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For $u\\in D(\\mathcal{E})_{loc}$, denote $A_t^{[u]}:=\\tilde{u}(X_{t})-\\tilde{u}(X_{0})$ and $F^{[u]}_t:=\\sum_{0<s\\leq t}(\\tilde u(X_{s})-\\tilde u(X_{s-}))1_{\\{|\\tilde u(X_{s})-\\tilde u(X_{s-})|>1\\}}$, where $\\tilde{u}$ is a quasi-continuous version of $u$. We show that there exist a unique locally square integrable martingale additive functional $Y^{[u]}$ and a unique continuous local additive functional $Z^{[u]}$ of zero quadratic variation such that $$A_t^{[u]}=Y_t^"},"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":"1406.2351","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-06-09T20:49:34Z","cross_cats_sorted":[],"title_canon_sha256":"23e76c4a47fb9da789f809764801d2e91ae583e72b4438fea8ac62d94d62dd75","abstract_canon_sha256":"0eacb7d60d201caa10107717c939cc857e309e5849f5700a7ed40afb7feb5a8c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:01.853929Z","signature_b64":"CpvBVumLIXuOrOhyrj9gU55VjHZEC2pypvhM7Z0u7kS3SMVysm547dALkcB8wAPqllbjLltl+osSrG5LKUsHDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"407be72e03629f6249da30c3446490c4fcfa3d39976cfd0a703aad3f92dcee9e","last_reissued_at":"2026-05-18T02:50:01.853435Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:01.853435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stochastic Calculus for Markov Processes Associated with Semi-Dirichlet Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chuan-Zhong Chen, Li Ma, Wei Sun","submitted_at":"2014-06-09T20:49:34Z","abstract_excerpt":"Let $(\\mathcal{E},D(\\mathcal{E}))$ be a quasi-regular semi-Dirichlet form and $(X_t)_{t\\geq0}$ be the associated Markov process. For $u\\in D(\\mathcal{E})_{loc}$, denote $A_t^{[u]}:=\\tilde{u}(X_{t})-\\tilde{u}(X_{0})$ and $F^{[u]}_t:=\\sum_{0<s\\leq t}(\\tilde u(X_{s})-\\tilde u(X_{s-}))1_{\\{|\\tilde u(X_{s})-\\tilde u(X_{s-})|>1\\}}$, where $\\tilde{u}$ is a quasi-continuous version of $u$. We show that there exist a unique locally square integrable martingale additive functional $Y^{[u]}$ and a unique continuous local additive functional $Z^{[u]}$ of zero quadratic variation such that $$A_t^{[u]}=Y_t^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.2351","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":"1406.2351","created_at":"2026-05-18T02:50:01.853506+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.2351v1","created_at":"2026-05-18T02:50:01.853506+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.2351","created_at":"2026-05-18T02:50:01.853506+00:00"},{"alias_kind":"pith_short_12","alias_value":"IB56OLQDMKPW","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"IB56OLQDMKPWESO2","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"IB56OLQD","created_at":"2026-05-18T12:28:33.132498+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/IB56OLQDMKPWESO2GDBUIZEQYT","json":"https://pith.science/pith/IB56OLQDMKPWESO2GDBUIZEQYT.json","graph_json":"https://pith.science/api/pith-number/IB56OLQDMKPWESO2GDBUIZEQYT/graph.json","events_json":"https://pith.science/api/pith-number/IB56OLQDMKPWESO2GDBUIZEQYT/events.json","paper":"https://pith.science/paper/1406.2351"},"agent_actions":{"view_html":"https://pith.science/pith/IB56OLQDMKPWESO2GDBUIZEQYT","download_json":"https://pith.science/pith/IB56OLQDMKPWESO2GDBUIZEQYT.json","view_paper":"https://pith.science/paper/1406.2351","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.2351&json=true","fetch_graph":"https://pith.science/api/pith-number/IB56OLQDMKPWESO2GDBUIZEQYT/graph.json","fetch_events":"https://pith.science/api/pith-number/IB56OLQDMKPWESO2GDBUIZEQYT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IB56OLQDMKPWESO2GDBUIZEQYT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IB56OLQDMKPWESO2GDBUIZEQYT/action/storage_attestation","attest_author":"https://pith.science/pith/IB56OLQDMKPWESO2GDBUIZEQYT/action/author_attestation","sign_citation":"https://pith.science/pith/IB56OLQDMKPWESO2GDBUIZEQYT/action/citation_signature","submit_replication":"https://pith.science/pith/IB56OLQDMKPWESO2GDBUIZEQYT/action/replication_record"}},"created_at":"2026-05-18T02:50:01.853506+00:00","updated_at":"2026-05-18T02:50:01.853506+00:00"}