{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:FC5SWS75CJMAC2UAXJUL77ZZOB","short_pith_number":"pith:FC5SWS75","canonical_record":{"source":{"id":"1003.0661","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-03-02T19:45:49Z","cross_cats_sorted":[],"title_canon_sha256":"be4e52dea7d4c2762f8db2e9cd5796d4fcb9388ee2387c643c8f78a288eee831","abstract_canon_sha256":"ac89f7fb38cbf8f502ee78ed0844d8700d5a48c1f1c38ab48d6e88d0a19e4838"},"schema_version":"1.0"},"canonical_sha256":"28bb2b4bfd1258016a80ba68bfff397059152afe4fb0cf2d00d36a78d92dbeb3","source":{"kind":"arxiv","id":"1003.0661","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.0661","created_at":"2026-05-18T02:24:38Z"},{"alias_kind":"arxiv_version","alias_value":"1003.0661v2","created_at":"2026-05-18T02:24:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.0661","created_at":"2026-05-18T02:24:38Z"},{"alias_kind":"pith_short_12","alias_value":"FC5SWS75CJMA","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"FC5SWS75CJMAC2UA","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"FC5SWS75","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:FC5SWS75CJMAC2UAXJUL77ZZOB","target":"record","payload":{"canonical_record":{"source":{"id":"1003.0661","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-03-02T19:45:49Z","cross_cats_sorted":[],"title_canon_sha256":"be4e52dea7d4c2762f8db2e9cd5796d4fcb9388ee2387c643c8f78a288eee831","abstract_canon_sha256":"ac89f7fb38cbf8f502ee78ed0844d8700d5a48c1f1c38ab48d6e88d0a19e4838"},"schema_version":"1.0"},"canonical_sha256":"28bb2b4bfd1258016a80ba68bfff397059152afe4fb0cf2d00d36a78d92dbeb3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:38.344384Z","signature_b64":"whMW49uHQjZ6JP2lbsgijuD1MfUP0Ve0NtU8V/yY1VdQnnC3itzDg5MDGChsSYM2sgkHVWu7O+H91YCyYHmFDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28bb2b4bfd1258016a80ba68bfff397059152afe4fb0cf2d00d36a78d92dbeb3","last_reissued_at":"2026-05-18T02:24:38.343848Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:38.343848Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.0661","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":"2026-05-18T02:24:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IRZCLRZcfIxEoEkFzedhKVp8MbY735CX8aP+WZduDvbHueQAy/07x0+eZiIHmyAXlMgAcTlSHWdjxwi79LMWCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T17:15:58.293152Z"},"content_sha256":"6293d5733f60fe6390dfb1c5d8ecf75687f71a51bdfc6994c9aaf1b7a5ed34ee","schema_version":"1.0","event_id":"sha256:6293d5733f60fe6390dfb1c5d8ecf75687f71a51bdfc6994c9aaf1b7a5ed34ee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:FC5SWS75CJMAC2UAXJUL77ZZOB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Almost sure asymptotics for the maximum local time in Brownian environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Roland Diel (MAPMO)","submitted_at":"2010-03-02T19:45:49Z","abstract_excerpt":"We study the asymptotic behaviour of the maximum local time L*(t) of the Brox's process, the diffusion in Brownian environment. Shi proved that the maximum speed of L*(t) is surprisingly, at least t log(log(log t)) whereas in the discrete case it is t. We show here that t log(log(log t)) is the proper rate and we prove that for the minimum speed the rate is the same as in the discrete case namely t/log(log(log t))."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.0661","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":"2026-05-18T02:24:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R0QAUxQIfwqJibBvxVv0HWabKLz3F08M/nnmbuxFlVs1bpqh5h4xoVeiXOqGa+C/y9NBaUmA7QZLQqwp3q4KBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T17:15:58.293509Z"},"content_sha256":"af612da3bae597b03e2ce687e9d06ee258a1328fd2656434e10f7898f5575a06","schema_version":"1.0","event_id":"sha256:af612da3bae597b03e2ce687e9d06ee258a1328fd2656434e10f7898f5575a06"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FC5SWS75CJMAC2UAXJUL77ZZOB/bundle.json","state_url":"https://pith.science/pith/FC5SWS75CJMAC2UAXJUL77ZZOB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FC5SWS75CJMAC2UAXJUL77ZZOB/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":"2026-06-24T17:15:58Z","links":{"resolver":"https://pith.science/pith/FC5SWS75CJMAC2UAXJUL77ZZOB","bundle":"https://pith.science/pith/FC5SWS75CJMAC2UAXJUL77ZZOB/bundle.json","state":"https://pith.science/pith/FC5SWS75CJMAC2UAXJUL77ZZOB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FC5SWS75CJMAC2UAXJUL77ZZOB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:FC5SWS75CJMAC2UAXJUL77ZZOB","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":"ac89f7fb38cbf8f502ee78ed0844d8700d5a48c1f1c38ab48d6e88d0a19e4838","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-03-02T19:45:49Z","title_canon_sha256":"be4e52dea7d4c2762f8db2e9cd5796d4fcb9388ee2387c643c8f78a288eee831"},"schema_version":"1.0","source":{"id":"1003.0661","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.0661","created_at":"2026-05-18T02:24:38Z"},{"alias_kind":"arxiv_version","alias_value":"1003.0661v2","created_at":"2026-05-18T02:24:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.0661","created_at":"2026-05-18T02:24:38Z"},{"alias_kind":"pith_short_12","alias_value":"FC5SWS75CJMA","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"FC5SWS75CJMAC2UA","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"FC5SWS75","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:af612da3bae597b03e2ce687e9d06ee258a1328fd2656434e10f7898f5575a06","target":"graph","created_at":"2026-05-18T02:24:38Z","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 study the asymptotic behaviour of the maximum local time L*(t) of the Brox's process, the diffusion in Brownian environment. Shi proved that the maximum speed of L*(t) is surprisingly, at least t log(log(log t)) whereas in the discrete case it is t. We show here that t log(log(log t)) is the proper rate and we prove that for the minimum speed the rate is the same as in the discrete case namely t/log(log(log t)).","authors_text":"Roland Diel (MAPMO)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-03-02T19:45:49Z","title":"Almost sure asymptotics for the maximum local time in Brownian environment"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.0661","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":"sha256:6293d5733f60fe6390dfb1c5d8ecf75687f71a51bdfc6994c9aaf1b7a5ed34ee","target":"record","created_at":"2026-05-18T02:24:38Z","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":"ac89f7fb38cbf8f502ee78ed0844d8700d5a48c1f1c38ab48d6e88d0a19e4838","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-03-02T19:45:49Z","title_canon_sha256":"be4e52dea7d4c2762f8db2e9cd5796d4fcb9388ee2387c643c8f78a288eee831"},"schema_version":"1.0","source":{"id":"1003.0661","kind":"arxiv","version":2}},"canonical_sha256":"28bb2b4bfd1258016a80ba68bfff397059152afe4fb0cf2d00d36a78d92dbeb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"28bb2b4bfd1258016a80ba68bfff397059152afe4fb0cf2d00d36a78d92dbeb3","first_computed_at":"2026-05-18T02:24:38.343848Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:38.343848Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"whMW49uHQjZ6JP2lbsgijuD1MfUP0Ve0NtU8V/yY1VdQnnC3itzDg5MDGChsSYM2sgkHVWu7O+H91YCyYHmFDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:38.344384Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.0661","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6293d5733f60fe6390dfb1c5d8ecf75687f71a51bdfc6994c9aaf1b7a5ed34ee","sha256:af612da3bae597b03e2ce687e9d06ee258a1328fd2656434e10f7898f5575a06"],"state_sha256":"424b60cbe15b3f05ebc98d64c79977637d23e73c97e712e67a3fc10b7f46d72f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cz+B5rpxP3zUoFjSueO8XLrlsRS9+zcSVzYbli9GcLJg/v8GAUGb9PiCMZbDW8ezo/OgQdv4x+5d0kFHh7w2AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T17:15:58.295458Z","bundle_sha256":"4df4740f38da20ed312e9252334c920fad12534318cc20f4d31df209ee71dd02"}}