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Let $\\alpha\\in(0,\\frac{1}{2}]$, $p\\in(\\frac{d}{1-\\alpha},\\infty)$ and $\\beta\\in[\\alpha,1]$, $q\\in(\\frac{d}{\\beta},\\infty)$. Assume $\\|({\\mathbb I}-\\Delta)^{-\\alpha/2}b\\|_p+\\|(-\\Delta)^{\\beta/2}\\sigma\\|_q<\\infty$. We show the existence and uniqueness of martingale solutions to the above S"},"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":"1710.10537","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-28T23:47:30Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"e2c44d81169093a6d84062cb3a9d2cdb35d2ca35a53f56c4b467607bdfaa84c4","abstract_canon_sha256":"afee686d441869a1983e2f87dd6741f8819435725af01b078879295c84d712c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:03.206336Z","signature_b64":"bR/dR5LVyWvde2oPmP8Fe++aXX6Kf7RD6NcGUTefdpMr1SbPUYptPeM+2bd/OBwa+AKpsZoQY1kjS82jCHq/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c492d8adef66f3da6a0b3196234e858d76afa1788ec0e2fa9200a2f089e3d2fc","last_reissued_at":"2026-05-18T00:19:03.205553Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:03.205553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heat kernel and ergodicity of SDEs with distributional drifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Guohuan Zhao, Xicheng Zhang","submitted_at":"2017-10-28T23:47:30Z","abstract_excerpt":"In this paper we consider the following SDE with distributional drift $b$: $$ {\\rm d} X_t=\\sigma(X_t){\\rm d} B_t+b(X_t){\\rm d} t,\\ X_0=x\\in{\\mathbb R}^d, $$ where $\\sigma$ is a bounded continuous and uniformly non-degenerate $d\\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $\\alpha\\in(0,\\frac{1}{2}]$, $p\\in(\\frac{d}{1-\\alpha},\\infty)$ and $\\beta\\in[\\alpha,1]$, $q\\in(\\frac{d}{\\beta},\\infty)$. Assume $\\|({\\mathbb I}-\\Delta)^{-\\alpha/2}b\\|_p+\\|(-\\Delta)^{\\beta/2}\\sigma\\|_q<\\infty$. We show the existence and uniqueness of martingale solutions to the above S"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.10537","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":"1710.10537","created_at":"2026-05-18T00:19:03.205682+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.10537v2","created_at":"2026-05-18T00:19:03.205682+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.10537","created_at":"2026-05-18T00:19:03.205682+00:00"},{"alias_kind":"pith_short_12","alias_value":"YSJNRLPPM3Z5","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_16","alias_value":"YSJNRLPPM3Z5U2QL","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_8","alias_value":"YSJNRLPP","created_at":"2026-05-18T12:31:56.362134+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2009.10786","citing_title":"Quantitative heat kernel estimates for diffusions with distributional drift","ref_index":22,"is_internal_anchor":true},{"citing_arxiv_id":"2604.23883","citing_title":"Sharp pathwise nonuniqueness for additive SDEs","ref_index":58,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV","json":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV.json","graph_json":"https://pith.science/api/pith-number/YSJNRLPPM3Z5U2QLGGLCGTUFRV/graph.json","events_json":"https://pith.science/api/pith-number/YSJNRLPPM3Z5U2QLGGLCGTUFRV/events.json","paper":"https://pith.science/paper/YSJNRLPP"},"agent_actions":{"view_html":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV","download_json":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV.json","view_paper":"https://pith.science/paper/YSJNRLPP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.10537&json=true","fetch_graph":"https://pith.science/api/pith-number/YSJNRLPPM3Z5U2QLGGLCGTUFRV/graph.json","fetch_events":"https://pith.science/api/pith-number/YSJNRLPPM3Z5U2QLGGLCGTUFRV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV/action/storage_attestation","attest_author":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV/action/author_attestation","sign_citation":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV/action/citation_signature","submit_replication":"https://pith.science/pith/YSJNRLPPM3Z5U2QLGGLCGTUFRV/action/replication_record"}},"created_at":"2026-05-18T00:19:03.205682+00:00","updated_at":"2026-05-18T00:19:03.205682+00:00"}