{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:VSI2XJZWFGLH2Z5W4PZEGBFRHP","short_pith_number":"pith:VSI2XJZW","canonical_record":{"source":{"id":"2506.07119","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-06-08T13:03:23Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"ce18f1cc56a085d817eb3e4bc03f0af31a4c334c3df6e30184d46e70d91f7005","abstract_canon_sha256":"033de5a300d2eff7f65139692290676406c71adf1c221afd02e980d356b07a58"},"schema_version":"1.0"},"canonical_sha256":"ac91aba73629967d67b6e3f24304b13bf79caddb29d5b94887ac85b32baf4400","source":{"kind":"arxiv","id":"2506.07119","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.07119","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"arxiv_version","alias_value":"2506.07119v1","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.07119","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"pith_short_12","alias_value":"VSI2XJZWFGLH","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"pith_short_16","alias_value":"VSI2XJZWFGLH2Z5W","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"pith_short_8","alias_value":"VSI2XJZW","created_at":"2026-07-05T11:18:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:VSI2XJZWFGLH2Z5W4PZEGBFRHP","target":"record","payload":{"canonical_record":{"source":{"id":"2506.07119","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-06-08T13:03:23Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"ce18f1cc56a085d817eb3e4bc03f0af31a4c334c3df6e30184d46e70d91f7005","abstract_canon_sha256":"033de5a300d2eff7f65139692290676406c71adf1c221afd02e980d356b07a58"},"schema_version":"1.0"},"canonical_sha256":"ac91aba73629967d67b6e3f24304b13bf79caddb29d5b94887ac85b32baf4400","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:18:01.895941Z","signature_b64":"BkYs2FmC2sjAN8taRghWGeDTT42zywkalJWVpyV7oh4mrgkZv9fRyQtX43EZMHuHXBLT48/GLAnAg8yNt6GiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac91aba73629967d67b6e3f24304b13bf79caddb29d5b94887ac85b32baf4400","last_reissued_at":"2026-07-05T11:18:01.895438Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:18:01.895438Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.07119","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":"2026-07-05T11:18:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XADx5xWO+p9shXGhfqasyBGDh7MGbFE2v51cJie1+vqLWoVB3eDTAj6XHWIXNoRwVRMFHdbzaB7vTjxS/O8WAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T05:53:22.303777Z"},"content_sha256":"fdd7b8923e8548fe1c93274685e6fefcde6e158be905f81b23ff681937c7ccf8","schema_version":"1.0","event_id":"sha256:fdd7b8923e8548fe1c93274685e6fefcde6e158be905f81b23ff681937c7ccf8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:VSI2XJZWFGLH2Z5W4PZEGBFRHP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant measures for stochastic Burgers equation on unbounded domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DS","authors_text":"Zhenxin Liu, Zhiyuan Shi","submitted_at":"2025-06-08T13:03:23Z","abstract_excerpt":"In this paper, we investigate the stochastic damped Burgers equation with multiplicative noise defined on the entire real line. We demonstrate the existence and uniqueness of a mild solution to the stochastic damped Burgers equation and establish that the solution is uniformly bounded in time. Furthermore, by employing the uniform estimates on the tails of the solution, we obtain the tightness of a family of probability distributions of the solution. Subsequently, by applying the Krylov-Bogolioubov theorem, we establish the existence of invariant measures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07119","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/2506.07119/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":"2026-07-05T11:18:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v2lptIlEgZSfqK84LJdvrC2acazWexA3e+DbccUGH6l9vWzqZKHBIML1YDdHLKvOqIUjEGUxrizUQHJOY3zaAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T05:53:22.304303Z"},"content_sha256":"6a2b0b5f54999dd70e256b6dc8a4a498ccfdacf8b9f1167bb5a9ddf0fe4602ed","schema_version":"1.0","event_id":"sha256:6a2b0b5f54999dd70e256b6dc8a4a498ccfdacf8b9f1167bb5a9ddf0fe4602ed"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VSI2XJZWFGLH2Z5W4PZEGBFRHP/bundle.json","state_url":"https://pith.science/pith/VSI2XJZWFGLH2Z5W4PZEGBFRHP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VSI2XJZWFGLH2Z5W4PZEGBFRHP/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":"2026-08-09T05:53:22Z","links":{"resolver":"https://pith.science/pith/VSI2XJZWFGLH2Z5W4PZEGBFRHP","bundle":"https://pith.science/pith/VSI2XJZWFGLH2Z5W4PZEGBFRHP/bundle.json","state":"https://pith.science/pith/VSI2XJZWFGLH2Z5W4PZEGBFRHP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VSI2XJZWFGLH2Z5W4PZEGBFRHP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:VSI2XJZWFGLH2Z5W4PZEGBFRHP","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":"033de5a300d2eff7f65139692290676406c71adf1c221afd02e980d356b07a58","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-06-08T13:03:23Z","title_canon_sha256":"ce18f1cc56a085d817eb3e4bc03f0af31a4c334c3df6e30184d46e70d91f7005"},"schema_version":"1.0","source":{"id":"2506.07119","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.07119","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"arxiv_version","alias_value":"2506.07119v1","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.07119","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"pith_short_12","alias_value":"VSI2XJZWFGLH","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"pith_short_16","alias_value":"VSI2XJZWFGLH2Z5W","created_at":"2026-07-05T11:18:01Z"},{"alias_kind":"pith_short_8","alias_value":"VSI2XJZW","created_at":"2026-07-05T11:18:01Z"}],"graph_snapshots":[{"event_id":"sha256:6a2b0b5f54999dd70e256b6dc8a4a498ccfdacf8b9f1167bb5a9ddf0fe4602ed","target":"graph","created_at":"2026-07-05T11:18:01Z","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/2506.07119/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we investigate the stochastic damped Burgers equation with multiplicative noise defined on the entire real line. We demonstrate the existence and uniqueness of a mild solution to the stochastic damped Burgers equation and establish that the solution is uniformly bounded in time. Furthermore, by employing the uniform estimates on the tails of the solution, we obtain the tightness of a family of probability distributions of the solution. Subsequently, by applying the Krylov-Bogolioubov theorem, we establish the existence of invariant measures.","authors_text":"Zhenxin Liu, Zhiyuan Shi","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-06-08T13:03:23Z","title":"Invariant measures for stochastic Burgers equation on unbounded domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07119","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:fdd7b8923e8548fe1c93274685e6fefcde6e158be905f81b23ff681937c7ccf8","target":"record","created_at":"2026-07-05T11:18:01Z","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":"033de5a300d2eff7f65139692290676406c71adf1c221afd02e980d356b07a58","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-06-08T13:03:23Z","title_canon_sha256":"ce18f1cc56a085d817eb3e4bc03f0af31a4c334c3df6e30184d46e70d91f7005"},"schema_version":"1.0","source":{"id":"2506.07119","kind":"arxiv","version":1}},"canonical_sha256":"ac91aba73629967d67b6e3f24304b13bf79caddb29d5b94887ac85b32baf4400","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac91aba73629967d67b6e3f24304b13bf79caddb29d5b94887ac85b32baf4400","first_computed_at":"2026-07-05T11:18:01.895438Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:18:01.895438Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BkYs2FmC2sjAN8taRghWGeDTT42zywkalJWVpyV7oh4mrgkZv9fRyQtX43EZMHuHXBLT48/GLAnAg8yNt6GiCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:18:01.895941Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.07119","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fdd7b8923e8548fe1c93274685e6fefcde6e158be905f81b23ff681937c7ccf8","sha256:6a2b0b5f54999dd70e256b6dc8a4a498ccfdacf8b9f1167bb5a9ddf0fe4602ed"],"state_sha256":"a49253689334a96a9388d891f04326eca0609f02fef8d0fd1ba835ea636eb066"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w6Fp6klW4hcrsH2QNkEZ3Nl1LKZ160rDRmoEWjUhoLx/vrwZe7DR1mPHvCBAiLVjB9HRcqzkXl7esVV23KG9AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T05:53:22.309654Z","bundle_sha256":"4b7823b4dac6e75ac2a3cf617f09a959a0d961e11afa3ffd16e29cd2f07d2985"}}