{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:BHEOESAKMW5AC7HOPZTSVQGCNB","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":"c2d1c752364177d99cdb5ca90e36cca6858f6db4de2bb59ada29c554198cd9fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-24T01:41:41Z","title_canon_sha256":"eb8f58c511abae839b4734088c71cffcd827bec4b3ff7b60c348d3f2ed73124f"},"schema_version":"1.0","source":{"id":"1202.5361","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.5361","created_at":"2026-05-18T04:01:37Z"},{"alias_kind":"arxiv_version","alias_value":"1202.5361v1","created_at":"2026-05-18T04:01:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.5361","created_at":"2026-05-18T04:01:37Z"},{"alias_kind":"pith_short_12","alias_value":"BHEOESAKMW5A","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"BHEOESAKMW5AC7HO","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"BHEOESAK","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:215cc709a8bd781abf951dd671d452e0d76fd718f37857b3d48f22c717da91a6","target":"graph","created_at":"2026-05-18T04:01:37Z","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 a class of continuous time Markov chains on $\\Z^d$. These chains are the discrete space analogue of Markov processes with jumps. Under some conditions, we show that harmonic functions associated with these Markov chains are H\\\"{o}lder continuous.","authors_text":"Fangjun Xu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-24T01:41:41Z","title":"Regularity of harmonic functions for some Markov chains with unbounded range"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5361","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":"sha256:18155d912ac9b4b3b891c1c043623ed00b7372c7788b6b30657d8f9b9191a95d","target":"record","created_at":"2026-05-18T04:01:37Z","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":"c2d1c752364177d99cdb5ca90e36cca6858f6db4de2bb59ada29c554198cd9fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-24T01:41:41Z","title_canon_sha256":"eb8f58c511abae839b4734088c71cffcd827bec4b3ff7b60c348d3f2ed73124f"},"schema_version":"1.0","source":{"id":"1202.5361","kind":"arxiv","version":1}},"canonical_sha256":"09c8e2480a65ba017cee7e672ac0c2687453e8dd4f10e69deb350952a7bcd495","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09c8e2480a65ba017cee7e672ac0c2687453e8dd4f10e69deb350952a7bcd495","first_computed_at":"2026-05-18T04:01:37.418923Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:01:37.418923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xw2NFj+M6cD0LXwvh/UMNk4Mz0Qr/N5dHM7MGi0XQQC4WzLiFFAYk4/PR+h3AXYdhWfuodJzVZHkWebgsTaUAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:01:37.419676Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.5361","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:18155d912ac9b4b3b891c1c043623ed00b7372c7788b6b30657d8f9b9191a95d","sha256:215cc709a8bd781abf951dd671d452e0d76fd718f37857b3d48f22c717da91a6"],"state_sha256":"26265d4938c2e9812206ebe1516a111d1bbf866ea63950a7dcd707e41db2a641"}