{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:42KCJIFFDNSE64XVSBF7SY44WA","short_pith_number":"pith:42KCJIFF","canonical_record":{"source":{"id":"1711.05058","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-14T11:11:03Z","cross_cats_sorted":[],"title_canon_sha256":"03570527c0971e47c3f5ff36699f8c74d35fbf09b2d96ba686255314844f613e","abstract_canon_sha256":"9ee8f94b89aa0184a793a61fe5d13d52f492b55c02288285b0100247bbc41ce0"},"schema_version":"1.0"},"canonical_sha256":"e69424a0a51b644f72f5904bf9639cb0278412dcf446a51fd2bcecb0c0fc2493","source":{"kind":"arxiv","id":"1711.05058","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.05058","created_at":"2026-05-18T00:30:34Z"},{"alias_kind":"arxiv_version","alias_value":"1711.05058v1","created_at":"2026-05-18T00:30:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.05058","created_at":"2026-05-18T00:30:34Z"},{"alias_kind":"pith_short_12","alias_value":"42KCJIFFDNSE","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"42KCJIFFDNSE64XV","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"42KCJIFF","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:42KCJIFFDNSE64XVSBF7SY44WA","target":"record","payload":{"canonical_record":{"source":{"id":"1711.05058","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-14T11:11:03Z","cross_cats_sorted":[],"title_canon_sha256":"03570527c0971e47c3f5ff36699f8c74d35fbf09b2d96ba686255314844f613e","abstract_canon_sha256":"9ee8f94b89aa0184a793a61fe5d13d52f492b55c02288285b0100247bbc41ce0"},"schema_version":"1.0"},"canonical_sha256":"e69424a0a51b644f72f5904bf9639cb0278412dcf446a51fd2bcecb0c0fc2493","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:34.886992Z","signature_b64":"KX4hRRiE/X8z8SiM2oBduaoVL+aXfGIEryH6c+ZfJycTzPoYwSgn2rFP7uQbaPtKKGMeQvGIgZ1Od9QrwtQ/BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e69424a0a51b644f72f5904bf9639cb0278412dcf446a51fd2bcecb0c0fc2493","last_reissued_at":"2026-05-18T00:30:34.886281Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:34.886281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.05058","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:30:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BsFqo/1My3/0iH7ja3Qetq+exWKjmngGL5f6v9KHu3l62LaE1YaDl5NnIiQ3DKNrjTaxH08Irdw6+Ag/tZImCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T14:42:27.728941Z"},"content_sha256":"c51922b5885ce8b26e11879ba0be140ce24217fe52cf3123f66ac149b4530289","schema_version":"1.0","event_id":"sha256:c51922b5885ce8b26e11879ba0be140ce24217fe52cf3123f66ac149b4530289"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:42KCJIFFDNSE64XVSBF7SY44WA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On weak solutions of stochastic differential equations with sharp drift coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guangying Lv, Jiang-Lun Wu, Jinlong Wei","submitted_at":"2017-11-14T11:11:03Z","abstract_excerpt":"We extend Krylov and R\\\"{o}ckner's result \\cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\\times{\\mathbb R}^d\\rightarrow{\\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\\int_0^tb(s,X_s)ds+W_t,\\quad t\\in[0,T], \\, x\\in{\\mathbb R}^d,$$ where $\\{W_t\\}_{t\\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\\cdot)\\in\\mathcal{C}_q^0((0,T];L^p({\\mathbb R}^d))$ with $2/q+d/p=1$ for $p,q\\ge1$ and $\\|b_1(T-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05058","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:30:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AP4PlYxYF51qsmKL49CURAwWNskepa0OeNnqUhjgO5FdQXfC7KUHrCiikE3/16IWA07v/hiIFES2d1oQ1GTQBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T14:42:27.729720Z"},"content_sha256":"2698b0638fbe380b610d81e76cb5e7ddcc01824506047183d2e615622f557685","schema_version":"1.0","event_id":"sha256:2698b0638fbe380b610d81e76cb5e7ddcc01824506047183d2e615622f557685"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/42KCJIFFDNSE64XVSBF7SY44WA/bundle.json","state_url":"https://pith.science/pith/42KCJIFFDNSE64XVSBF7SY44WA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/42KCJIFFDNSE64XVSBF7SY44WA/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":"2026-05-25T14:42:27Z","links":{"resolver":"https://pith.science/pith/42KCJIFFDNSE64XVSBF7SY44WA","bundle":"https://pith.science/pith/42KCJIFFDNSE64XVSBF7SY44WA/bundle.json","state":"https://pith.science/pith/42KCJIFFDNSE64XVSBF7SY44WA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/42KCJIFFDNSE64XVSBF7SY44WA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:42KCJIFFDNSE64XVSBF7SY44WA","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":"9ee8f94b89aa0184a793a61fe5d13d52f492b55c02288285b0100247bbc41ce0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-14T11:11:03Z","title_canon_sha256":"03570527c0971e47c3f5ff36699f8c74d35fbf09b2d96ba686255314844f613e"},"schema_version":"1.0","source":{"id":"1711.05058","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.05058","created_at":"2026-05-18T00:30:34Z"},{"alias_kind":"arxiv_version","alias_value":"1711.05058v1","created_at":"2026-05-18T00:30:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.05058","created_at":"2026-05-18T00:30:34Z"},{"alias_kind":"pith_short_12","alias_value":"42KCJIFFDNSE","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"42KCJIFFDNSE64XV","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"42KCJIFF","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:2698b0638fbe380b610d81e76cb5e7ddcc01824506047183d2e615622f557685","target":"graph","created_at":"2026-05-18T00:30:34Z","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":"We extend Krylov and R\\\"{o}ckner's result \\cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\\times{\\mathbb R}^d\\rightarrow{\\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\\int_0^tb(s,X_s)ds+W_t,\\quad t\\in[0,T], \\, x\\in{\\mathbb R}^d,$$ where $\\{W_t\\}_{t\\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\\cdot)\\in\\mathcal{C}_q^0((0,T];L^p({\\mathbb R}^d))$ with $2/q+d/p=1$ for $p,q\\ge1$ and $\\|b_1(T-","authors_text":"Guangying Lv, Jiang-Lun Wu, Jinlong Wei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-14T11:11:03Z","title":"On weak solutions of stochastic differential equations with sharp drift coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05058","kind":"arxiv","version":1},"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:c51922b5885ce8b26e11879ba0be140ce24217fe52cf3123f66ac149b4530289","target":"record","created_at":"2026-05-18T00:30:34Z","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":"9ee8f94b89aa0184a793a61fe5d13d52f492b55c02288285b0100247bbc41ce0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-14T11:11:03Z","title_canon_sha256":"03570527c0971e47c3f5ff36699f8c74d35fbf09b2d96ba686255314844f613e"},"schema_version":"1.0","source":{"id":"1711.05058","kind":"arxiv","version":1}},"canonical_sha256":"e69424a0a51b644f72f5904bf9639cb0278412dcf446a51fd2bcecb0c0fc2493","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e69424a0a51b644f72f5904bf9639cb0278412dcf446a51fd2bcecb0c0fc2493","first_computed_at":"2026-05-18T00:30:34.886281Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:30:34.886281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KX4hRRiE/X8z8SiM2oBduaoVL+aXfGIEryH6c+ZfJycTzPoYwSgn2rFP7uQbaPtKKGMeQvGIgZ1Od9QrwtQ/BA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:30:34.886992Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.05058","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c51922b5885ce8b26e11879ba0be140ce24217fe52cf3123f66ac149b4530289","sha256:2698b0638fbe380b610d81e76cb5e7ddcc01824506047183d2e615622f557685"],"state_sha256":"f2bfc2f29f8bebcc1fa320da2066c1a4eea685e3d1df12b8a312d62c7892b86a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XdTcNYhWWWObqL11UsaYjkuswOEXIdzULt94nZDYcnTsc5CFcg2ZFjOiMr6U6peAzPsg9rJRdw5Ulenpt5ARBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T14:42:27.733810Z","bundle_sha256":"688648497f0b16eee4b12f9e6debad804bf8b7c22ae2ead00cf4d46ac1f524e8"}}