{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:DAMN4FE76KTPR2ZCH2MQT24HRT","short_pith_number":"pith:DAMN4FE7","schema_version":"1.0","canonical_sha256":"1818de149ff2a6f8eb223e9909eb878cdbdf4c66e4e9b0409ef56e5dc71b1710","source":{"kind":"arxiv","id":"1602.08796","version":2},"attestation_state":"computed","paper":{"title":"Quadratic covariations for the solution to a stochastic heat equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Litan Yan, Xianye Yu, Xichao Sun","submitted_at":"2016-02-29T02:13:53Z","abstract_excerpt":"Let $u(t,x)$ be the solution to a stochastic heat equation $$ \\frac{\\partial}{\\partial t}u=\\frac12\\frac{\\partial^2}{\\partial x^2}u+\\frac{\\partial^2}{\\partial t\\partial x}X(t,x),\\quad t\\geq 0, x\\in {\\mathbb R} $$ with initial condition $u(0,x)\\equiv 0$, where $X$ is a time-space white noise. This paper is an attempt to study stochastic analysis questions of the solution $u(t,x)$. In fact, the solution is a Gaussian process such that the process $t\\mapsto u(t,\\cdot)$ is a bi-fractional Brownian motion seemed a fractional Brownian motion with Hurst index $H=\\frac14$ for every real number $x$. How"},"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":"1602.08796","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2016-02-29T02:13:53Z","cross_cats_sorted":[],"title_canon_sha256":"3853448fa26831eaed384995a2a62463dd7302d75eb49aeb97a55db8f089367c","abstract_canon_sha256":"4e988d28babd99e9e0b61575c7601bc2169ac61e3229d72924f2a200d800fe7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:49.371588Z","signature_b64":"vOkyBIUdL5uThObebIeC78Y0R06mxSZ6cHyOyymEJBnOFEKWehOdXLei80uEoW1SkVc0O2pyAJzyOrDFEIb6DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1818de149ff2a6f8eb223e9909eb878cdbdf4c66e4e9b0409ef56e5dc71b1710","last_reissued_at":"2026-05-18T01:19:49.371163Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:49.371163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quadratic covariations for the solution to a stochastic heat equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Litan Yan, Xianye Yu, Xichao Sun","submitted_at":"2016-02-29T02:13:53Z","abstract_excerpt":"Let $u(t,x)$ be the solution to a stochastic heat equation $$ \\frac{\\partial}{\\partial t}u=\\frac12\\frac{\\partial^2}{\\partial x^2}u+\\frac{\\partial^2}{\\partial t\\partial x}X(t,x),\\quad t\\geq 0, x\\in {\\mathbb R} $$ with initial condition $u(0,x)\\equiv 0$, where $X$ is a time-space white noise. This paper is an attempt to study stochastic analysis questions of the solution $u(t,x)$. In fact, the solution is a Gaussian process such that the process $t\\mapsto u(t,\\cdot)$ is a bi-fractional Brownian motion seemed a fractional Brownian motion with Hurst index $H=\\frac14$ for every real number $x$. How"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.08796","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1602.08796","created_at":"2026-05-18T01:19:49.371219+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.08796v2","created_at":"2026-05-18T01:19:49.371219+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.08796","created_at":"2026-05-18T01:19:49.371219+00:00"},{"alias_kind":"pith_short_12","alias_value":"DAMN4FE76KTP","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"DAMN4FE76KTPR2ZC","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"DAMN4FE7","created_at":"2026-05-18T12:30:09.641336+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/DAMN4FE76KTPR2ZCH2MQT24HRT","json":"https://pith.science/pith/DAMN4FE76KTPR2ZCH2MQT24HRT.json","graph_json":"https://pith.science/api/pith-number/DAMN4FE76KTPR2ZCH2MQT24HRT/graph.json","events_json":"https://pith.science/api/pith-number/DAMN4FE76KTPR2ZCH2MQT24HRT/events.json","paper":"https://pith.science/paper/DAMN4FE7"},"agent_actions":{"view_html":"https://pith.science/pith/DAMN4FE76KTPR2ZCH2MQT24HRT","download_json":"https://pith.science/pith/DAMN4FE76KTPR2ZCH2MQT24HRT.json","view_paper":"https://pith.science/paper/DAMN4FE7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.08796&json=true","fetch_graph":"https://pith.science/api/pith-number/DAMN4FE76KTPR2ZCH2MQT24HRT/graph.json","fetch_events":"https://pith.science/api/pith-number/DAMN4FE76KTPR2ZCH2MQT24HRT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DAMN4FE76KTPR2ZCH2MQT24HRT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DAMN4FE76KTPR2ZCH2MQT24HRT/action/storage_attestation","attest_author":"https://pith.science/pith/DAMN4FE76KTPR2ZCH2MQT24HRT/action/author_attestation","sign_citation":"https://pith.science/pith/DAMN4FE76KTPR2ZCH2MQT24HRT/action/citation_signature","submit_replication":"https://pith.science/pith/DAMN4FE76KTPR2ZCH2MQT24HRT/action/replication_record"}},"created_at":"2026-05-18T01:19:49.371219+00:00","updated_at":"2026-05-18T01:19:49.371219+00:00"}