{"paper":{"title":"The stochastic porous media equation in $\\R^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Francesco Russo (UMA), Michael R\\\"ockner (SFB 701), Viorel Barbu","submitted_at":"2013-12-21T10:57:10Z","abstract_excerpt":"Existence and uniqueness of solutions to the stochastic porous media equation $dX-\\D\\psi(X) dt=XdW$ in $\\rr^d$ are studied. Here, $W$ is a Wiener process, $\\psi$ is a maximal monotone graph in $\\rr\\times\\rr$ such that $\\psi(r)\\le C|r|^m$, $\\ff r\\in\\rr$, $W$ is a coloured Wiener process. In this general case the dimension is restricted to $d\\ge 3$, the main reason being the absence of a convenient multiplier result in the space $\\calh=\\{\\varphi\\in\\mathcal{S}'(\\rr^d);\\ |\\xi|(\\calf\\varphi)(\\xi)\\in L^2(\\rr^d)\\}$, for $d\\le2$. When $\\psi$ is Lipschitz, the well-posedness, however, holds for all dim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.6234","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"}