{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:NEM6LC6T7VW2RBICRJWQVYHILF","short_pith_number":"pith:NEM6LC6T","schema_version":"1.0","canonical_sha256":"6919e58bd3fd6da885028a6d0ae0e8594cb22a2c0143836fa43b7020a1d5a7ce","source":{"kind":"arxiv","id":"2607.12140","version":1},"attestation_state":"computed","paper":{"title":"Causal Graphs, Markov Properties and Do-calculus for Stochastic Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"Joris M. Mooij, Philip Boeken","submitted_at":"2026-07-13T20:45:21Z","abstract_excerpt":"Stochastic differential equations (SDEs) are widely used to model continuous-time dynamical systems, but graphical causal models for them are not yet well-understood. We consider systems of causal SDEs that are equipped with an explicit causal semantics. We pose solvability conditions for systems of causal SDEs such that they have well-defined observational and interventional distributions - even after marginalisation - and provide a general class of Lipschitz semimartingale SDEs that satisfies these conditions. As core results we establish the $\\sigma$-separation Markov property and the do-ca"},"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":"2607.12140","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-13T20:45:21Z","cross_cats_sorted":["stat.ML","stat.TH"],"title_canon_sha256":"c6d6511e0a49e5119cf17e2551f3a4ab1ce4c441d79b97c3a4d7a28b5053df3e","abstract_canon_sha256":"e732b2878b97903f7db73f290c7898c73a4fda966fe10b48f3c2fb9e01410c92"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-15T00:21:35.222236Z","signature_b64":"cV9yI3lBh/s+pwnOkf69hIkhFTd5PCru5+pxnJjy8/EocVNzvyjM9Bxrr4CzlBl/LAJNpg8Acb375zoKxydUBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6919e58bd3fd6da885028a6d0ae0e8594cb22a2c0143836fa43b7020a1d5a7ce","last_reissued_at":"2026-07-15T00:21:35.221412Z","signature_status":"signed_v1","first_computed_at":"2026-07-15T00:21:35.221412Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Causal Graphs, Markov Properties and Do-calculus for Stochastic Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"Joris M. Mooij, Philip Boeken","submitted_at":"2026-07-13T20:45:21Z","abstract_excerpt":"Stochastic differential equations (SDEs) are widely used to model continuous-time dynamical systems, but graphical causal models for them are not yet well-understood. We consider systems of causal SDEs that are equipped with an explicit causal semantics. We pose solvability conditions for systems of causal SDEs such that they have well-defined observational and interventional distributions - even after marginalisation - and provide a general class of Lipschitz semimartingale SDEs that satisfies these conditions. As core results we establish the $\\sigma$-separation Markov property and the do-ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12140","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.12140/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.12140","created_at":"2026-07-15T00:21:35.221841+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.12140v1","created_at":"2026-07-15T00:21:35.221841+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.12140","created_at":"2026-07-15T00:21:35.221841+00:00"},{"alias_kind":"pith_short_12","alias_value":"NEM6LC6T7VW2","created_at":"2026-07-15T00:21:35.221841+00:00"},{"alias_kind":"pith_short_16","alias_value":"NEM6LC6T7VW2RBIC","created_at":"2026-07-15T00:21:35.221841+00:00"},{"alias_kind":"pith_short_8","alias_value":"NEM6LC6T","created_at":"2026-07-15T00:21:35.221841+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF","json":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF.json","graph_json":"https://pith.science/api/pith-number/NEM6LC6T7VW2RBICRJWQVYHILF/graph.json","events_json":"https://pith.science/api/pith-number/NEM6LC6T7VW2RBICRJWQVYHILF/events.json","paper":"https://pith.science/paper/NEM6LC6T"},"agent_actions":{"view_html":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF","download_json":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF.json","view_paper":"https://pith.science/paper/NEM6LC6T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.12140&json=true","fetch_graph":"https://pith.science/api/pith-number/NEM6LC6T7VW2RBICRJWQVYHILF/graph.json","fetch_events":"https://pith.science/api/pith-number/NEM6LC6T7VW2RBICRJWQVYHILF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF/action/storage_attestation","attest_author":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF/action/author_attestation","sign_citation":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF/action/citation_signature","submit_replication":"https://pith.science/pith/NEM6LC6T7VW2RBICRJWQVYHILF/action/replication_record"}},"created_at":"2026-07-15T00:21:35.221841+00:00","updated_at":"2026-07-15T00:21:35.221841+00:00"}