{"paper":{"title":"On the chaotic character of the stochastic heat equation, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniel Conus, Davar Khoshnevisan, Mathew Joseph, Shang-Yuan Shiu","submitted_at":"2011-11-21T04:25:16Z","abstract_excerpt":"Consider the stochastic heat equation $\\partial_t u = (\\frac{\\varkappa}{2})\\Delta u+\\sigma(u)\\dot{F}$, where the solution $u:=u_t(x)$ is indexed by $(t,x)\\in (0, \\infty)\\times\\R^d$, and $\\dot{F}$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-$|x|$ fixed-$t$ behavior of the solution $u$ in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function $f$ of the noise is of Riesz type, that is $f(x)\\propto \\|x\\|^{-\\alpha}$, then the \"fl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4728","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"}