{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:CASN473CAAGRLZS272GSWJZBYF","short_pith_number":"pith:CASN473C","schema_version":"1.0","canonical_sha256":"1024de7f62000d15e65afe8d2b2721c162ea33400c86e5411595d8e0dec638c1","source":{"kind":"arxiv","id":"1105.5016","version":1},"attestation_state":"computed","paper":{"title":"A geometric interpretation of the transition density of a symmetric L\\'evy Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"N. Jacob, R.L. Schilling, S. Landwehr, V. Knopova","submitted_at":"2011-05-25T13:18:06Z","abstract_excerpt":"We study for a class of symmetric L\\'evy processes with state space $\\rn$ the transition density $p_t(x)$ in terms of two one-parameter families of metrics, $(d_t)_{t>0}$ and $(\\delta_t)_{t>0}$. The first family of metrics describes the diagonal term $p_t(0)$; it is induced by the characteristic exponent $\\psi$ of the L\\'evy process by $d_t(x,y)=\\sqrt{t\\psi(x-y)}$. The second and new family of metrics $\\delta_t$ relates to $\\sqrt{t\\psi}$ through the formula\n  $$\n  \\exp(-\\delta_t^2(x,y))\n  = \\Ff[\\frac{e^{-t\\psi}}{p_t(0)}](x-y)\n  $$\n  where $\\Ff$ denotes the Fourier transform. Thus we obtain the"},"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":"1105.5016","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-05-25T13:18:06Z","cross_cats_sorted":[],"title_canon_sha256":"a07c468d9a885cd5b9e71f62008f9e71b7183bbcaed62c59ce32ce9c171a0603","abstract_canon_sha256":"44fe6c7549f47f5c20f64488fdcf94f073c94131c3ccb731045258095b9a7bdb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:20.936768Z","signature_b64":"fJ/nX4Dgvo//sFWiwnwiQUPNfZd7gEzWEu9O3kDOph7brCNo3oRjaR0AWmdGdijbW4sZR39KexHQvVhUJheZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1024de7f62000d15e65afe8d2b2721c162ea33400c86e5411595d8e0dec638c1","last_reissued_at":"2026-05-18T04:21:20.936292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:20.936292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A geometric interpretation of the transition density of a symmetric L\\'evy Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"N. Jacob, R.L. Schilling, S. Landwehr, V. Knopova","submitted_at":"2011-05-25T13:18:06Z","abstract_excerpt":"We study for a class of symmetric L\\'evy processes with state space $\\rn$ the transition density $p_t(x)$ in terms of two one-parameter families of metrics, $(d_t)_{t>0}$ and $(\\delta_t)_{t>0}$. The first family of metrics describes the diagonal term $p_t(0)$; it is induced by the characteristic exponent $\\psi$ of the L\\'evy process by $d_t(x,y)=\\sqrt{t\\psi(x-y)}$. The second and new family of metrics $\\delta_t$ relates to $\\sqrt{t\\psi}$ through the formula\n  $$\n  \\exp(-\\delta_t^2(x,y))\n  = \\Ff[\\frac{e^{-t\\psi}}{p_t(0)}](x-y)\n  $$\n  where $\\Ff$ denotes the Fourier transform. Thus we obtain the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.5016","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":"1105.5016","created_at":"2026-05-18T04:21:20.936388+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.5016v1","created_at":"2026-05-18T04:21:20.936388+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.5016","created_at":"2026-05-18T04:21:20.936388+00:00"},{"alias_kind":"pith_short_12","alias_value":"CASN473CAAGR","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"CASN473CAAGRLZS2","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"CASN473C","created_at":"2026-05-18T12:26:26.731475+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/CASN473CAAGRLZS272GSWJZBYF","json":"https://pith.science/pith/CASN473CAAGRLZS272GSWJZBYF.json","graph_json":"https://pith.science/api/pith-number/CASN473CAAGRLZS272GSWJZBYF/graph.json","events_json":"https://pith.science/api/pith-number/CASN473CAAGRLZS272GSWJZBYF/events.json","paper":"https://pith.science/paper/CASN473C"},"agent_actions":{"view_html":"https://pith.science/pith/CASN473CAAGRLZS272GSWJZBYF","download_json":"https://pith.science/pith/CASN473CAAGRLZS272GSWJZBYF.json","view_paper":"https://pith.science/paper/CASN473C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.5016&json=true","fetch_graph":"https://pith.science/api/pith-number/CASN473CAAGRLZS272GSWJZBYF/graph.json","fetch_events":"https://pith.science/api/pith-number/CASN473CAAGRLZS272GSWJZBYF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CASN473CAAGRLZS272GSWJZBYF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CASN473CAAGRLZS272GSWJZBYF/action/storage_attestation","attest_author":"https://pith.science/pith/CASN473CAAGRLZS272GSWJZBYF/action/author_attestation","sign_citation":"https://pith.science/pith/CASN473CAAGRLZS272GSWJZBYF/action/citation_signature","submit_replication":"https://pith.science/pith/CASN473CAAGRLZS272GSWJZBYF/action/replication_record"}},"created_at":"2026-05-18T04:21:20.936388+00:00","updated_at":"2026-05-18T04:21:20.936388+00:00"}