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We construct a Banach space ${\\mathscr H}$ of measurable functions such that the quadratic covariation $[f(B),B]$ and the integral $\\int_{\\mathbb R}f(x){\\mathscr L}(dx,t)$ exist provided $f\\in {\\mathscr H}$. Moreover, the Bouleau-Yor identity $$ [f(B),B]_t=-2^{1-K}\\int_{\\mathbb R}f(x){\\mathscr L}(dx,t),\\qquad t\\geq 0, $$ holds for all $f\\in {\\mathscr H}$."},"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":"1212.6347","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.PR","submitted_at":"2012-12-27T11:16:03Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"eaaa008f68cd14b2e6445162599d60835f91f4f28b01373e2b976b8c966ae212","abstract_canon_sha256":"b06033d76c31a5bde6268030f5f49e62457856d357f4f9be8a45ebb8f773783a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:52:23.180534Z","signature_b64":"R0P20KPf3f8wrIQhQhXwV6ChhNr+yO0k6d/6o1n+e5cgu5K8bgvISOcPlT8ce0hgHi4Y6b3HBLe5ZqLFnjs0Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"602ed1985c30430f0068de114f4d0b4062a4d73e47a1e4784283fbde38d0cf02","last_reissued_at":"2026-05-18T01:52:23.179988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:52:23.179988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Bouleau-Yor identity for a bi-fractional Brownian motion","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Bo Gao, Junfeng Liu, Litan Yan","submitted_at":"2012-12-27T11:16:03Z","abstract_excerpt":"Let $B$ be a bi-fractional Brownian motion with indices $H\\in (0,1),K\\in (0,1]$, $2HK=1$ and let ${\\mathscr L}(x,t)$ be its local time process. We construct a Banach space ${\\mathscr H}$ of measurable functions such that the quadratic covariation $[f(B),B]$ and the integral $\\int_{\\mathbb R}f(x){\\mathscr L}(dx,t)$ exist provided $f\\in {\\mathscr H}$. Moreover, the Bouleau-Yor identity $$ [f(B),B]_t=-2^{1-K}\\int_{\\mathbb R}f(x){\\mathscr L}(dx,t),\\qquad t\\geq 0, $$ holds for all $f\\in {\\mathscr H}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.6347","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1212.6347","created_at":"2026-05-18T01:52:23.180064+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.6347v1","created_at":"2026-05-18T01:52:23.180064+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.6347","created_at":"2026-05-18T01:52:23.180064+00:00"},{"alias_kind":"pith_short_12","alias_value":"MAXNDGC4GBBQ","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MAXNDGC4GBBQ6ADI","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MAXNDGC4","created_at":"2026-05-18T12:27:14.488303+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/MAXNDGC4GBBQ6ADI3YIU6TILIB","json":"https://pith.science/pith/MAXNDGC4GBBQ6ADI3YIU6TILIB.json","graph_json":"https://pith.science/api/pith-number/MAXNDGC4GBBQ6ADI3YIU6TILIB/graph.json","events_json":"https://pith.science/api/pith-number/MAXNDGC4GBBQ6ADI3YIU6TILIB/events.json","paper":"https://pith.science/paper/MAXNDGC4"},"agent_actions":{"view_html":"https://pith.science/pith/MAXNDGC4GBBQ6ADI3YIU6TILIB","download_json":"https://pith.science/pith/MAXNDGC4GBBQ6ADI3YIU6TILIB.json","view_paper":"https://pith.science/paper/MAXNDGC4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.6347&json=true","fetch_graph":"https://pith.science/api/pith-number/MAXNDGC4GBBQ6ADI3YIU6TILIB/graph.json","fetch_events":"https://pith.science/api/pith-number/MAXNDGC4GBBQ6ADI3YIU6TILIB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MAXNDGC4GBBQ6ADI3YIU6TILIB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MAXNDGC4GBBQ6ADI3YIU6TILIB/action/storage_attestation","attest_author":"https://pith.science/pith/MAXNDGC4GBBQ6ADI3YIU6TILIB/action/author_attestation","sign_citation":"https://pith.science/pith/MAXNDGC4GBBQ6ADI3YIU6TILIB/action/citation_signature","submit_replication":"https://pith.science/pith/MAXNDGC4GBBQ6ADI3YIU6TILIB/action/replication_record"}},"created_at":"2026-05-18T01:52:23.180064+00:00","updated_at":"2026-05-18T01:52:23.180064+00:00"}