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arXiv · 2311.06886

Hitting probabilities and uniformly $S$-transient subgraphs

Emily Dautenhahn·Laurent Saloff-Coste
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Abstract

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).

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Emily Dautenhahn, Laurent Saloff-Coste. 2025-01-14. Hitting probabilities and uniformly $S$-transient subgraphs. https://doi.org/10.1214/24-ejp1141

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