{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:FCUS3IEWLVQOWB7PMJE3YEFJXF","short_pith_number":"pith:FCUS3IEW","schema_version":"1.0","canonical_sha256":"28a92da0965d60eb07ef6249bc10a9b961e38492515fcdc95dce2321dee3b595","source":{"kind":"arxiv","id":"1307.0046","version":2},"attestation_state":"computed","paper":{"title":"A probabilistic solution to the Stroock-Williams equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Goran Peskir","submitted_at":"2013-06-28T22:54:00Z","abstract_excerpt":"We consider the initial boundary value problem \\begin{eqnarray*}u_t=\\mu u_x+\\tfrac{1}{2}u_{xx}\\qquad (t>0,x\\ge0),\\\\u(0,x)=f(x)\\qquad (x\\ge0),\\\\u_t(t,0)=\\nu u_x(t,0)\\qquad (t>0)\\end{eqnarray*} of Stroock and Williams [Comm. Pure Appl. Math. 58 (2005) 1116-1148] where $\\mu,\\nu\\in \\mathbb{R}$ and the boundary condition is not of Feller's type when $\\nu<0$. We show that when $f$ belongs to $C_b^1$ with $f(\\infty)=0$ then the following probabilistic representation of the solution is valid: \\[u(t,x)=\\mathsf{E}_x\\bigl[f(X_t)\\bigr]-\\mathsf{E}_x\\biggl[f'(X_t)\\int_0^{\\ell_t^0(X)}e^{-2(\\nu-\\mu)s}\\,ds\\big"},"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":"1307.0046","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-28T22:54:00Z","cross_cats_sorted":[],"title_canon_sha256":"2d9586f79fce0cd50614b059917d8acd4bf0e8c3d3b3ca681c37c157f6896f9f","abstract_canon_sha256":"7c281df978a53d8bbcbdfcccc3d6d4480b3887a0140e38618e8991a87c09557a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:45.545106Z","signature_b64":"aZ02GPb/ziYDWYw8kS6vh7KX2P/iIN+1GrrmB2FJZokouS0c/wP1aa3cd9+kypbLcILCnY3XTQupDUW5J8+DDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28a92da0965d60eb07ef6249bc10a9b961e38492515fcdc95dce2321dee3b595","last_reissued_at":"2026-05-18T02:43:45.544674Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:45.544674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A probabilistic solution to the Stroock-Williams equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Goran Peskir","submitted_at":"2013-06-28T22:54:00Z","abstract_excerpt":"We consider the initial boundary value problem \\begin{eqnarray*}u_t=\\mu u_x+\\tfrac{1}{2}u_{xx}\\qquad (t>0,x\\ge0),\\\\u(0,x)=f(x)\\qquad (x\\ge0),\\\\u_t(t,0)=\\nu u_x(t,0)\\qquad (t>0)\\end{eqnarray*} of Stroock and Williams [Comm. Pure Appl. Math. 58 (2005) 1116-1148] where $\\mu,\\nu\\in \\mathbb{R}$ and the boundary condition is not of Feller's type when $\\nu<0$. We show that when $f$ belongs to $C_b^1$ with $f(\\infty)=0$ then the following probabilistic representation of the solution is valid: \\[u(t,x)=\\mathsf{E}_x\\bigl[f(X_t)\\bigr]-\\mathsf{E}_x\\biggl[f'(X_t)\\int_0^{\\ell_t^0(X)}e^{-2(\\nu-\\mu)s}\\,ds\\big"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0046","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":"1307.0046","created_at":"2026-05-18T02:43:45.544737+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.0046v2","created_at":"2026-05-18T02:43:45.544737+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0046","created_at":"2026-05-18T02:43:45.544737+00:00"},{"alias_kind":"pith_short_12","alias_value":"FCUS3IEWLVQO","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"FCUS3IEWLVQOWB7P","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"FCUS3IEW","created_at":"2026-05-18T12:27:45.050594+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/FCUS3IEWLVQOWB7PMJE3YEFJXF","json":"https://pith.science/pith/FCUS3IEWLVQOWB7PMJE3YEFJXF.json","graph_json":"https://pith.science/api/pith-number/FCUS3IEWLVQOWB7PMJE3YEFJXF/graph.json","events_json":"https://pith.science/api/pith-number/FCUS3IEWLVQOWB7PMJE3YEFJXF/events.json","paper":"https://pith.science/paper/FCUS3IEW"},"agent_actions":{"view_html":"https://pith.science/pith/FCUS3IEWLVQOWB7PMJE3YEFJXF","download_json":"https://pith.science/pith/FCUS3IEWLVQOWB7PMJE3YEFJXF.json","view_paper":"https://pith.science/paper/FCUS3IEW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.0046&json=true","fetch_graph":"https://pith.science/api/pith-number/FCUS3IEWLVQOWB7PMJE3YEFJXF/graph.json","fetch_events":"https://pith.science/api/pith-number/FCUS3IEWLVQOWB7PMJE3YEFJXF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FCUS3IEWLVQOWB7PMJE3YEFJXF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FCUS3IEWLVQOWB7PMJE3YEFJXF/action/storage_attestation","attest_author":"https://pith.science/pith/FCUS3IEWLVQOWB7PMJE3YEFJXF/action/author_attestation","sign_citation":"https://pith.science/pith/FCUS3IEWLVQOWB7PMJE3YEFJXF/action/citation_signature","submit_replication":"https://pith.science/pith/FCUS3IEWLVQOWB7PMJE3YEFJXF/action/replication_record"}},"created_at":"2026-05-18T02:43:45.544737+00:00","updated_at":"2026-05-18T02:43:45.544737+00:00"}