{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:BJQUBPGEHJVP5IA7O7KJ72V753","short_pith_number":"pith:BJQUBPGE","canonical_record":{"source":{"id":"1606.05843","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-19T08:35:52Z","cross_cats_sorted":[],"title_canon_sha256":"8355b0e8ec035f98ce94a11418294af7d59c6554e62d95aaed1c7dcdb5ea72d2","abstract_canon_sha256":"4d6a6837b0f5a734d495bc3f91ef59599c994756be813f7d732604f40afe615a"},"schema_version":"1.0"},"canonical_sha256":"0a6140bcc43a6afea01f77d49feabfeedf64a2a1e9d8205d6b5c6dd67316a93e","source":{"kind":"arxiv","id":"1606.05843","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.05843","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"arxiv_version","alias_value":"1606.05843v3","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05843","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"pith_short_12","alias_value":"BJQUBPGEHJVP","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"BJQUBPGEHJVP5IA7","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"BJQUBPGE","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:BJQUBPGEHJVP5IA7O7KJ72V753","target":"record","payload":{"canonical_record":{"source":{"id":"1606.05843","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-19T08:35:52Z","cross_cats_sorted":[],"title_canon_sha256":"8355b0e8ec035f98ce94a11418294af7d59c6554e62d95aaed1c7dcdb5ea72d2","abstract_canon_sha256":"4d6a6837b0f5a734d495bc3f91ef59599c994756be813f7d732604f40afe615a"},"schema_version":"1.0"},"canonical_sha256":"0a6140bcc43a6afea01f77d49feabfeedf64a2a1e9d8205d6b5c6dd67316a93e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:20.958427Z","signature_b64":"bD1GMZmxpfJJET9xPQXDHIobYpY+ZI8Ie2+5JxokewjzdKOeEaYGAr7SkUHa+q+opCOiOBjbkBeSqkuZ/UGQAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a6140bcc43a6afea01f77d49feabfeedf64a2a1e9d8205d6b5c6dd67316a93e","last_reissued_at":"2026-05-18T00:46:20.957962Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:20.957962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.05843","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:46:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RZfDX6DqcykZLfEzou+9oz6rgi+XIOzBx5UaajU2cmEtnpjBZFhTuNSUTqraPU008VZ8SUEcbpM4bW4i9uBgBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:57:18.778046Z"},"content_sha256":"9f08acab71dc4d1bab44a1f03e257914b87c588bb024d481a05070351ac7208d","schema_version":"1.0","event_id":"sha256:9f08acab71dc4d1bab44a1f03e257914b87c588bb024d481a05070351ac7208d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:BJQUBPGEHJVP5IA7O7KJ72V753","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distribution-Dependent SDEs for Landau Type Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang","submitted_at":"2016-06-19T08:35:52Z","abstract_excerpt":"The distribution-dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a non-linear PDE. Due to the distribution-dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a non-linear semigroup $P_t^*$ on the space of probability measu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05843","kind":"arxiv","version":3},"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":"2026-05-18T00:46:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E0v7xKrSobrfUP/8z+DbzRj2jFoieormu6dOtMeWVOFWQsl46nDZ+2+M4kWNGJKUppZl/60ccA/pBU6AP7zoDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T04:57:18.778413Z"},"content_sha256":"49bbf0c202bed00e54a8a30980e50c49d4cee6dbc8f642e1165f86e1c0092790","schema_version":"1.0","event_id":"sha256:49bbf0c202bed00e54a8a30980e50c49d4cee6dbc8f642e1165f86e1c0092790"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BJQUBPGEHJVP5IA7O7KJ72V753/bundle.json","state_url":"https://pith.science/pith/BJQUBPGEHJVP5IA7O7KJ72V753/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BJQUBPGEHJVP5IA7O7KJ72V753/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":"2026-06-01T04:57:18Z","links":{"resolver":"https://pith.science/pith/BJQUBPGEHJVP5IA7O7KJ72V753","bundle":"https://pith.science/pith/BJQUBPGEHJVP5IA7O7KJ72V753/bundle.json","state":"https://pith.science/pith/BJQUBPGEHJVP5IA7O7KJ72V753/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BJQUBPGEHJVP5IA7O7KJ72V753/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:BJQUBPGEHJVP5IA7O7KJ72V753","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":"4d6a6837b0f5a734d495bc3f91ef59599c994756be813f7d732604f40afe615a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-19T08:35:52Z","title_canon_sha256":"8355b0e8ec035f98ce94a11418294af7d59c6554e62d95aaed1c7dcdb5ea72d2"},"schema_version":"1.0","source":{"id":"1606.05843","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.05843","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"arxiv_version","alias_value":"1606.05843v3","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05843","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"pith_short_12","alias_value":"BJQUBPGEHJVP","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"BJQUBPGEHJVP5IA7","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"BJQUBPGE","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:49bbf0c202bed00e54a8a30980e50c49d4cee6dbc8f642e1165f86e1c0092790","target":"graph","created_at":"2026-05-18T00:46:20Z","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":"The distribution-dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a non-linear PDE. Due to the distribution-dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a non-linear semigroup $P_t^*$ on the space of probability measu","authors_text":"Feng-Yu Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-19T08:35:52Z","title":"Distribution-Dependent SDEs for Landau Type Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05843","kind":"arxiv","version":3},"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":"sha256:9f08acab71dc4d1bab44a1f03e257914b87c588bb024d481a05070351ac7208d","target":"record","created_at":"2026-05-18T00:46:20Z","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":"4d6a6837b0f5a734d495bc3f91ef59599c994756be813f7d732604f40afe615a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-19T08:35:52Z","title_canon_sha256":"8355b0e8ec035f98ce94a11418294af7d59c6554e62d95aaed1c7dcdb5ea72d2"},"schema_version":"1.0","source":{"id":"1606.05843","kind":"arxiv","version":3}},"canonical_sha256":"0a6140bcc43a6afea01f77d49feabfeedf64a2a1e9d8205d6b5c6dd67316a93e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a6140bcc43a6afea01f77d49feabfeedf64a2a1e9d8205d6b5c6dd67316a93e","first_computed_at":"2026-05-18T00:46:20.957962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:20.957962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bD1GMZmxpfJJET9xPQXDHIobYpY+ZI8Ie2+5JxokewjzdKOeEaYGAr7SkUHa+q+opCOiOBjbkBeSqkuZ/UGQAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:20.958427Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.05843","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9f08acab71dc4d1bab44a1f03e257914b87c588bb024d481a05070351ac7208d","sha256:49bbf0c202bed00e54a8a30980e50c49d4cee6dbc8f642e1165f86e1c0092790"],"state_sha256":"bb96f78f8a3e4a4da3dda4863b211eeae75e873b06d604e51dbf3f07172cbd63"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PByaaAE0r7MQOaD0Ct+xetqoKVpon38Kdw+4h2dEkOy3uv6/d/TfWnS2x9bFWggt0W1OkXTR79kM6juogVZVBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T04:57:18.780417Z","bundle_sha256":"81cd11be062be82e7e43e2d63d9f2e3005c0da1c5b1caf1540f49b3d190b49ec"}}