{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:H2BS73AGP6CPZNCKUCFZ7YHXZI","short_pith_number":"pith:H2BS73AG","canonical_record":{"source":{"id":"1202.2563","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-12T20:21:41Z","cross_cats_sorted":["math-ph","math.MP","math.PR","physics.data-an"],"title_canon_sha256":"f3c02034d7c8334ee5cc179257cb31dc4de136ccae29749c49a6bacf310c2e57","abstract_canon_sha256":"0ce69c2b46673c53f488e7d6547a1c266279b80290b3c412b580ee8a27efc617"},"schema_version":"1.0"},"canonical_sha256":"3e832fec067f84fcb44aa08b9fe0f7ca0fe78817c32324639cede01c5fa45dc2","source":{"kind":"arxiv","id":"1202.2563","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.2563","created_at":"2026-05-18T01:58:30Z"},{"alias_kind":"arxiv_version","alias_value":"1202.2563v1","created_at":"2026-05-18T01:58:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.2563","created_at":"2026-05-18T01:58:30Z"},{"alias_kind":"pith_short_12","alias_value":"H2BS73AGP6CP","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"H2BS73AGP6CPZNCK","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"H2BS73AG","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:H2BS73AGP6CPZNCKUCFZ7YHXZI","target":"record","payload":{"canonical_record":{"source":{"id":"1202.2563","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-12T20:21:41Z","cross_cats_sorted":["math-ph","math.MP","math.PR","physics.data-an"],"title_canon_sha256":"f3c02034d7c8334ee5cc179257cb31dc4de136ccae29749c49a6bacf310c2e57","abstract_canon_sha256":"0ce69c2b46673c53f488e7d6547a1c266279b80290b3c412b580ee8a27efc617"},"schema_version":"1.0"},"canonical_sha256":"3e832fec067f84fcb44aa08b9fe0f7ca0fe78817c32324639cede01c5fa45dc2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:30.886565Z","signature_b64":"H4/bdFa3ggiY2K6IpikqwFE8plafHAFVl+VlqgYX2hCkRKzzLIVwTxG3yXaVDmu7kUegJo2x6IMbgLCGebcOCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e832fec067f84fcb44aa08b9fe0f7ca0fe78817c32324639cede01c5fa45dc2","last_reissued_at":"2026-05-18T01:58:30.886181Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:30.886181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.2563","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":"2026-05-18T01:58:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MfNPMHPS3FkZCSODIaDuRm5RbKF7FFrFvRcRJWOY2SbaoqWnQqf7wL+GwmSfDWUqJBm1M50MKKrPrrSQ2mB0BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T16:02:40.130228Z"},"content_sha256":"fdb525e67c426e1ea40759ec5c6f51ba0701ea28de69b613a87aec871c8cb34a","schema_version":"1.0","event_id":"sha256:fdb525e67c426e1ea40759ec5c6f51ba0701ea28de69b613a87aec871c8cb34a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:H2BS73AGP6CPZNCKUCFZ7YHXZI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Levy processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR","physics.data-an"],"primary_cat":"math.DS","authors_text":"Jinqiao Duan, Xu Sun","submitted_at":"2012-02-12T20:21:41Z","abstract_excerpt":"The Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and are thus widely used to quantify random phenomena such as uncertainty propagation. For dynamical systems driven by non-Gaussian L\\'evy processes, however, it is difficult to obtain explicit forms of Fokker-Planck equations because the adjoint operators of the associated infinitesimal generators usually do not have exact formulation. In the present paper, Fokker- Planck equations are derived in terms of infinite series for nonlinear stochastic differential equations with non-Gaussian"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2563","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":""},"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":"2026-05-18T01:58:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BN9XObvtvHu9ZLlaHUb3Rm4H1UY0sHKbij36ORW6a3aKzy0KcSvv1mLI8i1qtgwlkf+V3Ksx8O3H6nPlBCROAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T16:02:40.130675Z"},"content_sha256":"b804407c334fee50c7dd0950985b761465cd526b28fee867a76ff40742865c6e","schema_version":"1.0","event_id":"sha256:b804407c334fee50c7dd0950985b761465cd526b28fee867a76ff40742865c6e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H2BS73AGP6CPZNCKUCFZ7YHXZI/bundle.json","state_url":"https://pith.science/pith/H2BS73AGP6CPZNCKUCFZ7YHXZI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H2BS73AGP6CPZNCKUCFZ7YHXZI/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":"2026-06-05T16:02:40Z","links":{"resolver":"https://pith.science/pith/H2BS73AGP6CPZNCKUCFZ7YHXZI","bundle":"https://pith.science/pith/H2BS73AGP6CPZNCKUCFZ7YHXZI/bundle.json","state":"https://pith.science/pith/H2BS73AGP6CPZNCKUCFZ7YHXZI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H2BS73AGP6CPZNCKUCFZ7YHXZI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:H2BS73AGP6CPZNCKUCFZ7YHXZI","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":"0ce69c2b46673c53f488e7d6547a1c266279b80290b3c412b580ee8a27efc617","cross_cats_sorted":["math-ph","math.MP","math.PR","physics.data-an"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-12T20:21:41Z","title_canon_sha256":"f3c02034d7c8334ee5cc179257cb31dc4de136ccae29749c49a6bacf310c2e57"},"schema_version":"1.0","source":{"id":"1202.2563","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.2563","created_at":"2026-05-18T01:58:30Z"},{"alias_kind":"arxiv_version","alias_value":"1202.2563v1","created_at":"2026-05-18T01:58:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.2563","created_at":"2026-05-18T01:58:30Z"},{"alias_kind":"pith_short_12","alias_value":"H2BS73AGP6CP","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"H2BS73AGP6CPZNCK","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"H2BS73AG","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:b804407c334fee50c7dd0950985b761465cd526b28fee867a76ff40742865c6e","target":"graph","created_at":"2026-05-18T01:58:30Z","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"},"paper":{"abstract_excerpt":"The Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and are thus widely used to quantify random phenomena such as uncertainty propagation. For dynamical systems driven by non-Gaussian L\\'evy processes, however, it is difficult to obtain explicit forms of Fokker-Planck equations because the adjoint operators of the associated infinitesimal generators usually do not have exact formulation. In the present paper, Fokker- Planck equations are derived in terms of infinite series for nonlinear stochastic differential equations with non-Gaussian","authors_text":"Jinqiao Duan, Xu Sun","cross_cats":["math-ph","math.MP","math.PR","physics.data-an"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-12T20:21:41Z","title":"Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Levy processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2563","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:fdb525e67c426e1ea40759ec5c6f51ba0701ea28de69b613a87aec871c8cb34a","target":"record","created_at":"2026-05-18T01:58:30Z","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":"0ce69c2b46673c53f488e7d6547a1c266279b80290b3c412b580ee8a27efc617","cross_cats_sorted":["math-ph","math.MP","math.PR","physics.data-an"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-12T20:21:41Z","title_canon_sha256":"f3c02034d7c8334ee5cc179257cb31dc4de136ccae29749c49a6bacf310c2e57"},"schema_version":"1.0","source":{"id":"1202.2563","kind":"arxiv","version":1}},"canonical_sha256":"3e832fec067f84fcb44aa08b9fe0f7ca0fe78817c32324639cede01c5fa45dc2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e832fec067f84fcb44aa08b9fe0f7ca0fe78817c32324639cede01c5fa45dc2","first_computed_at":"2026-05-18T01:58:30.886181Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:58:30.886181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H4/bdFa3ggiY2K6IpikqwFE8plafHAFVl+VlqgYX2hCkRKzzLIVwTxG3yXaVDmu7kUegJo2x6IMbgLCGebcOCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:58:30.886565Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.2563","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fdb525e67c426e1ea40759ec5c6f51ba0701ea28de69b613a87aec871c8cb34a","sha256:b804407c334fee50c7dd0950985b761465cd526b28fee867a76ff40742865c6e"],"state_sha256":"c12de950a9ca1a14fd406df6fe6f52c96c130c84518c9c8a788cdc46019a0d7b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qc3rtWLoV7bQv+210wi9zMCxJ2hCpsrogrZ6Ro0zpVx+YHoB2nnYYc+6FmVTInq8rWC3CsFvK3O0Zpqqq86QBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T16:02:40.132923Z","bundle_sha256":"2c857a80010bc93f19abe193f6024f07d03de1e5235d36b492ed2cc3ec29ef95"}}