{"paper":{"title":"On the chaotic character of the stochastic heat equation, before the onset of intermitttency","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.PR","authors_text":"Daniel Conus, Davar Khoshnevisan, Mathew Joseph","submitted_at":"2011-04-01T14:51:49Z","abstract_excerpt":"We consider a nonlinear stochastic heat equation $\\partial_tu=\\frac{1}{2}\\partial_{xx}u+\\sigma(u)\\partial_{xt}W$, where $\\partial_{xt}W$ denotes space-time white noise and $\\sigma:\\mathbf {R}\\to \\mathbf {R}$ is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_0$: under suitable conditions on $u_0$ and $\\sigma$, $\\sup_{x\\in \\mathbf {R}}u_t(x)$ is a.s. finite when $u_0$ has compact support, whereas with probability one, $\\limsup_{|x|\\to\\infty}u_t(x)/({\\log}|x|)^{1/6}>0$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.0189","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"}