{"paper":{"title":"Nonlinear stochastic time-fractional slow and fast diffusion equations on $\\mathbb{R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Nualart, Le Chen, Yaozhong Hu","submitted_at":"2015-09-25T15:45:29Z","abstract_excerpt":"This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \\[ \\left(\\partial^\\beta+\\frac{\\nu}{2}(-\\Delta)^{\\alpha/2}\\right)u(t,x) = I_t^\\gamma\\left[\\rho(u(t,x))\\dot{W}(t,x)\\right],\\quad t>0,\\: x\\in\\mathbb{R}^d, \\] where $\\dot{W}$ is the space-time white noise, $\\alpha\\in(0,2]$, $\\beta\\in(0,2)$, $\\gamma\\ge 0$ and $\\nu>0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang's "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07763","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"}