{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:W2Y22W4EY32AHI7JJ6Z3ANAPW4","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":"4a66ddbfbf643541f7b8a8dbd90cabb9946df2a277398f32343823ed5e37333f","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-03-22T07:35:16Z","title_canon_sha256":"8b74f7a36fdc5515841a8f97a8b13a7f5934072d35768ed981dea65328409c12"},"schema_version":"1.0","source":{"id":"2403.15006","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.15006","created_at":"2026-07-05T07:59:29Z"},{"alias_kind":"arxiv_version","alias_value":"2403.15006v1","created_at":"2026-07-05T07:59:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.15006","created_at":"2026-07-05T07:59:29Z"},{"alias_kind":"pith_short_12","alias_value":"W2Y22W4EY32A","created_at":"2026-07-05T07:59:29Z"},{"alias_kind":"pith_short_16","alias_value":"W2Y22W4EY32AHI7J","created_at":"2026-07-05T07:59:29Z"},{"alias_kind":"pith_short_8","alias_value":"W2Y22W4E","created_at":"2026-07-05T07:59:29Z"}],"graph_snapshots":[{"event_id":"sha256:1cdcc9393224c97d2802bc105b74f80a7b3a7be05f1b52e8cb3e134c80bc95c1","target":"graph","created_at":"2026-07-05T07:59:29Z","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/2403.15006/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The goal of these lecture notes is to present recent results regarding the large-scale behaviour of critical and super-critical non-linear stochastic PDEs, that fall outside the realm of the theory of Regularity Structures. These include the two-dimensional Anisotropic KPZ equation, the stochastic Burgers equation in dimension $d\\ge 2$ and the stochastic Navier-Stokes equation with divergence-free noise in dimension $d=2$. Rather than providing complete proofs, we try to emphasise the main ideas, and some crucial aspects of our approach: the role of the generator equation and of the Fluctuatio","authors_text":"Fabio Toninelli (Technical University of Vienna), Giuseppe Cannizzaro (University of Warwick)","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-03-22T07:35:16Z","title":"Lecture notes on stationary critical and super-critical SPDEs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.15006","kind":"arxiv","version":1},"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":"sha256:7042bc9480bec5a5c9259432ab49081360f92107060a0f0d6afcb128a034d970","target":"record","created_at":"2026-07-05T07:59:29Z","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":"4a66ddbfbf643541f7b8a8dbd90cabb9946df2a277398f32343823ed5e37333f","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-03-22T07:35:16Z","title_canon_sha256":"8b74f7a36fdc5515841a8f97a8b13a7f5934072d35768ed981dea65328409c12"},"schema_version":"1.0","source":{"id":"2403.15006","kind":"arxiv","version":1}},"canonical_sha256":"b6b1ad5b84c6f403a3e94fb3b0340fb70d8e913d4affc626ea489cdb8b555ace","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b6b1ad5b84c6f403a3e94fb3b0340fb70d8e913d4affc626ea489cdb8b555ace","first_computed_at":"2026-07-05T07:59:29.070966Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:59:29.070966Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7azEp05r9rMrGQqPQczB1/f10dEjZiCGBwBfmRqiAskAUAhLaupWdgBOXBCJOfZ/YsnEZzD4u+GQfAVDOWSCAw==","signature_status":"signed_v1","signed_at":"2026-07-05T07:59:29.071547Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.15006","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7042bc9480bec5a5c9259432ab49081360f92107060a0f0d6afcb128a034d970","sha256:1cdcc9393224c97d2802bc105b74f80a7b3a7be05f1b52e8cb3e134c80bc95c1"],"state_sha256":"6ecb105ba9279d9f3d23030332dbc96a259e054346a27b3493df3423145df547"}