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Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise."},"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":"2607.01611","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T02:29:01Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"fd2c93f23e4a4e4a3f582767f430ce9ac7e08ad4e423fd851fa29d6ce406fa79","abstract_canon_sha256":"c01598e2252e750f1096e9d47b28b0e364cb6093db33e3128b3cd6b23375890e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:03.320367Z","signature_b64":"vSGmUUf7Ec3M9vPN2IdnAjOXJ/j6gfM+ifvFJwZY5K7S9LIB2ON75SwAV90pGdmLuZeh6d1M4rlpXkoqWaKkCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e5fd7aaea9acd5b19a5e26292f75308c507ed80efb102d3ba68e769bf55f637","last_reissued_at":"2026-07-03T01:17:03.319950Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:03.319950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invariant Measure of the Camassa-Holm Equation with Linear Multiplicative Noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Pei Zheng, Wei Luo, Zhaoyang Yin","submitted_at":"2026-07-02T02:29:01Z","abstract_excerpt":"In this paper, we prove that the solution map of Camassa-Holm equation with linear multiplicative noise\n  $$\n  \\left\\{\n  \\begin{array}{l}\n  {\\rm d}u+(u\\partial_xu+\\partial_xP[u])\\,{\\rm d}t=\\beta u\\,{\\rm d}W,\n  u(0,x)=u_0(x),\n  P[u]=(1-\\partial_x^2)^{-1}\\left(u^2+\\frac 1 2(\\partial_x u)^2\\right)\n  \\end{array}\n  \\right.\n  $$\n  depends almost surely continuously on the deterministic initial data in $H^s$ for $s>3/2$. Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01611","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/2607.01611/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":"2607.01611","created_at":"2026-07-03T01:17:03.320006+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.01611v1","created_at":"2026-07-03T01:17:03.320006+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01611","created_at":"2026-07-03T01:17:03.320006+00:00"},{"alias_kind":"pith_short_12","alias_value":"LZP5PKXKTLGV","created_at":"2026-07-03T01:17:03.320006+00:00"},{"alias_kind":"pith_short_16","alias_value":"LZP5PKXKTLGVWGNF","created_at":"2026-07-03T01:17:03.320006+00:00"},{"alias_kind":"pith_short_8","alias_value":"LZP5PKXK","created_at":"2026-07-03T01:17:03.320006+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/LZP5PKXKTLGVWGNF4JRJF52TBD","json":"https://pith.science/pith/LZP5PKXKTLGVWGNF4JRJF52TBD.json","graph_json":"https://pith.science/api/pith-number/LZP5PKXKTLGVWGNF4JRJF52TBD/graph.json","events_json":"https://pith.science/api/pith-number/LZP5PKXKTLGVWGNF4JRJF52TBD/events.json","paper":"https://pith.science/paper/LZP5PKXK"},"agent_actions":{"view_html":"https://pith.science/pith/LZP5PKXKTLGVWGNF4JRJF52TBD","download_json":"https://pith.science/pith/LZP5PKXKTLGVWGNF4JRJF52TBD.json","view_paper":"https://pith.science/paper/LZP5PKXK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.01611&json=true","fetch_graph":"https://pith.science/api/pith-number/LZP5PKXKTLGVWGNF4JRJF52TBD/graph.json","fetch_events":"https://pith.science/api/pith-number/LZP5PKXKTLGVWGNF4JRJF52TBD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LZP5PKXKTLGVWGNF4JRJF52TBD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LZP5PKXKTLGVWGNF4JRJF52TBD/action/storage_attestation","attest_author":"https://pith.science/pith/LZP5PKXKTLGVWGNF4JRJF52TBD/action/author_attestation","sign_citation":"https://pith.science/pith/LZP5PKXKTLGVWGNF4JRJF52TBD/action/citation_signature","submit_replication":"https://pith.science/pith/LZP5PKXKTLGVWGNF4JRJF52TBD/action/replication_record"}},"created_at":"2026-07-03T01:17:03.320006+00:00","updated_at":"2026-07-03T01:17:03.320006+00:00"}