{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2000:X43W5CUIXWE4CRXFQ6LVQBI7QB","short_pith_number":"pith:X43W5CUI","canonical_record":{"source":{"id":"math/0005306","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2000-05-01T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"099085b407462dc74686d42390c981db5ee241c2fb61062e5dc0a813af8fad58","abstract_canon_sha256":"658b56b6804ecfe4e7c676991415cb721d4aba143ed2957175e2b4bf56de7cbb"},"schema_version":"1.0"},"canonical_sha256":"bf376e8a88bd89c146e5879758051f8043995ee974800707fc6c7d4e4033f25b","source":{"kind":"arxiv","id":"math/0005306","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0005306","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"arxiv_version","alias_value":"math/0005306v1","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0005306","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"pith_short_12","alias_value":"X43W5CUIXWE4","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"X43W5CUIXWE4CRXF","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"X43W5CUI","created_at":"2026-05-18T12:25:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2000:X43W5CUIXWE4CRXFQ6LVQBI7QB","target":"record","payload":{"canonical_record":{"source":{"id":"math/0005306","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2000-05-01T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"099085b407462dc74686d42390c981db5ee241c2fb61062e5dc0a813af8fad58","abstract_canon_sha256":"658b56b6804ecfe4e7c676991415cb721d4aba143ed2957175e2b4bf56de7cbb"},"schema_version":"1.0"},"canonical_sha256":"bf376e8a88bd89c146e5879758051f8043995ee974800707fc6c7d4e4033f25b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:38.543246Z","signature_b64":"pMTifvAXXtsug0YUhp0b16cHAGH9jhLcv52oMfsk9jIttjj+7PlMTBupAOWV3i8qqOuUGpUjFrcC4eW71SAOBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf376e8a88bd89c146e5879758051f8043995ee974800707fc6c7d4e4033f25b","last_reissued_at":"2026-05-18T01:05:38.542480Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:38.542480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0005306","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7zK3wIU9M3SYbbSpi71FjAEW1MAFzDwKL1a7YHaJ4acsTZjPXzj4M1W36P746eqYexZRul4vdPrB1fl5ZU5dAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:08:31.283489Z"},"content_sha256":"a719b82aff893c14d377f64b18dc6fa67bdb3f9c43e370c8067a9fe33ea7de83","schema_version":"1.0","event_id":"sha256:a719b82aff893c14d377f64b18dc6fa67bdb3f9c43e370c8067a9fe33ea7de83"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2000:X43W5CUIXWE4CRXFQ6LVQBI7QB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant measures for Burgers equation with stochastic forcing","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. E. Mazel, K. M. Khanin, Weinan E, Ya. G. Sinai","submitted_at":"2000-05-01T00:00:00Z","abstract_excerpt":"In this paper we study the following Burgers equation\n  du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t)\n  where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t.  We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimiz"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0005306","kind":"arxiv","version":1},"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":"2026-05-18T01:05:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"46X6QHzLkG59GvLd9CoruqeX7g8YrBCfuQ/KFKSdxXIFIvOLkyuir3KLKoJpXTWPVStxaGyHBjxCKDaZK1YwAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:08:31.283835Z"},"content_sha256":"a57abd1aa700b717728999ea9b00f8a4f1e65f5ce250299b92b811bf92ff496b","schema_version":"1.0","event_id":"sha256:a57abd1aa700b717728999ea9b00f8a4f1e65f5ce250299b92b811bf92ff496b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X43W5CUIXWE4CRXFQ6LVQBI7QB/bundle.json","state_url":"https://pith.science/pith/X43W5CUIXWE4CRXFQ6LVQBI7QB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X43W5CUIXWE4CRXFQ6LVQBI7QB/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":"2026-06-25T06:08:31Z","links":{"resolver":"https://pith.science/pith/X43W5CUIXWE4CRXFQ6LVQBI7QB","bundle":"https://pith.science/pith/X43W5CUIXWE4CRXFQ6LVQBI7QB/bundle.json","state":"https://pith.science/pith/X43W5CUIXWE4CRXFQ6LVQBI7QB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X43W5CUIXWE4CRXFQ6LVQBI7QB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2000:X43W5CUIXWE4CRXFQ6LVQBI7QB","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":"658b56b6804ecfe4e7c676991415cb721d4aba143ed2957175e2b4bf56de7cbb","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2000-05-01T00:00:00Z","title_canon_sha256":"099085b407462dc74686d42390c981db5ee241c2fb61062e5dc0a813af8fad58"},"schema_version":"1.0","source":{"id":"math/0005306","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0005306","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"arxiv_version","alias_value":"math/0005306v1","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0005306","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"pith_short_12","alias_value":"X43W5CUIXWE4","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"X43W5CUIXWE4CRXF","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"X43W5CUI","created_at":"2026-05-18T12:25:50Z"}],"graph_snapshots":[{"event_id":"sha256:a57abd1aa700b717728999ea9b00f8a4f1e65f5ce250299b92b811bf92ff496b","target":"graph","created_at":"2026-05-18T01:05:38Z","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":"In this paper we study the following Burgers equation\n  du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t)\n  where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t.  We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimiz","authors_text":"A. E. Mazel, K. M. Khanin, Weinan E, Ya. G. Sinai","cross_cats":[],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2000-05-01T00:00:00Z","title":"Invariant measures for Burgers equation with stochastic forcing"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0005306","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:a719b82aff893c14d377f64b18dc6fa67bdb3f9c43e370c8067a9fe33ea7de83","target":"record","created_at":"2026-05-18T01:05:38Z","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":"658b56b6804ecfe4e7c676991415cb721d4aba143ed2957175e2b4bf56de7cbb","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2000-05-01T00:00:00Z","title_canon_sha256":"099085b407462dc74686d42390c981db5ee241c2fb61062e5dc0a813af8fad58"},"schema_version":"1.0","source":{"id":"math/0005306","kind":"arxiv","version":1}},"canonical_sha256":"bf376e8a88bd89c146e5879758051f8043995ee974800707fc6c7d4e4033f25b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf376e8a88bd89c146e5879758051f8043995ee974800707fc6c7d4e4033f25b","first_computed_at":"2026-05-18T01:05:38.542480Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:38.542480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pMTifvAXXtsug0YUhp0b16cHAGH9jhLcv52oMfsk9jIttjj+7PlMTBupAOWV3i8qqOuUGpUjFrcC4eW71SAOBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:38.543246Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0005306","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a719b82aff893c14d377f64b18dc6fa67bdb3f9c43e370c8067a9fe33ea7de83","sha256:a57abd1aa700b717728999ea9b00f8a4f1e65f5ce250299b92b811bf92ff496b"],"state_sha256":"88bfea646b8c1f1a9d0a38a45ede7e66a86c30d3901f34a0ba26a929713aeedb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ay4Z2Hr7LABeXCsUMUaNQLfPc9nltOWrxWhISI03mqnbw2LaIhvhKNiPD3KVVdWgulojtFKVL42Pyh9Jw4FsCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T06:08:31.285811Z","bundle_sha256":"b7f6f4be2daeee7130d6f0886b67ffe43dd48a7c7bf4b4177174615c466c7ed5"}}