{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2005:XEHZXLEU3WVVNJ4DNRTMWMGFH6","short_pith_number":"pith:XEHZXLEU","schema_version":"1.0","canonical_sha256":"b90f9bac94ddab56a7836c66cb30c53f8c22e71bc7d85f21db4824f979032862","source":{"kind":"arxiv","id":"math/0501486","version":1},"attestation_state":"computed","paper":{"title":"Synchronous couplings of reflected Brownian motions in smooth domains","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Krzysztof Burdzy, Peter Jones, Zhen-Qing Chen","submitted_at":"2005-01-27T07:04:44Z","abstract_excerpt":"For every bounded planar domain $D$ with a smooth boundary, we define a `Lyapunov exponent' $\\Lambda(D)$ using a fairly explicit formula. We consider two reflected Brownian motions in $D$, driven by the same Brownian motion (i.e., a `synchronous coupling'). If $\\Lambda(D)>0$ then the distance between the two Brownian particles goes to 0 exponentially fast with rate $\\Lambda (D)/(2|D|)$ as time goes to infinity. The exponent $\\Lambda(D)$ is strictly positive if the domain has at most one hole. It is an open problem whether there exists a domain with $\\Lambda(D)<0$."},"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":"math/0501486","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2005-01-27T07:04:44Z","cross_cats_sorted":[],"title_canon_sha256":"c6eec6797b2d496f4d10db3a8d74cb4a9b5b856d32be01b4ad3da1c1692eb4b3","abstract_canon_sha256":"2dba4c30c6bfc8572c986d905fc056d46dc06691fec0478052909b4b39ff7d34"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:39:33.894715Z","signature_b64":"8PA9dppLFy2TX2AI5QW8MY5My4AmXEgN/BrKml5bvxUTO2Pyn3gSv5WfPOkhZsY9ewVvK9PeckC6nRManp+qDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b90f9bac94ddab56a7836c66cb30c53f8c22e71bc7d85f21db4824f979032862","last_reissued_at":"2026-07-04T14:39:33.894339Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:39:33.894339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Synchronous couplings of reflected Brownian motions in smooth domains","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Krzysztof Burdzy, Peter Jones, Zhen-Qing Chen","submitted_at":"2005-01-27T07:04:44Z","abstract_excerpt":"For every bounded planar domain $D$ with a smooth boundary, we define a `Lyapunov exponent' $\\Lambda(D)$ using a fairly explicit formula. We consider two reflected Brownian motions in $D$, driven by the same Brownian motion (i.e., a `synchronous coupling'). If $\\Lambda(D)>0$ then the distance between the two Brownian particles goes to 0 exponentially fast with rate $\\Lambda (D)/(2|D|)$ as time goes to infinity. The exponent $\\Lambda(D)$ is strictly positive if the domain has at most one hole. It is an open problem whether there exists a domain with $\\Lambda(D)<0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0501486","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/math/0501486/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":"math/0501486","created_at":"2026-07-04T14:39:33.894400+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0501486v1","created_at":"2026-07-04T14:39:33.894400+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0501486","created_at":"2026-07-04T14:39:33.894400+00:00"},{"alias_kind":"pith_short_12","alias_value":"XEHZXLEU3WVV","created_at":"2026-07-04T14:39:33.894400+00:00"},{"alias_kind":"pith_short_16","alias_value":"XEHZXLEU3WVVNJ4D","created_at":"2026-07-04T14:39:33.894400+00:00"},{"alias_kind":"pith_short_8","alias_value":"XEHZXLEU","created_at":"2026-07-04T14:39:33.894400+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/XEHZXLEU3WVVNJ4DNRTMWMGFH6","json":"https://pith.science/pith/XEHZXLEU3WVVNJ4DNRTMWMGFH6.json","graph_json":"https://pith.science/api/pith-number/XEHZXLEU3WVVNJ4DNRTMWMGFH6/graph.json","events_json":"https://pith.science/api/pith-number/XEHZXLEU3WVVNJ4DNRTMWMGFH6/events.json","paper":"https://pith.science/paper/XEHZXLEU"},"agent_actions":{"view_html":"https://pith.science/pith/XEHZXLEU3WVVNJ4DNRTMWMGFH6","download_json":"https://pith.science/pith/XEHZXLEU3WVVNJ4DNRTMWMGFH6.json","view_paper":"https://pith.science/paper/XEHZXLEU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0501486&json=true","fetch_graph":"https://pith.science/api/pith-number/XEHZXLEU3WVVNJ4DNRTMWMGFH6/graph.json","fetch_events":"https://pith.science/api/pith-number/XEHZXLEU3WVVNJ4DNRTMWMGFH6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XEHZXLEU3WVVNJ4DNRTMWMGFH6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XEHZXLEU3WVVNJ4DNRTMWMGFH6/action/storage_attestation","attest_author":"https://pith.science/pith/XEHZXLEU3WVVNJ4DNRTMWMGFH6/action/author_attestation","sign_citation":"https://pith.science/pith/XEHZXLEU3WVVNJ4DNRTMWMGFH6/action/citation_signature","submit_replication":"https://pith.science/pith/XEHZXLEU3WVVNJ4DNRTMWMGFH6/action/replication_record"}},"created_at":"2026-07-04T14:39:33.894400+00:00","updated_at":"2026-07-04T14:39:33.894400+00:00"}