{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:GEX2J3ZOVEFQC4SOXFDJLLJ36M","short_pith_number":"pith:GEX2J3ZO","canonical_record":{"source":{"id":"2206.02240","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-05T19:05:11Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"41814c604a60cdb295d4dca50b3c5a7176c1ec7839050f8fc60b60a23581109a","abstract_canon_sha256":"b390125c901693da8f78e5b264b636a43355019463e23f3aa4c417a940130243"},"schema_version":"1.0"},"canonical_sha256":"312fa4ef2ea90b01724eb94695ad3bf300508e26497fbad039e6f3f6e6ff0a60","source":{"kind":"arxiv","id":"2206.02240","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2206.02240","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"arxiv_version","alias_value":"2206.02240v1","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.02240","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"pith_short_12","alias_value":"GEX2J3ZOVEFQ","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"pith_short_16","alias_value":"GEX2J3ZOVEFQC4SO","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"pith_short_8","alias_value":"GEX2J3ZO","created_at":"2026-07-05T04:29:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:GEX2J3ZOVEFQC4SOXFDJLLJ36M","target":"record","payload":{"canonical_record":{"source":{"id":"2206.02240","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-05T19:05:11Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"41814c604a60cdb295d4dca50b3c5a7176c1ec7839050f8fc60b60a23581109a","abstract_canon_sha256":"b390125c901693da8f78e5b264b636a43355019463e23f3aa4c417a940130243"},"schema_version":"1.0"},"canonical_sha256":"312fa4ef2ea90b01724eb94695ad3bf300508e26497fbad039e6f3f6e6ff0a60","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:29:15.722767Z","signature_b64":"VUyuxYit9EjH4YMXbp2LE60/L0sb3WmXzPuBrjXjCZSYYKYXChKzShh0Ws9NbEC5+Cw8phpPvOTH4Hks5UuyBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"312fa4ef2ea90b01724eb94695ad3bf300508e26497fbad039e6f3f6e6ff0a60","last_reissued_at":"2026-07-05T04:29:15.722325Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:29:15.722325Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2206.02240","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":"2026-07-05T04:29:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JsMQ7B7veBeUv0FGRmiFcFmtMl+WhWiuDZQSxwH8rA/ZUPAx3whKndIsCVBYVKMVMryFyJEqp6GT5GmkoUeoDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T11:04:21.892020Z"},"content_sha256":"581824df301cbafd96486e9ae19f462f20fdc60b40160b7c4eaa97c3b3add868","schema_version":"1.0","event_id":"sha256:581824df301cbafd96486e9ae19f462f20fdc60b40160b7c4eaa97c3b3add868"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:GEX2J3ZOVEFQC4SOXFDJLLJ36M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stationary measures for stochastic differential equations with degenerate damping","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"Jacob Bedrossian, Kyle Liss","submitted_at":"2022-06-05T19:05:11Z","abstract_excerpt":"A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in $\\mathbb R^n$ with a quadratic, conservative nonlinearity $B(x,x)$ and a linear damping term $-Ax$ which is degenerate in the sense that $\\mathrm{ker} A \\neq \\emptyset$. We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforwa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.02240","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/2206.02240/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":"2026-07-05T04:29:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a5WVTuuwU/LODrONzUrZM5U3yfy518+tnM+pY6sXhlcIzJXzgZEgSfeNDRtY8MZDP8YN5romXoa9U/qFr7YfDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T11:04:21.892996Z"},"content_sha256":"e1f96bc87fb46d26d56d448c49b4ef7db8dd161310cc482127588586b1a561b1","schema_version":"1.0","event_id":"sha256:e1f96bc87fb46d26d56d448c49b4ef7db8dd161310cc482127588586b1a561b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GEX2J3ZOVEFQC4SOXFDJLLJ36M/bundle.json","state_url":"https://pith.science/pith/GEX2J3ZOVEFQC4SOXFDJLLJ36M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GEX2J3ZOVEFQC4SOXFDJLLJ36M/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":"2026-08-08T11:04:21Z","links":{"resolver":"https://pith.science/pith/GEX2J3ZOVEFQC4SOXFDJLLJ36M","bundle":"https://pith.science/pith/GEX2J3ZOVEFQC4SOXFDJLLJ36M/bundle.json","state":"https://pith.science/pith/GEX2J3ZOVEFQC4SOXFDJLLJ36M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GEX2J3ZOVEFQC4SOXFDJLLJ36M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:GEX2J3ZOVEFQC4SOXFDJLLJ36M","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":"b390125c901693da8f78e5b264b636a43355019463e23f3aa4c417a940130243","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-05T19:05:11Z","title_canon_sha256":"41814c604a60cdb295d4dca50b3c5a7176c1ec7839050f8fc60b60a23581109a"},"schema_version":"1.0","source":{"id":"2206.02240","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2206.02240","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"arxiv_version","alias_value":"2206.02240v1","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.02240","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"pith_short_12","alias_value":"GEX2J3ZOVEFQ","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"pith_short_16","alias_value":"GEX2J3ZOVEFQC4SO","created_at":"2026-07-05T04:29:15Z"},{"alias_kind":"pith_short_8","alias_value":"GEX2J3ZO","created_at":"2026-07-05T04:29:15Z"}],"graph_snapshots":[{"event_id":"sha256:e1f96bc87fb46d26d56d448c49b4ef7db8dd161310cc482127588586b1a561b1","target":"graph","created_at":"2026-07-05T04:29:15Z","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/2206.02240/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in $\\mathbb R^n$ with a quadratic, conservative nonlinearity $B(x,x)$ and a linear damping term $-Ax$ which is degenerate in the sense that $\\mathrm{ker} A \\neq \\emptyset$. We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforwa","authors_text":"Jacob Bedrossian, Kyle Liss","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-05T19:05:11Z","title":"Stationary measures for stochastic differential equations with degenerate damping"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.02240","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":"sha256:581824df301cbafd96486e9ae19f462f20fdc60b40160b7c4eaa97c3b3add868","target":"record","created_at":"2026-07-05T04:29:15Z","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":"b390125c901693da8f78e5b264b636a43355019463e23f3aa4c417a940130243","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-05T19:05:11Z","title_canon_sha256":"41814c604a60cdb295d4dca50b3c5a7176c1ec7839050f8fc60b60a23581109a"},"schema_version":"1.0","source":{"id":"2206.02240","kind":"arxiv","version":1}},"canonical_sha256":"312fa4ef2ea90b01724eb94695ad3bf300508e26497fbad039e6f3f6e6ff0a60","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"312fa4ef2ea90b01724eb94695ad3bf300508e26497fbad039e6f3f6e6ff0a60","first_computed_at":"2026-07-05T04:29:15.722325Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:29:15.722325Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VUyuxYit9EjH4YMXbp2LE60/L0sb3WmXzPuBrjXjCZSYYKYXChKzShh0Ws9NbEC5+Cw8phpPvOTH4Hks5UuyBw==","signature_status":"signed_v1","signed_at":"2026-07-05T04:29:15.722767Z","signed_message":"canonical_sha256_bytes"},"source_id":"2206.02240","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:581824df301cbafd96486e9ae19f462f20fdc60b40160b7c4eaa97c3b3add868","sha256:e1f96bc87fb46d26d56d448c49b4ef7db8dd161310cc482127588586b1a561b1"],"state_sha256":"6164c5d756cde8828e7d8496a02d17535052b8dad91f0d1ee0cce02f127cebb2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SxHc+4M0HSJI7IX0N858jZMlth6GiP7LqjsDl6Z/PzSwlum770ob0Y4PbqyC3qRCGpSSgnxKvUtXlnikeLEWCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T11:04:21.899289Z","bundle_sha256":"af6fdcfbdd8444ccdecc9daf931d8cc75195bd6756503b8fb98eb20eb4e66d22"}}