2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146748The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.Probabilitytext