Randomized Stopping Times, Bounded Harmonic Functions on Groups, and Topological Realizations

Bounded harmonic functions on groups (and their alter ego, the Poisson boundary)—bounded functions such that f(x) is the average of f(xa), where a is chosen from some probability measure—are objects of key importance in random walks, dynamics, and probability. Two core techniques for understanding these ideas, used since the beginning of the subject, have been stopping times (randomized times when you stop your random walk) and realizations (classification of bounded harmonic functions by looking at a space where the random walk almost surely converges). I will discuss some new results linking these two concepts. This is joint work with Kunal Chawla.