{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:J22K4YHKSU32BGASDH75HWIY5B","short_pith_number":"pith:J22K4YHK","canonical_record":{"source":{"id":"1903.11349","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-03-27T11:17:25Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"f7268a5e63fc7d5b625cdded0adbfadf20d1866ae098893f77f671b4ce36011c","abstract_canon_sha256":"dc111ed57da5494840028ff250c79cb323b187bfd269bbd19184a49859d5ed03"},"schema_version":"1.0"},"canonical_sha256":"4eb4ae60ea9537a0981219ffd3d918e8676def819f12ff6bf10e8d3505ebc622","source":{"kind":"arxiv","id":"1903.11349","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.11349","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"arxiv_version","alias_value":"1903.11349v2","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.11349","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"pith_short_12","alias_value":"J22K4YHKSU32","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"pith_short_16","alias_value":"J22K4YHKSU32BGAS","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"pith_short_8","alias_value":"J22K4YHK","created_at":"2026-07-05T11:59:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:J22K4YHKSU32BGASDH75HWIY5B","target":"record","payload":{"canonical_record":{"source":{"id":"1903.11349","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-03-27T11:17:25Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"f7268a5e63fc7d5b625cdded0adbfadf20d1866ae098893f77f671b4ce36011c","abstract_canon_sha256":"dc111ed57da5494840028ff250c79cb323b187bfd269bbd19184a49859d5ed03"},"schema_version":"1.0"},"canonical_sha256":"4eb4ae60ea9537a0981219ffd3d918e8676def819f12ff6bf10e8d3505ebc622","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:59:01.464301Z","signature_b64":"PpyR5ULAwpYRNUQN7yO4izZq6AcIZ0/Pm7GHXSZD9KNIYU6ytgjI494iJh1764w+q9YXo6HccCgVvo+CMpltCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4eb4ae60ea9537a0981219ffd3d918e8676def819f12ff6bf10e8d3505ebc622","last_reissued_at":"2026-07-05T11:59:01.463802Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:59:01.463802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.11349","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":"2026-07-05T11:59:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/Bew36IjBgJrUs6XQjqNob8F2nLqueb2ILCxNt/ovnny2l7djDt9T5ROmIs572WzCMbz9gItGgF86aDZ1oQjDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T02:11:05.736809Z"},"content_sha256":"38349e3e414ca4c28cb555e68d54c46920c40ca93981c9795aa0188a31ffd9a8","schema_version":"1.0","event_id":"sha256:38349e3e414ca4c28cb555e68d54c46920c40ca93981c9795aa0188a31ffd9a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:J22K4YHKSU32BGASDH75HWIY5B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Monge-Kantorovich distance for PDEs: the coupling method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Beno\\^it Perthame (LJLL, MAMBA), Nicolas Fournier (LPSM UMR 8001)","submitted_at":"2019-03-27T11:17:25Z","abstract_excerpt":"We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This amounts to solve a well-chosen PDE posed on the Euclidian square of the physical space, i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.11349","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1903.11349/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T11:59:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c6xbKGBSRQhq78KPCYarRqvnQH+Qfh1gOkwYt4XUi2ICe23XnFNo0n+Wg/LAerJAOX7gpsT/AXeXGJMHqPAVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T02:11:05.737316Z"},"content_sha256":"77f37bad6fa8d47e120f7e1c145ff677111fe8563add75f6f2da83cf741d7d59","schema_version":"1.0","event_id":"sha256:77f37bad6fa8d47e120f7e1c145ff677111fe8563add75f6f2da83cf741d7d59"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J22K4YHKSU32BGASDH75HWIY5B/bundle.json","state_url":"https://pith.science/pith/J22K4YHKSU32BGASDH75HWIY5B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J22K4YHKSU32BGASDH75HWIY5B/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":"2026-08-17T02:11:05Z","links":{"resolver":"https://pith.science/pith/J22K4YHKSU32BGASDH75HWIY5B","bundle":"https://pith.science/pith/J22K4YHKSU32BGASDH75HWIY5B/bundle.json","state":"https://pith.science/pith/J22K4YHKSU32BGASDH75HWIY5B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J22K4YHKSU32BGASDH75HWIY5B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:J22K4YHKSU32BGASDH75HWIY5B","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":"dc111ed57da5494840028ff250c79cb323b187bfd269bbd19184a49859d5ed03","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-03-27T11:17:25Z","title_canon_sha256":"f7268a5e63fc7d5b625cdded0adbfadf20d1866ae098893f77f671b4ce36011c"},"schema_version":"1.0","source":{"id":"1903.11349","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.11349","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"arxiv_version","alias_value":"1903.11349v2","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.11349","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"pith_short_12","alias_value":"J22K4YHKSU32","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"pith_short_16","alias_value":"J22K4YHKSU32BGAS","created_at":"2026-07-05T11:59:01Z"},{"alias_kind":"pith_short_8","alias_value":"J22K4YHK","created_at":"2026-07-05T11:59:01Z"}],"graph_snapshots":[{"event_id":"sha256:77f37bad6fa8d47e120f7e1c145ff677111fe8563add75f6f2da83cf741d7d59","target":"graph","created_at":"2026-07-05T11:59:01Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1903.11349/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This amounts to solve a well-chosen PDE posed on the Euclidian square of the physical space, i","authors_text":"Beno\\^it Perthame (LJLL, MAMBA), Nicolas Fournier (LPSM UMR 8001)","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-03-27T11:17:25Z","title":"Monge-Kantorovich distance for PDEs: the coupling method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.11349","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":"sha256:38349e3e414ca4c28cb555e68d54c46920c40ca93981c9795aa0188a31ffd9a8","target":"record","created_at":"2026-07-05T11:59:01Z","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":"dc111ed57da5494840028ff250c79cb323b187bfd269bbd19184a49859d5ed03","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-03-27T11:17:25Z","title_canon_sha256":"f7268a5e63fc7d5b625cdded0adbfadf20d1866ae098893f77f671b4ce36011c"},"schema_version":"1.0","source":{"id":"1903.11349","kind":"arxiv","version":2}},"canonical_sha256":"4eb4ae60ea9537a0981219ffd3d918e8676def819f12ff6bf10e8d3505ebc622","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4eb4ae60ea9537a0981219ffd3d918e8676def819f12ff6bf10e8d3505ebc622","first_computed_at":"2026-07-05T11:59:01.463802Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:59:01.463802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PpyR5ULAwpYRNUQN7yO4izZq6AcIZ0/Pm7GHXSZD9KNIYU6ytgjI494iJh1764w+q9YXo6HccCgVvo+CMpltCA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:59:01.464301Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.11349","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:38349e3e414ca4c28cb555e68d54c46920c40ca93981c9795aa0188a31ffd9a8","sha256:77f37bad6fa8d47e120f7e1c145ff677111fe8563add75f6f2da83cf741d7d59"],"state_sha256":"8d44b2b5a0a5bda226beb33a6feb6021da85611653bc2127c3005184db61780a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OsO7kOxQMcsC18rRblu1r1YMaws49k2FBjAS8gRU0kAKBs2GN7/rrBWkdNEAt4UEmwmEHiqyGnlKIExOnnKSAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T02:11:05.743018Z","bundle_sha256":"704f18dfeb759b24c5ae0f2b0de9218d9269dfb7254e46376a76c6f549b17a70"}}