{"paper":{"title":"Asymptotics of the density of parabolic Anderson random fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Khoa L\\^e, Yaozhong Hu","submitted_at":"2018-01-10T14:20:59Z","abstract_excerpt":"We investigate the sharp density $\\rho(t,x; y)$ of the solution $u(t,x)$ to stochastic partial differential equation $\\frac{\\partial }{\\partial t} u(t,x)=\\frac12 \\Delta u(t,x)+u\\diamond \\dot W(t,x)$, where $\\dot W$ is a general Gaussian noise and $\\diamond$ denotes the Wick product. We mainly concern with the asymptotic behavior of $\\rho(t,x; y)$ when $y\\rightarrow \\infty$ or when $t\\to0+$. Both upper and lower bounds are obtained and these two bounds match each other modulo some multiplicative constants. If the initial datum is positive, then $\\rho(t,x;y)$ is supported on the positive half li"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03386","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"}