{"paper":{"title":"Champagne subdomains with unavoidable bubbles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ivan Netuka, Wolfhard Hansen","submitted_at":"2012-07-31T18:52:21Z","abstract_excerpt":"A champagne subdomain of a connected open set $U\\ne\\emptyset$ in $R^d$, $d\\ge 2$, is obtained omitting pairwise disjoint closed balls $\\bar{B}(x,r_x)$, $x\\in X$, the bubbles, where $X$ is an infinite, locally finite set in $U$. The union $A$ of these balls may be unavoidable, that is, Brownian motion, starting in $U\\setminus A$ and killed when leaving $U$, may hit $A$ almost surely or, equivalently, $A$ may have harmonic measure one for $U\\setminus A$.\n  Recent publications by Gardiner/Ghergu ($d\\ge 3$) and by Pres ($d=2$) give rather sharp answers to the question how small such a set $A$ may "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.7342","kind":"arxiv","version":1},"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"}