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Let $\\mathcal{X}$ be a random variable independent of the diffusion process $X(\\cdot)$ and distributed according to the process's invariant probability measure $\\mu(x)dx$. Denote by $\\mathcal{E}^\\mu$ the expectation with respect to $\\mathcal{X}$. Consider the expression $$ \\mathcal{E}^\\mu E_x\\tau_\\mathcal{X}=\\int_{-\\infty}^\\infty (E_x\\tau_y)\\mu(y)dy, \\ x\\in\\mathbb{R}. $$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1903.12005","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-03-28T14:29:04Z","cross_cats_sorted":[],"title_canon_sha256":"fc96a1e404178951d6c7625bcb26688b0ab177d2fc75925035c6918f7742216f","abstract_canon_sha256":"f3f8bd7f33c1ca6a2ee62aedc3be4bd3b57dbc0b6710b09ba48495ed8cfff4c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:58.394169Z","signature_b64":"JmS0BonaPghVGg1AjfbLM2hS6iNHFTUu2bRQ/8PIxbinjg5Gp0DBQXhBnIaNPQAUVyt81cJI5unHmCMw7CCKDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80fe0cd8050a023369b69fb7a53568d9799c892f004b53319081cefb122b6a41","last_reissued_at":"2026-05-17T23:49:58.393501Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:58.393501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kemeny's constant for one-dimensional diffusions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ross G. Pinsky","submitted_at":"2019-03-28T14:29:04Z","abstract_excerpt":"Let $X(\\cdot)$ be a non-degenerate, positive recurrent one-dimensional diffusion process on $\\mathbb{R}$ with invariant probability density $\\mu(x)$, and let $\\tau_y=\\inf\\{t\\ge0: X(t)=y\\}$ denote the first hitting time of $y$. Let $\\mathcal{X}$ be a random variable independent of the diffusion process $X(\\cdot)$ and distributed according to the process's invariant probability measure $\\mu(x)dx$. Denote by $\\mathcal{E}^\\mu$ the expectation with respect to $\\mathcal{X}$. Consider the expression $$ \\mathcal{E}^\\mu E_x\\tau_\\mathcal{X}=\\int_{-\\infty}^\\infty (E_x\\tau_y)\\mu(y)dy, \\ x\\in\\mathbb{R}. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.12005","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1903.12005","created_at":"2026-05-17T23:49:58.393592+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.12005v1","created_at":"2026-05-17T23:49:58.393592+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.12005","created_at":"2026-05-17T23:49:58.393592+00:00"},{"alias_kind":"pith_short_12","alias_value":"QD7AZWAFBIBD","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_16","alias_value":"QD7AZWAFBIBDG2NW","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_8","alias_value":"QD7AZWAF","created_at":"2026-05-18T12:33:27.125529+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F","json":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F.json","graph_json":"https://pith.science/api/pith-number/QD7AZWAFBIBDG2NWT632KNLI3F/graph.json","events_json":"https://pith.science/api/pith-number/QD7AZWAFBIBDG2NWT632KNLI3F/events.json","paper":"https://pith.science/paper/QD7AZWAF"},"agent_actions":{"view_html":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F","download_json":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F.json","view_paper":"https://pith.science/paper/QD7AZWAF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.12005&json=true","fetch_graph":"https://pith.science/api/pith-number/QD7AZWAFBIBDG2NWT632KNLI3F/graph.json","fetch_events":"https://pith.science/api/pith-number/QD7AZWAFBIBDG2NWT632KNLI3F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F/action/storage_attestation","attest_author":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F/action/author_attestation","sign_citation":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F/action/citation_signature","submit_replication":"https://pith.science/pith/QD7AZWAFBIBDG2NWT632KNLI3F/action/replication_record"}},"created_at":"2026-05-17T23:49:58.393592+00:00","updated_at":"2026-05-17T23:49:58.393592+00:00"}