{"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"}