{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:HA4N33XQ6ULHSFIIANOPR2QAGA","short_pith_number":"pith:HA4N33XQ","canonical_record":{"source":{"id":"1202.0618","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-03T07:39:47Z","cross_cats_sorted":[],"title_canon_sha256":"efc76cfa0abbdedfd1d66ccd652eb02283175e997a070abfbe4a00dbfdd373d0","abstract_canon_sha256":"9907455fd662b3ba7697056461fe5f791011e5434ed7c20180e8e92aace5ee4f"},"schema_version":"1.0"},"canonical_sha256":"3838ddeef0f516791508035cf8ea003036c55b4bc27e78a2b4581de954b655d9","source":{"kind":"arxiv","id":"1202.0618","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.0618","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"arxiv_version","alias_value":"1202.0618v2","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.0618","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"pith_short_12","alias_value":"HA4N33XQ6ULH","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HA4N33XQ6ULHSFII","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HA4N33XQ","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:HA4N33XQ6ULHSFIIANOPR2QAGA","target":"record","payload":{"canonical_record":{"source":{"id":"1202.0618","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-03T07:39:47Z","cross_cats_sorted":[],"title_canon_sha256":"efc76cfa0abbdedfd1d66ccd652eb02283175e997a070abfbe4a00dbfdd373d0","abstract_canon_sha256":"9907455fd662b3ba7697056461fe5f791011e5434ed7c20180e8e92aace5ee4f"},"schema_version":"1.0"},"canonical_sha256":"3838ddeef0f516791508035cf8ea003036c55b4bc27e78a2b4581de954b655d9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:10.659763Z","signature_b64":"7EnuL/YMmHpZjE23W6wwhZk93je+KjDeW7NtjEX0PxYZ86O/oSc5etW3zxtvoz2U62+zN08EeLexNzpxbyDsDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3838ddeef0f516791508035cf8ea003036c55b4bc27e78a2b4581de954b655d9","last_reissued_at":"2026-05-18T03:45:10.659007Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:10.659007Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.0618","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:45:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Agv9EsvykgGEY5OoKdgSk2966mL1ra0Ffpx6bYWbeYHpRTZdNqAZFug4HbBH4qDKc9et1gcbAxKGyNJiyeyBDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T15:40:05.486939Z"},"content_sha256":"556f6bf409f2ee2357a5f9dc71358d9bf1722d930b734780db015cf904d74902","schema_version":"1.0","event_id":"sha256:556f6bf409f2ee2357a5f9dc71358d9bf1722d930b734780db015cf904d74902"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:HA4N33XQ6ULHSFIIANOPR2QAGA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"2D- stochastic currents over the Wiener sheet","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ciprian Tudor (LPP), Franco Flandoli (DMA), Peter Imkeller","submitted_at":"2012-02-03T07:39:47Z","abstract_excerpt":"By using stochastic calculus for two-parameter processes and chaos expansion into multiple Wiener-It\\^o integrals, we define a 2D-stochastic current over the Brownian sheet. This concept comes from geometric measure theory. We also study the regularity of the stochastic current with respect to the randomness variable in the Watanabe spaces and with respect to the spatial variable in the deterministic Sobolev spaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0618","kind":"arxiv","version":2},"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":"2026-05-18T03:45:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5ursM+fAssq8WpMLdyQy+5r5PtQ+qD7JyJqMThW7MXXZRMKikFD4tiSa0UhD/IoevH4ttm6NUGxiOPBQyRfhCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T15:40:05.487528Z"},"content_sha256":"48fbe05b0c99f67b0a5386fd5c9ba2e926a75ec7b40ab4a57ed53899470a9f62","schema_version":"1.0","event_id":"sha256:48fbe05b0c99f67b0a5386fd5c9ba2e926a75ec7b40ab4a57ed53899470a9f62"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HA4N33XQ6ULHSFIIANOPR2QAGA/bundle.json","state_url":"https://pith.science/pith/HA4N33XQ6ULHSFIIANOPR2QAGA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HA4N33XQ6ULHSFIIANOPR2QAGA/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":"2026-05-30T15:40:05Z","links":{"resolver":"https://pith.science/pith/HA4N33XQ6ULHSFIIANOPR2QAGA","bundle":"https://pith.science/pith/HA4N33XQ6ULHSFIIANOPR2QAGA/bundle.json","state":"https://pith.science/pith/HA4N33XQ6ULHSFIIANOPR2QAGA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HA4N33XQ6ULHSFIIANOPR2QAGA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HA4N33XQ6ULHSFIIANOPR2QAGA","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":"9907455fd662b3ba7697056461fe5f791011e5434ed7c20180e8e92aace5ee4f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-03T07:39:47Z","title_canon_sha256":"efc76cfa0abbdedfd1d66ccd652eb02283175e997a070abfbe4a00dbfdd373d0"},"schema_version":"1.0","source":{"id":"1202.0618","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.0618","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"arxiv_version","alias_value":"1202.0618v2","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.0618","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"pith_short_12","alias_value":"HA4N33XQ6ULH","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HA4N33XQ6ULHSFII","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HA4N33XQ","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:48fbe05b0c99f67b0a5386fd5c9ba2e926a75ec7b40ab4a57ed53899470a9f62","target":"graph","created_at":"2026-05-18T03:45:10Z","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":"By using stochastic calculus for two-parameter processes and chaos expansion into multiple Wiener-It\\^o integrals, we define a 2D-stochastic current over the Brownian sheet. This concept comes from geometric measure theory. We also study the regularity of the stochastic current with respect to the randomness variable in the Watanabe spaces and with respect to the spatial variable in the deterministic Sobolev spaces.","authors_text":"Ciprian Tudor (LPP), Franco Flandoli (DMA), Peter Imkeller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-03T07:39:47Z","title":"2D- stochastic currents over the Wiener sheet"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0618","kind":"arxiv","version":2},"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":"sha256:556f6bf409f2ee2357a5f9dc71358d9bf1722d930b734780db015cf904d74902","target":"record","created_at":"2026-05-18T03:45:10Z","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":"9907455fd662b3ba7697056461fe5f791011e5434ed7c20180e8e92aace5ee4f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-03T07:39:47Z","title_canon_sha256":"efc76cfa0abbdedfd1d66ccd652eb02283175e997a070abfbe4a00dbfdd373d0"},"schema_version":"1.0","source":{"id":"1202.0618","kind":"arxiv","version":2}},"canonical_sha256":"3838ddeef0f516791508035cf8ea003036c55b4bc27e78a2b4581de954b655d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3838ddeef0f516791508035cf8ea003036c55b4bc27e78a2b4581de954b655d9","first_computed_at":"2026-05-18T03:45:10.659007Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:10.659007Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7EnuL/YMmHpZjE23W6wwhZk93je+KjDeW7NtjEX0PxYZ86O/oSc5etW3zxtvoz2U62+zN08EeLexNzpxbyDsDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:10.659763Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.0618","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:556f6bf409f2ee2357a5f9dc71358d9bf1722d930b734780db015cf904d74902","sha256:48fbe05b0c99f67b0a5386fd5c9ba2e926a75ec7b40ab4a57ed53899470a9f62"],"state_sha256":"648f17eef9caf83e1c33dfdf5ec5d4095780982c7bf49f3f693932292bbd5e0c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r6NVUi3nSF1koBeSjRV3KYrfvTS20I27Su37lShZ/VShlJEMAW2J82ltuHnA6IvmJSXe0g0vCx8kISiVKkDCBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T15:40:05.490461Z","bundle_sha256":"165e179bd496c4ff9d10196020b43605a64707114958954b67ee4e46b886b65f"}}