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We investigate the structure of $(P_t)_{t>0}$.\n  (i) Denote respectively by $(A,D(A))$ and $(\\hat A,D(\\hat A))$ the generator and the co-generator of $(P_t)_{t>0}$. Under the assumption that $C^{\\infty}_0(U)\\subset D(A)\\cap D(\\hat A)$, we give an explicit L\\'evy-Khintchine type representation of $A$ on $C^{\\infty}_0(U)$.\n  (ii) If $(P_t)_{t>0}$ is an analytic semigroup and hence is associated with a"},"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":"1303.3552","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-14T19:00:57Z","cross_cats_sorted":[],"title_canon_sha256":"86bb64de5c71a9575ad96f6e48e2def05390e3f2940a1f543dc14b7e2cdea59c","abstract_canon_sha256":"988f09b24857098151c23465facd71cecf7ec405f3c92db3bfce2e73701d71f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:16.956430Z","signature_b64":"OBYcyl8QXx6B9v3bEkSdnQkFTefPrJtp/JbI0GyqlP2rdmupc16h0rKLjixu5L0iVse5yaHTRTtF+LKG5VinCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4db9fb1fbc6973d1f2be966ecfe792462264546ecbeca0007c154ba7a2d2720","last_reissued_at":"2026-05-18T03:28:16.955699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:16.955699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Levy-Khintchine type representation of Dirichlet generators and Semi-Dirichlet forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jing Zhang, Wei Sun","submitted_at":"2013-03-14T19:00:57Z","abstract_excerpt":"Let $U$ be an open set of $\\mathbb{R}^n$, $m$ a positive Radon measure on $U$ such that ${\\rm supp}[m]=U$, and $(P_t)_{t>0}$ a strongly continuous contraction sub-Markovian semigroup on $L^2(U;m)$. We investigate the structure of $(P_t)_{t>0}$.\n  (i) Denote respectively by $(A,D(A))$ and $(\\hat A,D(\\hat A))$ the generator and the co-generator of $(P_t)_{t>0}$. Under the assumption that $C^{\\infty}_0(U)\\subset D(A)\\cap D(\\hat A)$, we give an explicit L\\'evy-Khintchine type representation of $A$ on $C^{\\infty}_0(U)$.\n  (ii) If $(P_t)_{t>0}$ is an analytic semigroup and hence is associated with a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3552","kind":"arxiv","version":2},"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":"1303.3552","created_at":"2026-05-18T03:28:16.955798+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.3552v2","created_at":"2026-05-18T03:28:16.955798+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3552","created_at":"2026-05-18T03:28:16.955798+00:00"},{"alias_kind":"pith_short_12","alias_value":"2TNZ7MP3Y2LT","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_16","alias_value":"2TNZ7MP3Y2LT2HZL","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_8","alias_value":"2TNZ7MP3","created_at":"2026-05-18T12:27:32.513160+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/2TNZ7MP3Y2LT2HZL5FTOZ7TZER","json":"https://pith.science/pith/2TNZ7MP3Y2LT2HZL5FTOZ7TZER.json","graph_json":"https://pith.science/api/pith-number/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/graph.json","events_json":"https://pith.science/api/pith-number/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/events.json","paper":"https://pith.science/paper/2TNZ7MP3"},"agent_actions":{"view_html":"https://pith.science/pith/2TNZ7MP3Y2LT2HZL5FTOZ7TZER","download_json":"https://pith.science/pith/2TNZ7MP3Y2LT2HZL5FTOZ7TZER.json","view_paper":"https://pith.science/paper/2TNZ7MP3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.3552&json=true","fetch_graph":"https://pith.science/api/pith-number/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/graph.json","fetch_events":"https://pith.science/api/pith-number/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/action/storage_attestation","attest_author":"https://pith.science/pith/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/action/author_attestation","sign_citation":"https://pith.science/pith/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/action/citation_signature","submit_replication":"https://pith.science/pith/2TNZ7MP3Y2LT2HZL5FTOZ7TZER/action/replication_record"}},"created_at":"2026-05-18T03:28:16.955798+00:00","updated_at":"2026-05-18T03:28:16.955798+00:00"}