Existence condition of strong stationary times for continuous time   Markov chains on discrete graphs
  We consider a random walk on a discrete connected graph having some infinite branches plus finitely many vertices with finite degrees. We find the generator of a strong stationary dual in the sense of Fill, and use it to find some equivalent condition to the existence of a strong stationary time. This strong stationary dual process lies in the set of connected compact sets of the compactification of the graph. When this graph is Z, this is simply the set of (possibly infinite) segments of Z. 
Guillaume Copros
math.PR