{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:ABBGWWTXOZSFHCTYE4OSJCXQXX","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":"519b41883cc5995d8c4d239af4c3bc2974b8f2dd4c5ffa7802b49a6192bbae26","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-25T14:42:46Z","title_canon_sha256":"a1848d83fb2e1d891d9c676c5962eb048b9507bb0ccb51d6493326b0f8a5283e"},"schema_version":"1.0","source":{"id":"0810.4609","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0810.4609","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"arxiv_version","alias_value":"0810.4609v4","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0810.4609","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"pith_short_12","alias_value":"ABBGWWTXOZSF","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"ABBGWWTXOZSFHCTY","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"ABBGWWTX","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:d3af6310c59efc4d050f5bde82218b8748f088fecedf24e9833afc0e59861249","target":"graph","created_at":"2026-05-18T04:39:17Z","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 formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbe","authors_text":"Szymon Peszat, Tomasz Komorowski, Tomasz Szarek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-25T14:42:46Z","title":"On ergodicity of some Markov processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.4609","kind":"arxiv","version":4},"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:d596f6716572fbb9903707d7c0c6ee05780d8125dbb33739501f789ba62e43d2","target":"record","created_at":"2026-05-18T04:39:17Z","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":"519b41883cc5995d8c4d239af4c3bc2974b8f2dd4c5ffa7802b49a6192bbae26","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-25T14:42:46Z","title_canon_sha256":"a1848d83fb2e1d891d9c676c5962eb048b9507bb0ccb51d6493326b0f8a5283e"},"schema_version":"1.0","source":{"id":"0810.4609","kind":"arxiv","version":4}},"canonical_sha256":"00426b5a777664538a78271d248af0bde284f4125ccc6e527013cb230c3f8753","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"00426b5a777664538a78271d248af0bde284f4125ccc6e527013cb230c3f8753","first_computed_at":"2026-05-18T04:39:17.049033Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:17.049033Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ynlDy8cOoIISdC+u1uIyd8c9koqITOoIx5dF4oMTiwZvs4BDEohLRD5mXZRdizG+h2glfLvrEt+U1gZe3d9gAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:17.049687Z","signed_message":"canonical_sha256_bytes"},"source_id":"0810.4609","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d596f6716572fbb9903707d7c0c6ee05780d8125dbb33739501f789ba62e43d2","sha256:d3af6310c59efc4d050f5bde82218b8748f088fecedf24e9833afc0e59861249"],"state_sha256":"e07d8a1c6bce481a3a401b394eaf42910737c69b3c3013f4e929dbf840ae0663"}