{"paper":{"title":"A counterexample to maximal $L_p$-regularity of the stochastic heat equation in polygons: the case $p>4$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Kyeong-Hun Kim","submitted_at":"2015-08-14T02:04:24Z","abstract_excerpt":"Let $D$ be a domain in $R^d$ and $u$ be the solution to the stochastic heat equation $$ du=\\Delta u dt+ g\\,dW_t, \\quad t>0, x\\in D, $$ with zero initial and boundary data. Here $W_t$ is a one-dimensional Wiener process on a probability space $\\Omega$. It has been proved (see below for references) that for any $p\\geq 2$ the inequality $$ \\|\\nabla u\\|_{L_p(\\Omega\\times [0,T]\\times D)} \\leq c \\|g\\|_{L_p(\\Omega\\times [0,T]\\times D)} $$ holds if $\\partial D\\in C^1$. In this note we prove that if $p>4$ then this inequality fails in any polygon in $R^2$ having an angle greater than or equal to $\\frac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03402","kind":"arxiv","version":3},"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"}