{"paper":{"title":"Nonuniqueness for a parabolic SPDE with $\\frac{3}{4}-\\varepsilon$-H\\\"older diffusion coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Carl Mueller, Edwin Perkins, Leonid Mytnik","submitted_at":"2012-01-13T09:00:19Z","abstract_excerpt":"Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \\[\\frac{\\partial u}{\\partial t}=\\frac{\\Delta}{2}u(t,x) +\\bigl|u(t,x)\\bigr|^{\\gamma}\\dot{W}(t,x),\\qquad u(0,x)=0.\\] Here $\\dot{W}$ is a space-time white noise on ${\\mathbb {R}}_+\\times {\\mathbb {R}}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<\\gamma<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2767","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"}