{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:Y4O6NXAGTMNUOZ5BI3TM3FRWX7","short_pith_number":"pith:Y4O6NXAG","canonical_record":{"source":{"id":"0811.2061","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-11-13T10:20:10Z","cross_cats_sorted":["math.FA","math.SP"],"title_canon_sha256":"8fd8ce618b28ccb7503edeb639ce9d76859a605394404638397640cdbfdbd88e","abstract_canon_sha256":"829919f32085b0c59e1264df52607e4bb036694ff719ba3e35d407e0096a9dd5"},"schema_version":"1.0"},"canonical_sha256":"c71de6dc069b1b4767a146e6cd9636bfc98a59d29efdf93eeb9e8b692fd80a3a","source":{"kind":"arxiv","id":"0811.2061","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0811.2061","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"arxiv_version","alias_value":"0811.2061v2","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0811.2061","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"pith_short_12","alias_value":"Y4O6NXAGTMNU","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"Y4O6NXAGTMNUOZ5B","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"Y4O6NXAG","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:Y4O6NXAGTMNUOZ5BI3TM3FRWX7","target":"record","payload":{"canonical_record":{"source":{"id":"0811.2061","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-11-13T10:20:10Z","cross_cats_sorted":["math.FA","math.SP"],"title_canon_sha256":"8fd8ce618b28ccb7503edeb639ce9d76859a605394404638397640cdbfdbd88e","abstract_canon_sha256":"829919f32085b0c59e1264df52607e4bb036694ff719ba3e35d407e0096a9dd5"},"schema_version":"1.0"},"canonical_sha256":"c71de6dc069b1b4767a146e6cd9636bfc98a59d29efdf93eeb9e8b692fd80a3a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:14.349600Z","signature_b64":"MYU7CZBU7/dsgOFhxi9ObipcrDdrxEh3YXgtRLx63OoIVanX1sj/eUdgOGINKOnGM912mb4RZEFKt1/bDnukDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c71de6dc069b1b4767a146e6cd9636bfc98a59d29efdf93eeb9e8b692fd80a3a","last_reissued_at":"2026-05-18T00:13:14.348957Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:14.348957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0811.2061","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:13:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CcIxk/Vv52jsEld7uidabA+M7LcNf/UtQZHza0c2d4zkO6ZpedUa/yfPS+Np5CGl6JbiX5CE97fRU7p/9h3GCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:02:08.823101Z"},"content_sha256":"d7a990d0ed27b0a1c0896324eef21e77176565dabef8677a144018c93da4d823","schema_version":"1.0","event_id":"sha256:d7a990d0ed27b0a1c0896324eef21e77176565dabef8677a144018c93da4d823"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:Y4O6NXAGTMNUOZ5BI3TM3FRWX7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.SP"],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang, Giuseppe Da Prato, Michael R\\\"ockner","submitted_at":"2008-11-13T10:20:10Z","abstract_excerpt":"We consider stochastic equations in Hilbert spaces with singular drift in the framework of [Da Prato, R\\\"ockner, PTRF 2002]. We prove a Harnack inequality (in the sense of [Wang, PTRF 1997]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure $\\mu$ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure $\\mu$ for non-continuous drifts."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.2061","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:13:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I6Cu/j1pSZPFFcYMPLWY3eSP9U1MAr+uURsR4OK/BqNA63P5pZewhZ1+U+JwwdtDkTbRGgd5JM5yyS2cDnLzAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:02:08.823471Z"},"content_sha256":"cce3653e904effe97df59117cdb04415198daa4128ff310838cedb3200ee9f56","schema_version":"1.0","event_id":"sha256:cce3653e904effe97df59117cdb04415198daa4128ff310838cedb3200ee9f56"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y4O6NXAGTMNUOZ5BI3TM3FRWX7/bundle.json","state_url":"https://pith.science/pith/Y4O6NXAGTMNUOZ5BI3TM3FRWX7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y4O6NXAGTMNUOZ5BI3TM3FRWX7/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":"2026-06-03T03:02:08Z","links":{"resolver":"https://pith.science/pith/Y4O6NXAGTMNUOZ5BI3TM3FRWX7","bundle":"https://pith.science/pith/Y4O6NXAGTMNUOZ5BI3TM3FRWX7/bundle.json","state":"https://pith.science/pith/Y4O6NXAGTMNUOZ5BI3TM3FRWX7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y4O6NXAGTMNUOZ5BI3TM3FRWX7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:Y4O6NXAGTMNUOZ5BI3TM3FRWX7","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":"829919f32085b0c59e1264df52607e4bb036694ff719ba3e35d407e0096a9dd5","cross_cats_sorted":["math.FA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-11-13T10:20:10Z","title_canon_sha256":"8fd8ce618b28ccb7503edeb639ce9d76859a605394404638397640cdbfdbd88e"},"schema_version":"1.0","source":{"id":"0811.2061","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0811.2061","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"arxiv_version","alias_value":"0811.2061v2","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0811.2061","created_at":"2026-05-18T00:13:14Z"},{"alias_kind":"pith_short_12","alias_value":"Y4O6NXAGTMNU","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"Y4O6NXAGTMNUOZ5B","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"Y4O6NXAG","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:cce3653e904effe97df59117cdb04415198daa4128ff310838cedb3200ee9f56","target":"graph","created_at":"2026-05-18T00:13:14Z","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":"We consider stochastic equations in Hilbert spaces with singular drift in the framework of [Da Prato, R\\\"ockner, PTRF 2002]. We prove a Harnack inequality (in the sense of [Wang, PTRF 1997]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure $\\mu$ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure $\\mu$ for non-continuous drifts.","authors_text":"Feng-Yu Wang, Giuseppe Da Prato, Michael R\\\"ockner","cross_cats":["math.FA","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-11-13T10:20:10Z","title":"Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.2061","kind":"arxiv","version":2},"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:d7a990d0ed27b0a1c0896324eef21e77176565dabef8677a144018c93da4d823","target":"record","created_at":"2026-05-18T00:13:14Z","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":"829919f32085b0c59e1264df52607e4bb036694ff719ba3e35d407e0096a9dd5","cross_cats_sorted":["math.FA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-11-13T10:20:10Z","title_canon_sha256":"8fd8ce618b28ccb7503edeb639ce9d76859a605394404638397640cdbfdbd88e"},"schema_version":"1.0","source":{"id":"0811.2061","kind":"arxiv","version":2}},"canonical_sha256":"c71de6dc069b1b4767a146e6cd9636bfc98a59d29efdf93eeb9e8b692fd80a3a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c71de6dc069b1b4767a146e6cd9636bfc98a59d29efdf93eeb9e8b692fd80a3a","first_computed_at":"2026-05-18T00:13:14.348957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:14.348957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MYU7CZBU7/dsgOFhxi9ObipcrDdrxEh3YXgtRLx63OoIVanX1sj/eUdgOGINKOnGM912mb4RZEFKt1/bDnukDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:14.349600Z","signed_message":"canonical_sha256_bytes"},"source_id":"0811.2061","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d7a990d0ed27b0a1c0896324eef21e77176565dabef8677a144018c93da4d823","sha256:cce3653e904effe97df59117cdb04415198daa4128ff310838cedb3200ee9f56"],"state_sha256":"fdc59a06bbb296106041f85638106a9f0d9986460d3a377bb977512fe7f24fd1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q9NiWL6R53/ZBkzFAvtnl7aruEUO8HBhmFiWTCRKoNFdJkmVVtThocvDsekgq1IOs3x1Jbki+zqAWei+WGdiBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T03:02:08.825398Z","bundle_sha256":"35d7ddc527d34327f5d26f742dc4c4b62b78a09ab872391bf045cf47007f6328"}}