{"paper":{"title":"A stochastic approach to a new type of parabolic variational inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Tianyang Nie","submitted_at":"2012-03-21T21:29:50Z","abstract_excerpt":"We study the following quasilinear partial differential equation with two subdifferential operators:\n  $${\\frac{\\partial u}{\\partial s}(s,x)} + (\\mathcal{L}u)(s,x,u(s,x),(\\nabla u(s,x))^\\ast\\sigma(s,x,u(s,x))) + f(s,x,u(s,x),(\\nabla u(s,x))^\\ast\\sigma(s,x,u(s,x))) \\in \\partial\\varphi(u(s,x)) + <\\partial\\psi(x),\\nabla u(s,x)>,\n  (s,x) \\in[0,T]\\times Dom\\psi,\nu(T,x) =g(x),\\quad x\\in Dom\\psi.$$\nwhere for $u\\in C^{1,2}\\big([0,T]\\times Dom\\psi\\big)$ and $(s,x,y,z)\\in [0,T]\\times Dom\\psi\\times Dom\\varphi\\times\\mathbb{R}^{1\\times d}$,\n  $$(\\mathcal{L}u)(s,x,y,z) := 1/2\\sum_{i,j=1}^n (\\sigma\\sigma^\\as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.4840","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"}