{"paper":{"title":"On the support of solutions to nonlinear stochastic heat equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Beom-seok Han, Jaeyun Yi, Kunwoo Kim","submitted_at":"2024-07-09T13:00:24Z","abstract_excerpt":"We investigate the strict positivity and the compact support property of solutions to the one-dimensional nonlinear stochastic heat equation: $$\\partial_t u(t,x) = \\frac{1}{2}\\partial^2_x u(t,x) + \\sigma(u(t,x))\\dot{W}(t,x), \\quad (t,x)\\in \\mathbf{R}_+\\times\\mathbf{R},$$ with nonnegative and compactly supported initial data $u_0$, where $\\dot{W}$ is the space-time white noise and $\\sigma:\\mathbf{R} \\to \\mathbf{R} $ is a continuous function with $\\sigma(0)=0$. We prove that (i) if $v/ \\sigma(v)$ is sufficiently large near $v=0$, then the solution $u(t,\\cdot)$ is strictly positive for all $t>0$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.06827","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.06827/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}