{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:KEPNB2LDY2EBQNX5EHMYPXFEY3","short_pith_number":"pith:KEPNB2LD","schema_version":"1.0","canonical_sha256":"511ed0e963c6881836fd21d987dca4c6f7e66eb7490248afda3607036b9b84ae","source":{"kind":"arxiv","id":"2507.14499","version":1},"attestation_state":"computed","paper":{"title":"Neural Brownian Motion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","cs.LG","math.OC","stat.ML"],"primary_cat":"math.PR","authors_text":"Qian Qi","submitted_at":"2025-07-19T06:09:52Z","abstract_excerpt":"This paper introduces the Neural-Brownian Motion (NBM), a new class of stochastic processes for modeling dynamics under learned uncertainty. The NBM is defined axiomatically by replacing the classical martingale property with respect to linear expectation with one relative to a non-linear Neural Expectation Operator, $\\varepsilon^\\theta$, generated by a Backward Stochastic Differential Equation (BSDE) whose driver $f_\\theta$ is parameterized by a neural network. Our main result is a representation theorem for a canonical NBM, which we define as a continuous $\\varepsilon^\\theta$-martingale with"},"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":"2507.14499","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2025-07-19T06:09:52Z","cross_cats_sorted":["cs.AI","cs.LG","math.OC","stat.ML"],"title_canon_sha256":"7069b7a98788b41e2bdb39798201f6ab902df5610f76426257ccb9c729b7f246","abstract_canon_sha256":"48050ebd3d1a89fac1099ccd1119bb9efc5214815831422421d307e478b81860"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:40:08.316744Z","signature_b64":"qjLICpydFPbbkZ1Cl/yRpa1RfFA/q0vpFIxw0zNmMbwHr9vtop31MevCnTs8A+oUbPLighGrsPj4mknenZ5EAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"511ed0e963c6881836fd21d987dca4c6f7e66eb7490248afda3607036b9b84ae","last_reissued_at":"2026-07-05T11:40:08.316255Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:40:08.316255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Neural Brownian Motion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","cs.LG","math.OC","stat.ML"],"primary_cat":"math.PR","authors_text":"Qian Qi","submitted_at":"2025-07-19T06:09:52Z","abstract_excerpt":"This paper introduces the Neural-Brownian Motion (NBM), a new class of stochastic processes for modeling dynamics under learned uncertainty. The NBM is defined axiomatically by replacing the classical martingale property with respect to linear expectation with one relative to a non-linear Neural Expectation Operator, $\\varepsilon^\\theta$, generated by a Backward Stochastic Differential Equation (BSDE) whose driver $f_\\theta$ is parameterized by a neural network. Our main result is a representation theorem for a canonical NBM, which we define as a continuous $\\varepsilon^\\theta$-martingale with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.14499","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/2507.14499/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":"2507.14499","created_at":"2026-07-05T11:40:08.316321+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.14499v1","created_at":"2026-07-05T11:40:08.316321+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.14499","created_at":"2026-07-05T11:40:08.316321+00:00"},{"alias_kind":"pith_short_12","alias_value":"KEPNB2LDY2EB","created_at":"2026-07-05T11:40:08.316321+00:00"},{"alias_kind":"pith_short_16","alias_value":"KEPNB2LDY2EBQNX5","created_at":"2026-07-05T11:40:08.316321+00:00"},{"alias_kind":"pith_short_8","alias_value":"KEPNB2LD","created_at":"2026-07-05T11:40:08.316321+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/KEPNB2LDY2EBQNX5EHMYPXFEY3","json":"https://pith.science/pith/KEPNB2LDY2EBQNX5EHMYPXFEY3.json","graph_json":"https://pith.science/api/pith-number/KEPNB2LDY2EBQNX5EHMYPXFEY3/graph.json","events_json":"https://pith.science/api/pith-number/KEPNB2LDY2EBQNX5EHMYPXFEY3/events.json","paper":"https://pith.science/paper/KEPNB2LD"},"agent_actions":{"view_html":"https://pith.science/pith/KEPNB2LDY2EBQNX5EHMYPXFEY3","download_json":"https://pith.science/pith/KEPNB2LDY2EBQNX5EHMYPXFEY3.json","view_paper":"https://pith.science/paper/KEPNB2LD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.14499&json=true","fetch_graph":"https://pith.science/api/pith-number/KEPNB2LDY2EBQNX5EHMYPXFEY3/graph.json","fetch_events":"https://pith.science/api/pith-number/KEPNB2LDY2EBQNX5EHMYPXFEY3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KEPNB2LDY2EBQNX5EHMYPXFEY3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KEPNB2LDY2EBQNX5EHMYPXFEY3/action/storage_attestation","attest_author":"https://pith.science/pith/KEPNB2LDY2EBQNX5EHMYPXFEY3/action/author_attestation","sign_citation":"https://pith.science/pith/KEPNB2LDY2EBQNX5EHMYPXFEY3/action/citation_signature","submit_replication":"https://pith.science/pith/KEPNB2LDY2EBQNX5EHMYPXFEY3/action/replication_record"}},"created_at":"2026-07-05T11:40:08.316321+00:00","updated_at":"2026-07-05T11:40:08.316321+00:00"}