{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:FFI4UL5MSOGLZ2GRQUPZALCHHA","short_pith_number":"pith:FFI4UL5M","canonical_record":{"source":{"id":"1708.08768","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-29T14:18:58Z","cross_cats_sorted":[],"title_canon_sha256":"27ab9cf5efdce77f9fee6921ad1d5a809d669509ac753094cd1182bcb8e01ed1","abstract_canon_sha256":"03239ed5b9f63c61fe51da0f0607aaa523bb7702d0dd81b46a6466b725e8e01e"},"schema_version":"1.0"},"canonical_sha256":"2951ca2fac938cbce8d1851f902c473827aea47be0119b29bca25023e68551fc","source":{"kind":"arxiv","id":"1708.08768","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.08768","created_at":"2026-05-18T00:32:05Z"},{"alias_kind":"arxiv_version","alias_value":"1708.08768v2","created_at":"2026-05-18T00:32:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.08768","created_at":"2026-05-18T00:32:05Z"},{"alias_kind":"pith_short_12","alias_value":"FFI4UL5MSOGL","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FFI4UL5MSOGLZ2GR","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FFI4UL5M","created_at":"2026-05-18T12:31:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:FFI4UL5MSOGLZ2GRQUPZALCHHA","target":"record","payload":{"canonical_record":{"source":{"id":"1708.08768","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-29T14:18:58Z","cross_cats_sorted":[],"title_canon_sha256":"27ab9cf5efdce77f9fee6921ad1d5a809d669509ac753094cd1182bcb8e01ed1","abstract_canon_sha256":"03239ed5b9f63c61fe51da0f0607aaa523bb7702d0dd81b46a6466b725e8e01e"},"schema_version":"1.0"},"canonical_sha256":"2951ca2fac938cbce8d1851f902c473827aea47be0119b29bca25023e68551fc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:05.489780Z","signature_b64":"uZEXp7Ysk4xDdSZD+XaE2+FDVbGgH7gjijirlzn5+yVno5T9LumZEX6OPIBwKpqawdruEtcFb5ATQgWvfPDlAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2951ca2fac938cbce8d1851f902c473827aea47be0119b29bca25023e68551fc","last_reissued_at":"2026-05-18T00:32:05.489108Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:05.489108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.08768","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":"2026-05-18T00:32:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NOOZTIwGPoNMNQCNhsJ/pdDjebxxoKbTUElheRjoSyFaNemJOvhR9CV+pMrvSsp5/yMNN/Vd/yKEpnQSt7QMBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T15:00:33.333706Z"},"content_sha256":"1de936b02db798cac7999e9633f486ea5d0396c0e9a209d2d065c668dc8b9ed3","schema_version":"1.0","event_id":"sha256:1de936b02db798cac7999e9633f486ea5d0396c0e9a209d2d065c668dc8b9ed3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:FFI4UL5MSOGLZ2GRQUPZALCHHA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonlinear Fokker-Planck equations driven by Gaussian linear multiplicative noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Michael R\\\"ockner, Viorel Barbu","submitted_at":"2017-08-29T14:18:58Z","abstract_excerpt":"Existence and uniqueness of a strong solution in $H^{-1}(\\mathbb R^d)$ is proved for the stochastic nonlinear Fokker-Planck equation $$dX-{\\rm div}(DX)dt-\\Delta\\beta(X)dt=X\\,dW \\mbox{ in }(0,T)\\times\\mathbb R^d,\\ X(0)=x,$$ via a corresponding random differential equation. Here $d\\geq 1$, $W$ is a Wiener process in $H^{-1}(\\mathbb R^d)$, $D\\in C^1(\\mathbb R^d,\\mathbb R^d)$ and $\\beta$ is a continuous monotonically increasing function. The solution exists for $x\\in L^1\\cap L^\\infty$ and preserves positivity. If $\\beta \\in L^1_{\\rm loc}(\\mathbb R)$, the solution is pathwise Lipschitz continuous w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.08768","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":"2026-05-18T00:32:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IxUnT2vXNPq4nhZTkHI+j314oG0LaL6KtNkhJDryqwt5okF86AtAMvj/3ptrWc1MCF8HD9KQxMTIe3nlXPQsBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T15:00:33.335355Z"},"content_sha256":"44cee67fa62d8913f523fb43467d4d9d194fd4c115fecc51278adf8cb468319a","schema_version":"1.0","event_id":"sha256:44cee67fa62d8913f523fb43467d4d9d194fd4c115fecc51278adf8cb468319a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FFI4UL5MSOGLZ2GRQUPZALCHHA/bundle.json","state_url":"https://pith.science/pith/FFI4UL5MSOGLZ2GRQUPZALCHHA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FFI4UL5MSOGLZ2GRQUPZALCHHA/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":"2026-08-16T15:00:33Z","links":{"resolver":"https://pith.science/pith/FFI4UL5MSOGLZ2GRQUPZALCHHA","bundle":"https://pith.science/pith/FFI4UL5MSOGLZ2GRQUPZALCHHA/bundle.json","state":"https://pith.science/pith/FFI4UL5MSOGLZ2GRQUPZALCHHA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FFI4UL5MSOGLZ2GRQUPZALCHHA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:FFI4UL5MSOGLZ2GRQUPZALCHHA","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":"03239ed5b9f63c61fe51da0f0607aaa523bb7702d0dd81b46a6466b725e8e01e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-29T14:18:58Z","title_canon_sha256":"27ab9cf5efdce77f9fee6921ad1d5a809d669509ac753094cd1182bcb8e01ed1"},"schema_version":"1.0","source":{"id":"1708.08768","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.08768","created_at":"2026-05-18T00:32:05Z"},{"alias_kind":"arxiv_version","alias_value":"1708.08768v2","created_at":"2026-05-18T00:32:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.08768","created_at":"2026-05-18T00:32:05Z"},{"alias_kind":"pith_short_12","alias_value":"FFI4UL5MSOGL","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FFI4UL5MSOGLZ2GR","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FFI4UL5M","created_at":"2026-05-18T12:31:15Z"}],"graph_snapshots":[{"event_id":"sha256:44cee67fa62d8913f523fb43467d4d9d194fd4c115fecc51278adf8cb468319a","target":"graph","created_at":"2026-05-18T00:32:05Z","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":"Existence and uniqueness of a strong solution in $H^{-1}(\\mathbb R^d)$ is proved for the stochastic nonlinear Fokker-Planck equation $$dX-{\\rm div}(DX)dt-\\Delta\\beta(X)dt=X\\,dW \\mbox{ in }(0,T)\\times\\mathbb R^d,\\ X(0)=x,$$ via a corresponding random differential equation. Here $d\\geq 1$, $W$ is a Wiener process in $H^{-1}(\\mathbb R^d)$, $D\\in C^1(\\mathbb R^d,\\mathbb R^d)$ and $\\beta$ is a continuous monotonically increasing function. The solution exists for $x\\in L^1\\cap L^\\infty$ and preserves positivity. If $\\beta \\in L^1_{\\rm loc}(\\mathbb R)$, the solution is pathwise Lipschitz continuous w","authors_text":"Michael R\\\"ockner, Viorel Barbu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-29T14:18:58Z","title":"Nonlinear Fokker-Planck equations driven by Gaussian linear multiplicative noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.08768","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":"sha256:1de936b02db798cac7999e9633f486ea5d0396c0e9a209d2d065c668dc8b9ed3","target":"record","created_at":"2026-05-18T00:32:05Z","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":"03239ed5b9f63c61fe51da0f0607aaa523bb7702d0dd81b46a6466b725e8e01e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-29T14:18:58Z","title_canon_sha256":"27ab9cf5efdce77f9fee6921ad1d5a809d669509ac753094cd1182bcb8e01ed1"},"schema_version":"1.0","source":{"id":"1708.08768","kind":"arxiv","version":2}},"canonical_sha256":"2951ca2fac938cbce8d1851f902c473827aea47be0119b29bca25023e68551fc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2951ca2fac938cbce8d1851f902c473827aea47be0119b29bca25023e68551fc","first_computed_at":"2026-05-18T00:32:05.489108Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:05.489108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uZEXp7Ysk4xDdSZD+XaE2+FDVbGgH7gjijirlzn5+yVno5T9LumZEX6OPIBwKpqawdruEtcFb5ATQgWvfPDlAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:05.489780Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.08768","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1de936b02db798cac7999e9633f486ea5d0396c0e9a209d2d065c668dc8b9ed3","sha256:44cee67fa62d8913f523fb43467d4d9d194fd4c115fecc51278adf8cb468319a"],"state_sha256":"c20a93612d535fbba3cb76e13ce5ef727b1fb820fb72c80557e69b8affacdee4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dxjGCLy/Q/hDtwFf5M6+wvRVO+foS4d5Wdm8SqmiNQOk7wnia76Hmqpi6lsKALdaFqVbLFVVB/IpyJlLl2YCDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T15:00:33.401424Z","bundle_sha256":"72b1a2185aee5d3ecc1c2e975ab7effb21bc30b465322385ce381984ed4a0237"}}