{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:LTDEXLBRFZHBSD4U6HQO6EHQ2E","short_pith_number":"pith:LTDEXLBR","canonical_record":{"source":{"id":"2607.01645","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T03:17:44Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"49108f4b03dd4e9b441d55b97990cc554040062183dc19cd8b26bee00d7691e4","abstract_canon_sha256":"20285b319029b0efd93e183877426e8102273296f73f3b4871394a4c974f9e4b"},"schema_version":"1.0"},"canonical_sha256":"5cc64bac312e4e190f94f1e0ef10f0d11c5264ea295795752ae27f84c8875c12","source":{"kind":"arxiv","id":"2607.01645","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01645","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01645v1","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01645","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"pith_short_12","alias_value":"LTDEXLBRFZHB","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"pith_short_16","alias_value":"LTDEXLBRFZHBSD4U","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"pith_short_8","alias_value":"LTDEXLBR","created_at":"2026-07-03T01:17:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:LTDEXLBRFZHBSD4U6HQO6EHQ2E","target":"record","payload":{"canonical_record":{"source":{"id":"2607.01645","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T03:17:44Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"49108f4b03dd4e9b441d55b97990cc554040062183dc19cd8b26bee00d7691e4","abstract_canon_sha256":"20285b319029b0efd93e183877426e8102273296f73f3b4871394a4c974f9e4b"},"schema_version":"1.0"},"canonical_sha256":"5cc64bac312e4e190f94f1e0ef10f0d11c5264ea295795752ae27f84c8875c12","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:04.358020Z","signature_b64":"Y6oGuE12IXnaGef9HG+RxOI14NSyAHHH9TggXPfNBlN5yKHckymcjjOxdePKeXeWViPWVwy1sAIYiyPbd4koDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5cc64bac312e4e190f94f1e0ef10f0d11c5264ea295795752ae27f84c8875c12","last_reissued_at":"2026-07-03T01:17:04.357654Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:04.357654Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.01645","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-03T01:17:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/6g+Tn0uXWUnS/lY3Gf3HBtKgR6cX0f0lRvoD8aLbbnWO0tMN9pGXUZ9ipdBBw1bGSajBi+eWpLWC2em2n0cBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T16:47:44.990499Z"},"content_sha256":"9677c67748b0c8498095dbe6068cb5363b3b6dd5fe4923a8c5df1e2154b15fba","schema_version":"1.0","event_id":"sha256:9677c67748b0c8498095dbe6068cb5363b3b6dd5fe4923a8c5df1e2154b15fba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:LTDEXLBRFZHBSD4U6HQO6EHQ2E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global Existence of Weak Martingale Solutions to 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":"2026-07-02T03:17:44Z","abstract_excerpt":"In this paper, we consider the global existence and properties of $H^1$ martingale solution to the Camassa-Holm equation with linear multiplicative noise under periodic boundary conditions. The solution is obtained as limit of regular viscous approximate solutions to parabolic SPDEs, which are constructed using the Galerkin approximations ans the stochastic compactness method. The proof of convergence to a solution argues via tightness of the laws of the viscous approximations and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. In particular, by means of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01645","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/2607.01645/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-03T01:17:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4pJoT5bHjzh50W5gdyGzJTbbBEuzSqBC554uiD+JEKOov4WcamaY5jpfrO+ckCVDXFqbBrU5b1O9rHHOEzLiDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T16:47:44.990869Z"},"content_sha256":"6a3e681b168561d6972762666cc813932967a45aeb73440105417860dbf3d118","schema_version":"1.0","event_id":"sha256:6a3e681b168561d6972762666cc813932967a45aeb73440105417860dbf3d118"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LTDEXLBRFZHBSD4U6HQO6EHQ2E/bundle.json","state_url":"https://pith.science/pith/LTDEXLBRFZHBSD4U6HQO6EHQ2E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LTDEXLBRFZHBSD4U6HQO6EHQ2E/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-03T16:47:44Z","links":{"resolver":"https://pith.science/pith/LTDEXLBRFZHBSD4U6HQO6EHQ2E","bundle":"https://pith.science/pith/LTDEXLBRFZHBSD4U6HQO6EHQ2E/bundle.json","state":"https://pith.science/pith/LTDEXLBRFZHBSD4U6HQO6EHQ2E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LTDEXLBRFZHBSD4U6HQO6EHQ2E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:LTDEXLBRFZHBSD4U6HQO6EHQ2E","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"20285b319029b0efd93e183877426e8102273296f73f3b4871394a4c974f9e4b","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T03:17:44Z","title_canon_sha256":"49108f4b03dd4e9b441d55b97990cc554040062183dc19cd8b26bee00d7691e4"},"schema_version":"1.0","source":{"id":"2607.01645","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01645","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01645v1","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01645","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"pith_short_12","alias_value":"LTDEXLBRFZHB","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"pith_short_16","alias_value":"LTDEXLBRFZHBSD4U","created_at":"2026-07-03T01:17:04Z"},{"alias_kind":"pith_short_8","alias_value":"LTDEXLBR","created_at":"2026-07-03T01:17:04Z"}],"graph_snapshots":[{"event_id":"sha256:6a3e681b168561d6972762666cc813932967a45aeb73440105417860dbf3d118","target":"graph","created_at":"2026-07-03T01:17:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.01645/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we consider the global existence and properties of $H^1$ martingale solution to the Camassa-Holm equation with linear multiplicative noise under periodic boundary conditions. The solution is obtained as limit of regular viscous approximate solutions to parabolic SPDEs, which are constructed using the Galerkin approximations ans the stochastic compactness method. The proof of convergence to a solution argues via tightness of the laws of the viscous approximations and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. In particular, by means of t","authors_text":"Pei Zheng, Wei Luo, Zhaoyang Yin","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T03:17:44Z","title":"Global Existence of Weak Martingale Solutions to the Camassa-Holm Equation with Linear Multiplicative Noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01645","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9677c67748b0c8498095dbe6068cb5363b3b6dd5fe4923a8c5df1e2154b15fba","target":"record","created_at":"2026-07-03T01:17:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"20285b319029b0efd93e183877426e8102273296f73f3b4871394a4c974f9e4b","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-02T03:17:44Z","title_canon_sha256":"49108f4b03dd4e9b441d55b97990cc554040062183dc19cd8b26bee00d7691e4"},"schema_version":"1.0","source":{"id":"2607.01645","kind":"arxiv","version":1}},"canonical_sha256":"5cc64bac312e4e190f94f1e0ef10f0d11c5264ea295795752ae27f84c8875c12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5cc64bac312e4e190f94f1e0ef10f0d11c5264ea295795752ae27f84c8875c12","first_computed_at":"2026-07-03T01:17:04.357654Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:04.357654Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Y6oGuE12IXnaGef9HG+RxOI14NSyAHHH9TggXPfNBlN5yKHckymcjjOxdePKeXeWViPWVwy1sAIYiyPbd4koDw==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:04.358020Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.01645","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9677c67748b0c8498095dbe6068cb5363b3b6dd5fe4923a8c5df1e2154b15fba","sha256:6a3e681b168561d6972762666cc813932967a45aeb73440105417860dbf3d118"],"state_sha256":"f6befb64815f5704778bda0113fdd60ecdd79b407027f43857c6b6694e0633c5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tl0DINZiccA2lRF1DJ6Fc90VBv1NfLP8qvt1uMJ6rOhVmb+6WXpNvIrOgDAsM4WZKL4g8XgRnutznFQJ1Y/PDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T16:47:44.992779Z","bundle_sha256":"bde607932b00ab01363b304a53162233b0e5ab927ed244adcc81e1d8cdea323a"}}