{
  "id": "2311.06886",
  "version": "v1",
  "published": "2023-11-12T16:18:52.000Z",
  "updated": "2023-11-12T16:18:52.000Z",
  "title": "Hitting probabilities and uniformly $S$-transient subgraphs",
  "authors": [
    "Emily Dautenhahn",
    "Laurent Saloff-Coste"
  ],
  "comment": "30 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "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).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-12T16:18:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60G50"
    ],
    "keywords": [
      "probability",
      "transient subgraphs",
      "hitting probabilities",
      "random walk",
      "graph satisfies two-sided gaussian estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}