{"paper":{"title":"Comparison principle for stochastic heat equation on $\\mathbb{R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jingyu Huang, Le Chen","submitted_at":"2016-07-14T05:31:26Z","abstract_excerpt":"We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\\mathbb{R}^d$ \\[ \\left(\\frac{\\partial }{\\partial t} -\\frac{1}{2}\\Delta \\right) u(t,x) = \\rho(u(t,x)) \\:\\dot{M}(t,x), \\] for measure-valued initial data, where $\\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\\rho$ is Lipschitz continuous. These results are obtained under the condition that $\\int_{\\mathbb{R}^d}(1+|\\xi|^2)^{\\alpha-1}\\hat{f}(\\text{d} \\xi)<\\infty$ for some $\\alpha\\in(0,1]$, where $\\hat{f}$ is the spectral measure of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.03998","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"}