{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:GODEHN3TFXFT5VZWTIOJBWZ6QR","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":"5b52d0029611c0edb79637785bfd2f66bbbfb9063a7aa5257960dda64422635b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-10-20T20:18:07Z","title_canon_sha256":"28b356c56c267f7fc982c26b1bc75694ac6c5c791420b8fd1dd958cfb638d3f5"},"schema_version":"1.0","source":{"id":"1510.06046","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.06046","created_at":"2026-05-18T01:29:38Z"},{"alias_kind":"arxiv_version","alias_value":"1510.06046v1","created_at":"2026-05-18T01:29:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.06046","created_at":"2026-05-18T01:29:38Z"},{"alias_kind":"pith_short_12","alias_value":"GODEHN3TFXFT","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GODEHN3TFXFT5VZW","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GODEHN3T","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:0abadab90e5a270b9988b02b1a91db2828dfa54bb0d1b609a5b0f8fac6a1f836","target":"graph","created_at":"2026-05-18T01:29:38Z","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 stochastic heat equation in the spatial domain $\\mathbb{R}^d$ subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies {\\it Dalang's condition}. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. As a first application of these moments bounds, we find the necessary and sufficient conditions for the solution to have phase transit","authors_text":"Kunwoo Kim, Le Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-10-20T20:18:07Z","title":"Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06046","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:fcc795550672402132ca12294b54b63e268c32863bc8eb6d71f5b1613bbb5272","target":"record","created_at":"2026-05-18T01:29:38Z","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":"5b52d0029611c0edb79637785bfd2f66bbbfb9063a7aa5257960dda64422635b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-10-20T20:18:07Z","title_canon_sha256":"28b356c56c267f7fc982c26b1bc75694ac6c5c791420b8fd1dd958cfb638d3f5"},"schema_version":"1.0","source":{"id":"1510.06046","kind":"arxiv","version":1}},"canonical_sha256":"338643b7732dcb3ed7369a1c90db3e84654bcdce84c52f6aa1820ced0908fd5d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"338643b7732dcb3ed7369a1c90db3e84654bcdce84c52f6aa1820ced0908fd5d","first_computed_at":"2026-05-18T01:29:38.848959Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:29:38.848959Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rr752kEP4lTzDnLyU1So1Dhp46h7j0M/SovUdRp8XXE225F8HeC/GoEf8blnRpfTW+d53WMMg6tmLopcOWCfAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:29:38.849584Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.06046","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fcc795550672402132ca12294b54b63e268c32863bc8eb6d71f5b1613bbb5272","sha256:0abadab90e5a270b9988b02b1a91db2828dfa54bb0d1b609a5b0f8fac6a1f836"],"state_sha256":"fa37957328a144666ffbf9d0e8b4c364c010304fc5ef4881ed9481308844db2d"}