{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:JLMXMYZTFI752FVGTXDS5REILB","short_pith_number":"pith:JLMXMYZT","canonical_record":{"source":{"id":"1711.03652","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-09T23:55:42Z","cross_cats_sorted":[],"title_canon_sha256":"c949e282d9335186376e06ec15cad0ec426017bc21f9bcba7d47944020fcdc8c","abstract_canon_sha256":"c9aa81412145e9a6536fc28558ebcbf5c2998aa02449ec8aa9ea0c79a3a7dc30"},"schema_version":"1.0"},"canonical_sha256":"4ad97663332a3fdd16a69dc72ec4885840aca4259bd211f7c3415b37b7d6c159","source":{"kind":"arxiv","id":"1711.03652","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.03652","created_at":"2026-05-17T23:40:20Z"},{"alias_kind":"arxiv_version","alias_value":"1711.03652v3","created_at":"2026-05-17T23:40:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.03652","created_at":"2026-05-17T23:40:20Z"},{"alias_kind":"pith_short_12","alias_value":"JLMXMYZTFI75","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JLMXMYZTFI752FVG","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JLMXMYZT","created_at":"2026-05-18T12:31:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:JLMXMYZTFI752FVGTXDS5REILB","target":"record","payload":{"canonical_record":{"source":{"id":"1711.03652","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-09T23:55:42Z","cross_cats_sorted":[],"title_canon_sha256":"c949e282d9335186376e06ec15cad0ec426017bc21f9bcba7d47944020fcdc8c","abstract_canon_sha256":"c9aa81412145e9a6536fc28558ebcbf5c2998aa02449ec8aa9ea0c79a3a7dc30"},"schema_version":"1.0"},"canonical_sha256":"4ad97663332a3fdd16a69dc72ec4885840aca4259bd211f7c3415b37b7d6c159","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:20.930283Z","signature_b64":"HoV1jAJLghjCKW/FCRwQMybkSGhdbJIGx0xM6uSPg9Z3cFwqHY+0CzgxA/yIeVb6eQIWIuHRldoOWPlu4wA/Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ad97663332a3fdd16a69dc72ec4885840aca4259bd211f7c3415b37b7d6c159","last_reissued_at":"2026-05-17T23:40:20.929582Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:20.929582Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.03652","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":"2026-05-17T23:40:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VN1z7RaTv/yVf0AucjF+elssG+NteZuFkGp6ycZqd3xYyg8JhJHSENFlWiDIrBL48WTna1A6Cixv7OpuC6ZqCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-13T04:05:57.229627Z"},"content_sha256":"6bc15fab3355cf6cf4fe4667d9df5c8b50d80c6ef7220304ce8262ea271dca5d","schema_version":"1.0","event_id":"sha256:6bc15fab3355cf6cf4fe4667d9df5c8b50d80c6ef7220304ce8262ea271dca5d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:JLMXMYZTFI752FVGTXDS5REILB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Geometric Ergodicity in a Weighted Sobolev Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Adithya Devraj, Ioannis Kontoyiannis, Sean Meyn","submitted_at":"2017-11-09T23:55:42Z","abstract_excerpt":"For a discrete-time Markov chain $\\{X(t)\\}$ evolving on $\\Re^\\ell$ with transition kernel $P$, natural, general conditions are developed under which the following are established:\n  1. The transition kernel $P$ has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space $L_\\infty^{v,1}$ of functions with norm, $$ \\|f\\|_{v,1} = \\sup_{x \\in \\Re^\\ell} \\frac{1}{v(x)} \\max \\{|f(x)|, |\\partial_1 f(x)|,\\ldots,|\\partial_\\ell f(x)|\\}, $$ where $v\\colon \\Re^\\ell \\to [1,\\infty)$ is a Lyapunov function and $\\partial_i:=\\partial/\\partial x_i$.\n  2. The Markov chain is geome"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03652","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":"2026-05-17T23:40:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sEumo3YfcV1lnM56VQ/BjOvvVWhpjSH7yQ0kYiMa7PtIjsBjzHA9/pyCoKW9xibAEOHwyMCRggO5BnuL0QyTBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-13T04:05:57.229970Z"},"content_sha256":"062f73854cb6474bb9f984e0eb40e037f98143d6199678ab9f22c3639ba747b6","schema_version":"1.0","event_id":"sha256:062f73854cb6474bb9f984e0eb40e037f98143d6199678ab9f22c3639ba747b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JLMXMYZTFI752FVGTXDS5REILB/bundle.json","state_url":"https://pith.science/pith/JLMXMYZTFI752FVGTXDS5REILB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JLMXMYZTFI752FVGTXDS5REILB/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":"2026-07-13T04:05:57Z","links":{"resolver":"https://pith.science/pith/JLMXMYZTFI752FVGTXDS5REILB","bundle":"https://pith.science/pith/JLMXMYZTFI752FVGTXDS5REILB/bundle.json","state":"https://pith.science/pith/JLMXMYZTFI752FVGTXDS5REILB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JLMXMYZTFI752FVGTXDS5REILB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:JLMXMYZTFI752FVGTXDS5REILB","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":"c9aa81412145e9a6536fc28558ebcbf5c2998aa02449ec8aa9ea0c79a3a7dc30","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-09T23:55:42Z","title_canon_sha256":"c949e282d9335186376e06ec15cad0ec426017bc21f9bcba7d47944020fcdc8c"},"schema_version":"1.0","source":{"id":"1711.03652","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.03652","created_at":"2026-05-17T23:40:20Z"},{"alias_kind":"arxiv_version","alias_value":"1711.03652v3","created_at":"2026-05-17T23:40:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.03652","created_at":"2026-05-17T23:40:20Z"},{"alias_kind":"pith_short_12","alias_value":"JLMXMYZTFI75","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JLMXMYZTFI752FVG","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JLMXMYZT","created_at":"2026-05-18T12:31:24Z"}],"graph_snapshots":[{"event_id":"sha256:062f73854cb6474bb9f984e0eb40e037f98143d6199678ab9f22c3639ba747b6","target":"graph","created_at":"2026-05-17T23:40:20Z","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":"For a discrete-time Markov chain $\\{X(t)\\}$ evolving on $\\Re^\\ell$ with transition kernel $P$, natural, general conditions are developed under which the following are established:\n  1. The transition kernel $P$ has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space $L_\\infty^{v,1}$ of functions with norm, $$ \\|f\\|_{v,1} = \\sup_{x \\in \\Re^\\ell} \\frac{1}{v(x)} \\max \\{|f(x)|, |\\partial_1 f(x)|,\\ldots,|\\partial_\\ell f(x)|\\}, $$ where $v\\colon \\Re^\\ell \\to [1,\\infty)$ is a Lyapunov function and $\\partial_i:=\\partial/\\partial x_i$.\n  2. The Markov chain is geome","authors_text":"Adithya Devraj, Ioannis Kontoyiannis, Sean Meyn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-09T23:55:42Z","title":"Geometric Ergodicity in a Weighted Sobolev Space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03652","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":"sha256:6bc15fab3355cf6cf4fe4667d9df5c8b50d80c6ef7220304ce8262ea271dca5d","target":"record","created_at":"2026-05-17T23:40:20Z","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":"c9aa81412145e9a6536fc28558ebcbf5c2998aa02449ec8aa9ea0c79a3a7dc30","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-09T23:55:42Z","title_canon_sha256":"c949e282d9335186376e06ec15cad0ec426017bc21f9bcba7d47944020fcdc8c"},"schema_version":"1.0","source":{"id":"1711.03652","kind":"arxiv","version":3}},"canonical_sha256":"4ad97663332a3fdd16a69dc72ec4885840aca4259bd211f7c3415b37b7d6c159","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ad97663332a3fdd16a69dc72ec4885840aca4259bd211f7c3415b37b7d6c159","first_computed_at":"2026-05-17T23:40:20.929582Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:20.929582Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HoV1jAJLghjCKW/FCRwQMybkSGhdbJIGx0xM6uSPg9Z3cFwqHY+0CzgxA/yIeVb6eQIWIuHRldoOWPlu4wA/Bg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:20.930283Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.03652","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6bc15fab3355cf6cf4fe4667d9df5c8b50d80c6ef7220304ce8262ea271dca5d","sha256:062f73854cb6474bb9f984e0eb40e037f98143d6199678ab9f22c3639ba747b6"],"state_sha256":"4a4661d700a4e252694d277937371abcc4bdec67cf2e723fa979257d41025b0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fm7QFYdMnsqSGivPeRDAMpD8XDZebv6vlem94T32hoOxs6Fq1wq9Z6zAhYmnaQnzQch8qQJAhvUANjB4Na1sDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-13T04:05:57.232344Z","bundle_sha256":"8ec5f6e2878863269e62c8b27df9f27cc435daf1639a5b905f5be82b3b824617"}}