{"paper":{"title":"Generation of fine transition layers and their dynamics for the stochastic Allen--Cahn equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dimitra Antonopoulou, Georgia Karali, Hiroshi Matano, Matthieu Alfaro","submitted_at":"2018-12-10T14:16:08Z","abstract_excerpt":"We study an $\\ep$-dependent stochastic Allen--Cahn equation with a mild random noise on a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$. Here $\\ep$ is a small positive parameter that represents formally the thickness of the solution interface, while the mild noise $\\xi^\\ep(t)$ is a smooth random function of $t$ of order $\\mathcal O(\\ep^{-\\gamma})$ with $0<\\gamma<1/3$ that converges to white noise as $\\ep\\rightarrow 0^+$. We consider initial data that are independent of $\\ep$ satisfying some non-degeneracy conditions, and prove that steep transition layers---or interfaces---develop within a very "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.03804","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"}