{"paper":{"title":"On the boundary of the zero set of super-Brownian motion and its local time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Edwin Perkins, Thomas Hughes","submitted_at":"2018-02-11T02:42:47Z","abstract_excerpt":"If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that $\\text{dim}(\\partial\\{x:X(t,x)>0\\})=2-2\\lambda_0\\in(0,1)$ a.s. on $\\{X_t\\neq 0\\}$, where $-\\lambda_0\\in(-1,-1/2)$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck process. This confirms a conjecture of Mueller, Mytnik and Perkins who proved the above with positive probability. To establish this result we derive some new basic properties of a recently introduced boundary local time and analyze the behaviour of $X(t,\\cdot)$ near the upper edge of its support. Numerical estimates of $\\lambda_0$ suggest that th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.03681","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"}