Hitting probabilities and uniformly $S$-transient subgraphs

We study the probability that a random walk started inside a subgraph of a larger graph exits that subgraph (or, equivalently, hits the exterior boundary of the subgraph). Considering the chance a random walk started in the subgraph never leaves the subgraph leads to a notion we call "survival" transience, or $S$-transience. In the case where the heat kernel of the larger graph satisfies two-sided Gaussian estimates, we prove an upper bound on the probability of hitting the boundary of the subgraph. Under the additional hypothesis that the subgraph is inner uniform, we prove a two-sided estimate for this probability. The estimate depends upon a harmonic function in the subgraph. We also provide two-sided estimates for related probabilities, such as the harmonic measure (the chance the walk exits the subgraph at a particular point on its boundary).