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<\\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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1203.4840","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-03-21T21:29:50Z","cross_cats_sorted":[],"title_canon_sha256":"144983cdb0a8138ae65168d58e77f91ba3546fa4dc2cfaac030f94d0be69d593","abstract_canon_sha256":"36adc22fdb6910835f884aff785d8f8e000500df39a2919b5122dbd16ef16b60"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:26.895224Z","signature_b64":"zCF9SJ7ZLMeKNCZ91XLfC02GgeHKTvO5QE9pt2DA9szmTSH5M2b5QU2bl8Crq+U60ChHrA0IjBESPgbSCyrEAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0432aec7fefc096f179ee1305d5701dc40f29a9c17aaa8ddace3121363a4e915","last_reissued_at":"2026-05-18T03:59:26.894630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:26.894630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 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