{"paper":{"title":"Nonlinear Fokker-Planck equations driven by Gaussian linear multiplicative noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Michael R\\\"ockner, Viorel Barbu","submitted_at":"2017-08-29T14:18:58Z","abstract_excerpt":"Existence and uniqueness of a strong solution in $H^{-1}(\\mathbb R^d)$ is proved for the stochastic nonlinear Fokker-Planck equation $$dX-{\\rm div}(DX)dt-\\Delta\\beta(X)dt=X\\,dW \\mbox{ in }(0,T)\\times\\mathbb R^d,\\ X(0)=x,$$ via a corresponding random differential equation. Here $d\\geq 1$, $W$ is a Wiener process in $H^{-1}(\\mathbb R^d)$, $D\\in C^1(\\mathbb R^d,\\mathbb R^d)$ and $\\beta$ is a continuous monotonically increasing function. The solution exists for $x\\in L^1\\cap L^\\infty$ and preserves positivity. If $\\beta \\in L^1_{\\rm loc}(\\mathbb R)$, the solution is pathwise Lipschitz continuous w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.08768","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"}