{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2002:TOQ7346JJDX2XTLD2TKWSGP57U","short_pith_number":"pith:TOQ7346J","canonical_record":{"source":{"id":"math/0207009","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2002-07-01T12:19:41Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"ec70ddaf2079fefdc6daace4cf15557086b3e3c1130971bb7b603df05f47bda9","abstract_canon_sha256":"c4b806e650299854b48cb421a4a2c58d72550624bef5791e8d3e3400117f23b8"},"schema_version":"1.0"},"canonical_sha256":"9ba1fdf3c948efabcd63d4d56919fdfd1e3b6ac400f578a271d23b53a7fa5cfb","source":{"kind":"arxiv","id":"math/0207009","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0207009","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0207009v1","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0207009","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"pith_short_12","alias_value":"TOQ7346JJDX2","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"TOQ7346JJDX2XTLD","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"TOQ7346J","created_at":"2026-05-18T12:25:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2002:TOQ7346JJDX2XTLD2TKWSGP57U","target":"record","payload":{"canonical_record":{"source":{"id":"math/0207009","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2002-07-01T12:19:41Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"ec70ddaf2079fefdc6daace4cf15557086b3e3c1130971bb7b603df05f47bda9","abstract_canon_sha256":"c4b806e650299854b48cb421a4a2c58d72550624bef5791e8d3e3400117f23b8"},"schema_version":"1.0"},"canonical_sha256":"9ba1fdf3c948efabcd63d4d56919fdfd1e3b6ac400f578a271d23b53a7fa5cfb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:28.592361Z","signature_b64":"b5D/Fvl4Vns447wKrRZNfL8XqbD5l4m8nmuilqmFS5tbdreR4OFwMa0wEnldqarQhoBoDo6kPdetiyea+gRBBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ba1fdf3c948efabcd63d4d56919fdfd1e3b6ac400f578a271d23b53a7fa5cfb","last_reissued_at":"2026-05-18T04:28:28.591958Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:28.591958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0207009","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":"2026-05-18T04:28:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GV/oHPLRs88wYhpMuxocT/fk1O3T1odr3wZpzZhPp1HBNBx6+kyZIwGNtUo1zbuqVr/7fg2xEAHiwKP8/ulTDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:42:27.430109Z"},"content_sha256":"ad049aa1b0ce1cb8f7573e04578fb1badf64fa058a32ca6f736837d7a31a9256","schema_version":"1.0","event_id":"sha256:ad049aa1b0ce1cb8f7573e04578fb1badf64fa058a32ca6f736837d7a31a9256"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2002:TOQ7346JJDX2XTLD2TKWSGP57U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some non-linear s.p.d.e.'s that are second order in time","license":"","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Carl Mueller, Robert C. Dalang","submitted_at":"2002-07-01T12:19:41Z","abstract_excerpt":"We extend Walsh's theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0207009","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":"2026-05-18T04:28:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7jOrpEw2GSKLMWMjZpUcL6J9HC2bWUrKbRLCYXkAY5RxMmJInn5ywgBX9L4JztfhJzvYb6w71vOPBoN3lxzbCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:42:27.430505Z"},"content_sha256":"547499d7fd4087ccb3b7316a684c0b1958fd68efc34a51d0da2b206765572e62","schema_version":"1.0","event_id":"sha256:547499d7fd4087ccb3b7316a684c0b1958fd68efc34a51d0da2b206765572e62"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TOQ7346JJDX2XTLD2TKWSGP57U/bundle.json","state_url":"https://pith.science/pith/TOQ7346JJDX2XTLD2TKWSGP57U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TOQ7346JJDX2XTLD2TKWSGP57U/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":"2026-06-01T04:42:27Z","links":{"resolver":"https://pith.science/pith/TOQ7346JJDX2XTLD2TKWSGP57U","bundle":"https://pith.science/pith/TOQ7346JJDX2XTLD2TKWSGP57U/bundle.json","state":"https://pith.science/pith/TOQ7346JJDX2XTLD2TKWSGP57U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TOQ7346JJDX2XTLD2TKWSGP57U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2002:TOQ7346JJDX2XTLD2TKWSGP57U","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":"c4b806e650299854b48cb421a4a2c58d72550624bef5791e8d3e3400117f23b8","cross_cats_sorted":["math.AP"],"license":"","primary_cat":"math.PR","submitted_at":"2002-07-01T12:19:41Z","title_canon_sha256":"ec70ddaf2079fefdc6daace4cf15557086b3e3c1130971bb7b603df05f47bda9"},"schema_version":"1.0","source":{"id":"math/0207009","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0207009","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0207009v1","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0207009","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"pith_short_12","alias_value":"TOQ7346JJDX2","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"TOQ7346JJDX2XTLD","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"TOQ7346J","created_at":"2026-05-18T12:25:51Z"}],"graph_snapshots":[{"event_id":"sha256:547499d7fd4087ccb3b7316a684c0b1958fd68efc34a51d0da2b206765572e62","target":"graph","created_at":"2026-05-18T04:28:28Z","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":"We extend Walsh's theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equatio","authors_text":"Carl Mueller, Robert C. Dalang","cross_cats":["math.AP"],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2002-07-01T12:19:41Z","title":"Some non-linear s.p.d.e.'s that are second order in time"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0207009","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:ad049aa1b0ce1cb8f7573e04578fb1badf64fa058a32ca6f736837d7a31a9256","target":"record","created_at":"2026-05-18T04:28:28Z","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":"c4b806e650299854b48cb421a4a2c58d72550624bef5791e8d3e3400117f23b8","cross_cats_sorted":["math.AP"],"license":"","primary_cat":"math.PR","submitted_at":"2002-07-01T12:19:41Z","title_canon_sha256":"ec70ddaf2079fefdc6daace4cf15557086b3e3c1130971bb7b603df05f47bda9"},"schema_version":"1.0","source":{"id":"math/0207009","kind":"arxiv","version":1}},"canonical_sha256":"9ba1fdf3c948efabcd63d4d56919fdfd1e3b6ac400f578a271d23b53a7fa5cfb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ba1fdf3c948efabcd63d4d56919fdfd1e3b6ac400f578a271d23b53a7fa5cfb","first_computed_at":"2026-05-18T04:28:28.591958Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:28.591958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b5D/Fvl4Vns447wKrRZNfL8XqbD5l4m8nmuilqmFS5tbdreR4OFwMa0wEnldqarQhoBoDo6kPdetiyea+gRBBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:28.592361Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0207009","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ad049aa1b0ce1cb8f7573e04578fb1badf64fa058a32ca6f736837d7a31a9256","sha256:547499d7fd4087ccb3b7316a684c0b1958fd68efc34a51d0da2b206765572e62"],"state_sha256":"f105e90cf44e2adeda276a18c1a526b5054bb63094c60dc106bc7692ca4f8f45"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"doHh+8PPBDEsfYI2/JnXQmDmEfJU9RFojkXCO8U+z73Rkra/pz7JTVQF0JSOTDLT8Ym1w0/5ws/Cen/BjNfQCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T04:42:27.432892Z","bundle_sha256":"f231ee63ba303fe1a8e29d89f29884a110b48315bb172d00da7183bf88aaf7d1"}}