Chutes and ladders in Markov chains

We investigate how the stationary distribution of a Markov chain changes wh en transitions from a single state are modified. In particular, adding a si ngle directed edge to nearest neighbor random walk on a finite discrete tor us in dimensions one, two, or three changes the stationary distribution lin early, logarithmically, or only locally. Related results are derived for bi rth and death chains approximating Bessel diffusions and for random walk on the Sierpinski gasket.