Stopping Rules and the Poisson Boundary of Random Walks on Countable Groups

In this paper, we study the terminal laws and Poisson boundaries obtained by stopping random walks on countable groups. For a transient walk, we give a direct Green-potential characterization of randomized stopping distributions and show that the terminal law determines the mean stopping time. On free groups, this yields finitely supported randomized stopping distributions at a positive total variation distance from every ordinary stopping distribution. We construct symmetric, fully supported recurrent measures of finite entropy on countable locally finite groups and obtain ordinary strict boundary enlargements when the group is not Choquet-Deny. For generating source and terminal measures of finite entropy, positive asymptotic entropy of the source forces the stopping time to have finite mean. A direct proof of the finite-mean entropy formula and boundary preservation then shows that a finite-entropy stopping chain has at most one strict enlargement. An exact stopped-word entropy identity shows that infinite-mean ordinary stopping with a finite-entropy terminal law has infinite conditional history entropy.