{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:SAL2XUO757QTOBOT7RN3DJ42JC","short_pith_number":"pith:SAL2XUO7","canonical_record":{"source":{"id":"2009.10786","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-09-22T19:53:41Z","cross_cats_sorted":[],"title_canon_sha256":"47f1aec08ad27d47c49cb6fa0a38a40ee658ae0fd23884af533c5bb34cc9dced","abstract_canon_sha256":"a7aa561800750c5d52539adb5bd768ff8bf65a34a80592be166d4e400bca5328"},"schema_version":"1.0"},"canonical_sha256":"9017abd1dfefe13705d3fc5bb1a79a48a1b647e2d85e5c2b6e0fce358078f636","source":{"kind":"arxiv","id":"2009.10786","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.10786","created_at":"2026-05-18T02:44:39Z"},{"alias_kind":"arxiv_version","alias_value":"2009.10786v2","created_at":"2026-05-18T02:44:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.10786","created_at":"2026-05-18T02:44:39Z"},{"alias_kind":"pith_short_12","alias_value":"SAL2XUO757QT","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"SAL2XUO757QTOBOT","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"SAL2XUO7","created_at":"2026-05-18T12:33:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:SAL2XUO757QTOBOT7RN3DJ42JC","target":"record","payload":{"canonical_record":{"source":{"id":"2009.10786","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-09-22T19:53:41Z","cross_cats_sorted":[],"title_canon_sha256":"47f1aec08ad27d47c49cb6fa0a38a40ee658ae0fd23884af533c5bb34cc9dced","abstract_canon_sha256":"a7aa561800750c5d52539adb5bd768ff8bf65a34a80592be166d4e400bca5328"},"schema_version":"1.0"},"canonical_sha256":"9017abd1dfefe13705d3fc5bb1a79a48a1b647e2d85e5c2b6e0fce358078f636","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:39.985619Z","signature_b64":"gwxKIzRzfJVxllU686YrRWCvUutSeFxDljJHMOT1RlI9BISa+TAXHWlemGqEfKTY+/eMPaWTf9wOLjQoRQSXCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9017abd1dfefe13705d3fc5bb1a79a48a1b647e2d85e5c2b6e0fce358078f636","last_reissued_at":"2026-05-18T02:44:39.985066Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:39.985066Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2009.10786","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/+xqEjbs71LQLwbM1Yb9QDcoNBosl9E1+BCbluwEwk8yvafQB2igHlzDm9GowPQuJDIRoDF33CECKtGGNx/lDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T14:55:44.583891Z"},"content_sha256":"75c08c93a5e3bbd82f41bac3ac3df1ef84f4e430343bc6c779817468c20dfb81","schema_version":"1.0","event_id":"sha256:75c08c93a5e3bbd82f41bac3ac3df1ef84f4e430343bc6c779817468c20dfb81"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:SAL2XUO757QTOBOT7RN3DJ42JC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative heat kernel estimates for diffusions with distributional drift","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Nicolas Perkowski, Willem van Zuijlen","submitted_at":"2020-09-22T19:53:41Z","abstract_excerpt":"We consider the stochastic differential equation on $\\mathbb{R}^d$ given by $$ \\, \\mathrm{d}X_t = b(t,X_t) \\, \\mathrm{d}t + \\, \\mathrm{d} B_t, $$ where $B$ is a Brownian motion and $b$ is considered to be a distribution of regularity $ > -\\frac12$. We show that the martingale solution of the SDE has a transition kernel $\\Gamma_t$ and prove upper and lower heat kernel bounds for $\\Gamma_t$ with explicit dependence on $t$ and the norm of $b$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.10786","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rk4iFnQGB9pLYi+JvsC1Z3uKt1zmk19pMAOJa9jKWzCP0aONg4HN7W0L2XlH3fl4RWdq230s18kkyKCKM0mHBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T14:55:44.584519Z"},"content_sha256":"866879ee19aeed778fab7a5a17da1a6fc740c825ec99d64523799f2ad4d8b78c","schema_version":"1.0","event_id":"sha256:866879ee19aeed778fab7a5a17da1a6fc740c825ec99d64523799f2ad4d8b78c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SAL2XUO757QTOBOT7RN3DJ42JC/bundle.json","state_url":"https://pith.science/pith/SAL2XUO757QTOBOT7RN3DJ42JC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SAL2XUO757QTOBOT7RN3DJ42JC/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":"2026-05-21T14:55:44Z","links":{"resolver":"https://pith.science/pith/SAL2XUO757QTOBOT7RN3DJ42JC","bundle":"https://pith.science/pith/SAL2XUO757QTOBOT7RN3DJ42JC/bundle.json","state":"https://pith.science/pith/SAL2XUO757QTOBOT7RN3DJ42JC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SAL2XUO757QTOBOT7RN3DJ42JC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:SAL2XUO757QTOBOT7RN3DJ42JC","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":"a7aa561800750c5d52539adb5bd768ff8bf65a34a80592be166d4e400bca5328","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-09-22T19:53:41Z","title_canon_sha256":"47f1aec08ad27d47c49cb6fa0a38a40ee658ae0fd23884af533c5bb34cc9dced"},"schema_version":"1.0","source":{"id":"2009.10786","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.10786","created_at":"2026-05-18T02:44:39Z"},{"alias_kind":"arxiv_version","alias_value":"2009.10786v2","created_at":"2026-05-18T02:44:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.10786","created_at":"2026-05-18T02:44:39Z"},{"alias_kind":"pith_short_12","alias_value":"SAL2XUO757QT","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"SAL2XUO757QTOBOT","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"SAL2XUO7","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:866879ee19aeed778fab7a5a17da1a6fc740c825ec99d64523799f2ad4d8b78c","target":"graph","created_at":"2026-05-18T02:44:39Z","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 consider the stochastic differential equation on $\\mathbb{R}^d$ given by $$ \\, \\mathrm{d}X_t = b(t,X_t) \\, \\mathrm{d}t + \\, \\mathrm{d} B_t, $$ where $B$ is a Brownian motion and $b$ is considered to be a distribution of regularity $ > -\\frac12$. We show that the martingale solution of the SDE has a transition kernel $\\Gamma_t$ and prove upper and lower heat kernel bounds for $\\Gamma_t$ with explicit dependence on $t$ and the norm of $b$.","authors_text":"Nicolas Perkowski, Willem van Zuijlen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-09-22T19:53:41Z","title":"Quantitative heat kernel estimates for diffusions with distributional drift"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.10786","kind":"arxiv","version":2},"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":"sha256:75c08c93a5e3bbd82f41bac3ac3df1ef84f4e430343bc6c779817468c20dfb81","target":"record","created_at":"2026-05-18T02:44:39Z","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":"a7aa561800750c5d52539adb5bd768ff8bf65a34a80592be166d4e400bca5328","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-09-22T19:53:41Z","title_canon_sha256":"47f1aec08ad27d47c49cb6fa0a38a40ee658ae0fd23884af533c5bb34cc9dced"},"schema_version":"1.0","source":{"id":"2009.10786","kind":"arxiv","version":2}},"canonical_sha256":"9017abd1dfefe13705d3fc5bb1a79a48a1b647e2d85e5c2b6e0fce358078f636","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9017abd1dfefe13705d3fc5bb1a79a48a1b647e2d85e5c2b6e0fce358078f636","first_computed_at":"2026-05-18T02:44:39.985066Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:39.985066Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gwxKIzRzfJVxllU686YrRWCvUutSeFxDljJHMOT1RlI9BISa+TAXHWlemGqEfKTY+/eMPaWTf9wOLjQoRQSXCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:39.985619Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.10786","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:75c08c93a5e3bbd82f41bac3ac3df1ef84f4e430343bc6c779817468c20dfb81","sha256:866879ee19aeed778fab7a5a17da1a6fc740c825ec99d64523799f2ad4d8b78c"],"state_sha256":"8460fc12594fff95aba355254cb36a7dc57e562314f95ba4a99b015eb61bf485"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"34Xqu1H+D7NZuF3sE1ZNOWN0uhDqb04AnlQ8Oa1zIqCZCGjQdrjQbmnqciHQRpi/dOXEGxWh6W8Gr1wpOqcnBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T14:55:44.587789Z","bundle_sha256":"744851c8c81ea293d96313b8108da297f94e32d614d3ba7087c6a072329abe2f"}}