{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:4MDTO35RQU3TKELBKNZD2JZ5W4","short_pith_number":"pith:4MDTO35R","canonical_record":{"source":{"id":"1209.4517","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-09-20T12:56:32Z","cross_cats_sorted":["math.MP","q-fin.RM"],"title_canon_sha256":"2938fe0e2844fc1ee7b58398e713e3445bbf3bb50ddc8dc92818d2e96d3f686c","abstract_canon_sha256":"fd4de8609fa253a3254cc93743ac6c8682a7687bf9f431890d45e05e023d0cfa"},"schema_version":"1.0"},"canonical_sha256":"e307376fb1853735116153723d273db72f5aef315afa7cdb00317275c9dc550a","source":{"kind":"arxiv","id":"1209.4517","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4517","created_at":"2026-05-18T03:30:55Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4517v3","created_at":"2026-05-18T03:30:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4517","created_at":"2026-05-18T03:30:55Z"},{"alias_kind":"pith_short_12","alias_value":"4MDTO35RQU3T","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4MDTO35RQU3TKELB","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4MDTO35R","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:4MDTO35RQU3TKELBKNZD2JZ5W4","target":"record","payload":{"canonical_record":{"source":{"id":"1209.4517","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-09-20T12:56:32Z","cross_cats_sorted":["math.MP","q-fin.RM"],"title_canon_sha256":"2938fe0e2844fc1ee7b58398e713e3445bbf3bb50ddc8dc92818d2e96d3f686c","abstract_canon_sha256":"fd4de8609fa253a3254cc93743ac6c8682a7687bf9f431890d45e05e023d0cfa"},"schema_version":"1.0"},"canonical_sha256":"e307376fb1853735116153723d273db72f5aef315afa7cdb00317275c9dc550a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:55.309389Z","signature_b64":"O+5dSbBKXaMgLblZEHyEqPdWgreXKnXeiCDYPU+v72Gs1WA3tEU/hb0zdE5bPT/+reXhwXh+548UPn6rTKdRDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e307376fb1853735116153723d273db72f5aef315afa7cdb00317275c9dc550a","last_reissued_at":"2026-05-18T03:30:55.308359Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:55.308359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.4517","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":"2026-05-18T03:30:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zAe+x8UWemBujdfyuFV+r4ECjnAQLdWwByNdKZ5XNHcyQjBVvprdkj4Rq6NX0ubXbtry7baw/mz+JJmc3ZzOCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:36:37.812148Z"},"content_sha256":"fb96125109715d9a51fd931a104d46b39b06ed4ac3b5a0010067df57c9abfd24","schema_version":"1.0","event_id":"sha256:fb96125109715d9a51fd931a104d46b39b06ed4ac3b5a0010067df57c9abfd24"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:4MDTO35RQU3TKELBKNZD2JZ5W4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ergodicity breaking in geometric Brownian motion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","q-fin.RM"],"primary_cat":"math-ph","authors_text":"Ole Peters, William Klein","submitted_at":"2012-09-20T12:56:32Z","abstract_excerpt":"Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average. A common tactic for bringing time averages closer to ensemble averages is diversification. In this letter we study the effects of diversification using the concept of ergodicity breaking."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4517","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":"2026-05-18T03:30:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r8EYku472Uf1iWU42tHBTjhUK+p9DB88lk01/uIeHWJg6XWRXkoFwoEGIiOcrBqDbguOQ17xKRWyqxdsVD1QBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:36:37.812519Z"},"content_sha256":"b06066a914943e9a662129bcc4a110a098d5876851a5b78f737f8b233d2c90fa","schema_version":"1.0","event_id":"sha256:b06066a914943e9a662129bcc4a110a098d5876851a5b78f737f8b233d2c90fa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4MDTO35RQU3TKELBKNZD2JZ5W4/bundle.json","state_url":"https://pith.science/pith/4MDTO35RQU3TKELBKNZD2JZ5W4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4MDTO35RQU3TKELBKNZD2JZ5W4/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":"2026-06-09T16:36:37Z","links":{"resolver":"https://pith.science/pith/4MDTO35RQU3TKELBKNZD2JZ5W4","bundle":"https://pith.science/pith/4MDTO35RQU3TKELBKNZD2JZ5W4/bundle.json","state":"https://pith.science/pith/4MDTO35RQU3TKELBKNZD2JZ5W4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4MDTO35RQU3TKELBKNZD2JZ5W4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:4MDTO35RQU3TKELBKNZD2JZ5W4","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":"fd4de8609fa253a3254cc93743ac6c8682a7687bf9f431890d45e05e023d0cfa","cross_cats_sorted":["math.MP","q-fin.RM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-09-20T12:56:32Z","title_canon_sha256":"2938fe0e2844fc1ee7b58398e713e3445bbf3bb50ddc8dc92818d2e96d3f686c"},"schema_version":"1.0","source":{"id":"1209.4517","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4517","created_at":"2026-05-18T03:30:55Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4517v3","created_at":"2026-05-18T03:30:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4517","created_at":"2026-05-18T03:30:55Z"},{"alias_kind":"pith_short_12","alias_value":"4MDTO35RQU3T","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4MDTO35RQU3TKELB","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4MDTO35R","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:b06066a914943e9a662129bcc4a110a098d5876851a5b78f737f8b233d2c90fa","target":"graph","created_at":"2026-05-18T03:30:55Z","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":"Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average. A common tactic for bringing time averages closer to ensemble averages is diversification. In this letter we study the effects of diversification using the concept of ergodicity breaking.","authors_text":"Ole Peters, William Klein","cross_cats":["math.MP","q-fin.RM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-09-20T12:56:32Z","title":"Ergodicity breaking in geometric Brownian motion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4517","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:fb96125109715d9a51fd931a104d46b39b06ed4ac3b5a0010067df57c9abfd24","target":"record","created_at":"2026-05-18T03:30:55Z","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":"fd4de8609fa253a3254cc93743ac6c8682a7687bf9f431890d45e05e023d0cfa","cross_cats_sorted":["math.MP","q-fin.RM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-09-20T12:56:32Z","title_canon_sha256":"2938fe0e2844fc1ee7b58398e713e3445bbf3bb50ddc8dc92818d2e96d3f686c"},"schema_version":"1.0","source":{"id":"1209.4517","kind":"arxiv","version":3}},"canonical_sha256":"e307376fb1853735116153723d273db72f5aef315afa7cdb00317275c9dc550a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e307376fb1853735116153723d273db72f5aef315afa7cdb00317275c9dc550a","first_computed_at":"2026-05-18T03:30:55.308359Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:30:55.308359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O+5dSbBKXaMgLblZEHyEqPdWgreXKnXeiCDYPU+v72Gs1WA3tEU/hb0zdE5bPT/+reXhwXh+548UPn6rTKdRDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:30:55.309389Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.4517","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb96125109715d9a51fd931a104d46b39b06ed4ac3b5a0010067df57c9abfd24","sha256:b06066a914943e9a662129bcc4a110a098d5876851a5b78f737f8b233d2c90fa"],"state_sha256":"e60c81966c56ccb5f032d4d80191bfd60e24e38768aab821f93a157d703da0e2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dIDNZhA3cqdMh+Fo0/Y9pp1w+plO8WhGdFf3Mpzue0s+QPBr4QaYAWcJJa6o8Z/uTpHHFAuFHZG31O40X1MnBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T16:36:37.814488Z","bundle_sha256":"86587b461dc40e43f5e2432f099b8ef0d1d919147eb824fba1426c0bfbe68f49"}}