{"paper":{"title":"A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Ivan Matic, James Nolen","submitted_at":"2012-06-13T20:25:32Z","abstract_excerpt":"We estimate the variance of the value function for a random optimal control problem. The value function is the solution $w^\\epsilon$ of a Hamilton-Jacobi equation with random Hamiltonian $H(p,x,\\omega) = K(p) - V(x/\\epsilon,\\omega)$ in dimension $d \\geq 2$. It is known that homogenization occurs as $\\epsilon \\to 0$, but little is known about the statistical fluctuations of $w^\\epsilon$. Our main result shows that the variance of the solution $w^\\epsilon$ is bounded by $O(\\epsilon/|\\log \\epsilon|)$. The proof relies on a modified Poincar\\'e inequality of Talagrand."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2937","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"}