{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2001:HK2RB3AAAMP33QX5ZMQDEO7BFY","short_pith_number":"pith:HK2RB3AA","canonical_record":{"source":{"id":"math/0112315","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2001-12-31T18:17:29Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"1f0ae779d9d7761baf590172d4587a9a3667f067fc80e79676091615841bbe29","abstract_canon_sha256":"4b27085466fc537ca610ab9f685b1cb4959217f56f200d2641eab452191b83a6"},"schema_version":"1.0"},"canonical_sha256":"3ab510ec00031fbdc2fdcb20323be12e24a46459d754bac665974d72cf5b0e15","source":{"kind":"arxiv","id":"math/0112315","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0112315","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0112315v3","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0112315","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"pith_short_12","alias_value":"HK2RB3AAAMP3","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"HK2RB3AAAMP33QX5","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"HK2RB3AA","created_at":"2026-05-18T12:25:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2001:HK2RB3AAAMP33QX5ZMQDEO7BFY","target":"record","payload":{"canonical_record":{"source":{"id":"math/0112315","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2001-12-31T18:17:29Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"1f0ae779d9d7761baf590172d4587a9a3667f067fc80e79676091615841bbe29","abstract_canon_sha256":"4b27085466fc537ca610ab9f685b1cb4959217f56f200d2641eab452191b83a6"},"schema_version":"1.0"},"canonical_sha256":"3ab510ec00031fbdc2fdcb20323be12e24a46459d754bac665974d72cf5b0e15","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:28.604988Z","signature_b64":"loqXf+YdicAf9K66EH3lzbnrS2ZcMRSvBAd97f8CpPDe8AW/1d0coI5YZP3FaJ5ZEhiu7nEVKx3KTmlNo/NJAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ab510ec00031fbdc2fdcb20323be12e24a46459d754bac665974d72cf5b0e15","last_reissued_at":"2026-05-18T04:28:28.604581Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:28.604581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0112315","source_version":3,"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":"0Dki9FMahcSEEJp1DizjzWX39HxowcEYpbPga27AGAPNFuUD3F0COfLnYbHbS+9D0aXiMUG9XDS6S3owBYOSBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:15:55.811217Z"},"content_sha256":"5fb781c02eeb7391ccb105da0893a2fbbc724972d996aaf277ef50b9f0cf7bf0","schema_version":"1.0","event_id":"sha256:5fb781c02eeb7391ccb105da0893a2fbbc724972d996aaf277ef50b9f0cf7bf0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2001:HK2RB3AAAMP33QX5ZMQDEO7BFY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hitting properties of a random string","license":"","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Carl Mueller, Roger Tribe","submitted_at":"2001-12-31T18:17:29Z","abstract_excerpt":"We consider Funaki's model of a random string taking values in R^d. It is specified by the following stochastic PDE, du = u_{xx} + W, where W=W(x,t) is two-parameter white noise, also taking values in R^d. We study hitting properties, double points, and recurrence. The main difficulty is that the process has the Markov property in time, but not in space. We find: (1) The string hits points if d<6. (2) For fixed t, there are points x,y such that u(t,x)=u(t,y) iff d < 4. (3) There exist points t,x,y such that u(t,x)=u(t,y) iff d < 8. (4) There exist points s,t,x,y such that u(t,x)=u(s,y) iff d <"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0112315","kind":"arxiv","version":3},"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":"gYCRxdf7b9i99tGjTe1z1ZYtIe3CVamhpaJl/Er1q4P5TdJlbmCINrABGjKOykhRxDFkZfb/J1mjnO/SK13sCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:15:55.811886Z"},"content_sha256":"64f51b44c87c677e8aefc991dd7b5c5899a230c516bc42d7fdd7137b82261611","schema_version":"1.0","event_id":"sha256:64f51b44c87c677e8aefc991dd7b5c5899a230c516bc42d7fdd7137b82261611"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HK2RB3AAAMP33QX5ZMQDEO7BFY/bundle.json","state_url":"https://pith.science/pith/HK2RB3AAAMP33QX5ZMQDEO7BFY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HK2RB3AAAMP33QX5ZMQDEO7BFY/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":"2026-05-27T20:15:55Z","links":{"resolver":"https://pith.science/pith/HK2RB3AAAMP33QX5ZMQDEO7BFY","bundle":"https://pith.science/pith/HK2RB3AAAMP33QX5ZMQDEO7BFY/bundle.json","state":"https://pith.science/pith/HK2RB3AAAMP33QX5ZMQDEO7BFY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HK2RB3AAAMP33QX5ZMQDEO7BFY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2001:HK2RB3AAAMP33QX5ZMQDEO7BFY","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":"4b27085466fc537ca610ab9f685b1cb4959217f56f200d2641eab452191b83a6","cross_cats_sorted":["math.AP"],"license":"","primary_cat":"math.PR","submitted_at":"2001-12-31T18:17:29Z","title_canon_sha256":"1f0ae779d9d7761baf590172d4587a9a3667f067fc80e79676091615841bbe29"},"schema_version":"1.0","source":{"id":"math/0112315","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0112315","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"arxiv_version","alias_value":"math/0112315v3","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0112315","created_at":"2026-05-18T04:28:28Z"},{"alias_kind":"pith_short_12","alias_value":"HK2RB3AAAMP3","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"HK2RB3AAAMP33QX5","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"HK2RB3AA","created_at":"2026-05-18T12:25:50Z"}],"graph_snapshots":[{"event_id":"sha256:64f51b44c87c677e8aefc991dd7b5c5899a230c516bc42d7fdd7137b82261611","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 consider Funaki's model of a random string taking values in R^d. It is specified by the following stochastic PDE, du = u_{xx} + W, where W=W(x,t) is two-parameter white noise, also taking values in R^d. We study hitting properties, double points, and recurrence. The main difficulty is that the process has the Markov property in time, but not in space. We find: (1) The string hits points if d<6. (2) For fixed t, there are points x,y such that u(t,x)=u(t,y) iff d < 4. (3) There exist points t,x,y such that u(t,x)=u(t,y) iff d < 8. (4) There exist points s,t,x,y such that u(t,x)=u(s,y) iff d <","authors_text":"Carl Mueller, Roger Tribe","cross_cats":["math.AP"],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2001-12-31T18:17:29Z","title":"Hitting properties of a random string"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0112315","kind":"arxiv","version":3},"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":"sha256:5fb781c02eeb7391ccb105da0893a2fbbc724972d996aaf277ef50b9f0cf7bf0","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":"4b27085466fc537ca610ab9f685b1cb4959217f56f200d2641eab452191b83a6","cross_cats_sorted":["math.AP"],"license":"","primary_cat":"math.PR","submitted_at":"2001-12-31T18:17:29Z","title_canon_sha256":"1f0ae779d9d7761baf590172d4587a9a3667f067fc80e79676091615841bbe29"},"schema_version":"1.0","source":{"id":"math/0112315","kind":"arxiv","version":3}},"canonical_sha256":"3ab510ec00031fbdc2fdcb20323be12e24a46459d754bac665974d72cf5b0e15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3ab510ec00031fbdc2fdcb20323be12e24a46459d754bac665974d72cf5b0e15","first_computed_at":"2026-05-18T04:28:28.604581Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:28.604581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"loqXf+YdicAf9K66EH3lzbnrS2ZcMRSvBAd97f8CpPDe8AW/1d0coI5YZP3FaJ5ZEhiu7nEVKx3KTmlNo/NJAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:28.604988Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0112315","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5fb781c02eeb7391ccb105da0893a2fbbc724972d996aaf277ef50b9f0cf7bf0","sha256:64f51b44c87c677e8aefc991dd7b5c5899a230c516bc42d7fdd7137b82261611"],"state_sha256":"463e0fe6e8211c915bd85a2f67a6e850089f680d46262f0a3c945ca340a1426f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dVelOj5jGtoudkLDkCGwwudU+yLhA7MtehKllRfq1DQW+mtxnZMV30HlqPcUZVwGRd3v/fCXLPAEzG9GI/USAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T20:15:55.814890Z","bundle_sha256":"5bff2c9b34ba38e8f9391b0d524c120483492eede94993c0feff4e0f700932fe"}}