logspace algorithm for solving s-t connectivity on directed graphs if (i) we are given a stationary distribution for random walk on the graph and (ii) the random walk which starts at the source vertex $s$ has polynomial mixing time. This result generalizes the recent deterministic logspace algorithm for s-t connectivity on undirected graphs (ECCC TR 04-94). It identifies knowledge of the stationary distribution as the gap between the s-t connectivity problems we know how to solve in logspace ($\L$) and those that capture all of randomized logspace ($\RL$).