{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:AXZNGUEXAPDFWJGL4L6MYAQ72N","short_pith_number":"pith:AXZNGUEX","schema_version":"1.0","canonical_sha256":"05f2d3509703c65b24cbe2fccc021fd34bf5302862933ca06c2afa8e069f7586","source":{"kind":"arxiv","id":"0909.4816","version":1},"attestation_state":"computed","paper":{"title":"Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jeremy Quastel, Marton Balazs, Timo Seppalainen","submitted_at":"2009-09-25T22:55:21Z","abstract_excerpt":"We consider the stochastic heat equation $\\partial_tZ= \\partial_x^2 Z - Z \\dot W$ on the real line, where $\\dot W$ is space-time white noise. $h(t,x)=-\\log Z(t,x)$ is interpreted as a solution of the KPZ equation, and $u(t,x)=\\partial_x h(t,x)$ as a solution of the stochastic Burgers equation. We take $Z(0,x)=\\exp\\{B(x)\\}$ where $B(x)$ is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist $0< c_1\\le c_2 <\\infty$ such that $c_1t^{2/3}\\le \\Var (\\log Z(t,x))\\le c_2 t^{2/3}.$ Analogous results are obtained for some mom"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0909.4816","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-09-25T22:55:21Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"5e8e15b137f0d5beef94a3d4667d0288300d50c2fa31e5e1e5dba6089a61a74e","abstract_canon_sha256":"e157bb32505abe8196aa1804255ab51351169d1cea7d4688729f2a4d1f64988c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:47.247640Z","signature_b64":"C5YL/wSx56/giqt+LJCw5cRo93Ks+EloVB42nJvSrW+3qENcyfKo75uWiyJ2N8oXcCowSJDgS128BxBH2sfAAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05f2d3509703c65b24cbe2fccc021fd34bf5302862933ca06c2afa8e069f7586","last_reissued_at":"2026-05-18T04:10:47.247249Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:47.247249Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jeremy Quastel, Marton Balazs, Timo Seppalainen","submitted_at":"2009-09-25T22:55:21Z","abstract_excerpt":"We consider the stochastic heat equation $\\partial_tZ= \\partial_x^2 Z - Z \\dot W$ on the real line, where $\\dot W$ is space-time white noise. $h(t,x)=-\\log Z(t,x)$ is interpreted as a solution of the KPZ equation, and $u(t,x)=\\partial_x h(t,x)$ as a solution of the stochastic Burgers equation. We take $Z(0,x)=\\exp\\{B(x)\\}$ where $B(x)$ is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist $0< c_1\\le c_2 <\\infty$ such that $c_1t^{2/3}\\le \\Var (\\log Z(t,x))\\le c_2 t^{2/3}.$ Analogous results are obtained for some mom"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.4816","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0909.4816","created_at":"2026-05-18T04:10:47.247303+00:00"},{"alias_kind":"arxiv_version","alias_value":"0909.4816v1","created_at":"2026-05-18T04:10:47.247303+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.4816","created_at":"2026-05-18T04:10:47.247303+00:00"},{"alias_kind":"pith_short_12","alias_value":"AXZNGUEXAPDF","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_16","alias_value":"AXZNGUEXAPDFWJGL","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_8","alias_value":"AXZNGUEX","created_at":"2026-05-18T12:25:58.837520+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N","json":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N.json","graph_json":"https://pith.science/api/pith-number/AXZNGUEXAPDFWJGL4L6MYAQ72N/graph.json","events_json":"https://pith.science/api/pith-number/AXZNGUEXAPDFWJGL4L6MYAQ72N/events.json","paper":"https://pith.science/paper/AXZNGUEX"},"agent_actions":{"view_html":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N","download_json":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N.json","view_paper":"https://pith.science/paper/AXZNGUEX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0909.4816&json=true","fetch_graph":"https://pith.science/api/pith-number/AXZNGUEXAPDFWJGL4L6MYAQ72N/graph.json","fetch_events":"https://pith.science/api/pith-number/AXZNGUEXAPDFWJGL4L6MYAQ72N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N/action/storage_attestation","attest_author":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N/action/author_attestation","sign_citation":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N/action/citation_signature","submit_replication":"https://pith.science/pith/AXZNGUEXAPDFWJGL4L6MYAQ72N/action/replication_record"}},"created_at":"2026-05-18T04:10:47.247303+00:00","updated_at":"2026-05-18T04:10:47.247303+00:00"}