{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:L54JW3CATLRUJF4KOMTPKVVS6E","short_pith_number":"pith:L54JW3CA","canonical_record":{"source":{"id":"1408.3571","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-15T15:50:28Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"c505d7ef896e5d89c255ea237a7b3d9f1615ad9b5fef4412f685e4a6f7de2149","abstract_canon_sha256":"62432d99060f06e1515c13e754a48534cd5ff83f770032e9e7d436ec6eca04d2"},"schema_version":"1.0"},"canonical_sha256":"5f789b6c409ae344978a7326f556b2f13517fbe5054f8a97b96f1444fb91103f","source":{"kind":"arxiv","id":"1408.3571","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.3571","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"arxiv_version","alias_value":"1408.3571v1","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3571","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"pith_short_12","alias_value":"L54JW3CATLRU","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"L54JW3CATLRUJF4K","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"L54JW3CA","created_at":"2026-05-18T12:28:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:L54JW3CATLRUJF4KOMTPKVVS6E","target":"record","payload":{"canonical_record":{"source":{"id":"1408.3571","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-15T15:50:28Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"c505d7ef896e5d89c255ea237a7b3d9f1615ad9b5fef4412f685e4a6f7de2149","abstract_canon_sha256":"62432d99060f06e1515c13e754a48534cd5ff83f770032e9e7d436ec6eca04d2"},"schema_version":"1.0"},"canonical_sha256":"5f789b6c409ae344978a7326f556b2f13517fbe5054f8a97b96f1444fb91103f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:05.478308Z","signature_b64":"5tc3vxenTM7bpX0U6eHptKqglgNhncH0PILRDIoNo4GbjABx/yHD4WAVtjj4qbMgZ4K5Sdyq0GHn5/+VCQDBAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f789b6c409ae344978a7326f556b2f13517fbe5054f8a97b96f1444fb91103f","last_reissued_at":"2026-05-18T02:45:05.477864Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:05.477864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.3571","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":"2026-05-18T02:45:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kSi08Fn+zZlEpZNP81jBr/6NQnfNaASVTB/zJbkerzuA8n8R+ab8WuhbF9Nyh6KWQFJWLTyzhYmaqFNGm260CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T16:10:09.598752Z"},"content_sha256":"0e4f66e880d0bc6cf8ab917fd825b0825721015f2a5e9d3b2c08dfbf414d177a","schema_version":"1.0","event_id":"sha256:0e4f66e880d0bc6cf8ab917fd825b0825721015f2a5e9d3b2c08dfbf414d177a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:L54JW3CATLRUJF4KOMTPKVVS6E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Brownian motion on stationary random manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DG","authors_text":"Pablo Lessa","submitted_at":"2014-08-15T15:50:28Z","abstract_excerpt":"We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3571","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":"2026-05-18T02:45:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"utXloR/TV6sJxTA7svk6xfosD9ZeZtD1DuCf7AeX0TwZAn6lppKZVzC2l8JN/JMcczQIefSG4ucjK/pWpIeVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T16:10:09.599090Z"},"content_sha256":"38a8eefeb37f2827f4b882035e23fee4a1c71f2d1d8df8f5d354f24c144a1fd6","schema_version":"1.0","event_id":"sha256:38a8eefeb37f2827f4b882035e23fee4a1c71f2d1d8df8f5d354f24c144a1fd6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L54JW3CATLRUJF4KOMTPKVVS6E/bundle.json","state_url":"https://pith.science/pith/L54JW3CATLRUJF4KOMTPKVVS6E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L54JW3CATLRUJF4KOMTPKVVS6E/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":"2026-07-01T16:10:09Z","links":{"resolver":"https://pith.science/pith/L54JW3CATLRUJF4KOMTPKVVS6E","bundle":"https://pith.science/pith/L54JW3CATLRUJF4KOMTPKVVS6E/bundle.json","state":"https://pith.science/pith/L54JW3CATLRUJF4KOMTPKVVS6E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L54JW3CATLRUJF4KOMTPKVVS6E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:L54JW3CATLRUJF4KOMTPKVVS6E","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":"62432d99060f06e1515c13e754a48534cd5ff83f770032e9e7d436ec6eca04d2","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-15T15:50:28Z","title_canon_sha256":"c505d7ef896e5d89c255ea237a7b3d9f1615ad9b5fef4412f685e4a6f7de2149"},"schema_version":"1.0","source":{"id":"1408.3571","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.3571","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"arxiv_version","alias_value":"1408.3571v1","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3571","created_at":"2026-05-18T02:45:05Z"},{"alias_kind":"pith_short_12","alias_value":"L54JW3CATLRU","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"L54JW3CATLRUJF4K","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"L54JW3CA","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:38a8eefeb37f2827f4b882035e23fee4a1c71f2d1d8df8f5d354f24c144a1fd6","target":"graph","created_at":"2026-05-18T02:45:05Z","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 introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volu","authors_text":"Pablo Lessa","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-15T15:50:28Z","title":"Brownian motion on stationary random manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3571","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":"sha256:0e4f66e880d0bc6cf8ab917fd825b0825721015f2a5e9d3b2c08dfbf414d177a","target":"record","created_at":"2026-05-18T02:45:05Z","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":"62432d99060f06e1515c13e754a48534cd5ff83f770032e9e7d436ec6eca04d2","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-15T15:50:28Z","title_canon_sha256":"c505d7ef896e5d89c255ea237a7b3d9f1615ad9b5fef4412f685e4a6f7de2149"},"schema_version":"1.0","source":{"id":"1408.3571","kind":"arxiv","version":1}},"canonical_sha256":"5f789b6c409ae344978a7326f556b2f13517fbe5054f8a97b96f1444fb91103f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f789b6c409ae344978a7326f556b2f13517fbe5054f8a97b96f1444fb91103f","first_computed_at":"2026-05-18T02:45:05.477864Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:45:05.477864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5tc3vxenTM7bpX0U6eHptKqglgNhncH0PILRDIoNo4GbjABx/yHD4WAVtjj4qbMgZ4K5Sdyq0GHn5/+VCQDBAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:45:05.478308Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.3571","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e4f66e880d0bc6cf8ab917fd825b0825721015f2a5e9d3b2c08dfbf414d177a","sha256:38a8eefeb37f2827f4b882035e23fee4a1c71f2d1d8df8f5d354f24c144a1fd6"],"state_sha256":"78ff272f6f0ebf60d311b2063a86efb80c3cb6eab6f6afffd8d504f690fb57b6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K6lfn8bgtBhHW63WyIbN775CSMwU5HzJCxPOU1R4vX2K13JFsfqgRMcXn7bSYA32YXRhLBxbRCaGn+IaBw/eCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T16:10:09.600946Z","bundle_sha256":"c2feb2928e970f40f6c1183b8c5ddf604adf44735eaa8c374273e21fd7f58888"}}