{"paper":{"title":"Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Robert C. Dalang, T. Mountford","submitted_at":"2017-02-27T08:29:04Z","abstract_excerpt":"We first consider the additive Brownian motion process $(X(s_1,s_2),\\ (s_1,s_2) \\in \\mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set $\\{(s_1,s_2)\\in \\mathbb{R}^2: X(s_1,s_2) >0\\}$ is equal to $$\n  \\frac{1}{4}\\left(1 + \\sqrt{13 + 4 \\sqrt{5}}\\right) \\simeq 1.421\\, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.08183","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"}