{"paper":{"title":"Random walks and induced Dirichlet forms on self-similar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ka-Sing Lau, Shi-Lei Kong, Ting-Kam Leonard Wong","submitted_at":"2016-04-19T06:18:44Z","abstract_excerpt":"Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\\partial_HX$ with the Gromov metric is H\\\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< \\lambda <1$ on $(X, {\\mathfrak E})$. We show that the Martin boundary ${\\mathcal M}$ can be identified with $\\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05440","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}