{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:GXTMNFEANU2IMSJWCZXYELREDQ","short_pith_number":"pith:GXTMNFEA","canonical_record":{"source":{"id":"1409.8031","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-29T08:56:06Z","cross_cats_sorted":[],"title_canon_sha256":"c833f67c513398419d349a6b6d30c7a86bbd4dfb9327b1257d5cc2595bdb7c7c","abstract_canon_sha256":"0d7bb795d2ac6428c1c5fe371bedce2c03744e4cfb6c2f11a16087ffa1b026a4"},"schema_version":"1.0"},"canonical_sha256":"35e6c694806d34864936166f822e241c07dd44bc766e70fe5532e7ab157bb2db","source":{"kind":"arxiv","id":"1409.8031","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.8031","created_at":"2026-05-18T02:20:36Z"},{"alias_kind":"arxiv_version","alias_value":"1409.8031v2","created_at":"2026-05-18T02:20:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.8031","created_at":"2026-05-18T02:20:36Z"},{"alias_kind":"pith_short_12","alias_value":"GXTMNFEANU2I","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GXTMNFEANU2IMSJW","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GXTMNFEA","created_at":"2026-05-18T12:28:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:GXTMNFEANU2IMSJWCZXYELREDQ","target":"record","payload":{"canonical_record":{"source":{"id":"1409.8031","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-29T08:56:06Z","cross_cats_sorted":[],"title_canon_sha256":"c833f67c513398419d349a6b6d30c7a86bbd4dfb9327b1257d5cc2595bdb7c7c","abstract_canon_sha256":"0d7bb795d2ac6428c1c5fe371bedce2c03744e4cfb6c2f11a16087ffa1b026a4"},"schema_version":"1.0"},"canonical_sha256":"35e6c694806d34864936166f822e241c07dd44bc766e70fe5532e7ab157bb2db","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:36.492886Z","signature_b64":"JiBtsDeyRMenFiZulxfFh6tHCPxhkKwatgqfOl8TESDqpARxkisSjwYBdkUm1MohhxzM+YpFnyGstoWNggssDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35e6c694806d34864936166f822e241c07dd44bc766e70fe5532e7ab157bb2db","last_reissued_at":"2026-05-18T02:20:36.492255Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:36.492255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.8031","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":"2026-05-18T02:20:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B7LQzxpTMC4BFvubqIos2Qj4S2YGo5xjD8AqWwq3B6d1VkZaMLLDTFbyZjhuETuhq7Vc3yF2dEocP6sBKYQrBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T01:10:50.106310Z"},"content_sha256":"8eac1e498113084fee7cef93ab80ff9ff8b443c004142d480034fb346a038ad0","schema_version":"1.0","event_id":"sha256:8eac1e498113084fee7cef93ab80ff9ff8b443c004142d480034fb346a038ad0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:GXTMNFEANU2IMSJWCZXYELREDQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Absolute continuity for SPDEs with irregular fundamental solution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Andr\\'e S\\\"u{\\ss}, Marta Sanz-Sol\\'e","submitted_at":"2014-09-29T08:56:06Z","abstract_excerpt":"For the class of stochastic partial differential equations studied in [Conus-Dalang,2008], we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space. The proof relies on the method developed in [Debussche-Romito, 2014]. The result can be applied to the solution of the stochastic wave equation with multiplicative noise, Lipschitz coefficients and any spatial dimension $d\\ge 1$, and also to the heat equation. This provides an extension of the results proved in [Sanz-Sol\\'e and S\\\"u\\ss, 2013]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8031","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":"2026-05-18T02:20:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DAXPUHbeGlV16bt/V+yF77CXvFTJzwBiXk0btGhVIio/TwGs8au+/sb5H3IgDz6jXFbbo4eNatIHrKfe7DEYBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T01:10:50.106667Z"},"content_sha256":"ecdc0549172a789b7bfe07d9fd6b32c10edfba24f1bd1afd74af4f56dcb34d2d","schema_version":"1.0","event_id":"sha256:ecdc0549172a789b7bfe07d9fd6b32c10edfba24f1bd1afd74af4f56dcb34d2d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GXTMNFEANU2IMSJWCZXYELREDQ/bundle.json","state_url":"https://pith.science/pith/GXTMNFEANU2IMSJWCZXYELREDQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GXTMNFEANU2IMSJWCZXYELREDQ/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":"2026-06-04T01:10:50Z","links":{"resolver":"https://pith.science/pith/GXTMNFEANU2IMSJWCZXYELREDQ","bundle":"https://pith.science/pith/GXTMNFEANU2IMSJWCZXYELREDQ/bundle.json","state":"https://pith.science/pith/GXTMNFEANU2IMSJWCZXYELREDQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GXTMNFEANU2IMSJWCZXYELREDQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:GXTMNFEANU2IMSJWCZXYELREDQ","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":"0d7bb795d2ac6428c1c5fe371bedce2c03744e4cfb6c2f11a16087ffa1b026a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-29T08:56:06Z","title_canon_sha256":"c833f67c513398419d349a6b6d30c7a86bbd4dfb9327b1257d5cc2595bdb7c7c"},"schema_version":"1.0","source":{"id":"1409.8031","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.8031","created_at":"2026-05-18T02:20:36Z"},{"alias_kind":"arxiv_version","alias_value":"1409.8031v2","created_at":"2026-05-18T02:20:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.8031","created_at":"2026-05-18T02:20:36Z"},{"alias_kind":"pith_short_12","alias_value":"GXTMNFEANU2I","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GXTMNFEANU2IMSJW","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GXTMNFEA","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:ecdc0549172a789b7bfe07d9fd6b32c10edfba24f1bd1afd74af4f56dcb34d2d","target":"graph","created_at":"2026-05-18T02:20:36Z","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":"For the class of stochastic partial differential equations studied in [Conus-Dalang,2008], we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space. The proof relies on the method developed in [Debussche-Romito, 2014]. The result can be applied to the solution of the stochastic wave equation with multiplicative noise, Lipschitz coefficients and any spatial dimension $d\\ge 1$, and also to the heat equation. This provides an extension of the results proved in [Sanz-Sol\\'e and S\\\"u\\ss, 2013].","authors_text":"Andr\\'e S\\\"u{\\ss}, Marta Sanz-Sol\\'e","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-29T08:56:06Z","title":"Absolute continuity for SPDEs with irregular fundamental solution"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8031","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":"sha256:8eac1e498113084fee7cef93ab80ff9ff8b443c004142d480034fb346a038ad0","target":"record","created_at":"2026-05-18T02:20:36Z","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":"0d7bb795d2ac6428c1c5fe371bedce2c03744e4cfb6c2f11a16087ffa1b026a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-29T08:56:06Z","title_canon_sha256":"c833f67c513398419d349a6b6d30c7a86bbd4dfb9327b1257d5cc2595bdb7c7c"},"schema_version":"1.0","source":{"id":"1409.8031","kind":"arxiv","version":2}},"canonical_sha256":"35e6c694806d34864936166f822e241c07dd44bc766e70fe5532e7ab157bb2db","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"35e6c694806d34864936166f822e241c07dd44bc766e70fe5532e7ab157bb2db","first_computed_at":"2026-05-18T02:20:36.492255Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:20:36.492255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JiBtsDeyRMenFiZulxfFh6tHCPxhkKwatgqfOl8TESDqpARxkisSjwYBdkUm1MohhxzM+YpFnyGstoWNggssDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:20:36.492886Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.8031","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8eac1e498113084fee7cef93ab80ff9ff8b443c004142d480034fb346a038ad0","sha256:ecdc0549172a789b7bfe07d9fd6b32c10edfba24f1bd1afd74af4f56dcb34d2d"],"state_sha256":"fdaae723e116db6c65e624b354efd7efef17fdf5805b7852d0ac68706c1770f5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"67mCFVtcTWJ9MMWnyY4pXLQBTtEycwZbaAjS6//Lo1Vt3ME6CdZTlll93PR9b2nvYlLwfFA3VbUxZWQqxvpjDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T01:10:50.108681Z","bundle_sha256":"74005ce3b1e88f1c40257a5dac04852ca6b77ed784585aa7cff069b7ee814ce6"}}