{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:BGMZB6AVPMS3P2FZVPREOEZZS4","short_pith_number":"pith:BGMZB6AV","canonical_record":{"source":{"id":"1611.03909","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-11T23:19:09Z","cross_cats_sorted":[],"title_canon_sha256":"92d466241dddd642bfc799c4bd78064ffb1f8d35e68be80ea2f99c719756f862","abstract_canon_sha256":"8d795bc3247cb8a7fab233d628f711bd527aed97355a599c432ab91ae6d50e2c"},"schema_version":"1.0"},"canonical_sha256":"099990f8157b25b7e8b9abe24713399738cbecca1734fe91474321f4a75203dd","source":{"kind":"arxiv","id":"1611.03909","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03909","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03909v1","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03909","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"pith_short_12","alias_value":"BGMZB6AVPMS3","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"BGMZB6AVPMS3P2FZ","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"BGMZB6AV","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:BGMZB6AVPMS3P2FZVPREOEZZS4","target":"record","payload":{"canonical_record":{"source":{"id":"1611.03909","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-11T23:19:09Z","cross_cats_sorted":[],"title_canon_sha256":"92d466241dddd642bfc799c4bd78064ffb1f8d35e68be80ea2f99c719756f862","abstract_canon_sha256":"8d795bc3247cb8a7fab233d628f711bd527aed97355a599c432ab91ae6d50e2c"},"schema_version":"1.0"},"canonical_sha256":"099990f8157b25b7e8b9abe24713399738cbecca1734fe91474321f4a75203dd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:18.105747Z","signature_b64":"UyIgEBOgcUOEPipW1PFMyjSc5IOKlSAPJL/PK8lnJuH/d5U1QtWB7bMBBc9EYdnHQ6P0vQ6yW7ugtHrCmrdLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"099990f8157b25b7e8b9abe24713399738cbecca1734fe91474321f4a75203dd","last_reissued_at":"2026-05-18T00:59:18.105123Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:18.105123Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.03909","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:59:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J+mWSc37QkIEhMNKalynG4ugXWSS5bsIibIMjte4qU+9+EkE2pg11nNtbIy/EuPeiYZHkIIKSlNccmnSAlgVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:53:33.732125Z"},"content_sha256":"958b751ae74be393ed4711503b0fa2fd2fdbd048ca46b7003b27c3469a34dbb3","schema_version":"1.0","event_id":"sha256:958b751ae74be393ed4711503b0fa2fd2fdbd048ca46b7003b27c3469a34dbb3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:BGMZB6AVPMS3P2FZVPREOEZZS4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regularity and strict positivity of densities for the nonlinear stochastic heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Nualart, Le Chen, Yaozhong Hu","submitted_at":"2016-11-11T23:19:09Z","abstract_excerpt":"In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove $u(t,x)\\in \\mathbb{D}^\\infty$ for measure-valued initia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03909","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:59:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GNTaAcMLlYwJmw7PtJ0rPg7tThU7LAlJ6ZgjngkfmBTuBtSng/oOHpS4s98mXE+ycMRsADNFIcLOYjbj18h8DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:53:33.732491Z"},"content_sha256":"ba020f2406a62f80faf8ce6f1c9f4f09d34c02d009e00b265ab3f1e86e527e3f","schema_version":"1.0","event_id":"sha256:ba020f2406a62f80faf8ce6f1c9f4f09d34c02d009e00b265ab3f1e86e527e3f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BGMZB6AVPMS3P2FZVPREOEZZS4/bundle.json","state_url":"https://pith.science/pith/BGMZB6AVPMS3P2FZVPREOEZZS4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BGMZB6AVPMS3P2FZVPREOEZZS4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T10:53:33Z","links":{"resolver":"https://pith.science/pith/BGMZB6AVPMS3P2FZVPREOEZZS4","bundle":"https://pith.science/pith/BGMZB6AVPMS3P2FZVPREOEZZS4/bundle.json","state":"https://pith.science/pith/BGMZB6AVPMS3P2FZVPREOEZZS4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BGMZB6AVPMS3P2FZVPREOEZZS4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:BGMZB6AVPMS3P2FZVPREOEZZS4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8d795bc3247cb8a7fab233d628f711bd527aed97355a599c432ab91ae6d50e2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-11T23:19:09Z","title_canon_sha256":"92d466241dddd642bfc799c4bd78064ffb1f8d35e68be80ea2f99c719756f862"},"schema_version":"1.0","source":{"id":"1611.03909","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03909","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03909v1","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03909","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"pith_short_12","alias_value":"BGMZB6AVPMS3","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"BGMZB6AVPMS3P2FZ","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"BGMZB6AV","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:ba020f2406a62f80faf8ce6f1c9f4f09d34c02d009e00b265ab3f1e86e527e3f","target":"graph","created_at":"2026-05-18T00:59:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove $u(t,x)\\in \\mathbb{D}^\\infty$ for measure-valued initia","authors_text":"David Nualart, Le Chen, Yaozhong Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-11T23:19:09Z","title":"Regularity and strict positivity of densities for the nonlinear stochastic heat equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03909","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:958b751ae74be393ed4711503b0fa2fd2fdbd048ca46b7003b27c3469a34dbb3","target":"record","created_at":"2026-05-18T00:59:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8d795bc3247cb8a7fab233d628f711bd527aed97355a599c432ab91ae6d50e2c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-11T23:19:09Z","title_canon_sha256":"92d466241dddd642bfc799c4bd78064ffb1f8d35e68be80ea2f99c719756f862"},"schema_version":"1.0","source":{"id":"1611.03909","kind":"arxiv","version":1}},"canonical_sha256":"099990f8157b25b7e8b9abe24713399738cbecca1734fe91474321f4a75203dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"099990f8157b25b7e8b9abe24713399738cbecca1734fe91474321f4a75203dd","first_computed_at":"2026-05-18T00:59:18.105123Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:59:18.105123Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UyIgEBOgcUOEPipW1PFMyjSc5IOKlSAPJL/PK8lnJuH/d5U1QtWB7bMBBc9EYdnHQ6P0vQ6yW7ugtHrCmrdLAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:59:18.105747Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.03909","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:958b751ae74be393ed4711503b0fa2fd2fdbd048ca46b7003b27c3469a34dbb3","sha256:ba020f2406a62f80faf8ce6f1c9f4f09d34c02d009e00b265ab3f1e86e527e3f"],"state_sha256":"5921c2907b8dcd71ae3e4976cafad8cbbe6f1544986fcf3c6aa0444292a54a5d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1bdnoHmFTHEHvwGnno+b/GAyMHBEgysTwYrUlWQiyVHiUs1xtK5jbcKXIu+WKBfsM7RY9AQDRIQe1okPK6x1Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T10:53:33.734409Z","bundle_sha256":"1473c82dc30dea0198bae12db23b3d5450ba116cf7b0ca53bf6dfb5f48d522f9"}}